Advances in Chemical Physics, Vol.133, Part B. Fractals, Diffusion, and Relaxation (Wiley 2006)

FRACTALS, DIFFUSION, AND RELAXATION IN DISORDERED COMPLEX SYSTEMS ADVANCES IN CHEMICAL PHYSICS VOLUME 133 PART B Edited...

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FRACTALS, DIFFUSION, AND RELAXATION IN DISORDERED COMPLEX SYSTEMS ADVANCES IN CHEMICAL PHYSICS VOLUME 133 PART B

Edited By WILLIAM T. COFFEY AND YURI P. KALMYKOV Series Editor STUART A. RICE Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois

AN INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC.

FRACTALS, DIFFUSION, AND RELAXATION IN DISORDERED COMPLEX SYSTEMS A SPECIAL VOLUME OF ADVANCES IN CHEMICAL PHYSICS VOLUME 133 PART B

EDITORIAL BOARD

BRUCE J. BERNE, Department of Chemistry, Columbia University, New York, New York, U.S.A. KURT BINDER, Institut fu¨r Physik, Johannes Gutenberg-Universita¨t Mainz, Mainz, Germany A. WELFORD CASTLEMAN, JR., Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania, U.S.A. DAVID CHANDLER, Department of Chemistry, University of California, Berkeley, California, U.S.A. M. S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford, U.K. WILLIAM T. COFFEY, Department of Microelectronics and Electrical Engineering, Trinity College, University of Dublin, Dublin, Ireland F. FLEMING CRIM, Department of Chemistry, University of Wisconsin, Madison, Wisconsin, U.S.A. ERNEST R. DAVIDSON, Department of Chemistry, Indiana University, Bloomington, Indiana, U.S.A. GRAHAM R. FLEMING , Department of Chemistry, University of California, Berkeley, California, U.S.A. KARL F. FREED, The James Franck Institute, The University of Chicago, Chicago, Illinois, U.S.A. PIERRE GASPARD, Center for Nonlinear Phenomena and Complex Systems, Brussels, Belgium ERIC J. HELLER, Institute for Theoretical Atomic and Molecular Physics, HarvardSmithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A. ROBIN M. HOCHSTRASSER, Department of Chemistry, The University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A. R. KOSLOFF, The Fritz Haber Research Center for Molecular Dynamics and Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel RUDOLPH A. MARCUS, Department of Chemistry, California Institute of Technology, Pasadena, California, U.S.A. G. NICOLIS, Center for Nonlinear Phenomena and Complex Systems, Universite´ Libre de Bruxelles, Brussels, Belgium THOMAS P. RUSSELL, Department of Polymer Science, University of Massachusetts, Amherst, Massachusetts, U.S.A. DONALD G. TRUHLAR , Department of Chemistry, University of Minnesota, Minneapolis, Minnesota, U.S.A. JOHN D. WEEKS, Institute for Physical Science and Technology and Department of Chemistry, University of Maryland, College Park, Maryland, U.S.A. PETER G. WOLYNES, Department of Chemistry, University of California, San Diego, California, U.S.A.

FRACTALS, DIFFUSION, AND RELAXATION IN DISORDERED COMPLEX SYSTEMS ADVANCES IN CHEMICAL PHYSICS VOLUME 133 PART B

Edited By WILLIAM T. COFFEY AND YURI P. KALMYKOV Series Editor STUART A. RICE Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois

AN INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC.

Copyright # 2006 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Catalog Number: 58-9935 ISBN-13 ISBN-10 ISBN-13 ISBN-10 ISBN-13 ISBN-10

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CONTRIBUTORS TO VOLUME 133 ELI BARKAI, Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46566, USA; and Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel ALEXANDER BRODIN, Experimentalphysik II, Universita¨t Bayreuth, D 95440 Bayreuth, Germany THOMAS BLOCHOWICZ, Institute fur Festko¨rperphysik, Technische Universita¨t Darmstadt, D 64289 Darmstadt, Germany SIMONE CAPACCIOLI, Dipartimento di Fisica and INFM, Universita` di Pisa, I-56127, Pisa, Italy; and CNR-INFM Center ‘‘SOFT: Complex Dynamics in Structured Systems,’’ Universita` di Roma ‘‘La Sapienza,’’ I-00185 Roma, Italy RICCARDO CASALINI, Naval Research Laboratory, Washington, DC 20375, USA; and Chemistry Department, George Mason University, Fairfax, Virginia 20030, USA ALEKSEI V. CHECHKIN, Institute for Theoretical Physics, National Science Center, Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine WILLIAM T. COFFEY, Department of Electronic and Electrical Engineering, School of Engineering, Trinity College, Dublin 2 Ireland YURI FELDMAN, Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel VSEVOLD Y. GONCHAR, Institute for Theoretical Physics, National Science Center, Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine PAOLO GRIGOLINI, Department of Physics, University of North Texas, Denton, Texas, 76203 USA; and Department of Physics, University of Pisa, Pisa, Italy YURI P. KALMYKOV, Laboratoire de Mathe´matiques et Physique des Syste`mes, Universite de Perpignan, 66860 Perpignan Cedex, France JOSEPH KLAFTER, School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel FRIEDRICH KREMER, Universita¨t Leipzig, Fakultat fu¨r Physik und Geowissenschaften, 04103 Leipzig, Germany

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MASARU KUNO, Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46566, USA; and Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel GENNADY MARGOLIN, Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46556, USA RALF METZLER, NORDITA–Nordic Institute for Theoretical Physics, DK-2100 Copenhagen Danish Denmark KIA L. NGAI Naval Research Laboratory, Washington, DC 20375, USA VITALY V. NOVIKOV, Odessa National Polytechnical University, 65044 Odessa, Ukraine MARIAN PALUCH, Institute of Physics, Silesian University, 40-007 Katowice, Poland NOE´LLE POTTIER, Matie`re et Syste`mes Complexes, UMR 7057 CNRS and Universite´ Paris 7—Denis Diderot, 75251 Paris Cedex 05, France VLADIMIR PROTASENKO, Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46566, USA; and Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel ALEXANDER PUZENKO, Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel C. M. ROLAND, Naval Research Laboratory, Washington, DC 20375, USA ERNST A. RO¨SSLER, Experimentalphysik II, Universita¨t Bayreuth, D 95440 Bayreuth, Germany YAROSLAV RYABOV, Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Maryland Center of Biomolecular Structure and Organization, University of Maryland, College Park, Maryland 20742-3360, USA ANATOLI SERGHEI, Fakultat fur Physik und Geowissenschaften, Universita¨t Leipzig, 04103 Leipzig, Germany SERGEY V. TITOV, Institute of Radio Engineering and Electronics of the Russian Academy of Seciences, Fryazino, Moscow Region, 141190, Russian Federation BRUCE J. WEST, Mathematical & Information Sciences Directorate, U.S. Army Research Ofﬁce, Research Triangle Park, North Carolina 27709, USA

INTRODUCTION Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field. STUART A. RICE

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PREFACE Fractals, Diffusion, and Relaxation in Disordered Complex Systems, which is the subject of the present anthology, may be said to have evolved in two stages: (1) in the course of conversations with Stuart Rice during a remarkably pleasant lunch at the University of Chicago following the Indianapolis meeting of the American Physical Society in March 2002 and (2) following the Royal Irish Academy Conference on Diffusion and Relaxation in Disordered Fractal Systems held in Dublin in September 2002 [1]. During each of these meetings, the necessity of reviewing the progress both experimental and theoretical which has been made in our understanding of physical systems with relaxation differing substantially from exponential behavior was recognized. Furthermore, it was considered that the Advances in Chemical Physics, in line with its stated aspirations and with its wide circulation, would provide an ideal means of attaining this goal. For the best part of three centuries the fractional calculus constituted a subject area mainly of interest to mathematicians. Indeed many great mathematicians such as Leibniz, L’Hoˆpital, Euler, Fourier, Abel, Liouville, Weierstrass, Riemann, Letnikov, Wiener, Le´vy, and Hardy, to name but a few, have contributed to its development (for a historical survey see Ref. 2). In contrast, applications of fractional calculus in other branches of science have appeared only sporadically—for example, the application to the propagation of disturbances on transmission lines in the context of Heaviside’s operational calculus and Kohlrausch’s stretched exponential decay law [2,14]. However, the situation radically changed toward the end of the last century following the appearance of the famous books of Benoit Mandelbrot on fractals [3]. Thus, over the past few decades, the fractional calculus has no longer been restricted to the realm of pure mathematics and probability theory [2,4]. Indeed many scientists have discovered that the behavior of a variety of complex systems (such as glasses, liquid crystals, polymers, proteins, biopolymers, living organisms, or even ecosystems) may be successfully described by fractional calculus; thus it appears that complex systems governed by fractional differential equations play a dominant role in both the exact and life sciences [5]. In particular in the context of applications in physics and chemistry, the fractional calculus allows one to describe complex systems exhibiting anomalous relaxation behavior in much the same way as the normal relaxation of simple systems [6]. Examples include charge transport in amorphous semiconductors, the spread ix

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of contaminants in underground water, relaxation in polymer systems, and tracer dynamics in both polymer networks and arrays of convection rolls, and so on [6]. In general, the diffusion and relaxation processes in such complex systems no longer follow Gaussian statistics so that the temporal evolution of these systems deviates from the corresponding standard laws (where the mean-square displacement of a particle is proportional to the time between observations) for normal diffusion such as exhibited by classical Brownian particles. Furthermore, following the development for complex systems of higher-order experimental resolutions or via a combination of different probe techniques, the deviations from the classical diffusion and relaxation laws have become ever more apparent. Thus the ever larger data windows that are becoming accessible bring ever more refinement to the experimental data [5], with the result that fractional diffusion and kinetic equations have become extremely powerful tools for the description of anomalous relaxation and diffusion processes in such systems. In the present anthology we have tried to present a comprehensive account of the present state of the subject. It is obvious, however, that we cannot survey completely such an enormous area of modern research, and inevitably many important topics will have been omitted. In order to remedy this defect, we remark that the interested reader can find additional information concerning anomalous diffusion and relaxation and applications of fractional calculus in physics, chemistry, biology, radio engineering, and so on, in various review articles and books, a selection of which is given in Refs. 5–23. Roughly speaking, the contents of the two-volume anthology may be divided into four experimental and seven theoretical chapters that may be described as follows. Chapter 1, ‘‘Dielectric Relaxation Phenomena in Complex Materials,’’ by Y. Feldman, A. Puzenko, and Y. Ryabov, concerns dielectric spectroscopy studies of the structure, dynamics, and macroscopic behavior of materials, which may broadly be described by the generic term complex systems. Complex systems constitute an almost universal class of materials including associated liquids, polymers, biomolecules, colloids, porous materials, doped ferroelectric crystals, and so on. These systems are characterized by a new ‘‘mesoscopic’’ length scale, intermediate between molecular and macroscopic. The mesoscopic structures of complex systems typically arise from fluctuations or competing interactions and exhibit a rich variety of static and dynamic behavior. This growing field is interdisciplinary; it complements solid-state and statistical physics, and it overlaps considerably with chemistry, chemical engineering, materials science, and even biology. A common theme in complex systems is that while such materials are disordered on the molecular scale and homogeneous on the macroscopic scale, they usually possess a certain degree of order on a intermediate, or mesocopic, scale due to

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the delicate balance of interaction and thermal effects. The authors demonstrate how dielectric spectroscopy studies of complex systems can be applied to determine both their structures and dynamics, how they both arise, and how both may influence the macroscopic behavior. The glass transition is an unsolved problem of condensed mater physics. This question is addressed in chapter 2 by T. Blochowicz, A. Brodin, and E. Ro¨ssler, entitled ‘‘Evolution of the Dynamic Susceptibility in Supercooled Liquids and Glasses.’’ The emergence of the mode coupling theory of the glass transition has prompted the compilation of a large body of information on the glass transition phenomenon as well as on the glassy state that is reviewed in this contribution. Thus this chapter focuses on describing the evolution of the dynamic susceptibility; that is, its characteristic changes while supercooling a molecular liquid. The authors provide information on the relevant molecular dynamics, and a comparison between experiment and theory is given. The phenomenon is essentially addressed from an experimental point of view, by simultaneously discussing the results from three different probe techniques, namely quasi-elastic light scattering, dielectric spectroscopy, and nuclear magnetic resonance spectroscopy. The application of each of the three methods allows one to investigate the dynamics in the 0- to 1-THz frequency range. The crossover from liquid dynamics at the highest temperatures to glassy dynamics at moderate temperatures as well as the crossover to solid-state behavior at the lowest temperatures near the glass transition temperature, is described in detail. In addition, some remarks on the evolution of the susceptibility down to cryogenic temperatures are given. The lesson to be drawn from this contribution is that an understanding of the dynamics of disordered systems can only be achieved by joint application of the various techniques covering a large frequency range. In many complex systems such as glasses, polymers, and proteins, temporal evolutions differ as we have seen from the conventional exponential decay laws (and are often much slower). Very slowly relaxing systems remain out of equilibrium over very long times, and they display aging effects so that the time scale of response and correlation functions increases with the age of the system (i.e., the time elapsed since its preparation): Older systems relax more slowly than younger ones. Chapter 3 by N. Pottier, entitled ‘‘Slow Relaxation, Anomalous Diffusion, and Aging in Equilibrated or Nonequilibrated Environments,’’ describes recent developments in the physics of slowly relaxing out of equilibrium systems. Questions specifically related to out-of-equilibrium dynamics, such as (1) aging effects and (2) their description by means of an effective temperature, are discussed in the framework of a simple model. A system well adapted to the analysis of these concepts is a diffusing particle in contact with an

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environment, which is either itself in equilibrium (thermal bath) or out of equilibrium (aging medium). In an aging environment, the diffusing particle acts as a thermometer: Independent measurements, at the same age of the medium, of the particle mobility and mean-square displacement yield the effective temperature of the medium. Time-dependent fluctuations in the spectra of individual molecules appear in many single-molecule experiments. Since the dynamics of a single molecule is typically strongly coupled to the dynamics of the local environment of that molecule, it is not unusual that the time trace of the intensity of a single molecule should exhibit stochastic behavior. It is frequently assumed that the process of photon emission is stationary and ergodic. In contrast, the correlation function of single nanocrystals (or quantum dots) is nonstationary and nonergodic; thus these systems exhibit statistical behaviour very different from other single emitting objects. In this context, G. Margolin, V. Protasenko, M. Kuno, and E. Barkai in Chapter 4, entitted ‘‘Power-Law Blinking Quantum Dots: Stochastic and Physical Models.’’ discuss simple models that may explain the nonergodic behaviour of nanocrystals. The authors use a stochastic model to discuss statistical properties of blinking nanocrystals and to illustrate the concept of non ergodicity and aging. They study intensity correlation functions and discuss ensemble average correlation functions for both capped and uncapped nanocrystals. Different modes of aging appear; that is, a nonvanishing dependence of the correlation functions on the age of the system exists, and this dependence has different functional forms in each of the two cases. The authors also discuss nonergodicity of intensity fluctuations for capped nanocrystals, comparing trajectory (time) and ensemble intensity mean values and correlation functions. They analyze experimental data and show that due to weak ergodicity breaking, the time-averaged intensity of blinking dots is a random variable even for long measurement times. The distribution of the time-averaged intensity is not centered around the ensemble-averaged intensity; instead the authors find very large fluctuations, in good agreement with the predictions of stochastic theory. The main purpose of Chapter 5 by P. Grigolini, ‘‘The Continuous-Time Random Walk Versus the Generalized Master Equation,’’ is to show that the interpretation of certain experimental results concerning the spectroscopy of blinking quantum dots and single molecules requires new theoretical methods. He argues that traditional methods of statistical mechanics, based on either the quantum or the classical Liouville equation—and thus based on densities—must be replaced by the continuous-time random walk model introduced by Montroll and Weiss in 1965. To justify this change, the author reviews the recent work done in deriving Le´vy anomalous diffusion from a Liouville equation formalism. He demonstrates that this method, which is

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satisfactory for Poisson statistics, cannot reproduce the numerical and experimental results in the non-Poisson case. Using the continuous-time random walk formalism, the author determines the generalized master equations that should arise from the Liouville method and also proves that such equation are characterized by aging. He shows that, in spite of making a given generalized master equation totally equivalent to the continuoustime random walk picture, an external field perturbing the generalized master equations yields effects distinctly different from those obtained by applying the same external field in the continuous-time random walk picture. Here there is no need for the reader to know a priori the projection approach to the generalized master equations, and the fundamentals of continuous-time random walk calculus needed are included in the chapter. Thus this chapter aims at being an elementary introduction to these techniques and thus will be accessible to both researchers and graduate and undergraduate students with no special knowledge of the formalism. In Chapter 6, entitled ‘‘Fractal Physiology, Complexity, and the Fractional Calculus,’’ B. J. West concentrates on describing the new area of medicine called fractal physiology and focuses on the complexity of the human body and the characterization of that complexity through fractal measures. It is demonstrated that not only various anatomical structures within the human body—such as the convoluted surface of the brain, the lining of the bowel, neural networks, and placenta—are fractal, but also the output of many other dynamical physiological systems. For example, the time series for the interbeat intervals of the heart, interbreath intervals, and interstride intervals have all been shown to be fractal or multifractal statistical processes. Consequently, the fractal dimension turns out to be a significantly better indicator of health than more traditional measures, such as heart rate, breathing rate, and average gait. The observation that human physiology is fractal was first made by the author and his collaborators in the 1980s, based on the analysis of the data sets mentioned above. Subsequently, it was determined that the appropriate methodology for describing the dynamics of fractal time series is the fractional calculus, using either the fractional Langevin equation or the fractional diffusion equation, both of which are discussed in a biomedical context. The general goal of this chapter is to understand how complex phenomena in human physiology can be faithfully described using dynamical models involving fractional stochastic differential equations. Now various structures—for example, aggregates of particles in colloids, certain binary solutions, polymers, composites, and so on—can be conceived as fractal. Materials with a fractal structure belong to a wide class of inhomogeneous media and may exhibit properties differing from those of uniform matter, like crystals, ordinary composites, or homogeneous

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fluids. Thus in Chapter 7 by V. Novikov, entitled ‘‘Physical Properties of Fractal Structures,’’ hierarchical structure models are applied to study the dielectric, conductive, and elastic properties of inhomogeneous media with a chaotic, fractal structure. The power of the fractional calculus is demonstrated using as example the derivation of certain known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimensionality of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of inhomogeneous media exhibiting nonexponential relaxation behavior is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. Different relaxation functions are derived assuming that the actual (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. It is predicted that at times shorter than the relaxation time at the lowest (primitive) self-similarity level, the relaxation should be of a classical, Debye-like type, irrespective of the pattern of nonclassical relaxation at longer times. The material described in this chapter can be used in the analysis of the frequency dependence of the dielectric permittivity, the conductivity, and the elastic parameters of various materials. Providing both a critical evaluation of characterization methods and a quantitative description of composition-dependent properties, the material surveyed is of particular interest to researchers in materials and polymer science. Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled ‘‘Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems,’’ provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker–Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole–Cole, Cole–Davidson, and Havriliak–Negami equations of anomalous dielectric relaxation from a microscopic model based on a

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kinetic equation, just as the Debye model. These kinetic equations are obtained from a generalization of the noninertial Fokker–Planck equation of conventional Brownian motion to fractional kinetics governed by the Cole– Cole, Cole–Davidson, and Havriliak–Negami relaxation mechanisms. As particular examples, approximate solutions of the fractional diffusion equation are derived for anomalous noninertial rotational diffusion in various potentials. It is shown that a knowledge of the effective relaxation times for normal rotational diffusion is sufficient to predict accurately the anomalous dielectric relaxation behavior of the system for all time scales of interest. Furthermore, the inertia-corrected Debye model of rotational Brownian motion of polar molecules is generalized to fractional dynamics (anomalous diffusion) using the fractional Klein–Kramers equation. The result can be considered as a generalization of the solution for the normal Brownian motion in a periodic cosine potential to fractional dynamics (giving rise to anomalous diffusion) and also represents a generalization of Fro¨hlich’s model of relaxation over a potential barrier. Chapter 9 by A.V. Chechkin, V.Y. Gonchar, J. Klafter, and R. Metzler, entitled ‘‘Fundamentals of Le´vy Flight Processes,’’ reviews recent developments in the fractional dynamics of Le´vy flights under the influence of an external force field and for non trivial boundary conditions—in particular, first passage time problems. The Le´vy flights are formulated in terms of a space-fractional Fokker–Planck equation, in which the usual Laplacian is replaced by the Riesz–Weyl fractional operator. The authors discuss the intriguing behavior of this type of random process in external fields; for example, for potentials of harmonic or softer types, the variance diverges and the stationary solution has the same Le´vy index as the external noise. In contrast, for steeper than harmonic potentials, the solution leaves the basin of attraction of Le´vy stable densities, and multimodal structures appear. The first passage time problem of Le´vy flights exhibits a universal character in the sense that the force-free first passage time density exhibits Sparre Andersen universality. This is discussed in detail, and it is compared to the problem of first arrival in Le´vy flights. The authors also address the question of the validity of Le´vy flights as a description of a physical system due to their diverging variance—for example, arguing that for a massive particle, dissipative nonlinearities may lead to a finite variance. Now on decreasing temperature or increasing pressure a noncrystallizing liquid will vitrify; that is, the structural relaxation time, , becomes so long that the system cannot attain an equilibrium configuration in the time available. Such theories as exist, including the well-known free volume and configurational entropy models, explain the glass transition by invoking a single quantity governing . Thus the dispersion of the structural relaxation is either so not addressed at all or else derived merely as afterthought and so

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is independent of . Thus, in these models the time dependence of the relaxation process bears no fundamental relation to and its dynamical properties. In Chapter 10 by K. Ngai, R. Casalini, S. Capaccioli, M. Paluch, and C. M. Roland, entitled ‘‘Dispersion of the Structural Relaxation and the Vitrification of Liquids,’’ the authors show from disparate experimental data that the dispersion (i.e., time dependence of the relaxation time or distribution of relaxation times) of the structural relaxation originating from many-molecule dynamics is a fundamental parameter governing and so controls its various properties. Large bodies of experimental data are presented or cited in order to support this conclusion in a convincing fashion. It appears that without considering dispersion as a fundamental physical entity at the outset of any theory of vitrification, many general experimental features of the molecular dynamics of supercooled liquids will remain unexplained. Glass-forming systems have been studied for decades using a variety of experimental tools measuring microscopic or macroscopic physical quantities. Thus the conjecture that the glass transition has an inherent length scale has led to numerous studies on confined glassy dynamics. In this context, thin polymer films are of special interest. Chapter 11 by F. Kremer and A. Serghei, entitled ‘‘Molecular Dynamics in Thin Polymer films,’’ contributes to this discussion. The authors address from an experimental point of view many interesting topics such as ensuring both reproducible preparation and reproducible measurements of thin polymer films, the influence of the molecular architecture of polymers on their dynamics in thin layers, the effect of confinement in thin polymer films giving rise to novel dynamic modes, methods for the determination of the glass transition temperature, and so on. The Guest Editors and authors are very grateful to the Series Editior, Stuart A. Rice, for the opportunity to produce this anthology. We would like to thank Dr. Sergey V. Titov and Ms Christine Moore for their excellent help in the preparation of the manuscripts. We would also like to thank Dr. David Burns and Dr. Michael Milligan of USAF, EOARD London for facilitating Window on Science visits to the United States during the course of which this project was conceived, as well as Professor Werner Blau of Trinity College Dublin for financial support from the HEA Ireland PRTLI Nanomaterials project and the Trinity College Dublin Trust. November 2005 WILLIAM T. COFFEY YURI P. KALMYKOV Dublin and Perpignan

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References 1. Proceedings available as special issue (Diffusion and Relaxation in Disordered Complex Systems) of Journal of Molecular Liquids 114, No. 1–3 (2004), guest editor W. T. Coffey. 2. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974. 3. B. B. Mandelbrot, Fractals: Form, Chance and Dimension, Freeman, San Francisco, 1982; The Fractal Geometry of Nature, Freeman, San Francisco, 1982. 4. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. 5. I. Sokolov, J. Klafter, and A. Blumen, Fractional Kinetics. Physics Today, Nov. 2002, p. 48. 6. R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, 1 (2000). 7. C. J. F. Bo¨ttcher and P. Bordewijk, Theory of Electric Polarization, Vol. 2, Elsevier, Amsterdam, 1973. 8. J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 195, 127 (1990). 9. N. G. McCrum, B. E. Read, and G. Williams, Anelastic and Dielectric Effects in Polymeric Solids Dover, New York, 1991. 10. M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Strange Kinetics, Nature 363, 31 (1993). 11. R. Richert and Blumen, eds., Disorder Effects on Relaxation Processes, Springer-Verlag, Berlin, 1994. 12. A. Bunde and S. Havlin, eds., Fractals in Disordered Systems, Springer-Verlag, Berlin, 1996. 13. A. K. Jonscher, Universal Relaxation Law, Chelsea Dielectric Press, London, 1996. 14. W. Paul and J. Baschnagel, Stochastic Processes from Physics to Finance, Springer-Verlag, Berlin, 1999. 15. R. Hilfer, ed., Applications of Fractional Calculus in Physics, World Scientiﬁc, River Edge, N J, 2000. 16. E. Donth, The Glass Transition: Relaxation Dynamics in Liquids and Disordered Materials, Springer-Verlag, Berlin, 2000. 17. G. M. Zaslavsky, Chaos, fractional kinetics and anomalous transport. Phys. Rep. 371, 461 (2002). 18. F. Kremer and A. Scho¨nhals, eds., Broadband Dielectric Spectroscopy, Springer, Berlin, 2002. 19. A. A. Potapov, Fractals in Radiophysics and Electromagnetic Detections, Logos, Moscow, 2002. 20. B. J. West, M. Bologna, and Grigonlini, Physics of Fractal Operators, Springer, New York, 2003. 21. W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron, The Langevin Equation, 2nd ed, World Scientiﬁc, Singapore, 2004. 22. R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent development in the description of anomalous transport by fractional dynamics. J. Phys. A: Math Gen 37, 1505 (2004) 23. J. Klafter and I. Sokolov, Anomalous diffusion spreads its wings. Physics Today, Aug. 2005 p. 29.

CONTENTS PART B CHAPTER 6 FRACTAL PHYSIOLOGY, COMPLEXITY, FRACTIONAL CALCULUS

AND THE

1

By Bruce J. West CHAPTER 7

PHYSICAL PROPERTIES OF FRACTAL STRUCTURES

93

By Vitaly V. Novikov CHAPTER 8 FRACTIONAL ROTATIONAL DIFFUSION AND ANOMALOUS DIELECTRIC RELAXATION IN DIPOLE SYSTEMS

285

By William T. Coffey, Yuri P. Kalmykov, and Sergey V. Titov CHAPTER 9

FUNDAMENTALS OF LE´VY FLIGHT PROCESSES

439

By Aleksei V. Chechkin, Vsevolod Y. Gonchar, Joseph Klafter, and Ralf Metzler CHAPTER 10 DISPERSION OF THE STRUCTURAL RELAXATION AND THE VITRIFICATION OF LIQUIDS

497

By Kia L. Ngai, Riccardo Casalini, Simone Capaccioli, Marian Paluch, and C.M. Roland CHAPTER 11

MOLECULAR DYNAMICS IN THIN POLYMER FILMS

595

By Friedrich Kremer and Anatoli Serghei

Author Index

633

Subject Index

671

xix

CONTENTS PART A CHAPTER 1

DIELECTIC RELAXATION PHENOMENA IN COMPLEX MATERIALS

1

By Yuri Feldman, Alexander Puzenko, and Yaroslav Ryabov CHAPTER 2 EVOLUTION OF THE DYNAMIC SUSCEPTIBILITY IN SUPER COOLED LIQUIDS AND GLASSES

127

By Thomas Blochowicz, Alexander Brodin, and Ernst Rossler CHAPTER 3 SLOW RELAXATION, ANOMALOUS DIFFUSION, AND AGING IN EQUILIBRATED OR NONEQUILIBRATED ENVIRONMENTS

257

By Noe¨lle Pottier CHAPTER 4 POWER-LAW BLINKING QUANTUM DOTS: STOCHASTIC PHYSICAL MODELS

AND

327

By Gennady Margolin, Vladimir Protasenko, Masaru Kuno, and Eli Barkai CHAPTER 5 THE CONTINUOUS-TIME RANDOM WALK VERSUS GENERALIZED MASTER EQUATION

THE

357

By Paolo Grigolini

Author Index

475

Subject Index

513

xxi

CHAPTER 6 FRACTAL PHYSIOLOGY, COMPLEXITY, AND THE FRACTIONAL CALCULUS BRUCE J. WEST Mathematical & Information Sciences Directorate, U.S. Army Research Office, Research Triangle Park, NC 27709, USA

CONTENTS I. Introduction II. Scaling in Physiological Time Series A. Allometric Aggregation Data Analysis B. Fractal Heartbeats C. Fractal Breathing D. Fractal Gait E. Fractal Neurons III. Dynamical Models of Scaling A. Scaling in Time Series 1. Simple Random Walks and Scaling 2. Fractional Random Walks and Scaling 3. Various Inverse B. Dichotomous Fluctuations with Memory 1. The Exact Solution 2. Early Time Behavior 3. Late Time Behavior C. Fractals, Multifractals, and Data Processing 1. Multifractal Special 2. Diffusion Entropy Analysis (DEA) IV. Fractional Dynamics A. Fractional Calculus 1. Derivative of a Fractal Function 2. Fractional Brownian Motion B. Fractional Langevin Equations 1. Physical/Physiological Models C. Fractional Diffusion Equations D. Langevin Equation with Le´vy Statistics

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

1

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bruce j. west

V. Summary, Conclusions, and Speculations References

I.

INTRODUCTION

The title of this chapter targets two central themes. The ﬁrst theme is a modern view of physiology that explicitly takes into account the complexity of living systems, since physiology is that branch of biology that deals with the functions and activities of life and of living matter such as organs, tissues, or cells. Complexity in this context incorporates the recent advances in physiology concerned with the applications of the concepts ranging from fractal geometry, fractal statistics, and nonlinear dynamics to the formation of a new kind of understanding in the life sciences. The second theme has to do with a parallel development on understanding the dynamics of fractal processes. For a number of years the study of fractals was restricted to the determination of the fractal dimension of structure—in particular, the static structure of objects and the stationary structure of time series. However, now we explore the dynamics of fractal processes using the fractional calculus, applying this dynamical approach to both regular and stochastic processes. In our discussion we motivate the applications of the fractional calculus using physiological time series. The fractal concept was formally introduced into the physical sciences by Benoit Mandelbrot over 20 years ago and has since then captured the imagination of a generation of scientists. Mandelbrot had, of course, been working on the development of the idea for over a decade before he was ﬁnally willing to expose his brainchild to the scrutiny of the scientiﬁc community at large. His monograph [1] brought together mathematical, experimental, and physical arguments that undermined the traditional picture of the physical world. It has been accepted that celestial mechanics and physical phenomena are, by and large, described by smooth, continuous, and unique functions, since before the time of Lagrange (1736–1813). This belief is part of the conceptual infrastructure of the physical sciences. The changes in physical processes are modeled by systems of dynamical equations, and the solutions to such equations are continuous and differentiable at all but a ﬁnite number of points. Therefore the phenomena being described by these equations were thought to have these properties of continuity and differentiability as well. Thus, the solutions to the equations of motion such as the Euler–Lagrange equations, or Hamilton’s equations, are analytic functions, and such functions were thought to represent physical phenomena in general. At the turn of the last century, there were two opposing points of view in physics regarding continuity: those held by the atomists and those held by the anti-atomists. The latter camp believed in the continuity of nature and saw no reason why matter should stop being divisible at the level of the atom and should, they reasoned, continue indeﬁnitely to smaller and smaller scales. The

fractal physiology, complexity, and the fractional calculus

3

atomists, on the other hand, with the successes of the periodic table and the kinetic theory of gases, had Boltzmann as their chief proponent. Boltzmann was such an ‘‘extreme’’ atomist that he did not even accept the continuity of time. In his St. Louis lecture in 1904 he stated [2] the following: Perhaps our equations are only very close approximations to average values that are made up of much finer elements and are not strictly differentiable.

It is the atomist’s view of the classical microscopic world and its inﬂuence on the macroscopic world that we endorse in this chapter. From the phenomenological side, Mandelbrot called the accuracy of the traditional perspective into question, by pointing to the failure of the equations of physics to explain such familiar phenomena as turbulence and phase transitions. In his books [1,3], Mandelbrot catalogued and described dozens of physical, social, and biological phenomena that cannot be properly described using the familiar tenets of dynamics from physics. The functions required to explain these complex phenomena have properties that for 100 years had been thought to be mathematically pathological. Mandelbrot argued that, rather than being pathological, these functions capture essential properties of reality and are therefore better descriptors of the physical world than the traditional analytic functions of theoretical physics. Living organisms are immeasurably more complicated than inanimate objects, so we do not have available fundamental laws and principles governing biological phenomena equivalent to those in physics. Some may object to this harsh characterization, but there are no equivalents of Newton’s Laws, Maxwell’s equations, and Boltzmann’s Principle in physiology. Part of the goal of biophysics, in fact, is to seek out and establish the existence of such biological laws and relate them to known physical laws, so that both the physical and biological aspects of living matter can be better understood. In this chapter our aim is much more modest than identifying such fundamental biophysical principles; it is merely to present a strategy for understanding a diverse set of complex phenomena in physiology and suggest that this strategy reveals an underlying symmetry that can be exploited. Schro¨dinger, in his book What is Life? [4], laid out his understanding of the connection between the world of microscopic and macroscopic based on the principles of equilibrium statistical physics. In that discussion he asked why atoms are so small relative to the dimension of the human body. The answer to this question is both immediate and profound. The high level of organization necessary for life is only possible in a macroscopic system; otherwise the order would be destroyed by microscopic (thermal) ﬂuctuations. A living system must be sufﬁciently large to maintain its integrity in the presence of thermal ﬂuctuations that disrupt its constitutive elements. Thus, macroscopic phenomena are characterized by averages over ensemble distribution functions characterizing

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bruce j. west

microscopic ﬂuctuations. Consequently, any strategy for understanding physiology must be based on the probabilistic description of complex phenomena and, as we shall see, on our understanding of phenomena lacking characteristic scales. There are three types of fractals that appear in the life sciences: geometrical fractals, which determine the spatial properties of the tree-like structures of the mammalian lung, arterial, and venous systems and other ramiﬁed structures [5]; statistical fractals, which determine the properties of the distribution of intervals in the beating of the mammalian heart, in breathing, in walking, and in the ﬁring of certain neurons; and ﬁnally dynamical fractals, which determine the dynamical properties of systems having a large number of characteristic time scales. In the complex systems found in physiology, the distinctions between these three kinds of fractals blur, but we focus our attention on the dynamical rather than the geometrical fractals, in part, because the latter have been reviewed in a number of places and we have little to add to that understanding. In this chapter we lay the foundation for how such concepts as complexity, fractals, diverging moments, nonlinear dynamics, and other related mathematical topics are used in understanding physiology. Of course, a number of books have been written about any one of these ideas—books for the research expert, books for the informed teacher, books for the struggling graduate student, and books for the intelligent lay person. Different authors stress different characteristics of complex phenomena, from the erratic data collected by clinical researchers to the ﬂuctuations generated by deterministic dynamical equations used to model such systems. Some authors have painted with broad brushstrokes, indicating only the panorama that these concepts reveal to us, whereas others have sketched with painstaking detail the structure of such phenomena and have greatly enriched those that could follow the arguments. Herein we view our efforts midway between the two because fractal physiology is still a work in progress and much of what we present may prove to be irrelevant, whereas some of it might be even more signiﬁcant than we can now appreciate. II.

SCALING IN PHYSIOLOGICAL TIME SERIES

We begin our discussion of physiological processes with an empirical study of the scaling behavior of time series obtained from the quantitative measurements of certain physiologic systems—for example, the cardiovascular and respiratory systems. We pursue this approach in order to convince the reader of the ubiquity of scaling in physiological time series. Once this property is established, we turn to the mathematical modeling of the mechanisms that generate such scaling. The attention is on scaling statistics because physiological measures are usually given to us in the form of time series. Whether it is the electrocardiogram (ECG) for the beating heart from which the interbeat intervals are extracted to determine heart rate variability (HRV) [6–8], the electrogastrogram (EGG) for gastric activity of

fractal physiology, complexity, and the fractional calculus

5

the stomach in which the contraction intervals are used to determine the gastric rate variability (GRV) [9], the stride intervals during walking that determines the stride rate variability (SRV) [10–13], or the interbreath interval which determines the breathing rate variability (BRV) [14], they all appear at ﬁrst sight to be random processes with no underlying pattern. However, upon processing the data, they reveal long-term memory indicative of fractal time series, as we review in Section III. A.

Allometric Aggregation Data Analysis

Note that the term scaling denotes a power-law relation between two variables x and y, y ¼ Axa

ð1Þ

as Barenblatt [15] explained in his excellent inaugural lecture delivered before the University of Cambridge on May 3, 1993. He points out that such scaling laws are not merely special cases of more general relations; they never appear by accident; they always reveal self-similarity, a very important property of the phenomenon being studied. In biology, Eq. (1) is historically referred to as an allometric relation between the two observables. Allometric relations are not new in science. Such relations were introduced into biology in the nineteenth century. Typically, an allometric relation interrelates two properties of a given organism. For example, the total mass of a deer y is proportional to the mass of the deer’s antlers x raised to a speciﬁc power a. Thus, on doubly logarithmic graph paper, this relation would yield a straight line with a slope given by the power-law index. Huxley summarized the experimental basis for this relation in his 1931 book, Problems of Relative Growth [16], and developed the mathematics to describe and explain allometric growth laws. In biological systems, he reasoned, two parts of an organism grow at different rates, but the rates are proportional to one another. Consequently, how rapidly one part of the organism grows can be related to how rapidly the other part of the organism grows, and the ratio of the two rates is constant. In this section the notion of an allometric relation is generalized to include measures of time series. In this view, y is interpreted to be the variance and x the average value of the quantity being measured. The fact that these two central measures of a time series satisfy an allometric relation implies that the underlying time series is a fractal random process and therefore scales. It was ﬁrst determined empirically that certain statistical data satisfy a power-law relation of the form given by Taylor [17] in Eq. (1), and this is where we begin our discussion of the allometric aggregation method of data analysis. Taylor was a scientist interested in biological speciation. For example, he was curious about how many species of beetle can be found over a given area of

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bruce j. west

land. He answered this question by sectioning off a large ﬁeld into plots and in each plot sampling the soil for the variety of beetles that were present. This enabled him to determine the distribution in the number of new species of beetle spatially distributed across the ﬁeld. From the distribution he could then and the variance in the number of determine the average number of species X species Var X. After this ﬁrst calculation, he partitioned his ﬁeld into smaller plots and redid his sampling, again determining the mean and standard deviation in the number of species at this increased resolution. This process was repeated a number of times, yielding a set of values of means and variances. In the ecological literature a graph of the logarithm of the variance versus the logarithm of the average value is called a power curve, which is linear in the logarithms of the two variables, and b is the slope of the curve. The algebraic form of the relation between the variance and mean is b Var X ¼ aX

ð2Þ

where the two parameters a and b determine how the variance and mean are related to one another. Taylor was able to exploit the curves obtained from data in a number of ways using the slope and intercept parameters [17]. If the slope of the curve and the intercept are both equal to one, a ¼ b ¼ 1, then the variance and mean are equal to one another. This equality is only true for a Poisson distribution, which, when it occurred, allowed him to interpret the number of species as being randomly distributed over the ﬁeld, with the number of species in any one plot being completely independent of the number of species in any other plot. If, however, the slope of the curve was less than unity, the number of species appearing in the plots was interpreted to be quite regular. The spatial regularity of the number of species, in this case, was compared with the trees in an orchard and given the name evenness. Finally, if the slope of the variance versus mean curve was greater than one, the number of new species was interpreted as being clustered in space, like disjoint herds of sheep grazing in a meadow. Of particular interest to us here was the mechanism that Taylor and Taylor [18] postulated to account for the experimentally observed allometric relation: We would argue that all spatial dispositions can legitimately be regarded as resulting from the balance between two fundamental antithetical sets of behaviour always present between individuals. These are, repulsion behaviour, which results from the selection pressure for individuals to maximise their resources and hence to separate, and attraction behaviour, which results from the selection pressure to make the maximum use of available resources and hence to congregate wherever these resources are currently most abundant.

fractal physiology, complexity, and the fractional calculus

7

Consequently, they postulated that it is the conﬂict between the attraction and repulsion, migration and congregation, which produces the interdependence (scaling) of the spatial variance and the average population density. We can now interpret Taylor’s observations more completely because the kind of clustering he observed in the spatial distribution of species number, when the slope of the power curve is greater than one, is consistent with an asymptotic inverse power-law distribution of the underlying data set. Furthermore, the clustering or clumping of events is due to the fractal nature of the underlying dynamics. Willis, some 40 years before Taylor, established the inverse power-law form of the number of species belonging to a given genus [19]. Willis used an argument associating the number of species with the size of the area they inhabit. It was not until the decade of the 1990s that it became clear to more than a handful of experts that the relationship between an underlying fractal process and its space ﬁlling character obeys a scaling law [1,3]. It is this scaling law that is reﬂected in the allometric relation between the variance and mean. It is possible to test the allometric relation of Taylor using computergenerated data. But before we do so, we note that Taylor and Woiwod [20] were able to extend the discussion from the stability of the population density in space, independent of time, to the stability of the population density in time, independent of space. Consequently, just as spatial stability, as measured by the variance, is a power function of the mean population density over a given area at all times, so too the temporal stability, as measured by the variance, is a power function of the mean population density over time at all locations. With this generalization in hand we can apply Taylor’s ideas to time series. The correlation of discrete time series data is here determined by grouping the data into aggregates of two or more of the original data points and calculating the mean and variance at each level of aggregation. Consider the jth data element of an aggregation of n-adjacent data points: ðnÞ

Yj

¼

n1 X

Ynjk

ð3Þ

k¼1

In terms of these new data the average is deﬁned by Y ðnÞ

1 X ðnÞ Y ¼ nY ð1Þ ½N=n j¼1 j ½N=n

ð4Þ

For example, when n ¼ 3 each value of the new variable, deﬁned by Eq. (3), consists of the sum of three non-overlapping original data points, and the number

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bruce j. west

of new data points is given by [N/3], where the brackets denote the closest integer value and N is the original number of data points. The variance, for a monofractal random time series, is similarly given by [21] Var Y ðnÞ ¼ n2H Var Y ð1Þ

ð5Þ

where the superscript (1) on the average variable indicates that it was determined using all the original data without aggregation, and the superscript (n) on the average variable indicates that it was determined using the aggregation of nadjacent data points and H is referred to as the Hurst exponent. Thus, comparing Eq. (5) with Eq. (4), we obtain the allometric relation given by Eq. (2): Var Y

ðnÞ

Y ðnÞ Y ð1Þ

2H

Y ð1Þ

¼ aY ðnÞ

b

ð6Þ

with the parameters given by the theoretical values a¼

Var Y ð1Þ ðY ð1Þ Þb

and

b ¼ 2H

ð7Þ

It is well established that the exponent in such scaling equations is related to the fractal dimension [21] D of the underlying time series by D ¼ 2 H, so that D ¼ 2 b=2

ð8Þ

A simple mono-fractal time series, therefore, satisﬁes the power-law relation of the allometric form given by Eq. (2). This allometric aggregation technique has been applied to a number of data sets implementing the method of linear regression using the equation log Var Y ðnÞ ¼ log a þ b log Y ðnÞ

ð9Þ

Fitting the parameters a and b in Eq. (9) to time series data gives the best leastsquares estimates of the parameters a and b in the allometric relation, respectively. We ﬁnd that the Gaussian distribution is a more useful exemplar for time series than is the Poisson distribution used by Taylor. In Fig. 1 we apply Eq. (9) to one million computer-generated data points with Gaussian statistics. The far left dot in Fig. 1 contains all the data in the calculation of the aggregated mean and variance so that n ¼ 1 in Eq. (9); the next point to the right in the ﬁgure

fractal physiology, complexity, and the fractional calculus

9

Figure 1. The logarithm of the variance is plotted versus the logarithm of the mean for the successive aggregation of 106 computer-generated random data points with Gaussian statistics. The slope of the curve is essentially one, determined by a linear regression using Eq. (9), so the fractal dimension of the time series is 1.5 using Eq. (8).

contains the nearest-neighbor data points added together to deﬁne a data set with 500,000 data points from which to calculate the aggregated mean and variance. Next we take the original data and add the three nearest-neighbor data points to deﬁne a data set with 333,333 data, and so on. The jth data element after aggregating n nearest-neighbor data points is given by Eq. (3). Consequently, this process of aggregating the data is equivalent to decreasing the resolution of the time series, and as the resolution is systematically decreased, the adopted measure, the relationship between the mean and variance, reveals an underlying property of the time series. The increase in the variance with increasing mean for increasing aggregation number shown in the ﬁgure is not an arbitrary pattern. The relationship indicates that the aggregated uncorrelated data points are interconnected. The original data points are not necessarily correlated, but the addition of data in the aggregation process induces a correlation, one that is completely predictable. The induced correlation is linear if the original data are uncorrelated, but not linear if the original data are correlated. The aggregated variance versus the aggregated mean falls along a straight line in Fig. 1 with a slope of b ¼ 1 for the uncorrelated random process with computer-generated Gaussian statistics. Therefore, in the case of Gaussian statistics, with b ¼ 1, we have D ¼ 1:5 corresponding to the fractal dimension of Brownian motion. In the same way, a completely correlated time series would

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bruce j. west

have b ¼ 2, so that D ¼ 1. The fractal dimension for most time series fall somewhere between the two extremes; the closer the fractal dimension is to one, the more regular the process; the closer the fractal dimension is to 1.5, the more it is like an uncorrelated random process. The Gaussian process depicted in Fig. 1 is certainly a mono-fractal in the sense we have deﬁned it here; that is, the computer-generated time series is characterized by a single fractal dimension. We point out here that the allometric aggregation method is just one of many procedures designed to take advantage of the scaling properties of the central moments of time series. We refer to such methods collectively as ﬁnite variance statistical methods (FVSM). However, it should be emphasized that not all time series that scale have ﬁnite variance. Time series having Le´vy a-stable statistics exemplify processes with diverging variance, but they are described by probability density functions that scale. We review these matters after some discussion of the scaling properties of physiological time series. B.

Fractal Heartbeats

The mechanisms producing the observed variability in the size of a heart’s interbeat intervals apparently arise from a number of sources. The sinus node (the heart’s natural pacemaker) receives signals from the autonomic (involuntary) portion of the nervous system which has two major branches: the parasympathetic, whose stimulation decreases the ﬁring rate of the sinus node, and the sympathetic, whose stimulation increases the ﬁring rate of the sinus node pacemaker cells. The inﬂuence of these two branches produces a continual tug-of-war on the sinus node, one decreasing and the other increasing the heart rate. It has been suggested that it is this tug-of-war that produces the ﬂuctuations in the heart rate of healthy subjects, but alternate suggestions will be pursued subsequently. Consequently, heart rate variability (HRV) provides a window through which we can observe the heart’s ability to respond to normal disturbances that can affect its rhythm. The clinician focuses on retaining the balance in regulatory impulses from the vagus nerve and sympathetic nervous system and in this effort requires a robust measure of that balance [22]. A quantitative measure of HRV time series, such as the fractal dimension, serves this purpose. HRV time series have become very well known over the past two decades as a quantitative indicator of autonomic activity. The medical community became interested in developing such an indicator of heart rate because experiments indicated a relationship between lethal arrhythmias and such activity. The importance of HRV to medicine became widely apparent when a task force was formed by the Board of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology and was charged with the responsibility of developing the standards of measurement, physiological interpretation and clinical use of HRV. The task force published their ﬁndings in

fractal physiology, complexity, and the fractional calculus

11

1996 [22]. It is one of the few times that the members of such a task force were drawn from the ﬁelds of mathematics, engineering, physiology and clinical medicine in recognition of the complexity of the phenomenon they were charged to investigate and actually worked together. When heart rate is atypical, say 120 bpm, in contrast to its usual 60 bpm, quantifying the variation in heart rate becomes very important. The degree of deviation from normality is determined by the interpretation of the size of the variation and how it is used to identify associated patterns. There are a number of ways to assess HRV, some 16 in all, each related to scaling in one way or another and most being FVSM, but we do not want to go into all of them here. Instead we identify the quantity that is the most revealing of the nature of HRV, but again we shall not go into a detailed discussion of the many ways of estimating this quantity. Instead we present a single explanatory technique that is relatively straightforward and that allows us to introduce the measure of interest. We use the allometric aggregation technique on real data to relate the variance and mean, as we discussed. Now we apply the allometric aggregation approach to the beat-to-beat intervals shown in Fig. 2, a typical HRV time series for a healthy young adult male. The data points in the ﬁgure are connected to aid in visualizing how the time intervals between heartbeats are changing. It is evident that the variation in the time intervals between heartbeats is relatively small, the mean being 1.0 s and the standard deviation being 0.06 s. This modest variance supports the frequently used Heartbeat Intervals 1.1

Seconds

1.05 1 0.95 0.9 0.85 0.8 0

100

200 300 Interval Number

400

500

Figure 2. The time series of heartbeat intervals of a healthy young adult male is shown. It is clear that the variation in the time interval between beats is relatively modest, but certainly not negligible.

12

bruce j. west Interbeat Interval

0

Log standard deviation

–0.2

–0.4

regular

–0.6

random

–0.8

–1

–1.2 0

0.2

0.4

0.6 0.8 Log average

1

1.2

Figure 3. The logarithm of the standard deviation is plotted versus the logarithm of the average value for the heartbeat interval time series for a young adult male, using sequential values of the aggregation number. The solid line is the best ﬁt to the aggregated data points and yields a fractal dimension of D ¼ 1:24 midway between the curve for a regular process and that for an uncorrelated random process as indicated by the dashed curves.

medical term ‘‘normal sinus rhythm.’’ So the question of what is learned by applying the allometric aggregation technique to these data and constructing the standard deviation and mean as a function of aggregation number, becomes important. In Fig. 3 the logarithm of the standard deviation is plotted versus the logarithm of the mean value for the HRV time series depicted in Fig. 2. Note that we use the standard deviation in the ﬁgure and not the variance used in the discussion of Taylor’s Law. We use the standard deviation because we are primarily interested in whether the time series is fractal or not and not particularly in the actual value of the fractal dimension. At the leftmost position the data point indicates the standard deviation and mean, using all the data points. Moving from left to right, the next data point is constructed from the time series with two nearest-neighbor data points added together, and the procedure is repeated moving right until the rightmost data point has 20 nearestneighbor data points added together. The solid line is the best linear representation of the scaling and intercepts most of the data points with a positive slope of 0.76. We can see that the slope of the HRV data is midway

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between the dashed curves depicting an uncorrelated random process (slope ¼ 1/2) and one that is deterministically regular (slope ¼ 1). We emphasize that the conclusions we draw here are not from this single ﬁgure or set of data presented; these are only representative of a much larger body of work. The conclusions are based on a large number of similar observations [23,24] made using a variety of data processing techniques, all of which yield results consistent with the scaling of the HRV time series indicated in Fig. 3. So we conclude that the heartbeat intervals do not form an uncorrelated random sequence. Instead we see that the HRV time series is a statistical fractal, indicating that the heartbeats have a long-time memory. The implications of this long-time memory concerning the underlying physiological control system will be taken up later when we discuss the mathematical models of these phenomena. Phenomena obeying a scaling relation, such as shown for the HRV time series data in Fig. 3, are said to be self-similar. The fact that the standard deviation and mean values change as a function of aggregation number implies that the magnitudes of the standard deviation and mean values depend on the size of the ruler used to measure the time interval. Recall that this is one of the deﬁning characteristics of fractal curves; the length of the curve becomes inﬁnite as the size of the ruler goes to zero. The dependence of the mean and standard deviation on the ruler size, for a self-similar time series, implies that the statistical process is fractal and consequently deﬁnes a fractal dimension for the HRV time series. The average scaling exponent obtained by Peng et al. [25] for a group of 10 healthy subjects having a mean age of 44 years, using 10,000 data points for each subject, was a ¼ 0.19 for the difference in heartbeat interval time series. They interpreted this value to be consistent with a theoretical value of a ¼ 0, which they conjectured would be obtained for an inﬁnitely long time series. The latter scaling implies that the scaling exponent for the beat intervals themselves would be 1.0. However, all data sets are ﬁnite, and it was determined that the asymptotic scaling coefﬁcients for the heartbeat interval time series of healthy young adults lie in the interval 0.7 a 1.0. The value of the scaling coefﬁcient obtained using much shorter time series and the relatively simple processing technique of allometric aggregation is consistent with these results. We also comment on certain speculations made by Peng et al. [25] regarding their analysis of the set of HRV time series; speculations that have been supported by subsequent research. They suggested that the scaling behavior is adaptive for two reasons: (i) The long-range correlations serve as an organizing principle for highly complex, nonlinear processes that generate ﬂuctuations on a wide range of time scales.

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(ii) The lack of a characteristic scale helps prevent excessive mode-locking that would restrict the functional responsiveness of the organism. C.

Fractal Breathing

Breathing is a function of the lungs, whereby the body takes in oxygen and expels carbon dioxide. The smooth muscles in the bronchial tree are innervated by sympathetic and parasympathetic ﬁbers, much like the heart, and produce contractions in response to stimuli such as increased carbon dioxide, decreased oxygen, and deﬂation of the lungs. Fresh air is transported through some 20 generations of bifurcating airways of the lung, during inspiration, down to the alveoli in the last four generations of the bronchial tree. At this tiny scale there is a rich capillary network that interfaces with the bronchial tree for the purpose of exchanging gases with the blood. In this section we are interested in the dynamics of breathing; the apparently regular breathing as you sit quietly reading this paper. Here evolution’s design of the lung may be closely tied to way in which the lung carries out its function. It is not by accident that the cascading branches of the bronchial tree become smaller and smaller, nor is it good fortune alone that ties the dynamics of our every breath to this biological structure. We argue that, like the heart, the lung is made up of fractal processes, some dynamic and others now static. However, both kinds of processes lack a characteristic scale, and a simple argument establishes that such lack of scale has evolutionary advantages [26]. An early application of fractal analysis was made by Szeto et al. [27] to fetal lamb breathing. The changing patterns of breathing in 17 fetal lambs and the clusters of faster breathing rates, interspersed with periods of relative quiescence, suggested to them that the breathing process was self-similar. The physiological property of self-similarity implies that the structure of the mathematical function describing the time series is repeated on progressively ﬁner time scales. Clusters of faster rates were seen within the fetal breathing data, what Dawes et al. [28] called breathing episodes. When the time series were examined on even ﬁner time scales, clusters could be found within these clusters, and the signature of this scaling behavior emerged as an inverse power-law distribution of time intervals. Consequently, the fractal scaling was found to reside in the statistical properties of the ﬂuctuations and not in the geometrical properties of the dynamic variable. As with the heart, the variability of breathing rate using breath-to-breath time intervals is denoted by breathing rate variability (BRV), to maintain a consistent notation. Examples of HRV and BRV time series data on which scaling calculations are based are shown in Fig. 4. A typical BRV time series for a senior citizen at rest is shown at the top of the ﬁgure; the simultaneous HRV time series for the same person is depicted at the bottom of the ﬁgure. Because

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Figure 4. Typical time series from one of the 18 subjects in the study conducted by West et al. [14], while at rest, is shown for the interbreath intervals (BRV) and the interbeat intervals (HRV) time series.

heart rate is higher than respiration rate, in the same measurement epoch there is a factor of ﬁve more data for the HRV time series than there is for the BRV time series. These data were collected under the supervision of Dr. Richard Moon, the Director of the Hyperbaric Laboratory at Duke Medical Center. Looking at these two time series together, one is struck by how different they appear. It is not apparent that both physiological phenomena scale in essentially the same way, but they do [14]. The allometric aggregation method applied to the various time series obtained by West et al. [14] yields the typical results depicted in Fig. 5, where the logarithms of the aggregated variance versus the aggregated means are plotted for the HRV and BRV data depicted in Fig. 4. At the extreme left of each graph in Fig. 5 (m ¼ 1), all the data points are used to calculate the variance and mean, and at the extreme right the aggregated quantities use m ¼ 10 data points. Note that we stop the aggregation at 10 points because of

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bruce j. west Interbreath Interval

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Figure 5. A typical ﬁt to the aggregated variance versus the aggregated mean for BRV and HRV time series obtained by West et al. [14]. The points are calculated from the data and the solid curve is the best least-square ﬁt to the data. The upper curve is the ﬁt to the BRV data (slope ¼ 0.86), and the lower curve is the best ﬁt to the HRV data (slope ¼ 0.80). It is evident from these two graphs that the allometric relation given by Eq. (9) does indeed ﬁt both data sets extremely well and lies well within the regular and random boundaries, indicated by the dashed curves.

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the small number of data in the breathing sequence. The solid curve at the top of Fig. 5 is the best least-square ﬁt to the aggregated BRV data and has a slope of 0.86, which is the scaling index. A similar graph is constructed for the HRV data in the lower curve, where we obtain a slope of 0.80 for the scaling index. The scaling index of both the HRV and BRV time series increase with increasing levels of exercise, but the data are not shown here, mainly because we can’t show everything all the time. The 18 subjects in the experiment rode a stationary bicycle with various levels of load on the wheels to mimic cycling uphill. The breathing rate, breathing volume, and heart rate were monitored for each of the individuals in the study. The consistent result was that as the level of exercise increased, the amount of variability in both HRV and BRV decreased, indicating that the associated time series were becoming more ordered. This increase in scaling index was determined to be statistically signiﬁcant [14]. The scaling indices and fractal dimensions obtained from these curves are consistent with the results obtained by other researchers. Such observations regarding the self-similar nature of breathing time series have been used in a medical setting to produce a revolutionary way of utilizing mechanical ventilators. Historically, ventilators have been used to facilitate breathing after an operation and have a built-in frequency of ventilation. The single-frequency ventilator design has recently been challenged by Mutch et al. [29], who have used an inverse power-law spectrum of respiratory rate to drive a variable ventilator. They demonstrated that this way of supporting breathing produces an increase in arterial oxygenation over that produced by conventional control-mode ventilators. This comparison indicates that the fractal variability in breathing is not the result of happenstance, but is an important property of respiration. A reduction in variability of breathing reduces the overall efﬁciency of the respiratory system. Altemeier et al. [30] measured the fractal characteristics of ventilation and determined that not only are local ventilation and perfusion highly correlated, but they scale as well. Finally, Peng et al. [31] analyzed the BRV time series for 40 healthy adults and found that under supine, resting, and spontaneous breathing conditions, the time series scale. This result implies that human BRV time series have ‘‘long-range (fractal) correlations across multiple time scales.’’ Some particularly long records of breathing were made by Kantelhardt et al. [32], for 29 young adults participating in a sleep study where their breathing was recorded during REM sleep, non-REM sleep, and periods of being awake. They showed that the stages of healthy sleep (deep, light, and REM) have different autonomic regulation of breathing. During deep or light sleep, both of which are non-REM, the scaling index was determined to be a ¼ 1/2 and consequently the BRV time series was essentially an uncorrelated random process. On the other hand, during REM sleep and during periods of wakefulness, the scaling index was

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found to be in the interval 0.9 a 0.8, so that the BRV time series has a longtime memory. Webber [33] also investigated the nonlinear physiology of breathing, pointing out that the central rhythm and pattern generators within the brainstem/ spinal cord are subject to afferent feedback and suprapontine feed-forward inputs of varying coupling strengths. In short, there are multiple signals that inﬂuence one another in the respiratory controller, and the dynamics are nonlinear; that is, a given response is not proportional to a given stimulus. But putting this aside for the moment, Webber applied a number of techniques from nonlinear dynamics systems theory to the analysis of breathing. He recorded 4500 consecutive respiratory cycles for the spontaneous breathing patterns of anaesthetized, unrestrained rats (10 in all). He then constructed phase-space plots using continuous thoracic pressure ﬂuctuations. His plots reveal patterns in the breathing data that are consistent with chaos and fractal time series. D. Fractal Gait Walking is one of those things we do without giving it much thought, day in and day out. We walk conﬁdently with a smooth pattern of strides and without apparent variation in gait. This seeming lack of pattern is remarkable considering that the motion of walking is created by the loss of balance, as pointed out by Leonardo da Vinci (1452–1515) in his treatise on painting. da Vinci considered walking to be a sequence of fallings; consequently, it should come as no surprise that there is variability in this sequence of falling intervals, even if such variability is usually masked. The regular gait cycle, so apparent in everyday experience, is not as regular as we believed. Gait is no more regular than is normal sinus rhythm or breathing. The subtle variability in the stride characteristics of normal locomotion were ﬁrst discovered by the nineteenth-century experimenter Vierordt [34], but his ﬁndings were not exploited for over 120 years. The random variability he observed was so small that the biomechanical community has historically considered these ﬂuctuations to be an uncorrelated random process. In practice this means that the ﬂuctuations in gait were thought to contain no information about the underlying motorcontrol process. The followup experiments to quantify the degree of irregularity in walking was ﬁnally done in the middle of the last decade by Hausdorff et al. [35] and involved observations of healthy individuals, as well as of subjects having certain diseases that affect gait and also the elderly. Additional experiments and analyses that both veriﬁed and extended the earlier results were subsequently done by West and Grifﬁn [11,36]. Human gait is a complex process, since the locomotor system synthesizes inputs from the motor cortex, the basal ganglia, and the cerebellum, as well as

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feedback from vestibular, visual, and proprioceptive sources. The remarkable feature of this complex phenomenon is that although the stride pattern is stable in healthy individuals, the duration of the gait cycle is not ﬁxed. Like normal sinus rhythm in the beating of the heart, where the intervals between successive beats change, the time interval for a gait cycle ﬂuctuates in an erratic way from step to step. The gait studies carried out to date concur that the ﬂuctuations in the stride-interval time series exhibit long-time inverse power-law correlations indicating that the phenomenon of walking is a self-similar fractal activity. Walking consists in a sequence of steps, and the corresponding time series is made up of the time intervals for these steps. These steps may be partitioned into two phases: a stance phase and a swing phase. It has been estimated, using blood ﬂow to skeletal muscles, that the stance phase muscles consume three times the energy as do the swing phase muscles, independently of speed [37]. The stance phase is initiated when a foot strikes the ground and ends when it is lifted. The swing phase is initiated when the foot is lifted and ends when it strikes the ground again. The time to complete each phase varies with the stepping speed. A stride interval is the length of time from the start of one stance phase to the start of the next stance phase. It is the variability in the time series made from these intervals that is probed in this analysis of the stride interval time series. One deﬁnition of the gait cycle or stride interval is the time between successive heel strikes of the same foot [10]. An equivalent deﬁnition of the stride interval uses successive maximum extensions of the knee of either leg [36]. The stride interval time series for a typical subject is shown in Fig. 6, where it is seen that the variation in time interval is on the order of 3–4%, indicating that the stride pattern is very stable. The stride interval time series is referred to as stride rate variability (SRV) for name consistency with the other two time series we have discussed. It is the stability of SRV that has historically led investigators to decide that not much could go wrong by assuming that the stride interval is constant and that the ﬂuctuations are merely biological noise. The second set of data in Fig. 6 is computer-generated Gaussian noise, having the same mean and standard deviation as the experimental data. Note that it is not easy to distinguish between the two data sets. However, the experimental data ﬂuctuate around the mean gait interval and, although small, are nonnegligible because they indicate an underlying complex structure and, as we show, these ﬂuctuations cannot be treated as an uncorrelated random noise. Using a 15-minute SRV time series, from which the data depicted in Fig. 6 were taken, we apply the allometric aggregation procedure to determine the relation between the standard deviation and mean of the time series as shown in Fig. 7. In the latter ﬁgure, the curve for the SRV data is, as we did with the other data sets, contrasted with an uncorrelated random process (slope ¼ 0.5) and a regular deterministic process (slope ¼ 1.0). The slope of the data curve is 0.70,

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Figure 6. Real data compared with computer-generated data. At the top, the time interval between strides for the ﬁrst 500 steps made by a typical walker in an experiment [36] is depicted. At the bottom a computer-generated time series having uncorrelated Gaussian statistics is shown, with the same mean and variance as in the data shown at the top.

midway between the two extremes of regularity and uncorrelated randomness. So, as in the cases of HRV and BRV time series, we again ﬁnd the erratic physiological time series to represent a random fractal process. In a previous section we argued that if the power-law index, the slope of the aggregated variance versus aggregated average curve on a log–log graph, is greater than one, then the data are clustered. In the SRV context, indicated by a slope greater than the random dashed line, this clustering indicates that the intervals between strides change in clusters and not in a statistically uniform manner over time. This result suggests that the walker does not smoothly adjust his/her stride from step to step. Rather, there are a number of steps over which adjustments are made followed by a number of steps over which the changes in stride are completely random. The number of steps in the adjustment process and the number of steps between adjustment periods are not independent. The

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Figure 7. The SRV data for a typical walker in the experiment [36] is used to construct the aggregated variance and mean as indicated by the dots. The logarithm of the aggregated variance is plotted versus the logarithm of the aggregated mean, starting with all the data points at the lower left to the aggregation of 20 data points at the upper right. The SRV data curve lies between the extremes of uncorrelated random noise (lower dashed curve) and a regular deterministic process (upper dashed curve) with a fractal dimension of D ¼ 1.30.

results of a substantial number of stride interval experiment supports the universality of this interpretation. The SRV time series for 16 healthy adults were downloaded from PhysioNet and the allometric aggregation procedure carried out. Each of the curves looked more or less like that in Fig. 7, with the experimental curve being closer to the indicated regular or the random limits (dashed curves). On average, the 16 individuals have fractal dimensions for gait in the interval 1.2 D 1.3 and an average correlation on the order of 40% [38]. The fractal dimension obtained from the analysis of an entirely different dataset, obtained using a completely different protocol, yields consistent results [36]. The narrowness of the interval around the fractal dimension suggests that this quantity may be a good quantitative measure of an individual’s dynamical variability. We suggest the use of the fractal dimension as a quantitative measure of how well the motor control system is doing in regulating locomotion. Furthermore, excursions

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outside the narrow interval of fractal dimension values for apparently healthy individuals may be indicative of hidden pathologies. Further analysis was done on the SRV time series of 50 children, also downloaded from PhysioNet. One of the features of the time series were large excursions in the fractal dimension for children under the age of ﬁve, 1.12 D 1.36, unlike the narrower range of values for mature adults. The interval expands from 0.08 for adults to 0.24 for children, a factor of three decrease of the interval from childhood to adulthood. It is clear that the average fractal dimension over each group is the same, approximately 1.24, but the range of variation in the fractal dimension decreases signiﬁcantly with age [38]. This would seem to make the fractal dimension an increasingly reliable indicator of the health of the motorcontrol system with advancing age. It should not go unnoticed that people use pretty much the same control system when they are standing still, maintaining balance, as they do when they are walking. This observation would lead one to suspect that the body’s slight movements around the center of mass of the body—that ﬁctitious point at which all of the body’s mass is located in any simple model of locomotion—would have the same statistical behavior as that observed during walking. These tiny movements are called postural sway in the literature and have given rise to papers with such interesting titles as ‘‘Random walking during quiet standing’’ by Collins and De Lucca [39]. It has been determined that postural sway is really chaotic [40], so one might expect that there exists a relatively simple dynamical model for balance regulation that can be used in medical diagnosis. Here again the fractal dynamics can be determined from the scaling properties of postural sway time series, and it is determined that a decrease of postural stability is accompanied by an increase of fractal dimension. E.

Fractal Neurons

Up to this point the discussion has been focused primarily on time series generated by various complex physiological phenomena. Now let us examine a class of basic building blocks used to construct these phenomena, the individual neurons in the various control systems of the body. The neuron is in most respects quite similar to other cells in that it contains a nucleus and cytoplasm. However, it is distinctive in that long, threadlike tendrils emerge from the cell body, and those numerous projections branch out into still ﬁner extensions. These are the dendrites that form a branching tree of ever more slender threads not unlike the fractal trees discussed by West and Deering [5]. One such thread does not branch and often extends for several meters even though it is still part of a single cell. This is the axon that is the nerve ﬁber in the typical nerve. Excitations in the dendrites always travel toward the cell body in a living system, whereas excitations in the axon always travel away from the cell body.

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The activity of a nerve is invariably accompanied by electrical phenomena. The ﬁrst systematic observation of this effect was made by Luigi Galvani in 1791. He observed that when a muscle (frog’s leg) was touched with a scalpel and a spark was drawn nearby, but not in direct physical contact with the scalpel, the muscle contracted. In 1852 the German physician/physicist Helmholtz ﬁrst measured the speed of the nerve impulse by stimulating a nerve at different points and recording the time it took the muscle to which it was connected to respond. Electrical impulses are observed along the corresponding axon, whether it is an external excitation of a nerve or the transmission of a message from the brain. The properties of individual nerves seem to be ubiquitous because there is apparently no fundamental difference in structure, chemistry, or function between the neurons and their interconnections in humans and those in a squid, a snail, or a leach. However, neurons do vary in size, position, shape, pigmentation, ﬁring patterns, and chemical substances by which they transmit information to other cells. There are two different ways in which neurons can be fractal. The ﬁrst way is through their geometrical structure. The shape of nerve cells may be fractal in space just as observed for the cardiac conduction system and the architecture of the lung [5]. The fractal dimension has been used to classify the different shapes of neurons and to suggest mechanisms of growth responsible for these shapes. The second way neurons can be fractal is through the time intervals between the action potentials recorded from nerves, again as was observed for the interbeat interval distribution in cardiac time series, the interbreath interval distribution for breathing, and the interstride interval for walking discussed herein. The statistical properties of these time intervals have the same strange properties observed earlier, in that collecting more data does not improve the accuracy of the statistical distribution of the intervals measured for some neurons as characterized by the width of the distribution. The realization that the statistics of these intervals are fractal helps in understanding these surprising properties. Bassingthwaighte et al. [21] review how the fractal dimension can be used to classify neurons into different functional types. The statistical properties of the time intervals between action potentials display three different types of distributions: (1) Some neurons are well described by a Poisson process, in which the probability that there is an action potential in a time interval t is proportional to t. The durations of subsequent intervals between action potentials are statistically independent of one another. (2) Some neurons ﬁre almost, but not exactly, at a constant rhythm. The time series for the action potentials in this case are well described by a Gaussian model, where the intervals have a small dispersion around the mean value. (3) Some neurons show self-similar fractal patterns on the scaled interval distributions. In the last case, neurons occasionally have very long intervals between action potentials. These long intervals occur with sufﬁcient frequency

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so that they can be the most important in the determination of the average interval length. As more and more data are collected, longer and longer intervals are found. Thus, the average interval increases with the duration of the data record analyzed, in such a way that if an inﬁnitely long record could be analyzed, then the average interval would be inﬁnite. Such inﬁnite moments are characteristic of a type of stable statistical fractal process called a Le´vy process [5]. Gernstein and Mandelbrot [41] were the ﬁrst to quantitatively analyze the scaling properties of neuronal action potentials, and from their analysis a number of conclusions were drawn. The ﬁrst conclusion concerns the fact that, as more data were analyzed, the values found for the average interval and the variance in the average increase. That is, in the limit of an inﬁnitely long data record, both the mean and its variance could become inﬁnite. Since the variance is, in principle, inﬁnite, the averages measured for different segments of the same data record may be markedly different. It is commonly thought that if moments, such as the average, measured from data are constant in time, then the parameters of the process that generate the data are constant, and that if these moments vary, then the parameters that generate the data also vary. This commonly held notion is wrong, as stressed by Gernstein and Mandelbrot. Processes, especially fractal processes, where the generating mechanism remains unchanged can yield time series whose moments, such as the average, vary with time. The second conclusion concerns the fact that as additional data are analyzed, increasingly long intervals are found. Hence the inclusion of these intervals increases rather than decreases the variance in the measured distribution. That is, the statistical irregularities in the measured distribution become larger as more data are collected. As stated by Gernstein and Mandelbrot [41]: Thus, in contradiction to our intuitive feelings, increasing the length of available data for such processes does not reduce the irregularity and does not make the sample mean or sample variance converge.

Action potentials can be recorded from the primary auditory neurons that transmit information about sound and from the pulse vestibular neurons that transmit information about head position to the brain. Fractal behavior is ubiquitous in these and other such sensory systems [42]. Without including the references given by Teich et al. [42], we quote their review of the evidence for this observation: Its presence has been observed in cat striate-cortex neural spike trains, and in the spike train of a locust visual interneuron, the descending contralateral movement detector. It is present in the auditory system of a number of species; primary auditory (VIII-nerve) nerve fiber in the cat, the chinchilla, and the chicken all exhibit fractal behavior. It is present at many biological levels, from the

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microscopic to the macroscopic; examples include ion-channel behavior, neurotransmitter exocytosis at the synapse, spike trains in rabbit somatosensorycortex neurons, spike trains in mesencephalic reticular-formation neurons, and even the sequence of human heartbeats. In almost all cases the upper limit of the observed time over which fractal correlations exist is imposed by the duration of the recording.

It would probably be considered bad form in a discussion of neurons and action potentials if we did not comment on the electrical properties of the human brain. It has been known for well over a century that the activity of a nerve is based on electrical phenomena and that the mammalian brain generates a small but measurable electrical signal. The electroencephalograms (EEGs) of small animals were measured by Caton in 1875, and those of humans were measured by Berger in 1925. The mathematician N. Wiener thought that generalized harmonic analysis would provide the mathematical tools necessary to penetrate the mysterious relations between EEG time series and the functioning of the brain. The progress along this path has been slow, and the understanding and interpretation of EEG’s remains quite elusive. After 130 years, one can only determine intermittent correlations between the activity of the brain and that found in EEG records. There is no taxonomy of EEG patterns that delineates the correspondence between those patterns and brain activity. It probably bears repeating that the traditional methods of analyzing EEG time series rely on the paradigm that all temporal variations consist of a superposition of harmonic and periodic vibrations, in the tradition of Wiener. The attractor reconstruction technique reinterprets the time series as a multidimensional geometrical object generated by a deterministic dynamical process in phase space. If the dynamics are reducible to deterministic laws, then the phase portraits of the system converge toward a ﬁnite region of phase space containing an attractor. Figure 8 shows EEG time series for a variety of brain states, including quiet resting with eyes open and closed, three stages of sleep, and a petit mal epileptic seizure. Adjacent to each of these time series is depicted a projection of the EEG attractor onto a two-dimensional subspace using the attractor reconstruction technique [43]. The brain wave activity of an individual during various stages of sleep is depicted in Fig. 8. Here the standard division of sleeping into four stages is used. In stage one, the individual drifts in and out of sleep. In stage two, the slightest noise will arouse the sleeper, whereas in stage three a loud noise is required. The ﬁnal stage, level four, is one of deep sleep. This is the normal ﬁrst sequence of stages one goes through during a sleep cycle. Afterwards the cycle is reversed back through stages three and two, at which time dreams set in and the individual manifests rapid eye movement (REM). The dream state is followed by stage two, after which the initial sequence begins again. It is clear that whatever the form of the cognitive attractor, if such an object exists, it is not static but varies with the

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Figure 8. Typical episodes of the electrical activity of the human brain as recorded from the electroencephalogram (EEG) time series together with the corresponding phase space portraits. The portraits are two-dimensional projections of the actual attractors. (From Babloyantz and Destexhe [43] with permission.)

level of sleep. In fact, the fractal dimension decreases as sleep deepens, from a fractal dimension of eight during REM to half that in deep sleep level four [43]. The dimension drops further, to a value of approximately two, during petit mal epileptic seizure. The seizure corresponds to highly organized discharges between the right and left hemispheres of the brain. III.

DYNAMICAL MODELS OF SCALING

The physiological time series processed in the previous section clearly show that the complex phenomena supporting life, although they appear to be random, do in fact scale in time. This scaling indicates that the ﬂuctuations that occur on multiple time scales are tied together, and the way we understand such interdependency in the physical sciences is through underlying mechanisms

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that are coupled one to the other. This coupling is typically done through the equations of motion governing the dynamical description of the process. Unfortunately, we do not have the dynamic equations to describe the physiologic phenomena in which we are interested. Therefore we must take a more phenomenological tack and develop mathematical models to explain the data based on heuristic reasoning. However, to start we need to know the formal properties of the models we propose to use. The best physical model is the simplest one that can ‘‘explain’’ all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. The continuous limit of a simple random walk model leads to a stochastic dynamic equation, ﬁrst discussed in physics in the context of diffusion by Paul Langevin. The random force in the Langevin equation [44], for a simple dichotomous process with memory, leads to a diffusion variable that scales in time and has a Gaussian probability density. A long-time memory in such a random force is shown to produce a non-Gaussian probability density for the system response, but one that still scales. Finally we show that physiologic time series are not mono-fractal, but have a fractal dimension that changes over time. The time series are multifractal, and as such they have a spectrum of dimensions. We review the procedure for constructing the multifractal spectrum and apply the technique to the SRV time series data obtained in our walking experiment [36] as a typical example of physiologic variability. A.

Scaling in Time Series

Time series analysis is the backdrop against which most theoretical models are developed in the life sciences and their analysis employs the traditional engineering assumption of signal plus noise. The signal plus noise model postulates that the time series variable X(t) consists of a slowly varying part S(t) and a randomly ﬂuctuating part x(t): XðtÞ ¼ SðtÞ þ xðtÞ

ð10Þ

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bruce j. west

The slow, regular variation of the time series S is called the signal, and the rapid erratic ﬂuctuations represented by x is called the noise. The implication of this separation of effects is that S(t) contains only information about the system of interest, whereas x(t) is a property of the environment and does not contain any information about the system. In this model the noise can therefore be removed, by means of such techniques as ﬁltering, without inﬂuencing what can be learned about the system. In more complex phenomena the separation of effects implied by Eq. (10) may no longer be appropriate. The low-frequency, slowly varying part of the spectrum may be coupled to, and exchange energy with, the high-frequency, rapidly varying part of the spectrum; a fact that often results in fractal statistical processes. For these latter processes the traditional view of a deterministic, predictable signal given by the smooth part of the time series, on which random, unpredictable noise is superposed, distorts the dynamics of the underlying process, see, for example, Biodynamics [38] for a complete discussion. Herein we study techniques that purport to isolate and separate the deterministic part from the scaling part of the time series, without distorting the mutual inﬂuence of the low-frequency and high-frequency components of the same time series. The science of complexity, in so far as it can be said to be a science, has relinquished the signal plus noise paradigm for a different perspective. Physiological time series invariably contain ﬂuctuations, so that when sampled N times the data set {Xj}, j ¼ 1, . . . , N, appears to be a sequence of random points. Examples of such data are the interbeat intervals of the human heart, interstride intervals of human gait, brain wave data from EEGs and interbreath intervals, to name a few. The processing of time series in each of these cases has made use of random walk concepts in both the processing of the data and in the interpretation of the results. So let us review some of what is known about random walks. 1.

Simple Random Walks and Scaling

We deﬁne the variable of interest as Xj, where j ¼ 0,1,2, . . . indexes the time step, and in the simplest model a step is taken in each increment of time, which for convenience we set to one. The operator B lowers the index by one unit such that BXj ¼ Xj1 so that a simple random walk can be written ð1 BÞXj ¼ xj

ð11Þ

where xj is þ1 or 1 and is selected according to some random process characterized by the probability density pðxÞ. The solution to this discrete equation is given by the position of the walker after N steps, the sum over the sequence of steps XðNÞ ¼

N X j¼1

xj

ð12Þ

fractal physiology, complexity, and the fractional calculus

29

and the total number of steps N can be interpreted as the total time t over which the walk unfolds, since we have set the time increment to one. For N sufﬁciently large and a symmetric probability density pðxÞ with a ﬁnite width, the central limit theorem determines that the statistics of the dynamic variable X(t) are Gaussian: 2 1 x pN ðxÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e 2hXðNÞ2 i 2phXðNÞ2 i

ð13Þ

Assuming that the random steps are statistically independent hxj xk i ¼ hx2 idjk , we have for the second moment of the diffusion variable hXðtÞ2 i ¼

N X N X

hxj xk i ¼ hx2 iN ! 2Dt

ð14Þ

j¼1 k¼1

In the continuum limit the second moment increases linearly with time and in direct proportion to the diffusion coefﬁcient, so that the probability density becomes the familiar Gaussian distribution for Einstein diffusion: 1 2 pðx; tÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ex =2Dt 4pDt

ð15Þ

Of particular interest to us here is the scaling property of the Gaussian distribution, Eq. (15). The joint probability distribution for two statistically independent processes X1 and X2, occurring in sequential time intervals, each of length t, is the product of the separate probability densities. Consequently the probability density for the aggregate process X ¼ X1 þ X2 , occurring in the time interval 2t, is obtained by integrating over the two-variable constrained integral: 1 ð

Pðx; 2tÞ ¼

1 ð

dx2 Pðx1 ; tÞPðx2 ; tÞdðx x1 x2 Þ

dx1 1

ð16Þ

1

Integrating out the delta function constraint and substituting the Gaussian distribution from Eq. (15) into Eq. (16) yields 1 ð

pðx; 2tÞ ¼

2

dx1 1

eðxx1 Þ =2Dt ex2 =2Dt ex =2Dð2tÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4pDt 4pDt 4pDð2tÞ 2

2

ð17Þ

30

bruce j. west

Thus, when viewed with only half the time resolution, that being 2t rather than t, the increments of the Brownian particle position are still zero-centered Gaussian random. More generally, whatever the number of the microscopic time steps between observations M, one always ﬁnds that the increments in the particle position constitute a zero-centered Gaussian process with a variance that increases linearly with M. The above property means that one can sample the process of Brownian motion at any level of resolution and still observe a zero-centered Gaussian process. The time series obtained by sampling the process at every time t, or at every time bt, where b is an integer, would be statistically indistinguishable. This is the scale invariance of Brownian motion. This scaling property is manifest by writing ^x ¼ l1=2 x and ^t ¼ lt, yielding the scaling result pð^x; ^tÞ ¼ l1=2 pðx; tÞ

ð18Þ

so that the distribution for the random variable l1=2 XðltÞ is the same as that for X(t). This scaling relation establishes that the random irregularities are generated at each scale in a statistically identical manner; that is, if the ﬂuctuations are known in a given time interval, they can be determined in a second larger time interval by scaling. This is the property manifest in the allometric aggregation method used in the previous section. In the simple random walk the steps are statistically independent of one another. The simplest generalization of this model is to make each step dependent on the preceding step in such a way that the second moment is hXðtÞ2 i ¼ 2Dt2H

ð19Þ

where H 6¼ 1/2 corresponds to anomalous diffusion. A value of H < 1/2 is interpreted as an antipersistent process in which case a step in one direction is preferentially followed by a reversal of direction. A value of H > 1/2 is interpreted as a persistent process in which case a step in one direction is preferentially followed by another step in the same direction. A value of H ¼ 1/2 is interpreted as ordinary diffusion in which case the steps are statistically independent of one another. This interpretation of anomalous diffusion would be compatible with the concept of environmental noise and the signal plus noise paradigm. In the science of complexity the system response X(t) is expected to depart from the totally random condition of the simple random walk model, since such ﬂuctuations are expected to have memory and correlation. In the physics literature, anomalous diffusion has been associated with phenomena with longtime memory such that the autocorrelation function is Cðt1 ; t2 Þ ¼ hXðt1 ÞXðt2 Þi / jt1 t2 jb

ð20Þ

fractal physiology, complexity, and the fractional calculus

31

and the brackets denote an average over an ensemble of realizations of ﬂuctuations in the random variable. Here the power-law index is given by b ¼ 2H 2. Note that the two-point correlation function depends only on the time difference, thus, the underlying process is stationary in time. The Fourier transform of the autocorrelation function yields the spectrum, which in this case has the inverse power-law form SðoÞ /

1 obþ1

ð21Þ

and could also have been determined by applying a Tauberian theorem to Eq. (20). These power-law properties of the spectrum and the autocorrelation function, as well as a number of other properties involving long-time memory, are clearly discussed for discrete time series by Beran [45]. 2.

Fractional Random Walks and Scaling

One way of introducing long-term memory into a random walk model is by means of fractional differences. The concept of fractional differences is most readily introduced through the shift operator introduced in the previous subsection. Following Hosking [46], we deﬁne a fractional difference process as ð1 BÞa Xj ¼ xj

ð22Þ

and the exponent a is not an integer. As it stands, Eq. (22) is just a formal deﬁnition without content. To make this equation usable, we must determine how to represent the operator acting on Xj, and this is done using the binomial expansion [45,46]. The inverse operator in the formal solution of Eq. (22), Xj ¼ ð1 BÞa xj

ð23Þ

has the binomial series expansion a

ð1 BÞ

¼

1 X a k¼0

k

ð1Þk Bk

ð24Þ

Expressing the binomial coefﬁcient as the ratio of gamma function in the solution given in Eq. (23), we obtain after some algebra [23] Xj ¼ ¼

1 X

ðk þ aÞ Bk x j ðk þ 1ÞðaÞ k¼0

1 X k¼0

k xjk

ð25Þ

32

bruce j. west

The solution to the fractional diffusion equation is clearly dependent on ﬂuctuations that have occurred in the remote past; note the time lag k in the index on the ﬂuctuations and the fact that it can be arbitrarily large. The extent of the inﬂuence of these distant ﬂuctuations on the system response is determined by the relative size of the coefﬁcients in the series. Using Stirling’s approximation on the gamma functions determines the size of the coefﬁcients in Eq. (25) as the ﬂuctuations recede into the past, that is, as k ! 1 we obtain k

ðk þ a 1Þkþa1 kk ða 1Þ

k

¼

ka1 ða 1Þ!

ð26Þ

since k a. Thus, the strength of the contributions to Eq. (25) decrease with increasing time lag as an inverse power law asymptotically in the time lag as long as a < 1=2. The spectrum of the time series in Eq. (25) is obtained using its discrete Fourier transform ðp 1 ^ o do eiko X ð27Þ Xk ¼ 2p p

in the discrete convolution form of the solution, Eq. (25), to obtain ^ o^ ^o ¼ xo X

ð28Þ

D E D E ^ o j2 ^ o j2 ¼ j^ SðoÞ ¼ jX x o j2 j

ð29Þ

yielding the power spectrum

The strength of the ﬂuctuations is assumed to be constant—that is, independent of the frequency. On the other hand, the Fourier transform of the strength parameter is given by o ¼

1 X

k eiko ¼

k¼0

¼

1 ð1 eio Þa

1 X ðk þ a 1Þ! io k ðe Þ k!ða 1Þ! k¼0

ð30Þ

so that rearranging terms in Eq. (30) and substituting that expression into Eq. (29), we obtain 1 SðoÞ / ð31Þ ð2sino=2Þ2a

fractal physiology, complexity, and the fractional calculus

33

for the spectrum of the fractional-differenced white noise process. In the lowfrequency limit we therefore obtain the inverse power-law spectrum SðoÞ /

1 o2a

ð32Þ

Thus, since the fractional-difference dynamics are linear, the system response is Gaussian, the same as the statistics for the white noise process on the right-hand side of Eq. (22). However, whereas the spectrum of ﬂuctuations is ﬂat, since it is white noise, the spectrum of the system response is inverse power law. From these analytic results we conclude that Xj is analogous to fractional Brownian motion. The analogy is complete if we set a ¼ H 1=2 so that the spectrum in Eq. (32) can be expressed as SðoÞ /

1 o2H1

as

o!0

ð33Þ

Taking the inverse Fourier transform of the exact expression in Eq. (31) yields the autocorrelation coefﬁcient [23] rk ¼

hXj Xjþk i ð1 aÞ 2a1 k ðaÞ hXj2 i ð1:5 HÞ 2H2 k ðH 0:5Þ

ð34Þ

as the lag time increases without limit k ! 1. The probability density function (pdf) for the fractional-difference diffusion process in the continuum limit p(x,t) satisﬁes the scaling condition pðx; tÞ ¼

1 x F td td

ð35Þ

where d ¼ H ¼ a 1=2. The deviation from ordinary statistical mechanics, and consequently the manifestation of complexity, is indicated by two distinct quantities. The ﬁrst indicator is the scaling parameter d departing from the ordinary value d ¼ 0:5, which it would have for a simple diffusion process. But for fractional Brownian motion the value of the scaling index can be quite different. A second indicator of the deviation from ordinary statistical mechanics is the function F(y) in Eq. (35) departing from the conventional Gaussian form. The scaling index is usually determined by calculating the second moment of a time series. This method of analysis is reasonable only when F(y) has the Gaussian form, or some other distribution with a ﬁnite second moment—that is, if it is a member of the FVSM class. If the scaling condition Eq. (35) is realized,

34

bruce j. west

it is convenient to measure the scaling parameter d by the method of Diffusion Entropy Analysis (DEA) [47], which, in principle, works independently of whether the second moment is ﬁnite or not. The DEA method affords many advantages, including that of being totally independent of a constant bias. However, before we review the DEA method, let us examine another way in which the diffusion variable may scale—that is, another mechanism to generate long-time memory. 3.

Various Inverse Power-Law Autocorrelation Functions

Consider the following form of the autocorrelation function, CðtÞ ¼ hXðtÞXðt þ tÞi ¼

1 ð1 þ jtja Þ

b=a

ð36Þ

where the random process X(t) is still Gaussian, see Gneiting and Schlather [48] for a complete discussion of this correlation function and its implications. Any combination of parameters 0 < a 2 and b > 0 is allowed, in which case (36) is referred to as the Cauchy class of correlation functions. Now consider the two asymptotic limits. In the short time limit t ! 0 we expand the autocorrelation function in a Taylor series and obtain the power-law form lim CðtÞ 1 jtja

t!0

ð37Þ

for 0 < a 2. The autocorrelation function in this case indicates realizations of the random function X(t) in an E-dimensional Euclidian space that has a fractal dimension given by D ¼ E þ 1 a=2

ð38Þ

with probability one [48]. In the one-dimensional case (E ¼ 1) the power spectrum corresponding to Eq. (25) is the inverse power law SðoÞ /

1 jojaþ1

as

o!1

ð39Þ

Consequently the inverse power-law spectrum obtained in this way has a slope related to the fractal dimension by a þ 1 ¼ 5 2D. At the long time extreme, the autocorrelation function can again be expanded in a Taylor series to yield the long-time memory indicated by the inverse powerlaw correlation function lim CðtÞ jtjb

t!1

ð40Þ

fractal physiology, complexity, and the fractional calculus

35

when 0 < b < 1. In this case we obtain, as found earlier, 2H ¼ 2 b, relating the scaling index with the Hurst exponent. Here again the Fourier transform of the autocorrelation function yields the power spectrum SðoÞ / jojb1 ¼

1 joj2H1

as o ! 0

ð41Þ

which is still an inverse power law for the index in the range 0 < b < 1, or equivalently the Hurst exponent in the same range. It cannot be too strongly emphasized that the fractal dimension and the Hurst exponent can vary independently of one another. From the example presented we see that the fractal dimension is a local property of time series ðt ! 0Þ, whereas the Hurst exponent is a global property of time series ðt ! 1Þ and, although not proven here, these are general properties of D and H [48]. Therefore, returning to our DEA argument, the scaling behavior d ¼ H would be the exception and not the rule for fractal stochastic processes. Note that these two scaling exponents are obtained by comparing the scaling of the second moment with that of the probability distribution (DEA). Most methods for determining the scaling index of time series, including the allometric aggregation method, rely on the central moments of the time series having ﬁnite values (FVSM). So let us consider the case where such moments do not exist and therefore are of no help in determining the scaling exponents. B.

Dichotomous Fluctuations with Memory

The time interval between steps were assumed to be a constant ﬁnite value in the simple random walk model. If, however, we explicitly take the limit where the time interval vanishes, then the discrete walk is replaced with a continuous rate. We begin our discussion of the scaling of statistical processes by considering one of the simplest stochastic rate equations and follow the development of Allegrini et al. [49]. dXðtÞ ¼ xðtÞ dt

ð42Þ

where x(t) is a two-state random process taking the values W. If fðx; x; R; tÞ is the phase-space distribution function, then the equation of evolution corresponding to the dynamical equation (42) is qfðx; x; R; tÞ ¼ qt

q ^ fðx; x; R; tÞ ^ x þ qx

ð43Þ

36

bruce j. west

^ is an operator where we are adopting a quantum-like formalism. Thus, characterizing the dynamics of the x-process and ^x is an operator having the eigenvalues W, namely, ^ xji ¼ Wji: ð44Þ The underlying process generating x(t) need not be speciﬁed, but one realization of it could be a Hamiltonian system with a set of variables R. These latter variables can be inﬁnitely many so as to result in the relaxation of the correlation properties of the system. At equilibrium, the two states jþi and ji must have the same statistical weight. Thus we assume that the bath equilibrium corresponds to the state 1 jp0 i ¼ pﬃﬃﬃ fjþi þ jigðRÞ 2

ð45Þ

where ðRÞ denotes the equilibrium distribution of the variables responsible for the stochastic dynamics of the variable x. The state jp0 i is one of the eigenstates ^ In fact, we set of the operator . ^ jmi ¼ m jmi

ð46Þ

and jp0 i ¼ jm ¼ 0i, 0 ¼ 0. Within this quantum-like formalism the variable x(t), as mentioned earlier, corresponds to the operator ^x. This operator, applied to the equilibrium state jp0 i, yields the excited state jp1 i ¼

^ xjp0 i ðRÞ W

ð47Þ

Thus the operator ^ x does not affect the distribution of R but has the effect of making the transitions jþi þ ji ! jþi ji and jþi ji ! jþi þ ji, without affecting the other bath variables. The ‘‘excited’’ state jp1 i is not an ^ but it is a linear combination of the states jmi, with m 6¼ 0. The eigenstate of , ^ operator applied to the ‘‘excited’’ state jp1 i has the effect of relaxing it through coupling the state jp1 i to inﬁnitely many other eigenstates jmi. The autocorrelation function hxxðtÞi, within this quantum-like formalism, reads ^ ^xjp0 i hxxðtÞi ¼ hp0 j^ xexp t

ð48Þ

^ this correlation function On the basis of the properties of the operators ^ x and , can also be expressed as X hxxðtÞi ¼ W 2 hp1 jmihmjp1 iexp m t ð49Þ m6¼0

fractal physiology, complexity, and the fractional calculus

37

It is convenient to deﬁne the phase space distribution function sm ðtÞ hmjfðx; x; R; tÞi

ð50Þ

with m ¼ 0,1,2, . . . . We are interpreting the distribution f of Eq. (43) as a sort of ket vector jfi. By multiplying Eq. (43) on the left by the states jmi, we obtain 1 X qs0 ðx; tÞ qsm ðx; tÞ ¼ W am qt qx m6¼0

ð51Þ

1 X qsm ðx; tÞ qs0 ðx; tÞ ¼ W m sm ðx; tÞ am qt qx m6¼0

ð52Þ

and, for m>0,

with am ¼ hmj^ xjp0 i. Let us make the assumption that at t ¼ 0 all the sm ’s but the one with m ¼ 0 vanish. This condition is equivalent to assuming the spatial distribution is statistically independent of the ‘‘velocity’’ distribution and results in an equation of motion without an inhomogeneous term. By solving Eq. (52) and placing the solution into Eq. (51), we obtain ð 1 X qs0 ðx; tÞ q q2 s0 ðx; tÞ ¼ W2 am jam j2 dt0 exp½m ðt t0 Þ qt qx qx2 m6¼0 t

ð53Þ

0

From now on we shall focus on the reduced density matrix s0 ðx; tÞ and for the sake of simplicity we omit the subscript zero. Using Eq. (48), we can rewrite Eq. (53) in the form ðt qsðx; tÞ q2 sðx; t0 Þ ¼ dt0 hxðtÞxðt0 Þi qt qx2

ð54Þ

0

where again the brackets denote an average over an ensemble of realizations of the statistical ﬂuctuations. In Section IV the form of Eq. (54) is determined to be that for a fractional diffusion equation when the two-point correlation function is appropriately chosen. In the case where the correlation function in Eq. (54) is an exponential hxðtÞxðt0 Þi ¼ hx2 iegt

ð55Þ

38

bruce j. west

taking the time derivative of Eq. (54) yields 2 q2 sðx; tÞ qsðx; tÞ 2 q sðx; tÞ hx þ g i ¼0 qt2 qt qx2

ð56Þ

This is the celebrated telegrapher’s equation, whose phenomenological pedigree dates back to Maxwell. His (Maxwell’s) argument was to include relaxation into the wave equation and did not require the invocation of microscopic dynamics. However, his use of dissipation was compatible with the action of inﬁnitely many degrees of freedom in the medium supporting the wave motion. The equation of motion for the Liouville density from Eq. (54) is ðt ðt 2 2 0 0 qsðx; tÞ 0 0 q sðx; t Þ 0 0 q sðx; t t Þ ¼ dt x ðt t Þ ¼ dt ðt Þ x qt qx2 qx2 0

ð57Þ

0

In the case when the correlation function x is integrable, using the last term of the equality of Eq. (57), we can make use of the Markov approximation. This approximation is based on replacing the second derivative in space containing the time argument (t t0 ) with the second derivative in space containing the time argument t and extending the upper bound of the time integration from t to inﬁnity. This can be justiﬁed by expanding the probability density in a Taylor series in time and neglecting all but the ﬁrst term. In this, the Markov approximation in Eq. (57) reduces to the ordinary diffusion equation qsðx; tÞ q2 sðx; tÞ ¼D qt qx2

ð58Þ

where the diffusion coefﬁcient D is given by D ¼ W 2 tC

ð59Þ

and the correlation time is given by 1 ð

tC ¼

dt0 ðt0 Þ

ð60Þ

0

Notice that in the Continuous-Time Random Walk (CTRW) as used in Klafter et al. [50], in the case where the waiting time distribution is exponential, cðtÞ ¼ a exp½at, the same evolution for the probability density p(x,t) and the phase-space distribution sðx; tÞ occurs as that resulting from Eq. [57]. This can

fractal physiology, complexity, and the fractional calculus

39

be established by noticing that in the case of the dichotomous variable x used here, the waiting-time distribution is related to the correlation function by the exact relation [51] 1 ð 1 ðt0 tÞcðt0 Þdt0 ð61Þ x ðtÞ ¼ tW t

where tW denotes the mean sojourn time. In the exponential case this sojourn time becomes identical to the correlation time; that is, since x ð0Þ ¼ 1, in the exponential case a ¼ 1=tW and tC ¼ tW . 1.

The Exact Solution

Here we are interested in the asymptotic behavior of the exact solution to Eq. (57) and we follow the analysis of Bologna et al. [52]. The most direct way to determine these properties is to take the Laplace transform in time and Fourier transform in space to obtain the Fourier–Laplace transform of the Liouville density 1 ^ ~ ðk; sÞ ¼ ð62Þ s ~ s þ x ðsÞk2 where we have imposed the initial conditions sðx; tÞjt¼0 ¼ dðxÞ

and

qsðx; tÞ ¼0 qt t¼0

ð63Þ

The inverse Fourier transform of Eq. (62) yields s sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jxjpﬃﬃﬃﬃﬃﬃ ~ ðsÞ x s e ~ ðx; sÞ ¼ s ~ x ðsÞ 2s

ð64Þ

which we can integrate over space to obtain 1 ð

~ ðx; sÞdx ¼ s 1

1 s

ð65Þ

indicating the conservation of normalization over time. To go beyond the formal solution in Eq. (64), we must specify the autocorrelation function. We select an inverse power-law autocorrelation function, x ðtÞ ¼ W 2

Tb ðT þ tÞb

ð66Þ

40

bruce j. west

with 0 < b < 1 and T is a positive constant. The Laplace transform of the autocorrelation function given by Eq. (66) becomes ~ x ðsÞ ¼ ð1 bÞTW ½esT EsT b1 ðsTÞ1b 2

ð67Þ

where the generalized exponential function is deﬁned by [23,53] Egx

1 X

x ng ðn þ 1 gÞ n¼0

ð68Þ

and we subsequently deﬁne the generalized exponential using the fractional derivative operator. 2.

Early Time Behavior

Let us ﬁrst consider the behavior of the autocorrelation function at early times. In this domain, t ! 0, we have s ! 1, so that the generalized exponential becomes sT Eb1 esT

1 ð1 bÞðsTÞb

ð69Þ

~ x ðsÞ W 2 =s so that the Laplace which when substituted into Eq. (67) yields transform of the early time solution for the phase-space equation of evolution is ~ ðx; sÞ s

ejxjs=W 2W

ð70Þ

The inverse Laplace transform of Eq. (70) yields the delta function for the phasespace distribution function 1 jxj d t sðx; tÞ 2W W

ð71Þ

Thus, for times shorter than T, the evolution of the Liouville density consists of two peaks traveling in opposite directions at the same speed, W. Note that this is the same early-time solution one would obtain for the solution to the telegrapher’s equation [51]. 3.

Late Time Behavior

Now let us now consider the time asymptotic behavior of the exact solution. In the late time domain t ! 1, we have s ! 0, so examining the behavior of

fractal physiology, complexity, and the fractional calculus Eq. (67) in this domain we obtain ~ x ðsÞ ð1 bÞTW ðsTÞ1b

2

"

ðsTÞ1b 1 ð1 bÞ

41

# ð72Þ

Note that as s ! 0 the leading term in this expansion diverges for b < 1, corresponding to the fact that there is no correlation time for this process. Inserting this expression for the Laplace transform of the correlation function into Eq. (64), keeping only the diverging term, yields

~ ðx; sÞ ¼ s

jxjs1b=2 exp ð1 bÞWT b=2 2ð1 bÞWðsTÞb=2

ð73Þ

The inverse Laplace transform of Eq. (73) yields the phase-space distribution function n 1 X 1 ð1Þn jxj ð74Þ sðx; tÞ 2hxit1b=2 n¼0 n!ð1 ðn þ 1Þð1 b=2ÞÞ hxit1b=2 where the average of the system variable is hxi ¼ WT b=2 ð1 bÞ

ð75Þ

as had been obtained previously [54]. Straightforward dimensional analysis indicates that the space variable in Eq. (74) scales as x ta where a ¼ 1 b=2

ð76Þ

as had also been found by other authors [55]. To further support this conclusion, note that in the asymptotic limit s ! 0, Eq. (62) yields 1 ^ ~ ðk; sÞ ¼ s ð77Þ s þ constant sb1 k2 The scaling condition x ta implies k ¼ sa , which when inserted into the righthand-side term of Eq. (77) makes the left-hand side of the same equation proportional to 1/s when the scaling condition of (76) applies. We note that 1/s is the Laplace transform of a constant in accordance with the fact that scaling is a reﬂection of stationarity. For this reason we are inclined to believe that the density perspective yields in the asymptotic limit a unique scaling and that our solution correctly reﬂects this condition.

42

bruce j. west C.

Fractals, Multifractals, and Data Processing

The salient property of mathematical random fractals processes is the existence of long-time correlations, here measured by the correlation index r, which can be related to the fractal dimension by [21] r ¼ 232D 1

ð78Þ

Successive increments of mathematical fractal random processes are independent of the time step. Here D ¼ 1:5 corresponds to a completely uncorrelated random process r ¼ 0, such as Brownian motion, and D ¼ 1:0 corresponds to a completely correlated process r ¼ 1, such as a regular curve. Studies of various physiologic time series have shown the existence of strong long-time correlations in healthy subjects and demonstrated the breakdown of these correlations in disease; see, for example, the review by West [56]. Complexity decreases with convergence of the Hurst exponent H to the value 0.5 or equivalently of the fractal dimension to the value 1.5. Conversely, system complexity increases as a single fractal dimension expands into a spectrum of dimensions. 1.

Multifractal Spectrum

The spectrum of fractal dimensions can be calculated in a number of ways. One way is to cover the time axis with cells of size d such that the time is given by t ¼ Nd and N 1. Following Falconer [57] we can deﬁne the partition function Zðq; dÞ

X

mðCj Þd

ð79Þ

j

where Cj is the jth box in the d-coordinate mesh that intersects with the measure m. We can construct the measure using the time series obtained from the physiologic interval data. This measure is made by aggregating the observed time intervals {tj}, j ¼ 1,2, . . . , N, where tj denotes the time interval between the end points of stride j 1 and j, Tðn; dÞ ¼

n X

tj

ð80Þ

j¼1

such that T(n, d) is interpreted as a random walk trajectory. In this way we can construct the partition function in Eq. (79) using jTð j þ n; dÞ Tð j; dÞj mðCj Þ ¼ Nn P jTðk þ n; dÞ Tðk; dÞj k¼1

ð81Þ

fractal physiology, complexity, and the fractional calculus

43

where the integer n lags the trajectory by n steps. The typical scaling behavior of the partition function in the limit of vanishing grid scale [57] is Zðq; dÞ dtðqÞ

ð82Þ

where t(q) is the mass exponent [58]. The mass exponent is related to the generalized dimension D(q) by the relation tðqÞ ¼ ð1 qÞDðqÞ

ð83Þ

where D(0) is the fractal or box counting dimension, D(1) is the information dimension, and D(2) is the correlation dimension [57,58]. The q-moment therefore accentuates different aspects of the underlying dynamical process. For q > 0, the partition function emphasizes large ﬂuctuations and strong singularities through the generalized dimensions, whereas for q < 0, the partition function stresses the small ﬂuctuations and the weak singularities. This property of the partition function deserves a cautionary note because the negative moments can easily become unstable, introducing artifacts into the calculation. Thus the interpretation of the trajectory approach must be judged with some caution for q < 0. A mono-fractal time series is characterized by a single fractal dimension. In general, time series have a local Ho¨lder exponent h that varies over the course of the trajectory and is related to the fractal dimension by D ¼ 2 h [57]. Note that for an inﬁnitely long time series the Ho¨lder exponent h and the Hurst exponent H are identical; however, for a time series of ﬁnite length they need not be the same. We stress that the fractal dimension and the Ho¨lder exponent are local quantities, whereas the Hurst exponent is a global quantity; consequently the relation D ¼ 2 H is only true for an inﬁnitely long time series. The function f(h), called the multifractal or singularity spectrum, describes how the local Ho¨lder (fractal) exponents contribute to such time series. Here h and f are independent variables, as are q and t. The general formalism of Legrendre transform pairs interrelates these two sets of variables by the relation [58], f ðqÞ ¼ qh þ tðqÞ

ð84Þ

The local Ho¨lder exponent h varies with the q-dependent mass exponent through the equality dtðqÞ hðqÞ ¼ ¼ t0 ðqÞ ð85Þ dq so the singularity spectrum can be written as f ðhðqÞÞ ¼ qt0 ðqÞ þ tðqÞ where the mass exponent and its derivative are determined by data.

ð86Þ

44

bruce j. west

Figure 9. (a) The mass exponent as a function of the q-moment obtained from a numerical ﬁt to the partition function using Eq. (87) for a typical walker. (b) The singularity spectrum f(h) obtained from a numerical ﬁt to the mass exponent and its derivative using Eq. (86) for a typical walker [36].

As mentioned above, a time series is mono-fractal when the mass exponent is linear in q, otherwise the underlying process is multifractal. We apply the partition function measure to numerically evaluate tðqÞ ¼

lnZðq; dÞ lnd

ð87Þ

and the results are depicted in Fig. 9a. Rigorously speaking, the expression for the mass exponent requires d ! 0, but we cannot do that with data, so there is

fractal physiology, complexity, and the fractional calculus

45

TABLE I The Fitting Parameters for the Mass Exponent [Eq. (88)]a Walker

a0

1 2 3 4 5 6 7 8 9 10 Average

1.03 0.99 1.05 1.05 1.00 1.01 1.02 1.09 1.02 1.01 1.03 0.03

a1 1.26 1.14 1.32 1.26 1.12 1.07 1.17 1.29 1.14 1.17 1.19 0.08

a2 0.13 0.08 0.14 0.12 0.07 0.05 0.09 0.14 0.08 0.09 0.10 0.03

a The column a1 is the fractal dimension for the SRV time series. In each case these numbers agree with those obtained earlier using a different method [36].

always some error in the results. The signiﬁcance of that error remains to be determined. In Fig. 9 the mass exponent for a typical subject in the walking experiment [11] is shown and the individual mass exponents do not look too different from the one shown. It is clear from the ﬁgure that the mass exponent is not linear in the moment index q and therefore the SRV time series is multifractal. In Table I we record the ﬁtting coefﬁcients for each of the 10 SRV time series using the quadratic polynomial in the moments interval 4 q 4 tðqÞ ¼ a0 þ a1 q þ a2 q2

ð88Þ

The ﬁt to the data using Eq. (88) is indicated by the solid curve in Fig. 9a. A second method for determining the singularity spectrum, the one we use here, is to numerically determine both the mass exponent and its derivative. In this way we calculate the multifractal spectrum directly from the data using Eq. (86). It is clear from Fig. 9b that we obtain the canonical form of the spectrum; that is, f(h) is a convex function of the scaling parameter h. The peak of the spectrum is determined to be the fractal dimension, as it should. Here again we have an indication that the interstride interval time series describes a multifractal process. We stress that we are only using the qualitative properties of the spectrum for q < 0, due to the sensitivity of the numerical method to weak singularities. The singularity spectrum can now be determined using the Legendre transformation by at least two different methods. One technique is to use the ﬁtting equation substituted into Eq. (86). We do not do this here, but we note in

46

bruce j. west

passing that if Eq. (88) is inserted into Eq. (85), the fractal dimension is determined by the q ¼ 0 moment to be hð0Þ ¼ t0 ð0Þ ¼ a1

ð89Þ

The values of the parameter a1 listed in Table I agree with the fractal dimensions obtained earlier using a scaling argument for the same data [11,36]. The multifractal behavior of time series such as SRV, HRV, and BRV can be modeled using a number of different formalisms. For example, a random walk in which a multiplicative coefﬁcient in the random walk is itself made random becomes a multifractal process [59,60]. This approach was developed long before the identiﬁcation of fractals and multifractals and may be found in Feller’s book [61] under the heading of subordination processes. The multifractal random walks have been used to model various physiological phenomena. A third method, one that involves an integral kernel with a random parameter, was used to model turbulent ﬂuid ﬂow [62]. Here we adopt a version of the integral kernel, but one adapted to time rather than space series. The latter procedure is developed in Section IV after the introduction and discussion of fractional derivatives and integrals. 2.

Diffusion Entropy Analysis (DEA)

So far in this section we have focused on mathematical models that generate time series with scaling properties. In Section II we introduced a simple data aggregation procedure to reveal the scaling of physiologic time series. The allometric aggregation method offered some insight into the scaling of the underlying process, but now we turn our attention to a method that can reveal both the statistical and the correlation properties of a time series. To do this, we interpret the physiologic time series as the generator of a diffusion process and replace the random elements of the right-hand side of the simple random walk in Eq. (90) with the time series data. Thus, even though we do not know the a priori statistical properties of the data set xj , we can deduce them from the probability density function pðx; tÞ for the diffusion variable X(t). Note that X(t) is the dynamic variable that aggregates the time series data into a ‘‘random walk’’ trajectory. If the time series is stationary, the scaling property of the probability density function for the diffusive process is given by Eq. (35), where d is the scaling exponent. We now offer a way to independently determine the scaling exponent from the time series data using the Shannon entropy for a diffusive process pðx; tÞ: 1 ð

SðtÞ ¼

pðx; tÞln pðx; tÞ dx 1

ð90Þ

47

fractal physiology, complexity, and the fractional calculus

Now if the probability density function satisﬁes the scaling condition Eq. (35) substituting this functional form of pðx; tÞ into Eq. (90) yields 1 ð

SðtÞ ¼ 1

1 x 1 x ln d F d dx F td td t t

x using the transformation y ¼ d simpliﬁes this equation to t SðtÞ ¼ A þ dlnt

ð91Þ

and the constant A is determined by the time-independent distribution F(y) 1 ð

A¼

FðyÞlnFðyÞ dy 1

It is obvious from Eq. (91) that a graph of the entropy S(t) versus the logarithm of the time t yields a straight line with positive slope d. Consequently using time series data to generate a diffusive process we can construct a histogram of the probability density enabling us to numerically determine the scaling parameter using the entropy. This procedure is called diffusion entropy analysis [47] (DEA). The theoretical scaling index for ordinary diffusion is d ¼ H ¼ 1=2. To test this prediction using a known data set, we generate a diffusive trajectory from Eq. (90) using a computer-generated uncorrelated Gaussian time series for 104 data points on the right-hand side of the equation. We consider a time series with a maximum length of 200 data points and construct a histogram from the nearly 104 realizations of such a time series X(t) obtained using Eq. (90). The histogram constructed from the realizations of the trajectories is inserted into Eq. (90), and the resulting entropy is calculated as a function of time. Figure 10 shows that the entropy calculated this way increases linearly with the logarithm of time with a slope of 0.48, very close to the theoretical value of 0.50 one would obtain for an inﬁnitely long time series. Recall that fractional Brownian motion, with the distribution given by Eq. (19), satisﬁes the scaling relation [Eq. (35)] for the probability density. Consequently, we have the equality d ¼ H 6¼ 1=2, so that the scaling exponent d, determined by DEA, is given by the Hurst exponent H. We emphasize that this equality is not true in general, and it is quite possible that d 6¼ H, indicating that there is scaling in the time series, but the statistics need not be Gaussian. We examine this case now. Consider the limit distribution ﬁrst studied by Paul Le´vy for processes having diverging central moments and which consequently violate the central limit

48

bruce j. west

Figure 10. The entropy for an uncorrelated Gaussian random process generating random walk trajectories calculated using DEA is graphed versus the natural logarithm of the time. The data indicate a linear relationship between S(t) and ln t as predicted by Eq. (91).

theorem. The generalized central limit theorem yields the probability density for the symmetric stable Le´vy process in terms of the Fourier transform of the characteristic function [63] 1 ð

pL ðx; tÞ ¼ 1

dk ikx gtjkja e e 2p

ð92Þ

where the Le´vy index is in the interval 0 < a 2 and g > 0. The only explicit expression for the Le´vy stable distribution is an inﬁnite series whose lowestorder term is given by [64] pL ðx; tÞ /

t aþ1

jxj

ð93Þ

It is evident from the inverse power-law form of the probability density given by Eq. (93) that the second moment hx2 i of the Le´vy a-stable distribution diverges since a þ 1 < 3. Equally clear is the fact that the ﬁrst moment for this distribution diverges for a þ 1 < 2. The ﬁrst and second moments converge for a ¼ 2, in which case the Le´vy stable distribution becomes a Gaussian distribution and the central limit theorem again applies to the time series. We use the parameters l and k to scale the random walk phase-space variable x and time t in the Le´vy stable distribution [Eq. (92)] and obtain after

fractal physiology, complexity, and the fractional calculus some algebra

x 1 p ; 1 L lt1=a t1=a 1 x ¼ 1=a F 1=a t t

49

pL ðlx; ktÞ ¼

ð94Þ

when the parameters are related by k ¼ la. Thus, the Le´vy a-stable distribution satisﬁes the scaling relation in Eq. (35), so that the underlying process is fractal, but without memory. This lack of memory is a consequence of the Le´vy a-stable distribution being a Markov process [63,64]. The scaling behavior of a time series described by Le´vy a-stable statistics is determined using the DEA by substituting Eq. (92) into the equation for the entropy. The scaling index is determined to be d ¼ 1/a, but d in this case is not related to a Hurst exponent H. Therefore we obtain the remarkable result that the scaling index can be determined from statistical processes even when the central moments of such processes diverge and the traditional scaling methods fail. Using DEA, we have established that there are statistical processes for which d ¼ H and statistical processes for which d 6¼ H, both of which scale. However, there is a third class of processes for which the scaling index is a function of the Hurst exponent, but the relation is not one of their being equal. This third class is the Le´vy random walk process (Le´vy diffusion) introduced by Shlesinger et al. [65] in their discussion of the application of Le´vy statistics to the understanding of turbulent ﬂuid ﬂow. Let us distinguish between a Le´vy ﬂight and a Le´vy walk. In a Le´vy ﬂight the jumps taken by the random ﬂyer each take the same amount of time, so that the statistical properties of the trajectories are determined solely by the length of the steps. If the distribution of step lengths is inverse power law, with index less than three, then using the generalized central limit theorem the resulting distribution is Le´vy [Eq. (92)]. If the step-distribution index is greater than three, the distribution converges to that of Gauss and the usual central limit theorem is recovered. A Le´vy walk, on the other hand, takes into account the fact that longer steps take longer times to complete than do shorter steps. The recognition of this simple fact ties the distribution of step sizes to the distribution of time intervals, which in the case of turbulence was determined by the ﬂuctuations in the ﬂuid velocity [62]. In the present example the continuum form of the Le´vy walk process is described by Eq. (42), with the autocorrelation function for the random driver being given by the inverse power law Eq. (66) and W is the constant speed of the walker. The asymptotic form of the second moment for this process is ðt

ðt

2

hXðtÞ i / dt1 dt2 x ðjt1 t2 jÞ ¼ t2b 0

0

ð95Þ

50

bruce j. west

and in terms of the index for the autocorrelation function the Hurst exponent is 2H ¼ 2 b

ð96Þ

Using the relation between the waiting-time distribution function Eq. (61) and the autocorrelation function Eq. (66), we obtain the inverse power-law waitingtime distribution cðtÞ ¼

aT a ðT þ tÞaþ1

ð97Þ

where the autocorrelation index and the waiting-time index are related by b¼a1

ð98Þ

Consequently, by means of the delta function dðjxj WtÞ which ties space and time together, the resulting distribution for the diffusion process is Le´vy a-stable and the scaling parameter in Eq. (98) is the Le´vy index. Therefore, using Eq. (96) and the fact that d ¼ 1=a for a Le´vy a-stable process, we obtain the relation between exponents: d¼

1 3 2H

ð99Þ

Scafetta and Grigolini [47] established that the DEA scaling does, in fact, yield the scaling relation given by (99) for Le´vy diffusion. We give two examples of fractal time series. The ﬁrst is fractal Gaussian intermittent noise characterized by a long-time correlated waiting-time sequence, and the second is a Le´vy-walk intermittent noise. These examples were developed in an environmental context to explain the observed distribution of earthquakes in California [66]. The Gaussian intermittent noise has an autocorrelation function given by Eq. (20) and power-law index given by 2H 2. The statistics of this sequence is determined by a ﬁnite variance waiting time distribution function c(t) whose form may be, for example, that of a Gaussian, exponential, or Poisson. The diffusion generated by a fractal Gaussian intermittent noise is a particular type of fractional Brownian motion and satisﬁes the asymptotic scaling relation between indices d ¼ H. Figure 11A is based on a realization of this process using an exponential waiting-time distribution function. The parallel lines through the computer-generated data show the two scaling exponents, one determined by DEA, indicated by the entropy S(t) minus the constant term, and the other determined directly from the second moment, denoted by D(t). If the long-time correlations are destroyed by means of shufﬂing—that is, by

fractal physiology, complexity, and the fractional calculus

51

Figure 11. DEA and SDA for (A) a fractal Gaussian intermittent noise with c(t) ¼ exp[t/g] with g ¼ 25 and H ¼ d ¼ 0.75; the fractal Gaussian relation of equal exponents is satisﬁed. (B) A Le´vy-walk intermittent noise with cðtÞ / tm and m ¼ 2.5; note the bifurcation between H ¼ 0.75 and d ¼ 0.67 caused by the Le´vy-walk diffusion relation [66].

randomly interchanging the positions of the elements of the sequence—the new intermittent random time series is characterized by the value d ¼ H ¼ 0:50. This latter result is not shown. The Le´vy-walk intermittent noise is characterized by an uncorrelated waiting-time sequence and a Le´vy or an inverse power-law waiting-time distribution function such as given by Eq. (97) with 1 < a < 2. This interval for the scaling index insures that although the second moment diverges, the ﬁrst moment is ﬁnite. The presence of a Le´vy-walk process in a given time series can be detected by means of the asymptotic relation Eq. (99), which we refer to as

52

bruce j. west

the Le´vy-walk diffusion relation [66]. Figure 11B is based on a realization of this process using an inverse power-law waiting-time distribution function. This ﬁgure shows the scaling properties of a computer-generated random Le´vy-walk intermittent noise with a ¼ 1.5 that has H ¼ 0.75 and d ¼ 0.67 in agreement with the Le´vy-walk diffusion relation. We stress that the Le´vy-walk diffusion relation is fulﬁlled if the waiting times are uncorrelated, in which case any shufﬂing of the elements in data sequence would not alter the scaling exponents H and d. In fact, the superdiffusion scaling exponent 0.5 < d < H < 1 of a Le´vy-walk intermittent noise are related to the fatness of the waiting-time inverse power-law tail, as measured by the exponent a. Contrary to a fractal Gaussian intermittent noise, this Le´vy scaling does not imply a temporal correlation, or a historical memory, among events because the occurrence of future events is independent of the frequency of past events. It should be stressed that even though the second moment for a Le´vy walk is ﬁnite, the scaling index obtained from the second moment does not give the correct scaling properties of the time series. This is a word of caution regarding the application of FVSMs to determine the scaling properties of a time series. Even when the second moment has the form hXðtÞ2 i / t2m

ð100Þ

this does not mean that the index so determined has any interesting implications regarding the underlying dynamics of the time series. It is only after the statistics of the time series are determined, by using DEA or some other technique, that can one begin to interpret the anomalous diffusion equation, Eq. (100). IV.

FRACTIONAL DYNAMICS

In the late nineteenth century, most mathematicians felt that a continuous function must have a derivative ‘‘almost everywhere,’’ which means that the derivative of a function is singular only on a set of points whose total length (measure) vanishes. However, some mathematicians wondered if functions existed that were continuous, but did not have a derivative at any point (continuous everywhere but differentiable nowhere). The motivation for considering such pathological functions was initiated within mathematics and not in the physical or biological sciences, the insights of Boltzmann and Perrin notwithstanding. In 1872, Karl Weierstrass (1815–1897) gave a lecture to the Berlin Academy in which he presented functions that had the aforementioned continuity and nondifferentiability properties; consequently, these functions had the symmetry of self-similarity. Twenty-six years later, Ludwig Boltzmann, who elucidated the microscopic basis of entropy, said that physicists could have

fractal physiology, complexity, and the fractional calculus

53

invented such functions in order to treat collisions among molecules in gases and ﬂuids. Boltzmann had a great deal of experience thinking about such things as discontinuous changes of particle velocities that occur in kinetic theory and to wonder about their proper mathematical representation. He had spent many years trying to develop a microscopic theory of gases and he was successful in developing such a theory, only to have his colleagues reject his contributions. Although kinetic theory led to acceptable results (and provided a suitable microscopic deﬁnition of entropy), it was based on time-reversible dynamic equations; that is, entropy distinguishes the past from the future, whereas the equations of classical mechanics do not [2]. This basic inconsistency between analytic dynamics and thermodynamics remains unresolved today, although there are indications that the resolution of this old chestnut lies in microscopic chaos. It was assumed in the kinetic theory of gases that molecules are materially unchanged as a result of interactions with other molecules, and collisions are instantaneous events as would occur if the molecules were impenetrable and perfectly elastic. As a result, it seemed quite natural that the trajectories of molecules would sometimes undergo discontinuous changes. Robert Brown, in 1827, observed the random motion of a speck of pollen immersed in a water droplet. Discontinuous changes in the speed and direction of the motion of the pollen mote were observed, but the mechanism causing these changes was not understood. Albert Einstein published a paper in 1905 that, although concerned with diffusion in physical systems, ultimately explained the source of Brownian motion as being due to the net imbalance of the random collisions of the lighter particles of the medium with the surface of the pollen mote. Jean Baptiste Perrin, of the University of Paris, experimentally veriﬁed Einstein’s predictions and received the Nobel Prize for his work in 1926. Perrin [67], giving a physicist’s view of mathematics in 1913, stated that curves without derivatives are more common than those special, but interesting ones, like the circle, that have derivatives. In fact he was quite adamant in his arguments emphasizing the importance of nonanalytic functions for describing complex physical phenomena, such as Brownian motion. Thus, there are valid physical reasons for looking for these types of functions, but the scientiﬁc reasons became evident to the general scientiﬁc community only long after the mathematical discoveries made by Weierstrass. On the theoretical physics side, the Kolmogorov–Arnold–Moser (KAM) theory for conservative dynamical systems describes how the continuous trajectories of a particle break up into a chaotic sea of randomly disconnected points. Furthermore, the strange attractors of dissipative dynamical systems have a fractal dimension in phase space. Both these developments in classical dynamics—KAM theory and strange attractors—emphasize the importance of nonanalytic functions in the description of the evolution of deterministic nonlinear dynamical systems. We do not discuss the details of such dynamical systems herein, but refer the reader to a number of excellent books on the

54

bruce j. west

subject ranging from the mathematical rigorous, but readable [68], to provocative picture books [69], to extensive applications [23]. In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary deﬁnitions of fractional integrals and fractional derivatives starting from the limits of appropriately deﬁned sums. We ﬁnd that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70]: It is interesting to investigate whether fractional calculus, which generates the operation of derivation and integration to fractional order, can provide a possible calculus to deal with fractals. In fact there has been a surge of activity in recent times which supports this point of view. This possible connection between fractals and fractional calculus gives rise to various interesting questions. . . .

The separation of time scales in physical phenomena allows us to smooth over the microscopic ﬂuctuations and construct a differentiable representation of the dynamics on large space scales and long time scales. However, such smoothing is not always possible, examples of physical phenomena that resist this approach include turbulent ﬂuid ﬂow [71], the stress relaxation of viscoelastic materials such as plastics and rubber [72,73], and ﬁnally phase transitions [74,75]. Metaphorically, these complex phenomena, whose evolution cannot be described by ordinary differential equations of motion, leap and jump in unexpected ways to obtain food; they unpredictably twist and turn to avoid capture, and they suddenly change strategy to anticipate environmental changes. To understand these and other analogous processes in physiology, we ﬁnd that we must adopt a new type of modeling, one that is not in terms of ordinary or partial differential equations of motion. It is clear that the fundamental elements of complex physical phenomena, such as phase transitions, the deformation of plastics, and the stress relaxation of polymers, satisfy Newton’s laws. In these phenomena the evolution of individual particles are described by ordinary differential equations that control the dynamics of individual particle trajectories. It is equally clear that the connection between the fundamental laws of motion controlling the individual particle dynamics and the observed large-scale dynamics cannot be made in any straightforward way.

55

fractal physiology, complexity, and the fractional calculus

In previous sections we have investigated the scaling properties of processes described by certain stochastic differential equations. The scaling in the system response was a consequence of the inverse power-law correlation in the ﬂuctuations driving the system. The fractal statistics of the dynamic model in Section III are suggestive of the scaling observed in the physiologic time series presented in Section II. Of course this is not the only way the system variable, as characterized by measured time series, can manifest scaling. Another is through the internal dynamics of the system, and that is what we explore in this section. We construct a fractional Langevin equation in which the fractional derivatives give rise to the long-time memory in the system dynamics. It is determined that the solutions to such equations describe multifractal statistics, and we subsequently apply this model to a number of physiological phenomena, including cerebral blood ﬂow and migraines. A.

Fractional Calculus

It is useful to have in mind the formalism of the fractional calculus before embarking on the interpretation of models using this formalism to explain the complexity of physiological phenomena. What we call the fractional calculus dates back to a question L’Hoˆpital asked Leibniz in 1695, where in a letter he asked the meaning of the expression d n y=dxn if n ¼ 1=2, that is: . . .what if n is fractional?

Leibniz replied in part [76]: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Thus it follows that d 1=2 x will be equal to 2 dx : x . . . . John Bernoulli seems to have told you of my having mentioned to him a marvelous analogy which makes it possible to say in a way the successive differentials are in geometric progression. One can ask what would be a differential having as its exponent a fraction. You see that the result can be expressed by an infinite series. Although this seems removed from Geometry, which does not yet know of such fractional exponents, it appears that one day these paradoxes will yield useful consequences, since there is hardly a paradox without utility.

After 310 years of sporadic development, the fractional calculus is now becoming so sufﬁciently well developed and well known that books and articles are being devoted to its consequences in the physical sciences [53,77,78]. The simplest way to introduce fractional derivatives is to consider the ordinary derivative of a monomial, say the nth derivative of tm for m > n: Dnt ½tm ¼ mðm 1Þ . . . ðm n þ 1Þtmn m! tmn ¼ ðm nÞ!

ð101Þ

56

bruce j. west

where the operator Dt is the ordinary derivative. We can generalize the form of Eq. (101) by recognizing that the ratio of factorials can be expressed as the ratio of gamma functions: Dnt ½tm ¼

ðm þ 1Þ mn t ðm þ 1 nÞ

ð102Þ

We can extend these considerations to fractional derivatives by means of analogy. We deﬁne a real indexed derivative of a monomial t: Dat ½tb ¼

ðb þ 1Þ ba t ðb þ 1 aÞ

ð103Þ

where b þ 1 6¼ 0; 1; 2; . . . ; n; this is, the monomial index is not integervalued. With this analogy we can solve the ½-derivative problem posed to Leibniz. Consider the deﬁnition (103) for a ¼ b ¼ ½ yielding ð1=2 þ 1Þ t1=21=2 ð1=2 þ 1 1=2Þ ð1=2Þ 1 t ¼0 ¼ ð0Þ

Dt ½t1=2 ¼ 1=2

since gð0Þ ¼ 1. Thus, a particular function is effectively a constant with regard to a certain functional derivative. Consider a second example, this time with monomial index equal to zero b ¼ 0 so that we have the ½-derivative of a constant: ð0 þ 1Þ t1=2 ð0 þ 1 1=2Þ 1 ¼ pﬃﬃﬃﬃﬃ pt

1=2

Dt ½1 ¼

where we see that the constant is not a constant with regard to fractional derivatives. Finally, there is the ½-derivative of t, b ¼ 1: 1=2 Dt ½t

ð1 þ 1Þ 11=2 t ¼ ¼ ð1 þ 1=2Þ

the result obtained by Leibniz.

rﬃﬃﬃ t p

fractal physiology, complexity, and the fractional calculus

57

Another way to introduce fractional operators is by generalizing Cauchy’s formula for a n-fold integration over a ﬁxed time interval (a,t): tn1 ðt ðt ðt1 ð 1 ðnÞ n1 ðt xÞ f ðxÞdx ¼

f ðxn Þ dxn dx1 a Dt ½ f ðtÞ ð104Þ ðn 1Þ! a a

a

a

ðnÞ ½ a Dt

denotes the n-fold integration operation. The where the operator fractional integral analogue to this equation is deﬁned as ðaÞ

a Dt

½ f ðtÞ ¼

ðt 1 ðt xÞa1 f ðxÞ dx; ðaÞ

ta

ð105Þ

a

where the factorial has been replaced by the gamma function, the latter having an analytic continuation into the complex domain for noninteger and nonpositive values of its argument. The corresponding fractional derivative is given by ðaÞ a Dt

¼

d n ðanÞ aD dtn t

ð106Þ

where ½a þ 1 n ½a and the bracket denotes the integer value n closest to a. Consequently for a < 1=2 we have n ¼ 0. Equation (105) is the Riemann– Liouville (RL) formula for the fractional operator; it is the integral operator when a < 0 and it is the differential operator interpreted as (106) when a > 0. 1.

Derivative of a Fractal Function

Richardson, in his 1926 investigation of turbulence, observed that the velocity ﬁeld of the atmospheric wind is so erratic that it probably cannot be described by an analytic function [79]. He suggested a Weierstrass function as a candidate to represent the velocity ﬁeld, since the function is continuous everywhere, but nowhere differentiable, properties he observed in the wind-ﬁeld data. Here we investigate a generalization of the Weierstrass function in order to simplify some of the discussion: 1 X 1 ½1 cosðbn tÞ ð107Þ WðtÞ ¼ n a n¼1 under the conditions that b > a > 1. The generalized Weierstrass function (GWF) satisﬁes the scaling relation WðbtÞ ¼ aWðtÞ

ð108Þ

58

bruce j. west

resulting from a simple shift of the index n in the summation. Equation (108) has the form of a renormalization group scaling relation [5], which can be solved by assuming a solution of the form WðtÞ ¼

1 X

ð109Þ

A n t Hn

n¼1

Inserting Eq. (109) into Eq. (108) yields the equation for the scaling index: Hn ¼

ln a 2pn i ln b ln b

ð110Þ

where the scaling exponent is seen to be complex. This exponent has been related to a complex fractal dimension in the architecture of the human lung [80], in many other physiological systems [5,23], and in earthquakes, turbulence, and ﬁnancial crashes [81]. The GWF is a superposition of harmonic terms having increasing frequencies as powers of b with decreasing amplitudes as powers of 1/a. This function has a fractal dimension D if we choose a ¼ b2D , so that in terms of the fractal dimension we write the GWF as WðtÞ ¼

1 X

1 ½1 cosðbn tÞ ð2DÞn b n¼1

ð111Þ

The RL -fractional integral of the GWF is given by

W

ðaÞ

ðtÞ

ðaÞ ½WðtÞ 1 Dt

1 ¼ ðaÞ

ðt 1

WðxÞ ðt xÞ1a

dx

ð112Þ

for 0 < a < 1, which after some not so straightforward analysis [23,82] yields W ðaÞ ðtÞ ¼

1 X

1

n¼1

bð2DþaÞn

½1 cosðbn tÞ

ð113Þ

Similarly the RL-fractional derivative of the GWF is given by

W

ðaÞ

ðtÞ

ðaÞ 1 Dt ½WðtÞ

1 d ¼ ð1 aÞ dt

ðt 1

WðxÞ dx ðt xÞa

ð114Þ

fractal physiology, complexity, and the fractional calculus

59

for 0 < a < 1, which integrates to [23,82] W ðaÞ ðtÞ ¼

1 X

1 ½1 cosðbn tÞ ð2DaÞn b n¼1

ð115Þ

Consequently, we see that the fractional integral shifts the fractal dimension D ! D a and the fractional derivative shifts the fractal dimension D ! D þ a. These results can be interpreted by noticing that the fractional dimension gives information about the degree of irregularity of the function under analysis. Carrying out a fractional integral of the GWF implies decreasing its fractional dimension and therefore smooths the process, whereas carrying out the fractional derivative means increasing the fractional dimension and therefore making the process and its increments more irregular. What is most intriguing is the fact that a fractional operator acting on a fractal function yields another fractal function; the derivative does not diverge, as does an ordinary derivative of a fractal function, like that of Weierstrass. This suggests that the fractional calculus might be the appropriate method for characterizing the dynamics of complex phenomena, particularly those that are described by fractal functions. 2.

Fractional Brownian Motion

In the previous subsection it would have been possible to extend out discussion to random processes by including random phases in the deﬁnition of the GWF: ( ) 1 X 1 n WðtÞ ¼ Re ð116Þ 1 eib t eifn an n¼1 where the phase is a random quantity uniformly distributed on the interval (0,2p). Equation (116) would be one way to introduce a random function that has the desired scaling properties and we could discuss the dynamics of this process in time using the fractional operators. An alternative approach is to start with continuous functions and that is what we do now. Consider a function deﬁned by the Fourier transform 1 ð 1 xðoÞ do ð117Þ xðtÞ ¼ eiot ^ 2p 1

Operating on this function with the RL-fractional operator, with a lower limit of negative inﬁnity, deﬁnes a new function 1 ð 1 ðaÞ Fa ðtÞ 1 Dt ½xðtÞ ¼ ð118Þ eiot ðioÞa ^xðoÞ do 2p 1

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and the weighting in the integrand is obtained by operating on the exponential in ðaÞ the Fourier transform 1 Dt ½eiot ¼ eiot ðioÞa . If we now interpret the function Eq. (117) as a stochastic quantity, then we can evaluate the correlation of the function Eq. (118) at two time points separated by an interval t hFa ðtÞFa ðt

þ tÞi ¼

1 ð

1 ð2pÞ2

1 ð

do1 1

a ^ ^ do2 eiðo1 o2 Þt eio2 t oa 1 o2 hxðo1 Þx ðo2 Þi

1

ð119Þ where the brackets denote an average over an ensemble of realizations of random ﬂuctuations. If the random ﬂuctuations correspond to a Wiener process, the average in the integrand reduces to h^ xðo1 Þ^ x ðo2 Þi ¼ Cdðo1 o2 Þ

ð120Þ

and C is a constant. Substituting Eq. (120) into Eq. (119) and integrating over one of the frequencies yields for the autocorrelation function hFa ðtÞFa ðt

þ tÞi ¼

C ð2pÞ2

1 ð

doeiot o2a / t2a1

ð121Þ

1

If we make the association of the order of the fractional operator with the Hurst exponent a ¼ H 1=2

ð122Þ

we observe that the fractional index lies in the interval 1=2 a 1=2, because the Hurst exponent is conﬁned to the interval 0 < H 1. Therefore the solution to the fractal stochastic equation of motion ðaÞ 1 Dt ½Fa ðtÞ

¼ xðtÞ

ð123Þ

given by Eq. (115) has the same scaling properties as the dichotomous process with the inverse power-law correlations studied in Section III. Note that Eq. (118) is a colored noise representation of the dynamics expressed by Eq. (123). Even though the central moments of the solution to Eq. (123) scale in the present case in the same way as the moments did for the process in Section III, they are very different processes. The simplest way to see the difference is to note that the integral relation in Eq. (118) is linear so that the solution Fa ðtÞ and the random ﬂuctuations x(t) have the same statistics,

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61

which, by assumption, are Gaussian. Consequently the solution Fa ðtÞ is a realization of fractional Brownian motion with the fractional index restricted to the indicated region. This is certainly different from the exact solution given by the series expansion for the phase-space distribution function in Eq. (74). B.

Fractional Langevin Equations

Of course, the fractional calculus does not in itself constitute a physical/ biological theory; however, one requires such a theory in order to interpret the fractional derivatives and integrals in terms of physical/biological phenomena. We therefore follow a pedagogical approach and examine the simple relaxation process described by the rate equation d ðtÞ þ lðtÞ ¼ 0 dt

ð124Þ

where t > 0 and the relaxation rate l determines how quickly the process returns to its equilibrium state. The solution to Eq. (124) is given by ðtÞ ¼ ð0Þelt, which is unique in terms of the initial condition ð0Þ. An alternative way of writing Eq. (124) is in terms of the anti-derivative operator 1 d ðtÞ ð0Þ ¼ l ðtÞ dt which suggests, for its generalization, replacing the anti-derivative with the RLfractional integral operator ðaÞ

ðtÞ ð0Þ ¼ la 0 Dt

½ðtÞ

ð125Þ

where the lower limit of the fractional integral is zero, corresponding to the initial value problem. Operating on the left in Eq. (125) with the fractional derivative we obtain the generalization to the relaxation equation given by [83] ðaÞ 0 Dt ½ðtÞ

þ la ðtÞ ¼

ta ð0Þ ð1 aÞ

ð126Þ

and the initial value becomes an inhomogeneous term in this fractional relaxation equation of motion. Here the relaxation time is raised to the power a > 0 in order to maintain the correct dimensionality. Equations of the form (126) are mathematically well-deﬁned, and strategies for solving such equations have been developed by a number of investigators, particularly in the book by Miller and Ross [84] that is devoted almost

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exclusively to solving such equations when the index is rational. Here we make no such restriction and consider the Laplace transform of Eq. (126) to obtain ð0Þ sa ~ ðsÞ ¼ s la þ sa

ð127Þ

whose inverse Laplace transform is the solution to the fractional differential equation. Inverting Laplace transforms such as Eq. (127) is nontrivial and an excellent technique that overcomes many of the technical difﬁculties, implemented by Nonnenmacher and Metzler [83], involve the use of Fox functions [53]. The solution to the fractional relaxation equation is given by the series expansion for the standard Mittag–Lefﬂer function ðtÞ ¼ ð0ÞEa ððltÞa Þ ¼ ð0Þ

1 X

ð1Þk ðltÞka ð1 þ kaÞ k¼0

ð128Þ

which in the limit a ! 1 yields the exponential function lim Ea ððltÞa Þ ¼ elt

a!1

as it should, since under this condition (126) reduces to the ordinary relaxation rate equation Eq. (124). The Mittag–Lefﬂer function has interesting properties in both the short-time and the long-time limits. In the short-time limit it yields the Kohlrausch– Williams–Watts Law from stress relaxation in rheology given by lim Ea ððltÞa Þ ¼ eðltÞ

a

t!0

ð129Þ

also known as the stretched exponential. In the long-time limit it yields the inverse power law, known as the Nutting Law, lim Ea ððltÞa Þ ¼ ðltÞa

t!1

ð130Þ

Figure 12 displays the Mittag–Lefﬂer function as well as the two asymptotes, the dashed curve being the stretched exponential and the dotted curve the inverse power law. What is apparent from this discussion is the long-time memory associated with the fractional relaxation process, being an inverse power law rather than the exponential of ordinary relaxation. It is apparent that the Mittag– Lefﬂer function smoothly joins these two empirically determined asymptotic distributions.

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63

MITTAG–LEFFLER FUNCTION

0 –0.25

log–function

–0.5 –0.75 –1 –1.25 –1.5 –1.75 –6

–4

–2 log–variable

0

2

Figure 12. The solid curve is the Mittag–Lefﬂer function, the solution to the fractional relaxation equation. The dashed curve is the stretched exponential (Kohlrausch–Williams–Watts Law), and the dotted curve is the inverse power law (Nutting Law).

We can now generalize the fractional differential equation to include a random force xðtÞ and in this way obtain a fractional Langevin equation ðaÞ 0 Dt ½ðtÞ

þ la ðtÞ ¼

ta ð0Þ þ xðtÞ ð1 aÞ

ð131Þ

The solution to this equation is obtained using Laplace transforms as done previously: ~ xðsÞ ð0Þsa1 ~ ðsÞ ¼ a þ a a l þ sa l þs

ð132Þ

Note the difference in the s-dependence of the two coefﬁcients of the right-hand side of Eq. (132). The inverse Laplace transform of the ﬁrst term yields the Mittag–Lefﬂer function as found in the homogeneous case above. The inverse Laplace transform of the second term is the convolution of the random force and a stationary kernel. The kernel is given by the series Ea;b ðzÞ

1 X

zk ; ðak þ bÞ k¼0

a > 0; b > 0

ð133Þ

which is the generalized Mittag–Lefﬂer function. The function deﬁned by Eq. (133) reduces to the usual Mittag–Lefﬂer function when b ¼ 1, so that both

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bruce j. west

the homogeneous and inhomogeneous terms in the solution to the fractional Langevin equation can be expressed in terms of these series. Note that taking the average value of Eq. (131) and observing that the average of the random force is zero, we obtain ðaÞ 0 Dt ½hðtÞi

þ la hðtÞi ¼

ta ð0Þ ð1 aÞ

ð134Þ

which is clearly of the form of the fractional stress relaxation equation. The average response of the system is determined by the Mittag–Lefﬂer function; that is, the average has a long-time memory (inverse power law). The explicit inverse of Eq. (132) yields the solution [53] ðt

ðtÞ ¼ ð0ÞEa ððltÞ Þ þ ðt t0 Þa1 Ea;a ððltÞa Þxðt0 Þ dt0 a

ð135Þ

0

In the case a ¼ 1, the Mittag–Lefﬂer function becomes the exponential, so that the solution to the fractional Langevin equation reduces to that for an Ornstein– Uhlenbeck process ðtÞ ¼ ð0Þe

lt

ðt

0

þ elðtt Þ xðt0 Þ dt0

ð136Þ

0

as it should. The analysis of the autocorrelation function of Eq. (135) can be quite daunting and so we do not pursue it further here, but refer the reader to the literature [53,85]. A somewhat simpler problem is the fractional Langevin equation without dissipation. Consider the second moment of the solution to the Langevin equation when l ¼ 0 giving rise to h½ðt1 Þ ð0Þ½ðt1 Þ ð0Þi ¼

ðt1

1 ðaÞ

ðt2 dt1 dt2

2 0

0

hxðt1 Þxðt2 Þi ðt1 t1 Þ1a ðt2 t2 Þ1a ð137Þ

Here again we take the random force to have Gaussian statistics and to be delta correlated in time: hxðt1 Þxðt2 Þi ¼ Cdðt1 t2 Þ

ð138Þ

fractal physiology, complexity, and the fractional calculus

65

Inserting Eq. (138) into the expression for the autocorrelation function (137) and noting that the integral is symmetric in the times, the delta function restricts the integration to the lesser of the two times, so introducing the notation for the lesser time t < and greater time t > we obtain [53] h½ðt > Þ ð0Þ½ðt < Þ ð0Þi ¼

a 2Cta1 > t<

ðaÞ2

t< F 1; 1 a; 1 þ a : t>

ð139Þ in terms of the hypergeometric function. Note that although the statistics of the solution are Gaussian, they are also nonstationary, since the autocorrelation function depends on the lesser and the greater times separately and not on their difference. Of course, we can also use the general expression Eq. (139) to write the second moment of the solution at time t ¼ t < ¼ t > , h½ðtÞ ð0Þ2 i ¼ ¼

2Ct2a1

Fð1; 1 a; 1 þ a : 1Þ ðaÞ2 2Ct2a1

ð2a 1ÞðaÞ2

ð140Þ

where the second equality results from writing the hypergeometric function as the ratio of gamma functions. The time dependence of the second moment [Eq. (140)] agrees with that obtained for anomalous diffusion in Section III, if we make the identiﬁcation 2H ¼ 2a 1, where, since the fractional index is less than one, we have 1=2 H > 0. Consequently, the process described by the dissipation-free fractional Langevin equation is antipersistent. This antipersistent behavior of the time series was observed by Peng et al. [25] for the differences in time intervals between heart beats. They interpreted this result, as did a number of subsequent investigators, in terms of random walks with H < 1=2. However, we can see from Eq. (140) that the fractional Langevin equation without dissipation is an equally good, or one might say an equivalent, description of the underlying dynamics. The scaling behavior alone cannot distinguish between these two models; what is needed is the complete statistical distribution and not just the time-dependence (scaling behavior) of a moment. 1.

Physical/Physiological Models

A theoretical Langevin equation is generally constructed from a Hamiltonian model for a simple dynamical system coupled to the environment. The equations of motion for the coupled system are manipulated so as to eliminate the degrees

66

bruce j. west

of freedom of the environment from the dynamical description of the system. Only the initial state of the environment (heat bath) remains in the Langevin description, where the random nature of the driving force is inserted through the choice of distribution of the initial states of the bath. The simplest Langevin equation for a dynamical system open to the environment has the form dXðtÞ þ lXðtÞ ¼ xðtÞ dt

ð141Þ

where x(t) is a random force, l is a dissipation parameter and there exists a ﬂuctuation–dissipation relation connecting the two [86]. Of course we cannot completely interpret Eq. (141) until we specify the statistical properties of the ﬂuctuations, and for this we need to know the environment of the system. The random driver is typically assumed to be a Wiener process—that is, to have Gaussian statistics and no memory. When the system dynamics depends on what occurred earlier—that is, the environment has memory—Eq. (141) is no longer adequate and the Langevin equation must be modiﬁed. The generalized Langevin equation takes this memory into account through an integral term of the form ðt dXðtÞ þ Kðt t0 ÞXðt0 Þ dt0 ¼ xðtÞ dt

ð142Þ

0

where the memory kernel replaces the dissipation parameter and the ﬂuctuation– dissipation relation becomes generalized: KðtÞ ¼ hxðt þ tÞxðtÞi

ð143Þ

Both these Langevin equations are monofractal if the ﬂuctuations are monofractal, which is to say, the time series given by the trajectory X(t) is a fractal random process if the random force is a fractal random process. However, neither of these models is adequate for describing multifractal statistical processes as they stand. A number of investigators have recently developed multifractal random walk models to account for the multifractal character of various physiological phenomena, and here we introduce a variant of those discussions based on the fractional calculus. The most recent generalization of the Langevin equation incorporates memory into the system’s dynamics and has the simple form of Eq. (131) with the dissipation parameter set to zero: ðmÞ 0 Dt ½XðtÞ

ta X0 ¼ xðtÞ ð1 mÞ

ð144Þ

fractal physiology, complexity, and the fractional calculus

67

Equation (144) could also be obtained from the construction of a fractional Langevin equation by Lutz [87] for a free particle coupled to a fractal heat bath, when the inertial term is negligible. The formal solution to this fractional Langevin equation is ðt 1 xðt0 Þdt0 XðtÞ X0 ¼ ðmÞ ðt t0 Þ1m 0

which can be expressed in terms of the integral kernel: ðt

XðtÞ X0 ¼ Km ðt t0 Þxðt0 Þ dt0

ð145Þ

0

As mentioned earlier, the form of this relation for multiplicative stochastic processes and its association with multifractals has been noted in the phenomenon of turbulent ﬂuid ﬂow [61], through a space, rather than time, integration kernel. The random force term on the right-hand side of Eq. (145) is selected to be a zero-centered, Gaussian random variable and therefore to scale as [21] xðltÞ ¼ lH xðtÞ

ð146Þ

where the Hurst exponent is in the range 0 < H 1. In a similar way the kernel in Eq. (145) is easily shown to scale as Km ðltÞ ¼ lm Km ðtÞ

ð147Þ

so that the solution to the fractional Langevin equation scales as XðltÞ X0 ¼ lHþm ½XðtÞ X0

ð148Þ

In order to make the solution to the fractional Langevin equation a multifractal, we assume that the parameter a is a random variable. To construct the traditional measures of multifractal stochastic processes, we calculate the qth moment of the solution (148) by averaging over both the random force and the random parameter to obtain hjXðltÞ X0 jq i ¼ lðq1ÞH hlqm ihjXðtÞ X0 jq i ¼ hjXðtÞ X0 jq ilrðqÞ

ð149Þ

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bruce j. west

The scaling relation in Eq. (149) determines the qth order structure function exponent r(q). Note that when r(q) is linear in q the underlying process is monofractal, whereas when it is nonlinear in q the process is multifractal, because we can relate the structure function to the mass exponent [88]: rðqÞ ¼ 2 tðqÞ

ð150Þ

Consequently we have that r(0) ¼ H so that tð0Þ ¼ 2 H, as it should because of the well-known relation between the fractal dimension and the global Hurst exponent D0 ¼ 2 H. To determine the structure function exponent, we make an assumption about the statistics of the parameter a. We can always write the m-average as hlqm i ¼ heqZðlnlÞ i

ð151Þ

where Z(ln l) is the random variable. In this way the expression on the right-hand side of Eq. (151) is the Laplace transform of the probability density. We assume the random variable is an a-stable Le´vy process in which case the statistics of the multiplicative ﬂuctuations are given by the distribution 1 Pðx; sÞ ¼ 2p

1 ð

a

eikz ebsjkj dk

ð152Þ

1

with 0 < a 2. Inserting Eq. (152) into Eq. (151) and integrating over z yields the delta function dðk þ iqÞ, which, integrating over k, results in a

heqZðlnlÞ i ¼ ebjqj

lnl

a

¼ lbjqj

ð153Þ

so that comparing this result with Eq. (149) we obtain for the structure function exponent rðqÞ ¼ ðq 1ÞH bjqja

ð154Þ

Therefore the solution to the fractional Langevin equation corresponds to a monofractal process only in the case a ¼ 1 and q > 0; otherwise the process is multifractal. We restrict the remaining discussion to positive moments. Thus, we observe that when the memory kernel in the fractional Langevin equation is random, the solution consists of the product of two random quantities giving rise to a multifractal process. This is Feller’s subordination process. We apply this approach to the SRV time series data discussed in Section II and observe, for the statistics of the multiplicative exponent given by Le´vy statistics, the singularity spectrum as a function of the positive moments

fractal physiology, complexity, and the fractional calculus

69

Singularity Spectrum

1

0.95 0.9 0.85 0.8 0

0.5

1

1.5

2

2.5

3

q-moment Figure 13. The singularity spectrum for q > 0 obtained through the numerical ﬁt to the human gait data. The curve is the average over the 10 data sets obtained in the experiment [11].

shown by the points in Fig. 13. The solid curve in this ﬁgure is obtained from the analytic form of the singularity spectrum f ðqÞ ¼ 2 H ða 1Þbqa

ð155Þ

which is determined by substituting Eq. (154) into the equation for the singularity spectrum [Eq. (84)], through the relationship between exponents [Eq. (150)]. It is clear from Fig. 13 that the data are well ﬁt by the solution to the fractional Langevin equation with the parameter values a ¼ 1.45 and b ¼ 0.1, obtained through a mean-square ﬁt of Eq. (155) to the SRV time series data. The nonlinear form of the mass exponent in Fig. 9a, the convex form of the singularity spectrum f(h) in Fig. 9b, and the ﬁt to f(q) in Fig. 13, are all evidence that the interstride interval time series are multifractal. This analysis is further supported by the fact that the maxima of the singularity spectra coincide with the fractal dimensions determined using the scaling properties of the time series using the allometric aggregation technique. Of course, different physiologic processes generate different fractal time series, because the long-time memory of the underlying dynamical processes can be quite different. Physiological signals, such as cerebral blood ﬂow (CBF), are typically generated by complex self-regulatory systems that handle inputs with a broad range of characteristics. Ivanov et al. [89] established that healthy human heartbeat intervals, rather than being fractal, exhibit multifractal properties and uncovered the loss of multifractality for a life-threatening condition of congestive heart failure. West et al. [90] similarly determined that

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bruce j. west

CBF in healthy humans is also multifractal, and this multifractality is severely narrowed for people who suffer from migraines. Migraine headaches have been the bane of humanity for centuries, afﬂicting such notables as Caesar, Pascal, Kant, Beethoven, Chopin, and Napoleon. However, its etiology and pathomechanism have to date not been satisfactorily explained. It was demonstrated [90] that the characteristics of CBF time series signiﬁcantly differs between that of normal healthy individuals and migraineurs. Transcranial Doppler ultrasonography (TCD) enables high-resolution measurement of middle cerebral artery blood ﬂow velocity. Like the HRV, SRV, and BRV time series data, the time series of cerebral blood ﬂow velocity consists of a sequence of waveforms. These waveforms are inﬂuenced by a complex feedback system involving a number of variables, such as arterial pressure, cerebral vascular resistance, plasma viscosity, arterial oxygen, and carbon dioxide content, as well as other factors. Even though the TCD technique does not allow us to directly determine CBF values, it helps to clarify the nature and role of vascular abnormalities associated with migraine. In particular we present the multifractal properties of human middle cerebral artery ﬂow velocity, an example of which is presented below in Fig. 14 The dynamical aspects of cerebral blood ﬂow regulation were recognized by Zhang et al. [91]. Rossitti and Stephensen [92] used the relative dispersion (the ratio of the standard deviation to mean), of the middle cerebral artery ﬂow velocity time series to reveal its fractal nature; this is a technique closely related to the allometric aggregation introduced in Section II. West et al. [93] extended this line or research by taking into account the more general properties of fractal time series, showing that the beat-to-beat variability in the ﬂow velocity has a long-time memory and is persistent with the average scaling exponent 0.85 0.04, a value consistent with that found earlier for HRV time series. They also observed that cerebral blood ﬂow was multifractal in nature.

Figure 14.

Middle cerebral artery ﬂow velocity time series for a typical healthy subject [90].

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71

Figure 15. The average multifractal spectrum for middle cerebral blood ﬂow time series is depicted by f(h). (a) The spectrum is the average of 10 time series measurements from ﬁve healthy subjects (ﬁlled circles). The solid curve is the best least-squares ﬁt of the parameters to the predicted spectrum using Eq. (157). (b) The spectrum is the average of 14 time series measurements of eight migraineurs (ﬁlled circles). The solid curve is the best least-squares ﬁt to the predicted spectrum using Eq. (157). (Taken from [90].)

In Fig. 15 we compare the multifractal spectrum for middle cerebral artery blood ﬂow velocity time series for a healthy group of ﬁve subjects and a group of eight migraineurs [90]. A signiﬁcant change in the multifractal properties of the blood ﬂow time series is apparent. Namely, the interval for the multifractal distribution on the local scaling exponent is greatly constricted. This is reﬂected in the small value of the width of the multifractal spectrum for the migraineurs (0.013), which is almost three times smaller than the width for the control group (0.038); for both migraineurs with and without aura the distribution is centered at 0.81, the same as that of the control group, so the average scaling behavior would appear to be the same. However, the contraction of the spectrum suggests that the underlying process has lost its ﬂexibility. The biological advantage of

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bruce j. west

multifractal processes is that they are highly adaptive, so that in this case the brain of a healthy individual adapts to the multifractality of the interbeat interval time series. Here again we see that disease, in this case migraine, may be associated with the loss of complexity and consequently the loss of adaptability, thereby suppressing the normal multifractality of cerebral blood ﬂow time series. Thus, the reduction in the width of the multifractal spectrum is the result of excessive dampening of the cerebral ﬂow ﬂuctuations and is the manifestation of the signiﬁcant loss of adaptability and overall hyperexcitability of the underlying regulation system. West et al. [90] emphasize that hyperexcitability of the CBF control system seems to be physiologically consistent with the reduced activation level of cortical neurons observed in some transcranial magnetic simulation and evoked potential studies. Regulation of CBF is a complex dynamical process and remains relatively constant over a wide range of perfusion pressure via a variety of feedback control mechanisms, such as metabolic, myogenic, and neurally mediated changes in cerebrovascular impedance respond to changes in perfusion pressure. The contribution to the overall CBF regulation by different areas of the brain is modeled by the statistics of the fractional derivative parameter, which determines the multifractal nature of the time series. The source of the multifractality is over and above that produced by the cardiovascular system. The multifractal nature of CBF time series is here modeled using a fractional Langevin model. We again implement the scaling properties of the random force and the memory kernel to obtain Eq. (148) as the scaling of the solution to the fractional Langevin equation. Here when we calculate the qth moment of the solution we assume Gaussian, rather than the more general Le´vy, statistics. Consequently we obtain the quadratic function for the singularity spectrum f ðqÞ ¼ 2 H bq2

ð156Þ

which can be obtained from Eq. (155) by setting a ¼ 2. Another way to express Eq. (156) is b f ðhÞ ¼ f ðHÞ ðh HÞ2 4

ð157Þ

where we have used the fact that the fractal dimension is given by 2 H, which is the value of the function at h ¼ H. It seems that the changes in the cerebral autoregulation associated with migraine can strongly modify the multifractality of middle cerebral artery blood ﬂow. The constriction of the multifractal to monofractal behavior of the blood ﬂow depends on the statistics of the fractional derivative index. As the distribution of this parameter narrows down to a delta function, the nonlocal

fractal physiology, complexity, and the fractional calculus

73

inﬂuence of the mechanoreceptor constriction disappears. On the other hand, the cerebral autoregulation does not modify the monofractal properties characterized by the single global Hurst exponent, presumably that produced by the cardiovascular system. C.

Fractional Diffusion Equations

The change in time of a stationary stochastic process using the conditional transition probability density Pðx; tjx0 ; t0 Þ for the dynamical variable X(t) to lie in the range (x; x þ dx) conditional on X(t0 ) ¼ x0 is given by the chain condition ð Pðx; tjx0 ; t0 Þ ¼ Pðx; tjx0 ; t0 ÞPðx0 ; t0 jx0 ; t0 Þ dx0

ð158Þ

where is the domain of the variate. This equation is often used as the starting equation for the analysis of Brownian motion. Here Pðx; tjx0 ; t0 Þ is the probability that the process undergoes a transition from the initial value x0 to a ﬁnal value x at time t through a sequence of intermediate values. Equation (158) was introduced by Bachelier in 1900 in his Ph.D. thesis on speculation in the French stock market. The nonphysical application of this equation was probably the reason why his work went unnoticed for nearly 50 years, even though the mathematical content was equivalent to that found in the Einstein papers on diffusion published ﬁve and more years later. The chain condition is the general description of the evolution of the probability density for an inﬁnitely divisible stable process and the solution to which has the most general form of a Markov probability density. When the range of the variate is unbounded ¼ ð1; 1Þ and the process under consideration has translational invariance, so the probability density is independent of the origin of the coordinate system Pðx; tjx0 ; t0 Þ ¼ Pðx x0 ; t t0 Þ, the chain condition becomes ð Pðx x0 ; t t0 Þ ¼ Pðx x0 ; t t0 ÞPðx0 x0 ; t0 t0 Þ dx0

ð159Þ

The stationary chain condition (159) is more simply expressed in terms of characteristic functions, the Fourier transform of the probability density, as the product fðk; t t0 Þ ¼ fðk; t t0 Þfðk; t0 t0 Þ

ð160Þ

using the convolution property of Fourier transforms. Montroll and West [64] noticed that, since the probability density resulting from the characteristic function satisﬁes the product form, its solution yields an inﬁnitely divisible

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distribution. The most general form of the characteristic function for inﬁnitely divisible distributions was ﬁrst obtained by Paul Le´vy in 1937. Here we merely sketch how to obtain the general solution to Eq. (160). Take the logarithm of the equation to obtain logfðk; t t0 Þ ¼ logfðk; t t0 Þ þ logfðk; t0 t0 Þ

ð161Þ

from which it is clear that the characteristic function factors into a function of k, say g(k), and a function of time. In order for the intermediate time to vanish from the solution, the function of time must be linear. Thus, the form of the solution to Eq. (161) is fðk; tÞ ¼ egðkÞt

ð162Þ

Since the probability density is normalizable at all times, the real part of g(k) must be negative deﬁnite. In order for the characteristic function to retain the product form at all spatial scales, it must be inﬁnitely divisible. If we scale the Fourier variable k by a constant factor b, then in order for the probability density to be inﬁnitely divisible, g(k) must be homogeneous: ð163Þ

gðbkÞ ¼ ba gðkÞ The homogeneity requirement implies that gðkÞ ¼ bðaÞjkja

ð164Þ

where b(a) is a complex function dependent on the parameter a, with a positive deﬁnite real part. Thus, we have for the characteristic function fðk; tÞ ¼ ebðaÞjkj

a

t

ð165Þ

The symmetric solution to the chain condition is obtained by setting the constant in the exponential to be independent of a, b(a) ¼ b. The most general solution is obtained using

k ð166Þ bðaÞ ¼ b 1 þ iCoðk; aÞ jkj where C is a real parameter, o(k,a) is a real function, and the imaginary part of the coefﬁcient determines the skewness of the distribution. The functional form of the characteristic function in Eq. (165) gives

k fðk; tÞ ¼ exp btjkja 1 þ iCoðk; aÞ jkj

ð167Þ

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so that the inverse Fourier transform sets the conditions 0 < a 2, so that it is positive deﬁnite, b > 0 so that it is normalizable, and 1 C 1 indicating the degree of skewness. Finally the function o(k,a) is deﬁned by ( tanðap=2Þ if a 6¼ 1 ð168Þ oðk; aÞ ¼ 2 ln jkj if a¼ 1 p whose derivation can be found in Gnedenko and Kolmogorov [63]. The equation of evolution for the probability density is obtained by taking the time derivative of the characteristic function in Eq. (165): qfðk; tÞ ¼ bðaÞjkja fðk; tÞ qt

ð169Þ

The inverse Fourier transform of Eq. (169) yields qPðx; tÞ ¼ qt

1 ð

bðaÞeikx jkja fðk; tÞ

1

dk 2p

ð170Þ

which Gorenﬂo and Mainardi [94] identify with Le´vy–Feller diffusion through the Feller pseudodifferential operator Day, the Feller fractional derivative of order a: qPðx; tÞ ¼ Day ½Pðx; tÞ ð171Þ qt In this notation the inverse Fourier transform of the characteristic function in Eq. (167) is Green’s function for Eq. (171), but in a notation where h i fðk; t; a; yÞ ¼ exp tjkja eiypsignðkÞ=2

ð172Þ

and the Feller pseudodifferential operator acting with respect to the spatial variable x has the Fourier representation ^ ay ¼ jkja eiypsignðkÞ=2 D

ð173Þ

In the symmetric case where C ¼ 0 Eq. in (167), using the convolution property of the product of Fourier amplitudes in Eq. (170), we obtain [95] qPðx; tÞ b ¼ ða þ 1Þsinðap=2Þ qt p

1 ð

1

Pðx0 ; tÞ dx0 jx x0 jaþ1

ð174Þ

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Note that the integral term in Eq. (174) is the Reisz fractional derivative, ﬁrst applied in this context by Seshadri and West [96] and whose solution is the symmetric Le´vy distribution. It is worth stressing that in the last few years the approaches based on fractional derivatives, of which Eq. (174) is an early example [95], have received an ever-increasing interest, as shown by the excellent review articles by Metzler and Klafter [97] and Sokolov et al. [78]. The symmetric Le´vy distribution that solves Eq. (174) is 1 Pðx; tÞ ¼ 2p

1 ð

a

eikx ebtjkj dk

ð175Þ

1

which satisﬁes the scaling relation 1 1 Pðx; tÞ ¼ ga P ga x; gt

ð176Þ

as does the more general form of the Le´vy a-stable distribution. From Eq. (176) it is clear that the Le´vy or Le´vy–Feller diffusion diffusion process has the scaling, x td , with d ¼ 1=a

ð177Þ

as we demonstrated in Section III. This scaling is consistent with the process generated by the ﬂuctuations of the variable x, as veriﬁed by numerical simulation [98]. D.

Langevin Equation with Le´vy Statistics

We now examine the response of a linear dissipative system to Le´vy ﬂuctuations using the ordinary Langevin equation, dVðtÞ þ lVðtÞ ¼ xðtÞ dt

ð178Þ

where V(t) is the dynamical variable—say, the velocity of a particle—l is the dissipation parameter, and the ﬂuctuations are represented by a stationary differential Markov process whose statistics are assumed to be given by a symmetric Le´vy distribution 1 pðx; tÞ ¼ 2p

1 ð

1

a

eikx ebtjkj

ð179Þ

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As we know, if the random force had Gaussian statistics and was delta-correlated in time, we would have an Ornstein–Uhlenbeck process. The variance of the system response would increase linearly in time for early times and be constant at late times. However, when the random force is Le´vy-stable the second moment of the system response is inﬁnite. The linear dynamical equation can be formally integrated to yield ðt 0 VðVð0Þ; tÞ ¼ V0 elt þ elðtt Þ xðt0 Þ dt0

ð180Þ

0

where V0 ¼ Vð0Þ is the initial value of the velocity variable. West and Seshadri [95] used the phase-space equations to determine the conditional probability density for this process. However, it is somewhat easier to use the characteristic function [53], given by 1 ð

fðk; tjVð0ÞÞ ¼

eiku Pðu; tju0 Þ du

ð181Þ

1

to determine the complete dynamical properties of the system response. Another way to express the characteristic function is in terms of the solution to the Langevin equation fðk; tjVð0ÞÞ ¼ heikVðtÞ i ¼ exp½iku0 e

*

lt

" ðt #+ lðtt0 Þ 0 0 exp ik e xðt Þ dt

ð182Þ

0

where V(0) ¼ u0. Doob [99] has shown that a differential Le´vy process described by Eq. (179), for an arbitrary analytic function q(t), satisﬁes the equation *

" ðt " #+ # ðt 0 0 0 2 0 a 0 exp i qðt Þxðt Þdt ¼ exp s jqðt Þj dt 0

ð183Þ

0

Thus, the characteristic function in (182) can be evaluated to yield

fðk; tjVð0ÞÞ ¼ exp iku0 elt exp s2al ðtÞjkja

ð184Þ

where the time-dependent ‘‘variance’’ is s2al ðtÞ

s2 1 ealt al

ð185Þ

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which agrees with the result obtained by Doob [99] and also by West and Seshadri [95]. The conditional probability density is then given by 1 ð

Pðu; tju0 Þ ¼ 1

dk exp ikðu u0 elt Þ s2al ðtÞjkja 2p

ð186Þ

where the Fourier transform is taken with respect to the centered variable u u0 elt . Hence, the solution to the linear Langevin equation driven by a random force with Le´vy statistics has Le´vy-stable statistics in the variable u u0 elt with Le´vy index a and parameter given by Eq. (185). In the long time limit the characteristic function in Eq. (184) reduces to the asymptotic form

s2 fðk; 1jVð0ÞÞ ¼ exp jkja al

ð187Þ

a characteristic function that is independent of both time and the initial state of the system. At long times the probability distribution given by the inverse Fourier transform of Eq. (187) attains the steady-state form 1 Pss ðuÞ ¼ 2p

1 ð

a

dkeiku ebjkj

ð188Þ

1

where the Le´vy parameter is given by b ¼ s2 =al. Thus, the dissipation in the linear Langevin equation leads to a steady state in the presence of Le´vy ﬂuctuations. The variance of the system response is, however, inﬁnite for all t > 0 and, in particular, for the steady-state distribution given by Eq. (188). A scale-invariant biological process that has been shown to possess such Le´vy statistics is the human heartbeat time series, see Peng et al. [25]. The data consist of digitized electrocardiograms of beat-to-beat heart rate ﬂuctuations over approximately 24 hours or 105 beats recorded with an ambulatory monitor. The time series is constructed by recording the interval between adjacent beats as data, for example, let f(n) be the interval between the n and n þ 1 beat. A great deal of variability is observed in the interbeat interval as we discussed in Section II. Peng et al. [25] graph the histogram for the differences in the beatto-beat intervals I(n) ¼ f(n þ 1) f(n) and ﬁnd that this is a stationary Le´vy process such as given by Eq. (188) and shown in Fig. 16. They ﬁnd that the statistics of healthy and diseased (dilated cardiomyopathy) conditions are the same, that being Le´vy-stable with an index a ¼ 1.7; however, the spectra for the two cases are quite different.

fractal physiology, complexity, and the fractional calculus 1.0

79

(a)

P(I) / P(0)

0.8 0.6 0.4 0.2 0 –3

P(I) / P(0)

10 0

–2

–1

0 I / S.D.

1

2

3

(b)

10 –1

10 –2

10 –3 –3.0 –2.0 –1.0

0.0 1.0 I / S.D.

2.0

3.0

Figure 16. The histogram in the differences between interbeat intervals I for healthy (circles) and diseased (triangles) subjects P(I) is the probability of ﬁnding an interbeat increment in the range [I I=2; I þ I=2]. To facilitate comparison, we divide the variable I by the standard deviation of the increment data and rescale the probability with P(0). In Le´vy-stable distributions, a is related to the power-law exponent describing the distribution for large values of the variable, while the width of the distribution is characterized by b. Both histograms are well ﬁtted by a Le´vy-stable distribution with a ¼ 1:7 (solid line). The dashed line is a Gaussian distribution and is shown for comparison purposes only. Similar ﬁts were obtained for 8 of the 10 normal subjects with heart disease. The slow decay of Le´vy-stable distributions for large increment values may be of physiological importance and relate to the dynamics of the system: (a) linear–linear scale; (b) log–linear scale. (From Peng et al. [25] with permission.)

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The power spectrum S(f), the square of the Fourier transform of I(n), yields Sðf Þ f b , where b ¼ 1 2H and the mean-square level of the interbeat ﬂuctuations increases as n2H. Here again H ¼ 0:5 corresponds to Brownian motion, so that b ¼ 0 indicates the absence of correlations in the time series I(n) (‘‘white noise’’). They observed that for a diseased data set, b is approximately zero in the low-frequency regime, conﬁrming that the I(n) are not correlated over long times. On the other hand, they observed that for the healthy data set, b is approximately equal to 1, indicating a long-time correlation in the interbeat interval differences. The anticorrelated property of I(n) are consistent with a nonlinear feedback system that ‘‘kicks’’ the heart rate away from extremes. This tendency operates on a wide range of time scales, not on a beat-to-beat basis. The conclusion is that the different scaling pattern must be a consequence of the ordering of the differences, rather than their statistics, which is to say in the correlations produced by the underlying dynamics. The power-law spectrum has been observed in a number of dynamical systems having chaotic solutions, see, for example, Reichl [100]. Goldberger and West suggested that the observed spectrum may be a consequence of such dynamics and that one may interpret the modiﬁcations in the inverse power-law spectrum as being indicative of pathology and therefore of diagnostic and prognostic value. Loss of heart rate variability has already been described in numerous settings, including multiple sclerosis [101], fetal distress [102], bed-rest deconditioning [103], aging [104], and in certain patients at risk for sudden cardiac death [105]. Presumably, the more severe pathologies will be associated with the greatest loss of spectral power, which we have referred to as loss of spectral reserve [106]. V.

SUMMARY, CONCLUSIONS, AND SPECULATIONS

If one were to form a hierarchy of understanding of complex phenomena, it would surely start with physics as the most basic, expand into chemistry as large aggregates of atoms and molecules form, become biology as life is breathed into these chemical aggregates, and then form physiology as the phenomenology of human life is explored. Mathematical rigor is demanded at the base of this hierarchy, but mathematical models become less familiar and more suspect as we climb the ladder of complexity from physics to physiology. The standard in physics is the high-order accuracy to which scientists can now theoretically predict the measured value of the ﬁne structure constant or the gravitational constant. The comfortable mathematical models that physicists rely on are missing in the studies of biomedical phenomena, and even where such models exist they do not have the degree of agreement with data that physicists have come to expect. Thus, it might seem to some that the application of mathematical concepts such as fractals, scaling, inverse power-law distributions, and the fractional calculus to physiology, such as done in this chapter, is premature.

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However, unless one is willing to neglect the serious attempts that have been and are being made to apply these ideas to the understanding of physiology and medicine, and dismiss the entire activity as misguided, overviews, such as the one given herein, of how these ideas are being implemented are useful. We have seen in Section II that physiologic time series, such as the interbeat intervals of the human heart, the interstride intervals of human gait, and the interbreath intervals in human breathing, although apparently random, do in fact have long-time memory. This combination of randomness and order has been used as the deﬁning characteristic of complexity in this chapter. In a medical context, this complexity is encountered when attempting to understand physiological phenomena from a hotistic perspective, rather than looking at speciﬁc mechanisms. We have used the allometric aggregation technique to establish that such dynamic phenomena are complex, at least in the sense that they generate time series that are statistical fractals. The scaling behavior of such time series determine the overall properties such complex systems must have, much like the older analysis of errors and noise in physical systems. The historical view of complexity involved having a large number of variables, each variable making its individual contribution to the operation of the system and each variable responding in direct proportion to the changes in the other variables. The small differences in the contributions produced the ﬂuctuations in the observed outcome. The linear additive statistics of measurement error or biological noise is not applicable to complex medical phenomena discussed here. The elements in complex physiologic systems are too tightly coupled, so instead of a linear additive process, nonlinear multiplicative statistics more accurately represent the ﬂuctuations. In this chapter we examined how intersystem interactions in a generic physiologic system may give rise to the observed scaling. The individual mechanisms giving rise to the observed statistical properties in physiologic systems are very different, so we did not attempt to present a common source to explain the observed scaling in walking, breathing, and the beating heart. On the other hand, the physiologic time series for each of these phenomena scale in the same way, so that at a certain level of abstraction the separate mechanisms cease to be important and only the relations matter and not those things being related. It is the relation between blood ﬂow and heart function, between locomotion and postural balance, and between breathing and respiration, which are important. The thesis of complexity theory, insofar as such a theory can be said to exist, is that such relations have a common form for complex phenomena. This assumption is not so dramatic as it might ﬁrst appear. Consider that traditionally such relations have been assumed to be linear, in which case their control was assumed to be in direct proportion the disturbance. Linear control theory has been the backbone of homeostasis, but fails miserably in describing, for example, the full range of HRV from the running child to the sedate senior, with all the pathologies that await them along the way.

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The issue we ﬁnally address is how to control complexity. Such control is one of the goals of medicine—in particular, understanding and controlling physiologic networks in order to ensure their proper operation. We distinguish between homeostatic control and allometric control mechanisms. Homeostatic control is familiar and has as its basis a negative feedback character which is both local and instantaneous. Allometric control, on the other hand, is a relatively new concept that can take into account (a) long-term memory, (b) correlations that are inverse power law in time, and (c) long-range interactions in complex phenomena as manifest by inverse power-law distributions in the system variable. Allometric control introduces the fractal character into otherwise featureless random time series to enhance the robustness of physiologic networks by introducing either fractional Brownian motion or fractional Le´vy diffusion into the control of the network. It is not merely a new kind of control that is suggested by the scaling of physiologic time series. Scaling also suggests that the historical notion of disease, which has loss of regularity at its core, is inadequate for the treatment of dynamical diseases. Instead of loss of regularity, we identify the loss of variability with disease, so that a disease not only changes an average measure, such as heart rate, which it does in late stages, but is manifest in changes in heart rate variability at very early stages. Loss of variability implies a loss of physiologic control, and this loss of control is reﬂected in the change of fractal dimension—that is, in the scaling index of the corresponding time series [56]. The proper operation of physiologic processes is manifest through the scaling of appropriate time series. A measured function denoted by X(t) is said to be homogeneous when the time axis being scaled by a constant g yields the original function modiﬁed by an overall scale X(gt) ¼ X(t)=gH. This scaling behavior generalizes to time series when the measured function is stochastic, and the scaling relation is interpreted in terms of the probability density function rather than the dynamic function itself. This latter scaling is evident in Eq. (35). In one of the stochastic models discussed Section III, the scaling behavior of the process of interest is a consequence of the two-point stochastic process driving the system having an inverse power-law autocorrelation function. The scaling of the autocorrelation function is only approximate, in that lim x ðgtÞ ¼ x ðtÞ=gb

t!1

so the scaling arises asymptotically in the noise. The correlation in the noise however, gives rise to scaling in the exact solution to the phase-space equation of evolution given by Eq. (74): XðgtÞ ¼ XðtÞ=g1b=2

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The overall scaling exponent is therefore given by H ¼ 1 b=2 where H is the Hurst exponent for the measured time series. In Section II we argue for the ubiquity of such behavior in physiologic phenomena. Section III steps back from physiology to review various mathematical models that can generate the fractal time series uncovered in Section II for various physiologic time series using allometric aggregation. We began with a brief discussion of simple random walks and showed how the resulting stochastic process for diffusion has a second moment that scales linearly in time. The arguments were extended to fractional random walks to explain anomalous diffusion deﬁned by the second moment [Eq. (19)] scaling nonlinearly in time: hXðtÞ2 i / t2H This leads us to one of the standard, but often inappropriate, explanations of anomalous diffusion using fractional Brownian motion with the probability density

x2 exp 2Dt2H pðx; tÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4pDt2H It is clear that this distribution function satisﬁes the scaling relation Eq. (35) with the scaling index given by d ¼ H. We note that the variance calculated using fractional Brownian motion distribution is given by the equation for anomalous diffusion. However, we can see that this explanation of anomalous diffusion is not unique and there are multiple statistics that lead to this type of scaling. Consequently, we referred to all such models collectively as fractal stochastic processes and subsequently discussed alternative measures that can distinguish among them. An alternative to the random walk model in describing diffusion is the Langevin equation, where the microscopic dynamics are linked to a macroscopic rate equation through a stochastic driving force. A simple dichotomous driver with an inverse power-law memory having an index b was shown to yield an asymptotic system response that has the scaling given by Eq. (35) with scaling index d ¼ 1 b=2. However, these mathematically generated fractal random processes are not in themselves sufﬁcient to properly describe the physiological processes considered herein. The physiologic time series are shown to be multifractal rather than mono-fractal. An example of the multifractal character

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bruce j. west

of walking is shown in Fig. 9 and that of others (such as heartbeats) are referenced. A way to distinguish among different ways of generating anomalous diffusion processes is by using diffusion entropy analysis (DEA), where the scaling of the probability density Eq. (35) yields a scaling of the Shannon entropy given by Eq. (91). Consequently, we determined that there are time series for which the scaling index satisﬁes d ¼ H, such as fractional Brownian motion, and other time series for which d 6¼ H, such as the exact solution to the dichotomous Langevin equation. Also we observed that there is a third class of processes where d ¼ (3 2H)1, which is valid for Le´vy random walks [66] as distinct from Le´vy ﬂights. We discussed in Section IV how the fractional calculus could embody a number of the properties so prevalent in physiologic phenomena, not the least of which being that the evolution of a fractal processes can be described by a fractional differential equation. We showed that the evolution of fractal stochastic process can be described by a fractional Langevin equation in which a fractional differential equation is driven by a stochastic force. In particular, we demonstrated that the fractional calculus could provide a description of the dynamics of an anomalous diffusion process in which the long-time memory is not part of the stochastic driver, as it was in the earlier models, but is actually part of the system’s nonlocal dynamics through the fractional derivatives. The multifractal character of certain physiological time series, such as gait and cerebral blood ﬂow, is described by fractional Langevin equations with random indices. The multifractal spectrum is shown to be related to the statistical properties of these random indices. Finally, the fractional calculus was used to construct fractional diffusion equations. One such equation, in particular, models the evolution of the Le´vy astable probability density describing Le´vy diffusion, another mechanism for generating anomalous diffusion. It was shown that this probability density satisﬁes the scaling relation [Eq. (35)] with the Le´vy index a such that d ¼ 1=a. The dynamics of a Le´vy diffusion process, using a Langevin equation, were also considered. The probability density for a simple dissipative process being driven by Le´vy noise is also Le´vy but with a change in parameters. This is a possible alternate model of the ﬂuctuations in the interbeat intervals for the human heart shown to be Le´vy stable over a decade ago [25]. The well-being of the body’s system-of-systems is measured by the fractal scaling properties of the various dynamic subsystems, and such scaling determines how well the overall complexity is maintained. Once the perspective that disease is the loss of complexity has been adopted, the strategies presently used in combating disease must be critically examined. Life support equipment is one such strategy, but the tradition of such life support is to supply blood at the average rate, of the beating heart, to ventilate the lungs at their average rate,

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and so on. So how does the new perspective regarding disease inﬂuence the traditional approach to healing the body? Alan Mutch, of the University of Manitoba, argues that both blood ﬂow and ventilation are delivered in a fractal manner in both space and time in a healthy body. However, during critical illness, conventional life support devices deliver respiratory gases by mechanical ventilation or blood by cardiopulmonary bypass pump in a monotonously periodic fashion. This periodic driving overrides the natural a´periodic operation of the body. Mutch [107] speculates that these devices result in the loss of normal fractal transmission and consequently: . . . life support systems do more damage the longer they are required and are more problematic the sicker the patient . . . . We hypothesize that loss of fractal transmission moves the system through a critical point . . . to transform a cohesive whole to one where organ systems are no longer as well connected.

Disease as the loss of complexity is consistent with the view that complex phenomena have a multiplicity of failure modes. These failure modes result in phenomena changing character, invariably becoming simpler with an accompanying inability to carryout their function. A cascade of failures is not so much a consequence of the initiating event as it is the result of the state of the network when the event is initiated. It is, in part, the irreversibility of failure cascades that makes them so formidable. In medicine such failure cascades may be manifest as multiple organ dysfunction syndrome (MODS) that rapidly accumulates following a minor insult; MODS is the leading cause of death in intensive care units. As Buchman [108] points out: Despite timely and appropriate reversal of the enticing insult . . . many patients develop the syndrome. Mortality is proportional to the number and depth of system dysfunction and the mortality of MODS after (for example) repair of ruptured abdominal aortic aneurysm is little changed despite three decades of medical progress.

One of the consequences of the traditional view of disease is what Buchman [108] calls ‘‘ﬁx-the-number’’ imperative: If the bicarbonate level is low, give bicarbonate; if the urine output is low, administer a diuretic; if the bleeding patient has a sinking blood pressure, make the blood pressure normal. Unfortunately, such interventions are commonly ineffective and even harmful. For example, sepsis—which is a common predecessor of MODS—is often accompanied by hypocalcaemia. In controlled experimental conditions, administering calcium to normalize the laboratory value increases mortality.

Consequently, one’s ﬁrst choice of options, based on an assumed simple linear causal relationship between input and output as in homeostatsis, is probably wrong.

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A number of scientists [109] have demonstrated that the stability of hierarchal biological systems is a consequence of the interactions among the elements of the system. Furthermore, there is an increase in stability resulting from the nesting of systems within systems—organelles into cells, cells into tissue, tissues into organs, and so on, up from the microscopic to the macroscopic. Each system level confers additional stability on the overall fractal structure. The fractal nature of the system suggests a basic variability in the way systems are coupled together. For example, the interaction between cardiac and respiratory cycles is not constant, but adapts to the physiologic challenges being experienced by the body. A number of scientists have arrived at remarkably similar conclusions regarding the nature of disease that is quite different from the traditional one. Take, for example, the following observation made by Buchman [108]: . . . Herein, we have suggested that breakdown of network interactions may actually cause disease, and when this breakdown is widespread the clinical manifestation is the multiple organ dysfunction syndrome. If the hypothesis is correct, then network dysfunction might be expected at multiple levels of granularity, from organ systems to intracellular signal molecules. Restoration of network integrity may be a reasonable therapeutic goal, and a more permissive approach to clinical support (including algorithms that simulate biological variability) might facilitate restoration of network complexity that now appears essential to health.

Or those made by Mutch [107]: The layer upon layer of fractal redundancy in scale-free biological systems suggests that attack at one level does not place the organism at undue risk. But attack at vital transmission nodes can cause catastrophic failure of the system. The development of multiple organ dysfunction syndrome (MODS) in critically ill humans may be such a failure. Once devolved, death almost inevitably ensues. The similarity to concerted attack on vital Internet router nodes is evident. Patients managed by conventional non-fractal life support may sustain further unintentional attack on a devolving scale-free system due to loss of normal fractal transmission. Returning fractal transmission to life support devices may improve patient care and potentially offer benefit to the sickest of patients.

We conclude with a number of observations: 1. The empirical evidence overwhelmingly supports the interpretation of the time series analysis that complex physiologic phenomena are described by fractal stochastic processes. Furthermore, the fractal nature of these time series is not constant in time but changes with the vagaries of the interaction of the system with its environment, and therefore these phenomena are multifractal.

fractal physiology, complexity, and the fractional calculus

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2. The scaling index or fractal dimension marks the system’s response and can be used as an indicator of the system’s state of health. Since the fractal dimension is also a measure of the level of complexity, the change in dimension with disease suggests a new deﬁnition of disease as a loss of complexity, rather than the loss of regularity [56]. This observation was ﬁrst made by Goldberger and West, see, for example, Ref. [110]. 3. The fractal dynamics of complex physiologic systems can be modeled using the fractional rather than the ordinary calculus because the changes in the fractal functions necessary to describe physiologic complexity remain ﬁnite in the former formalism but diverge in the latter [53].

References 1. B. B. Mandelbrot, Fractals, Form, Chance and Dimension, W. H. Freeman, San Francisco, 1977. 2. E. Broda, Ludwig Boltzmann, Man-Physicist-Philosopher, Ox Bow Press, Woodbridge, 1983. 3. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, 1982. 4. E. Schro¨dinger, What Is Life? The Physical Aspects of the Living Cell, Cambridge University Press, London, 1943. 5. B. J. West and W. Deering, Fractal physiology for physicists: Le´vy statistics. Phys. Rep. 246, 1–100 (1994). 6. C. K. Peng, J. Mistus, J. M. Hausdorff, S. Havlin, H. E. Stanley, and A. L. Goldberger, Long-range anticorrelations and non-Gaussian behavior of the heartbeat. Phys. Rev. Lett. 70, 1343–46 (1993). 7. B. Suki, A. M. Alencar, U. Frey, P. Ch. Ivanov, S. V. Buldyrev. A. Majumdar, H. E. Stanley, C. A. Dawson, G. S. Krenz, and M. Mishima, Fluctuations, noise and scaling in the cardiopulmonary system. Fluctuations and Noise Lett. 3, R1–R25 (2003). 8. B. J. West, R. Zhang, A. W. Sanders, J. H. Zuckerman, and B. D. Levine, Fractal ﬂuctuations in transcranial Doppler signals. Phys. Rev. E 59, 3492 (1999). 9. B. J. West, A. Maciejewsk, M. Latka, T. Sebzda, and Z. Swierczynski, Wavelet analysis of scaling properties of gastric electrical activity. To appear in Am. J. Physiol. 10. J. M. Hausdorff, C.-K. Peng, Z. Ladin, J. Y. Ladin, J. Y. Wei, and A. L. Goldberger, Is walking a random walk? Evidence for long-range correlations in stride interval of human gait. J. Appl. Physiol. 78(1), 349–358 (1995). 11. L. Grifﬁn, D. J. West, and B. J. West, Random stride intervals with memory. J. Biol. Phys. 26, 185–202 (2000). 12. J. M. Hausdorff, L. Zemany, C.-K. Peng and A. L. Goldberger, Maturation of gait dynamics: stride-to-stride variability and its temporal organization in children. J. Appl. Physiol. 86, 1040– 1047 (1999). 13. N. Scafetta, L. Grifﬁn, and B. J. West, Holder exponent spectra for human gait. Physica A 328, 561–583 (2003). 14. B. J. West, L. A. Grifﬁn, H. J. Frederick, and R. E. Moon, The independently fractal nature of respiration and heart rate during exercise under normobaric and hyperbaric conditions. To appear in Respiratory Physiol. Neurobiol.

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15. G. I. Barenblatt, Scaling Phenomena in Fluid Mechanics, Cambridge University Press, Cambridge, 1994. 16. J. S. Huxley, Problems of Relative Growth, Dial Press, New York, 1931. 17. L. R. Taylor, Aggregation, variance and the mean. Nature 189, 732–735 (1961). 18. L. R. Taylor and R. A. J. Taylor, Aggregation, migration and population mechanics. Nature 265, 415–421 (1977). 19. J. C. Willis, Age and Area: A Study in Geograhical Distribution and Origin of Species, Cambridge University Press, New York, 1922. 20. L. R. Taylor and I. P. Woiwod, Temporal stability as a density-dependent species characteristic. J. Animal Ecol. 49, 209–224 (1980). 21. J. B. Bassingthwaighte, L. S. Liebovitch, and B. J. West, Fractal Physiology, Oxford University Press, New York, 1994. 22. Heart rate variability. Eur. Heart J. 17, 354–381 (1996). 23. B. J. West, Physiology, Promiscuity and Prophecy at the Millennium: A Tale of Tails, Studies of Nonlinear Phenomena in Life Science, Vol. 7, World Scientiﬁc, Hackensack, NJ, 1999. 24. B. Suki, A. M. Alencar, U. Frey, P. C. Ivanov, S. V. Buldyrev, A. Majumdar, H. E. Stanley, C. A. Dawson, G. S. Krenz, and M. Mishima, Fluctuations, noise and scaling in the cardio-pulmonary system. Fluctuations and Noise Lett. 3, R1–R25 (2003). 25. C. K. Peng, J. Mistus, J. M. Hausdorff, S. Havlin, H. E. Stanley, and A. L. Goldberger, Longrange anticorrelations and non-Gaussian behavior of the heartbeat. Phys. Rev. Lett. 70, 1343– 1346 (1993). 26. B. J. West, Physiology in fractal dimension: Error tolerance. Ann. Biomed. Eng. 18, 135–149 (1990). 27. H. H. Szeto, P. Y. Cheng, J. A. Decena, Y. Chen, Y. Wu, and G. Dwyer, Fractal properties of fetal breathing dynamics. Am. J. Physiol. 262 (Regulatory Integrative Comp. Physiol. 32), R141– R147 (1992). 28. G. S. Dawes, H. E. Cox, M. B. Leduc, E. C. Liggins, and R. T. Richards, Fractal properties of fetal breathing dynamics. J. Physiol. Lond. 220, 119–143 (1972). 29. W. A. C. Mutch, S. H. Harm, G. R. Lefevre, M. R. Graham, L. G. Girling, and S. E. Kowalski, Biologically variable ventilation increases arterial oxygenation over that seen with positive end-expiratory pressure alone in a porcine model of acute respiratory distress syndrome. Crit. Care Med. 28, 2457–2464 (2000). 30. W. A. Altemeier, S. McKinney, and R. W. Glenny, Fractal nature of regional ventilation distribution. J. Appl. Physiol. 88, 1551–1557 (2000). 31. C. K. Peng, J. Metus, Y. Li, C. Lee, J. M. Hausdorff, H. E. Stanley, A. L. Goldberger, and L. A. Lipsitz, Quantifying fractal dynamics of human respiration: Age and gender effects. Ann. Biomed. Eng. 30, 683–692 (2002). 32. J. W. Kantelhardt, T. Penzel, S. Rostig, H. F. Becker, S. Havlin, and A. Bunde, Breathing during REM and non-REM sleep: Correlated versus uncorrelated behaviour. Physica A 319, 447–457 (2003). 33. C. L. Webber, Rhythmogenesis of deterministic breathing patterns, in Rhythms in Physiological Systems, H. Haken and H. P. Koepchen, eds. Springer, Berlin, 1991, pp. 177–191. 34. Vierordt, Ueber das Gehen des Menschen in Gesunden und kranken Zustaenden nach Selbstregistrirender Methoden, Tuebingen, Germany, 1881. 35. J. M. Hausdorff, S. L. Mitchell, R. Firtion, C. K. Peng, M. E. Cudkowicz, J. Y. Wei, and A. L. Goldberger, Altered fractal dynamics of gait: Reduced stride-interval correlations with aging and Huntington’s disease. J. Appl. Physiol. 82, (1997).

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36. B. J. West and L. Grifﬁn, Allometric control of human gait. Fractals 6, 101–108 (1998); B. J. West and L. Grifﬁn, Allometric control, inverse power laws and human gait. Chaos, Solitons & Fractals 10, 1519–1527 (1999). 37. R. L. Marsh, D. J. Ellerby, J. A. Carr, H. T. Henry and C. I. Buchanan, Partitioning the energetics of walking and running: Swinging the limbs is expensive. Science 303, January (2004). 38. B. J. West and L. Grifﬁn, Biodynamics: Why the Wirewalker Doesn’t Fall, John Wiley & Sons, New York (2003). 39. J. J. Collins and C. J. De Lucca, Random walking during quiet standing. Phys. Rev. Lett. 73, 764–767 (1994). 40. J. W. Blaszczyk and W. Klonowski, Postural stability and fractal dynamics. Acta Neurobiol. Exp. 61, 105–112 (2001). 41. G. L. Gernstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J. 4, 41–68 (1968). 42. M. C. Teich, C. Heneghan, S. B. Lowen, T. Ozaki, and E. Kaplan, Fractal character of the neural spike train in the visual system of the cat, Opt. Soc. Am. 14, 529–546 (1997). 43. A. Babloyantz and A. Destexhe, Low-dimensional chaos in an instance of epilepsy. Proc. Natl. Acad. Sci. USA 83, 3515–3517 (1987). 44. P. Langevin, C. R. Acad. Sci. Paris, 530 (1908). 45. J. Beran, Statistics for Long-Memory Processes, Chapman & Hall, New York, 1994. 46. J. T. M. Hosking, Fractional differencing. Biometrika 68, 165–176 (1982). 47. N. Scafetta and P. Grigolini, Scaling detection in time series: Diffusion entropy analysis. Phys. Rev. E 66, 036130 (2002). 48. T. Gneiting and M. Schlather, Stochastic models that separate fractal dimension and the hurst effect. SIAM Rev. 46, 269–282 (2004). 49. P. Allegrini, P. Grigolini, and B. J. West, Dynamical approach to Le´vy processes. Phys. Rev. E 54, 4760 (1996). 50. J. Klafter, M. F. Shlesinger, and G. Zumofen, Phys. Today 49(2), 33 (1996); Lect. Notes in Phys. 519, 15 (1998). 51. T. Geisel, in Le´vy Flights and Related Topics in Physics, Proceedings, Nice, France; Editors, M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, eds., Lecture Notes in Physics 450, 153 (1995). 52. M. Bologna, P. Grigolini, and B. J. West, Strange kinetics: Conﬂict between density and trajectory description. Chem. Phys. 284, 115–128 (2002). 53. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, 2003. 54. E. Barkai, R. Metzler, and J. Klafter, Phys. Rev. E 61, 132 (2000). 55. R. Metzler and T. F. Nonnenmacher, Phys. Rev. E 57, 6409 (1998). 56. B. J. West, Where Medicine Went Wrong, Rediscovering the Path to Complexity, to be published (2005). 57. K. Falconer, Fractal Geometry, John Wiley & Sons, New York, 1990. 58. J. Feder, Fractals, Plenum Press, New York, 1988. 59. J. F. Muzy, E. Bacry, and A. Arnedo, Phys. Rev. E 47, 875 (1993). 60. B. J. West, M. Latka, M. Glaubic-Latka, and D. Latka, Multifractality of cerebral blood ﬂow. Physica A 318, 453–460 (2003).

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61. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, John Wiley and Sons, New York, 1966. 62. D. Schertzer, S. Lovejoy, F. Schmitt, Y. Chigirinskays, and D. Marsan, Multifractal cascade dynamics and turbulent intermittency. Fractals 5, 427 (1997). 63. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, MA, 1954. 64. E. Montroll and B. J. West, An enriched collection of stochastic processes, in Fluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds., North Holland Personal Library, Amsterdam, 1987; 1st ed. 1979. 65. M. Shlesinger, J. Klafter, and B. J. West, Le´vy dynamics of enhanced diffusion: Applications to turbulence. Phys. Rev. Lett. 58, 1100–1103 (1987). 66. N. Scafetta and B.J. West, Multiscale Comparative Analysis of Time Series and a Discussion on ‘Earthquake Conversations’ in California. Phys. Rev. Lett. 92, 138501 (2004). 67. J. Perrin, Mouvement brownien et moleculaire culaire. Ann. chim. Phys. VIII 18, 5–114; translated by F. Soddy as Brownian Movement and Molecular Reality, Taylor and Francis, London. 68. E. Ott, Chaos in Dynamical Systems, Cambridge University Press, New York, 1993. 69. R. J. Abraham and C. D. Shaw, Dynamics-The Geometry of Behavior, Part 1 (1982), Part 2 (1983), Part 3 (1985), and Part 4 (1988), Aerial Press, Santa Cruz, CA. 70. K. M. Kolwankar, Studies of Fractal Structures and Processes using Methods of the Fractional Calculus, unpublished thesis, University of Pune, 1997. 71. D. Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, Cambridge, 1989. 72. M. F. Shlesinger, Fractal time and 1/f noise in complex systems. Ann. N.Y. Acad. Sci. 504, 214 (1987). 73. Y. N. Rabotnov, Elements of Hereditary Solid Mechanics, MIR , Moscow, 1980. 74. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, Oxford, 1979. 75. P. Meakin, Fractals, Scaling and Growth Far from Equilibrium, Cambridge Nonlinear Science Series 5, Cambridge University Press, Cambridge, 1998. 76. This letter was translated by B. Mandelbrot and is contained in the Historical Sketches of his second book [3]. 77. R. Hilfer, ed., Applications of Fractional Calculus in Physics, World Scientiﬁc, Singapore, 2000. 78. I. Sokolov, J. Klafter, and A. Blumen, Fractional Kinetics. Phys. Today Nov. (2002). 79. L. F. Richardson, Atmospheric diffusion shown on a distance-neighbor graph. Proc. R. Soc. Lond. A 110, 709–737 (1926). 80. M. F. Shlesinger and B. J. West, Complex fractal dimension of the bronchial tree. Phys. Rev. Lett. 67, 2106–2109 (1991). 81. D. Sornette, Discrete scale invariance and complex dimensions. Phys. Rep. 297, 239–270 (1994). 82. A. Rocco and B. J. West, Fractional calculus and the evolution of fractal phenomena. Physica A 265, 535 (1999). 83. T. F. Nonnenmacher and R. Metzler, On the Riemann-Liouville fractional calculus and some recent applications. Fractals 3, 557 (1995).

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84. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993. 85. V. Kobelev and E. Romanov, Fractional Langevin equation to describe anomalous diffusion. Prog. Theor. Phys. Suppl. 139, 470–476 (2000). 86. See, for example, K. Lindenberg and B. J. West, The Nonequilibrium Statistical Mechanics of Open and Closed Systems, VCH, New York, 1990. 87. E. Lutz, Fractional Langevin equation. Phys. Rev. E 64, 051106 (2001). 88. B. Rajagopalon and D. G. Tarboton, Fractals 1, 6060 (1993). 89. P. C. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, H. E. Stanley, Multifractality in human heartbeat dynamics. Nature 399, 461 (1999). 90. B.J. West, M. Latka, M.Galaubic-Latka, and D. Latka, Multifactility of cerebral blood ﬂow. Physica A 318, 453–460 (2003). 91. R. Zhang, J. H. Zuckerman, C. Giller, and B. D. Levine, Am. J. Physiol. 274, H233 (1999). 92. S. Rossitti and H. Stephensen, Acta Physiol. Scand. 151, 191 (1994). 93. B. J. West, R. Zhang, A. W. Sanders, J. H. Zuckerman, and B. D. Levine, Fractal ﬂuctuations in transcranial Doppler signals. Phys. Rev. E 59, 3492 (1999). 94. R. Gorenﬂo and F. Mainardi, Feller fractional diffusion and Le´vy stable motion. Preprint 1999. 95. B. J. West and V. Seshadri, Linear systems with Le´vy ﬂuctuations. Physica A 113, 2030216 (1982). 96. V. Seshadri and B. J. West, Fractal dimensionality of Le´vy processes. Proc. Natl. Acad. Sci. USA 79, 4051 (1982). 97. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000). 98. M. Annunziato and P. Grigolini, The Markov approximation revisited: Inconsistency of the standard quantum Brownian motion model. Phys. Lett. A 269, 31 (2000). 99. J. L. Doob, Stochastic Processes, John Wiley & Sons, New York (1953). 100. L. E. Reichl, The Transition to Chaos, Springer, New York, 1992. 101. B. Neubauer and H. J. G. Gundersen, Analysis of heart rate variations in patients with multiple sclerosis. A simple measure of autonomic disturbances using an ordinary ECG. J. Neurol. Neuosurg. Psychiatry 41, 417–419 (1978). 102. V. Kariniemi and P. Amma¨la¨, Short-term variability of fetal heart rate during pregnancies with normal and insufﬁcient placental function. Am. J. Obster. Gynecol. 139, 33–37 (1981). 103. A. L. Goldberger, D. Goldwater, and V. Bhargava, Atrophine unmasks bed-rest deconditioning effect in healthy men: A spectral analysis of cardiac interbeat intervals. J. Appl. Physiol. 61, 1843–1848 (1986). 104. J. L. Waddington, M.J. MacCulloch and J. E. Sambrooks, Resting heartrate variability in man declines with age. Experientia 35, 1197–1198 (1979). 105. A. L. Goldberger, L. Findley, M. J. Blackburn, and A. J. Mandell, Nonliear dynamics of heart failure: implications of long-wavelength cardiopulmonary osciallations. Am. Heart J. 107, 612–615 (1984); G. A. Myers, G. J. Martin, and N. M. Magrid et al. Power spectral analysis of heart rate variability in sudden cardiac death: Comparison to other methods. IEEE Trans. Biomed. Eng. 33, 1149–1156 (1986). 106. A. L. Goldberger, V. Bhargava, B. J. West, and A. J. Mandell, On a mechanism of cardiac electrical stability: The fractal hypothesis. Biophys. J. 48, 525–528 (1985); A. L. Goldberger and B. J. West, Applications of nonlinear dynamics to clinical cardiology. Ann. NY Acad. Sci. 504, 195–213 (1987).

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107. A. Mutch, Health, ‘‘small-worlds’’, fractals and complex networks: An emerging ﬁeld. Med. Sci. Monit. 9, MT55–MT59 (2003). 108. T. G. Buchman, Physiologic failure: Multiple organ dysfunction syndrome. Preprint. 109. G. A. Chauvet, Hierarchical functional organization of formal biological systems: A dynamical approach. I. The increase of complexity by self-association increases the domain of stability of a biological system. Philos. Trans. R. Soc. Lond. B Biol. Sci. 339(1290), 425–44 (1993). 110. A. L. Goldberger, D. R. Rigney and B. J. West, Chaos and Fractals in Human Physiology, Scientiﬁc American, Feb., 42–49, (1990).

CHAPTER 7 PHYSICAL PROPERTIES OF FRACTAL STRUCTURES VITALY V. NOVIKOV Odessa National Polytechnical University, 65044 Odessa, Ukraine

CONTENTS I. II.

III.

IV.

Introduction Elements of Fractal Theory A. Continuous, Nowhere Differentiable Functions and Deterministic Fractals B. Fractal Sets C. Fractional Hausdorff–Besicovich Dimensions D. Multifractals E. Fractal Set Constructed on a Square Lattice F. Cayley Tree. Ultrametric space Chaotic Structures A. Percolation Systems 1. Percolation Cluster 2. Critical Indices 3. Renormalization-Group Transformations 4. Physical Properties B. Fractal Structure Model 1. Properties of Finite Lattices 2. Appendix. The probability functions Physical Properties A. Conductivity 1. Maxwell Model 2. The Effective Medium Theory 3. Variational Approach 4. Iterative Averaging Method for Conductivity B. Frequency Dependence of Dielectric Properties 1. Iterative Averaging Method for Dielectric Properties C. Galvanomagnetic Properties 1. Iterative Averaging Method for Hall’s Coefficient 2. Results and Discussion 3. Appendix. Galvanomagnetic Properties of the Cube Inside a Cube Cell

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

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94 D.

E. F.

G.

Elastic Properties 1. Iterative Averaging Method for Elastic Properties 2. Results of Calculation Negative Poisson’s Ratio 1. Results of Calculations of Poisson’s Ratio Frequency Dependence of Viscoelastic Properties 1. Iterative Averaging Method for Viscoelastic Properties 2. Results of Calculations for Viscoelastic Media 3. Negative Shear Modulus 4. Appendix. Fractal Model of Shear Stress Relaxation Relaxation and Diffusion Processes 1. Non-Debye Relaxation 2. Anomalous Diffusion 3. Distribution Function of a Brownian Particle with Memory 4. Inertial Effects of a Brownian Particle 5. Appendix. Derivative of Fractal Functions

References

I.

INTRODUCTION

Recently much attention has been paid to materials with a random structure. They possess an internal hierarchical organization depending on the scale; moreover, the physical properties of such materials depend on mesoscopic parameters. A mesoscopic inhomogeneous material can be considered as a subensemble of a wider class of inhomogeneous media. Materials with a fractal structure also belong to this class. Various structures can be thought of as fractal—for example, aggregates of particles in colloids, as well as the structures of certain binary solutions, polymers, and composites. Fractal structures are formed, in particular, in diffusion-controlled aggregation (polymerization). It is worth noting that media with fractal structure can exhibit properties different from those of uniform matter, like crystals, ordinary composites or homogeneous ﬂuids. In this chapter, numerical calculations for the elastic properties of a random medium are carried out using an iterative method of averaging developed by the author and his co-workers. This method is based on the results of fractal geometry and renormalization—group transformation methods. Our averaging method has been shown to be efﬁcient in the development of the physical properties of composites. The chapter consists of three main sections. In Section II the elements of fractal theory are given. In Section III the basis of percolation theory is described; moreover, a model of fractal structures conceived by us is described. Fractal growth models, constructed using small square or rectangular generating cells as representative structural elements, are considered. Fractal dimensions of structures generated on various unit cells ð2 1; 2 2; 2 3; 2 4; 3 1; 3 2; 3 3; 3 4; 4 1; 4 2; 4 3; 4 4Þ are calculated. Probability

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functions and critical indices for the percolation threshold and percolation cluster density are also derived. It is shown that use of anisotropic (rectangular) initial cells in place of ‘‘isotropic’’ (square) cells increases the range of possible values of critical indices characterizing the modeled systems. In the third Section IV the results of our calculations for physical properties of inhomogeneous media with a fractal structure are presented. Hierarchical structure models are applied to study the conductivity, elastic properties and Poisson’s ratio of a two-component inhomogeneous medium with a chaotic, fractal structure. Elastic properties of non-uniform, two-component systems are studied using a model of percolation on a simple cubic lattice. It is shown that as the ratio of the bulk modulus K of the components tends to zero, namely, K2 =K1 ! 0 (where 1, 2 denote the harder and softer phase, respectively), then Poisson’s ratio for the system tends to 0.2 at the percolation threshold of the harder phase in any individual Poisson ratio of the components. A qualitatively new, collective mechanism leading to negative Poisson’s ratio is suggested. Moreover, a Poisson ratio calculation for a composite with Hashin–Strikman structure has been carried out. The hierarchical structure model is generalized and applied to study the viscoelastic properties of a two-component inhomogeneous medium with chaotic, fractal structure. It is shown that just as the results obtained recently using the Hashin–Strikman model, the present model predicts the possibility of obtaining composites with an effective shear and dumping coefﬁcient much higher than those characterizing the individual component phases. The viscoelastic properties of the fractal medium, however, differ qualitatively from the properties of the Hashin–Strikman medium. In the following section the power of the fractional derivative technique is demonstrated using as example the derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of the microstructure of disordered media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. Different relaxation functions are derived assuming that the real (physical) ensemble of relaxation times is conﬁned between the upper and lower limits of self-similarity. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the relaxation should be of classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times. The analysis of diffusion for a Brownian particle, where the assumption that the

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Maxwellian distribution of velocities is instantaneously attained is abandoned, i.e. inertial effects are included, has also been carried out. The material described in this chapter can be used in the analysis of the dependence of the conductivity and elastic parameters of various polymeric materials. Providing both a critical evaluation of characterization methods and a quantitative description of composition-dependent properties the material given in this chapter should have broad appeal in both the academic and industrial sectors, being of particular interest to researchers in materials and polymer science. II.

ELEMENTS OF FRACTAL THEORY

Progress in the physics of disordered media—that is, in the physics of media with a random distribution of microheterogeneity—is mainly made via the solution of problems involving the connection between the microscopic structure and the macroscopic behavior. This problem properly belongs to the realm of the kinetic theory of matter and is analogous to the problem of locking in the theory of ﬂuids, hydrodynamic turbulence, the theory of phase transitions, and so on. Using the methods of classical statistical physics one may more or less rigorously solve problems where the system on a microscopic level is either in a state of complete chaos (perfect gas) or total order (solid perfectly crystalline bodies). In contrast, disordered media and processes in which there is neither crystalline order nor complete chaos on the microscopic level have not yet had an adequate description. This problem is connected with the condition that the macroscopic variables must considerably exceed the correlation scales of microscopic variables, a condition which is not met by disordered media. Consequently in order to describe such systems, fractal models and phased averaging on different scale levels (meso-levels) should be effective. The success of fractal models applied to the physics of disordered media may be explained ﬁrst of all by the fact that fractal forms are characteristic of a huge number of processes and structures because many diverse models of the formation and growth of disordered objects of disparate nature may ultimately be reduced to a transition model—namely a connected set and an unconnected set—and to a limited diffusive aggregation [1–6]. In the ﬁrst case a fractal percolation cluster is formed; in the second case a fractal aggregate is formed. A typical situation is realized in such cases – fractal structure properties appear on a range scale which is limited by the dimensions of the particles forming the aggregate at the beginning, and at the end, by the dimensions of the initial fractal clusters. Typical particle dimensions are 1–10 nm; aggregate dimensions are 10– 1000 nm. The fractal dimension, df , depends on the conditions of aggregate formation and, as a rule, lies within a range of df ¼ 2 2:9. Another class of materials with fractal structure are amorphous polymers. Here fractal properties manifest themselves on scales exceeding the dimensions

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of monomer molecules and are restricted at the top end to a scale of several tens of angstroms. A linear monomer molecule subject to random link shifts has already a fractal curve shape. Referring to ramiﬁed polymer molecules, these form fractal nets similar to those forming fractal aggregates in gels from macroscopic particles. The main characteristic of fractal structures is the dependence of their properties, C, on some linear scale, L: C La

ð1Þ

where a is a constant. The scale dependence of the properties is the result of the self-similarity property of a fractal structure. A percolation cluster is a typical fractal. It is formed in a ‘‘geometric phase transition’’ region when unconnected sets become connected. In actual media, this dependence is usually limited to the so-called intermediate asymptotic region, which is deﬁned as l0 L x

ð2Þ

where l0 is the lattice constant (microscopic constant) and x is the correlation length. In the range of scales L x, a microheterogeneous medium is homogeneous (the self-similarity property disappears) and can be characterized by the effective properties. Fractal structures have been examined, in particular, in diffusion-controlled aggregation process (polymerization) [7–9], in colloids (aggregates of particles) [10–13], and in percolation clusters [1–3]. The regularities revealed in the theory of fractals and percolation have turned out to be generally true for heterogeneous stochastic media and, in particular, for composite materials. Fractals are geometric objects, which have a fractal dimension and where the constituent small parts are similar to the whole object. Fractals became well known following the publications of Mandelbrot in 1977 [14,15]. Note that fractals (self-similar sets with fractal dimension) were ﬁrst studied and described by mathematicians long before the publications of Mandelbrot, when such fundamental deﬁnitions as function, line, surface, and shape were analyzed. In mathematics, fractals appear as a result of the opposition and unity of two ﬁelds of mathematics: One of these ﬁelds studies numbers (discrete objects), while the other studies shapes (continuous objects). Use of the concept of the fractal set allows one to examine the dependence of physical properties on the behavior of hierarchical structures. Such structures appear in stochastic inhomogeneous medium.

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Algorithms for the formation of fractal sets and the determination of fractal dimension have a signiﬁcant place in fractal theory. Simple models of fractal sets are considered below to illustrate formation algorithms and the calculation of the fractal dimension. A.

Continuous, Nowhere Differentiable Functions and Differentiable Fractals

In the second half of the nineteenth century, the theories of real numbers and sets were created (Weierstrass, Dedekind, Cantor [17]). These allow one to give a general and rigorous mathematical deﬁnition of a function. This deﬁnition can be formulated as follows. A function y ¼ f ðxÞ given on a set X is a rule according to which each element x from a set X may be mapped onto an element from set Y. It is known, however, that deﬁning mathematical notions more precisely is often rather risky because the visual connection between the object under investigation and its mathematical model can disappear in tandem with the increase of accuracy, clearness, and rigor of mathematical proofs. For example, the following deﬁnition of a function f ðxÞ can be used as a Dirichlet function: 0; if x is an irrational number f ðxÞ ¼ ð3Þ 1; if x is a rational number which is a function discontinuous at all points. This function cannot be realized, and mathematicians were unable to study such a function until the second half of the nineteenth century. Along with functions discontinuous at all points, continuous functions having no derivatives at any point were discovered—that is, functions which are almost everywhere continuous but nowhere differentiable. We consider a few nowhere differentiable functions. Some of them have such amazing properties that they have been given the names of the great mathematicians who invented them: Bolzano, Cantor, Peano, Weierstrass, Koch, Van der Waerden, Sierpinski, and others. Bolzano Function About 1830, Bolzano showed that continuous, nowhere differentiable functions exist [16, 17]. The manuscript of Bolzano’s was discovered only in 1920. So that the example of a nowhere differentiable function found by Weierstrass in 1871 was deemed the ﬁrst example of such a function. Nowadays many examples of nowhere differentiable functions exist. Let us consider a nowhere differentiable Bolzano function paying tribute to Bolzano as the ﬁrst scientist who formulated a nowhere differentiable function (Fig. 1).

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Figure 1. Constructing a graph of the Bolzano function: (a) Graph of function B0 (x) and B1 (x), (b) Graph of function B1 (x). and B2 (x), (c) Graph of function B2 (x) and B3 (x).

According to Bolzano, we shall call the B-operation an operation on two points, Aðx; yÞ and Bðx þ a; y þ bÞ, namely, a b a ; A23 x þ ; y A11 ðx; yÞ; A22 x þ ; y 4 2 2 3a b A24 x þ ; y þ ; A25 ð x þ a; y þ bÞ 4 2

ð4Þ

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Consider the function with graph the straight line segment connecting points A011 ð0; 0Þ; A022 ð0; 0Þ as B0 ðxÞ. Applying the B-operation to these points, we obtain the additional points 1 1 1 3 1 0 0 0 0 A21 ð0; 0Þ; A22 ; ; A23 ; 0 ; A24 ; ; A025 ð1; 1Þ ð5Þ 4 2 2 4 2 Then we deﬁne a function B1 ðxÞ whose graph is the broken line A021 ; A022 ; A023 ; A024 ; A025 : Using the B-operation at each pair of points, we again obtain the additional points 1 1 2 3 A31 ð0; 0Þ; A32 2 ; ; A33 2 ; 0 ; A34 2 ; 0 4 4 4 4 ð6Þ 4 1 5 3 6 1 A35 2 ; ; A36 2 ; ; A37 2 ; 4 2 4 2 4 2 Then a function B2 ðxÞ with graph the broken line A31 ; A32 , . . . , A37 is now deﬁned. Continuing this process, we ﬁnd a function Bn ðxÞ with graph the broken line with peaks at the points having abscissae (see Fig. 1a): 0;

1 2 4n 1 ; n ;...; n 4 4 4n

ð7Þ

It is now possible to deﬁne the graph of Bolzano function BðxÞ at the values of x: k ðk ¼ 0; 1; 2; . . . ; 4n ; n ¼ 0; 1; . . .Þ ð8Þ x¼ n 4 taking account of B

k l ¼ B p n 4 4p

ð9Þ

Thus, the graph of Bolzano function BðxÞ traverses the peaks of all broken lines Bn ðxÞðn ¼ 0; 1; 2; . . .Þ (Table I). Any value of x, if it is different from values (1.4), can be exhibited as the limit of a sequence of numbers. The limit is: lim Bðam Þ ¼ BðaÞ

m!1

ð10Þ

Thus, the Bolzano function BðxÞ is deﬁned on the whole segment [0, 1]. It can be rigorously proved that BðxÞ is a nowhere differentiable function [16, 17].

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101

TABLE I Values of the Argument of Bn ðxÞ n 0

X 0, 1

1

0; 14 ; 24 ; 34 ; 1

2 .. . n

0; 412 ; 422 ; 432 ; . . . ; 15 42 ; 1 .......................................................... ............................. n 0; 41n ; 42n ; . . . ; 4 41 n ;1

The Bolzano construction for obtaining a nowhere differentiable function can be simpliﬁed and made more graphic [18]. B.

Fractal Sets

Van der Waerden Function and Koch Snowﬂake If we divide segment [0,1] into four equal parts and construct an isosceles triangle without a base over the two central parts, we obtain a line which is the graph of a function y ¼ f1 ðxÞ (Fig. 2b). Next we construct the graph of the function y ¼ f2 ðxÞ. To do this, we divide segment [0,1] ﬁrst into four parts and then each part again into four parts. According to the previous algorithm, we construct an isosceles triangle on each 1/8 part of the segment [0,1] and obtain the graph of a function y ¼ f2 ðxÞ (Fig. 2c). Adding up these functions, y ¼ f1 ðxÞ þ f2 ðxÞ, we obtain the graph shown in Fig. 2d, Next, we divide each part into four parts again and construct 16 isosceles triangles. The graph so obtained, y ¼ f5 ðxÞ, is added to the previous construction. Continuing this process, we obtain a nowhere differentiable function (Van der Waerden function) [18].

Figure 2.

Generating the graph of a Van der Waerden function.

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Another example of a nowhere differentiable function is the Weierstrass function, deﬁned by [19] n X bk cos p ak x ð11Þ f ðxÞ ¼ lim n!1

k¼1

where 0 < b < 1, a is an odd whole number satisfying: 3 ab > 1 þ p 2 The following function is a little simpler: f ðxÞ ¼

x cosðp=xÞ;

if x 6¼ 0

0;

if x ¼ 0

ð12Þ

ð13Þ

It has inﬁnitely many maxima and minima in the segment [0,1]. Along with the discovery of nowhere differentiable functions, geometric shapes were created and it was difﬁcult to say whether they were lines, surfaces, or volumes. Koch’s ‘‘Snowﬂake’’ From an analogous procedure to that which has been used for the formation of the graph of Van der Waerden function from an isosceles triangle, it is possible to obtain Koch’s ‘‘snowﬂake’’ (Fig. 3). For this purpose, sides of an isosceles triangle are divided into three equal parts. Then the middle parts are rejected, and above them ‘‘small houses’’ are built up. In accordance with Koch’s ‘‘snowﬂake’’ 1n construction, the length of a link at the nth iteration step is equal to l ¼ n 3 , and the total length of the broken line n is equal to Ln ¼ 43 . Therefore, lim ln ¼ 0;

n!1

Figure 3.

lim Ln ¼ 1

n!1

Koch’s ‘‘snowﬂake.’’

ð14Þ

physical properties of fractal structures

103

…

(a)

…

(b)

Figure 4.

Constructing a Sierpinski carpet: (a) quadrangle; (b) triangle.

that is, a line of unlimited length results from the iteration procedure. This line has no tangent at any point. Sierpinski Carpet Consider a square with side 1. We divide it into nine equal squares and remove the central part. Then we repeat this procedure with each of the eight remaining squares (Fig. 4). Repeating the procedure n times (n ! 1), we obtain a shape that is called the Sierpinski carpet (Fig. 4). Cantor Set Consider a one-unit segment. We divide it into three equal parts and remove the central one. Each of the two remaining parts is also divided into three parts and the middle parts are removed (Fig. 5). Continuing this procedure n times, ﬁnd that the length of the link at the we n nth iteration stage is equal to ln ¼ 13 , and the total length of the segments remaining at the nth iteration stage is Ln ¼

n 2 3

ð15Þ

Thus, when n ! 1, we obtain Cantor ‘‘dust’’: lim ln ¼ 0;

n!1

lim Ln ¼ 0

n!1

ð16Þ

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Figure 5. Cantor set.

The function shown in Fig. 6 is connected with the Cantor set (Cantor ‘‘dust’’). This function has been called the ‘‘devil’s stairs.’’ Constructing the ‘‘Devil’s Stairs’’ This function is obtained as follows. Again, we divide segment [0,1] into three equal parts and assume that the value of the function y ¼ f ðxÞ equals 1=2 at all points of the middle part. Then the left and right thirds of the segment are again divided into three equal parts. Now the function y ¼ f ðxÞ is assumed to equal 1=4

Figure 6.

The ‘‘devil’s stairs.’’

physical properties of fractal structures

105

over the middle of the left part and 3=4 over the middle of the right part. Therefore, 1=4 is subtracted from 1=2 for the left part, and 1=4 is added to 1=2 for the right part; that is, increment in the function is 1=2. Each of the remaining segments are also divided into three equal parts. The function y ¼ f ðxÞ over the middle parts of the segments is assumed to equal 1=8; 3=8; 5=8; 7=8, respectively; that is, the increment in the function is equal to ð1=2Þ3 . Continuing this process, as n ! 1 we obtain a function that is deﬁned at all points of segment [0,1], except at the points belonging to the Cantor set (Cantor ‘‘dust,’’ Fig. 6). Here the jump in the function is lim ð1=2Þn ¼ 0

n!1

ð17Þ

This means that the function is continuous as n ! 1. Thus, the graph of the function y ¼ f ðxÞ looks like a staircase with an inﬁnite number of steps whose total length is 1 because the length of Cantor ‘‘dust’’ (where there are no steps) is zero. This function goes up by 1, although it only increases over the set of zero length and does not make any jumps. Jordan ﬁrst gave a deﬁnition of a line as the trajectory of a point moving continuously without any jumps. The Italian scientist Peano showed that it is possible to construct a curve, in Jordan’s sense, traversing all points of a square. Peano Function Peano constructed the representation of the set of points of a straight line segment on the set of points of a square (Fig. 7). In this, adjacent points of the square corresponded to adjacent points on the segment. That is, Peano constructed a line that encompassed all points of the square. It must be stated

Figure 7.

Constructing a Peano function.

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9

8

7

6

4

5

1

2

3

Figure 8. Constructing a Peano function.

that Peano’s curve passed through certain points of the square several times. It was proved that no continuous curve passing over all points of the square only once exists. Let us consider the square K in the plane: 0 x 1;

0y1

ð18Þ

We divide it into nine equal squares and enumerate them as shown in Figs. 8 and 9. The squares having immediately adjacent numbers have a common side.

9

8

7

52 51

53

4

6 54 55

56

58 57

59

1

2

3

Figure 9. Constructing a Peano function.

physical properties of fractal structures

107

The squares so constructed will be called ﬁrst rank squares. Each ﬁrst-rank square is divided into nine second-rank squares. The second-rank squares are numbered in such a way that square number one is adjacent to square number nine (see Fig. 8). This process can be continued inﬁnitely. The length of the nthrank square side will be ð1=3Þn . If point A belongs to the main square, K, it also belongs to at least one ﬁrst-rank square, at least one second-rank square, and so on (Fig. 7). If we take two different points, A and B, of the main square K, then they will belong to two different nrank squares beginning with some large n. Let us take some point Aðx; yÞ belonging to the main square K. Let s1 ; s2 , . . . , sn , . . . be the numbers of 1st, 2nd, . . . , nth-rank squares respectively to which point A belongs. Thus, at least one sequence of numbers s1 , s2 , . . . , sn , . . . corresponds to point A and, on the contrary, one point of square K corresponds to each such sequence. Then, two different sequences will correspond to two different points of square K, however even one point of the square may correspond to two different sequences. For example, the sequences 2; 3; 5; 6; 7; 1; 1; . . . ; 1 2; 3; 5; 5; 9; 9; 9; . . . ; 9 deﬁne the same point of square K. Consider segment [0,1]: 0 t 1 We can show that each point of segment [0,1] can be assigned a corresponding point of square K. We divide the segment [0,1] into nine equal parts, number them from left to right with 1; 2; . . . ; 9, and call them ﬁrst-rank segments. Each ﬁrst-rank segment is divided into nine equal second-rank segments that are also numbered from left to right 1,2,3,4,5,6,7,8,9. This procedure is carried out n times (n ! 1). Thus, at least one sequence of numbers s1 ; s2 ; . . . ; sn ; . . . (si ¼ 1; 2; . . . ; 9Þ will correspond to each A point of segment [0,1]; vice versa, a point of segment [0,1] corresponds to each point of the sequence. Now take some point B belonging to segment [0,1]. If it is not the end of an n-rank segment, then only one sequence a1 ; a2 ; . . . ; an ; . . . corresponds to it. If this point is the end of an n-rank segment, then two sequences will correspond to it, namely: 1. a1 ; a2 ; . . . ; an , 1; 1; . . . ; 1; . . . . 2. a1 ; a2 ; . . . ; an1 , 9; 9; . . . ; 9; . . . . Thus, each point B 2 ½0; 1 can be assigned the corresponding point A 2 K.

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Let a point B belonging to the segment have abscissa t, and point A belonging to the square have coordinates x and y. Then we have single-valued functions x ¼ jðtÞ and

y ¼ cðtÞ

ð19Þ

We prove that these functions are continuous. Thus, if we take two values t1 and t2 , such that jt1 t2 j <

1 9n

ð20Þ

then they belong in one or two adjacent n-rank segments. Then the corresponding points ðx1 ; y1 Þ and ðx2 ; y2 Þ of the square will be situated in one or two adjacent n-rank squares; consequently, the difference of the abscissae of these points cannot exceed 2=3n , that is, jj ðt1 Þ j ðt2 Þj <

2 3n

ð21Þ

Let e > 0. Let us select n such that 2=3n < e

ð22Þ

Taking d ¼ 91n , we realize that from the inequality jt1 t2 j < d

ð23Þ

jj ðt1 Þ j ðt2 Þj < e

ð24Þ

it follows that

Thus, an inﬁnitesimally small increment in the function corresponds to an inﬁnitesimally small increment of argument t; that is, jðtÞ is a continuous function. The functions x ¼ jðtÞ, y ¼ cðtÞ are called Peano functions; that is, the Peano curve can be rendered in parametric form by the equations x ¼ jðtÞ y ¼ cðtÞ

ð25Þ

This curve occupies the whole square K, that is, it traverses each point of the square. In order to justify our introduction of the Peano functions (which as will be shown below are nowhere differentiable) and to better our understanding of the

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109

Figure 10. Constructing the graph of a Peano function. (a) Using 10 points. (b) Using the points obtained by dividing each segment 1=9 t ðl þ 1Þ=9 into 9 equal parts.

proses whereby point of the segment may be taken to correspond with point of the square we consider the following. First the point t ¼ 0, of segment [0,1] is assumed to be a point of ﬁrst-rank segment number one, a point of second-rank segment number one, a point of n-rank segment number one, and so on. The corresponding point of square K must be in the ﬁrst-rank square number one, in the second-rank square with the same number, and so on. Therefore, it will be the point ð0; 0Þ (see Fig. 10). Now t ¼ 1=9. The point in the segment [0,1] belongs to the ﬁrst-rank segment and to the ninth segment of each of the following ranks. The corresponding point of square K must be in the ﬁrst-rank square and in the ninth square of each of the following ranks. Therefore, it will be the point (1/3,1/3). Then let us take t ¼ 2=9. The point of segment 1 lies in the second ﬁrst-rank segment and in the ninth segment of each of the following ranks. The corresponding point of square K will be the point ð0; 2=3Þ. Continuing in this way, we obtain Table II. Now it is possible to construct, 10 points of the graph of function x ¼ jðtÞ and by connecting these points with straight line segments obtain a broken line, which is the ﬁrst approximation to the curve x ¼ jðtÞ (Fig. 10). TABLE II t xðtÞ yðtÞ

0 0 0

1/9 1/3 1/3

2/9 0 2/3

3/9 1/3 1

4/9 2/3 2/3

5/9 1/3 1/3

6/9 2/3 0

7/9 1 1/3

8/9 2/3 2/3

1 1 1

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110

The second approximation to the curve x ¼ jðtÞ can be constructed in a similar way again dividing each segment 9i t iþ1 9 into nine equal parts. This approximation is shown in Fig. 10b, and it already indicates that the function x ¼ jðtÞ may be nowhere differentiable. Now we will prove that it is really so. First, we mention that the length of an n-rank square side is equal to 31n , and 1 the length of an n-rank a segment is equal to 9n ; that is, an interval of the change aþ1 of function x ¼ jðtÞ 3n ; 3n corresponds to each interval of the change of independent variable t 9an ; aþ1 9n We will take h such that jhj 92n , and jxðt0 þ hÞ xðt0 Þj where t0 belongs to interval we have the increment

a 9n

1 2 3n

ð26Þ

; aþ1 9n , where n is as large as possible. Therefore,

xðt0 þ hÞ xðt0 Þ 1 n 3 4 h

ð27Þ

xðt0 þ hÞ xðt0 Þ h

ð28Þ

Whence the ratio

does not have a ﬁnite limit; that is, the function x ¼ jðtÞ is not differentiable at an arbitrary point t0 . It turns out that none of the geometric objects examined above can be classed as curves (one-dimensional objects) or planes (two-dimensional objects). The Julia–Mandelbrot set belongs to these sets. Julia Sets The process of obtaining fractal sets at the transition from order to chaos can be regarded as an example of the change of boundaries between different regions which possess gravity centers (attractors) inﬂuencing the distribution of points in the region. Now the boundary constitutes a kind of order–disorder phase transition [20, 21]. We examine the simplest iteration process xnþ1 ¼ x2n Three variants are possible: If x0 < 1, then xn ! 0; n ! 1.

ð29Þ

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111

If x0 > 1, then xn ! 1; n ! 1. If x0 ¼ 1; then the xn lie on the unit circle. Therefore, the plane is divided into two domains of inﬂuence. The boundary of these domains is the circle. Everything becomes more interesting if complex representations are used instead of the former iteration functions. znþ1 ¼ f ðzn Þ;

zn ¼ xn þ iyn

ð30Þ

For such an iteration function scheme, a fractal attractor exists. The best known example for obtaining a fractal set is the square representation in the complex plane znþ1 ¼ z2n þ c

ð31Þ

f ðzn Þ ¼ z2n þ c

ð32Þ

that is,

where c ¼ a þ ib is a complex number. Therefore, xnþ1 ¼ x2n y2n þ a;

ynþ1 ¼ 2xn yn þ b

ð33Þ

The ﬁxed point of representation (30) is deﬁned as the root of the equation f ðzÞ ¼ z

ð34Þ

Developing f ðzÞ in a Taylor series about the ﬁxed point z0 , we obtain f ðzÞ ¼ f ðz0 Þ þ f 0 ðz0 Þðz z0 Þ þ

ð35Þ

f ðz0 Þ ¼ z0

ð36Þ

znþ1 ¼ f ðzn Þ

ð37Þ

Then, noting that

and

in the linear approximation we obtain znþ1 z0 ¼ ðz z0 Þ f 0 ðz0 Þ

ð38Þ

vitaly v. novikov

112 Thus:

(a) If jf 0 ðz0 Þj < 1, then the distance to the ﬁxed point z0 decreases as a result of iteration: jznþ1 z0 j < 1

ð39Þ

that is, z0 is a stable ﬁxed point. (b) If jf 0 ðz0 Þj > 1, then jznþ1 z0 j > 1

ð40Þ

The distance increases and z0 is an unstable ﬁxed point. Periodic points and cycles may consist of several points. So, for example, a cycle of period two consists of two points, z1 and z2 : f ðz1 Þ ¼ z2

and

f ðz2 Þ ¼ z1

ð41Þ

It is obvious that points z1 and z2 are ﬁxed points of the reﬂection: znþ1 ¼ f ðf ðzn ÞÞ

ð42Þ

f ðz1 Þ ¼ z2

ð43Þ

f ðf ðz2 ÞÞ ¼ z2

ð44Þ

f ðf ðz1 ÞÞ ¼ z1

ð45Þ

Also,

Concerning f ðz1 Þ ¼ z2 we obtain

Analogously,

If z0 is a ﬁxed point of period n, then it is a ﬁxed point of the function: f ðnÞ ðzÞ f ð. . . ðf ðf ðzÞÞÞ . . .Þ ¼ z0

ð46Þ

The point to which the iteration process converges is called an attractor in the complex plane znþ1 ¼ f ðzn Þ as n ! 1; that is, a stable ﬁxed point can act as an attractor.

ð47Þ

physical properties of fractal structures

Figure 11.

113

Julia sets.

If, for example, one chooses c ¼ 0:12375 þ i0:56508 in Eq. (30), then the sequence fzn þ 1g admits of three possibilities, however the inner attractor is not zero, and the boundary is no longer smooth (Fig. 11a). The boundary is a broken self-similar curve, a Julia set [14,15,21]. Given different values of c one can obtain different Julia sets (Fig. 11). The important feature of all geometric shapes examined in this chapter is their self-similarity, that is, scaling invariance. The dimensions of such geometric objects can be deﬁned using the Hausdorff–Besicovitch measure. Verhulst Dynamics Consider the model of growth of a bond set in f . Let p0 be the initial relative number of the complete bonds and pn their number after n iteration steps. The growth coefﬁcient, K, is K¼

pnþ1 1 pn

ð48Þ

If this is equal to a constant r, then the evolution law controlling the growth dynamics is pnþ1 ¼ f ðpn Þ ¼ ð1 þ rÞpn

ð49Þ

After n iteration steps, the number of complete bonds is pn ¼ ð1 þ rÞn p0 With a maximum value ðpn Þmax ¼ 1.

ð50Þ

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Verhulst assumed that K is proportional to 1 pn . He took K ¼ rð1 pn Þ, where r is the growth parameter. Hence, the evolution law is pnþ1 ¼ f ðpn Þ ¼ ð1 þ rÞpn rp2n

ð51Þ

If p0 ¼ 0 or 1, then pn does not change. If 0 < p0 1, then at r > 0 number p1 increases at the next iteration step p1 ﬃ p0 þ rp0 ¼ ð1 þ rÞp0

ð52Þ

So, p0 , p1 , . . . , pn :; . . . increase until 1. We examine the stability of the balanced condition for p0 6¼ 1; p0 < 1: For small deviations dn ¼ pn p0

ð53Þ

according to Eq. (51), on linearization we ﬁnd dnþ1 ð1 rÞdn

ð54Þ

Hence jdnþ1 j < jdn j, if 0 < r < 2. When r > 2, jdnþ1 j > jdn j, that is, the deviations increase, and the point p ¼ 1 is now unstable. For r ¼ 2:5, stable periodic oscillations occur. When r ¼ 2:570 the process becomes chaotic. The ratio dn ¼

rn rn1 rnþ1 rn

ð55Þ

converges to 4:669 . . . dn jn!1 ! 4; 669 . . .

ð56Þ

This regularity refers to the length interval of the parameter values at which periodic motion with some deﬁnite period is stable. These intervals are reduced at each reduplication of the period, the multiplier characterizing the reduction approaching the universal value: d ¼ 4; 669201660910 . . .

ð57Þ

which was ﬁrst published by Grossman and Tome in 1977 and is called the Feigenbaum number [20].

physical properties of fractal structures

115

In general, when we have the equation xnþ1 ¼ kxn ; ðk > 0Þ

ð58Þ

the solution xn ¼ xn ehn ;

h ¼ ln k

ð59Þ

exhibits instability with respect to perturbation of the initial condition. Whence ðaÞ k > 1; h > 0 and lim dn ¼ 1 n!1

ðbÞ k < 1; h < 0 C.

and

lim dn ¼ 0

n!1

ð60Þ ð61Þ

Fractional Hausdorff–Besicovich Dimensions

We now consider the deﬁnition of the dimensions of such common geometric objects as a straight line segment, a square, a cube, and so on. We divide the segment of length of L into Nn equal parts. Then each part of the division of length ln can be considered as a copy of the whole segment 1=Nn times reduced. It is obvious that N and ln are connected via the correlation (Table III) ð62Þ Nn ln ¼ L TABLE III Geometric Object

Unit

Straight line

ln

ln 2

Dimension 1 ln 3 n ¼ 3; Nn ¼ 3; ln ¼ ; d ¼ ¼1 3 lnð1=3Þ

1 ln 9 n ¼ 3; Nn ¼ 9; ln ¼ ; d ¼ ¼2 3 lnð1=3Þ

Square

ln 3

1 ln 27 n ¼ 3; Nn ¼ 27; ln ¼ ; d ¼ ¼3 3 lnð1=3Þ

Cube

d

lnf

Koch’s snowflake

n ¼ 3; Nn ¼ 64; ln ¼

3 1 ln 64 ; df ¼ ﬃ 1:26 3 lnð1=3Þ3

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116

Dividing the square with area S into Nn equal squares with areas l2n , the correlation becomes Nn l2n ¼ S

ð63Þ

Nn lnd ¼ M

ð64Þ

In general, for a set

where M is the measure of the set (of a geometric object). Hence, the dimension d of the geometric object with a ﬁnite measure M (M ¼ constant, in particular M ¼ 1) is deﬁned via Nðln Þand ln as d ¼ lim log M=logð1=ln Þ þ lim log Nðln Þ=logð1=ln Þ

ð65Þ

d ¼ lim log Nðln Þ=logð1=ln Þ

ð66Þ

ln !0 ðN!1Þ

ln !0

or ln !0

Let the fractal set f be in a Euclidean space with a dimension of d. It is possible to generalize the above result to a covering set f comprised of elementary geometric objects with unequal dimensions ln . Thus, the dimension procedure applied to sets of arbitrary metric space f generally consists of the following: First Nðln Þ is deﬁned—the minimum number of cubes with linear dimension ln < e ðe > 0Þ needed to cover the set f . Then the dimension of the set is deﬁned as the limit df ¼ lim log Nðln Þ=logð1=ln Þ e!0

ð67Þ

If the limit (67) exists, then df is called the Hausdorff, or Hausdorff– d Besicovitch, dimension. When covering the Cantor set with segments l nf , the covered area (Cantor set measure) is equal to Nðln Þ ldnf ¼ 1

ð68Þ

2n ð3n Þdf ¼ ð2 3df Þn ¼ 1

ð69Þ

or

It follows from Eq. (69) that df ¼ log3 2 ¼ 0; 63093.

physical properties of fractal structures

117

Thus, df is a fractional number and deﬁnes the Cantor set dimension. Thus, d the function of the set measure, Mf ¼ Nðln Þl nf represents the main characteristics of the fractal. We ﬁrst examine the result obtained more thoroughly. Let us ﬁnd the summed length of the remaining segments at the nth division of the Cantor set. We have 1 2 22 2n þ þ þ þ nþ1 þ Ln ¼ 1 3 32 33 3 " # 2 n 1 2 2 2 ¼1 1þ þ þ þ þ 3 3 3 3

ð70Þ

The expression in square brackets is a geometric series with partial sum n 2 Sn ¼ 3 1 3

ð71Þ

Thus, the summed length of the remaining segments, Ln , at the nth stage is n 2 Ln ¼ 3

ð72Þ

lim Ln ¼ 0

ð73Þ

Whence n!1

that is, due to the division of the unitary segment according to the above mentioned procedure, we obtain the remainder with a length of zero. The set obtained in this way is called a Cantor set, or Cantor ‘‘dust.’’ It follows from the construction that Cantor ‘‘dust’’ is inﬁnitely divisible and absolutely broken. It is obvious intuitively that after such a division some remainder whose length is not equal to zero must exist. Let us show that with a corresponding ‘‘ruler’’ used to measure the length of the remaining segment, the summed length of Cantor ‘‘dust’’ is not equal to zero. If we choose a ‘‘ruler’’ (a unit of d measurement) to measure the summed length of Cantor ‘‘dust’’ as l nf ¼ ð3n Þdf , then the summed length Ln (Cantor set measure) is Ln ¼

Nn lndf

n df

¼ 2 ð3 Þ ¼ n

2 3d f

n ð74Þ

vitaly v. novikov

118 and

lim Ln ¼ lim ðNn lndf Þ ¼ lim

n!1

n!1

n!1

2 3 df

n ð75Þ

Therefore, if we take df ¼ ln 2= ln 3, then we obtain the ﬁnite dimension of the Cantor dust summed length. Thus, the measure (measurement) of set f depends on the dimensions of the objects covering it. Measuring a set gives a ‘‘smart’’ result if the ‘‘ruler’’ used to measure the set corresponds to the geometry (dimensions) of the set. The idea that every fractal set needs its own ruler in order to be measured will be used when analyzing the physical properties of heterogeneous media. Generally, for the measure Mf of the set of points in space f with dimension d 8 9 0 . . . if . . . d > df < = Mf ¼ lim Nn ðln Þldn ¼ constant . . . if . . . d ¼ df ð76Þ n!1 : ; 1; . . . if . . . d < df The main feature of the deﬁnition in Eq. (76) is that max ln ! 0. Whence in general the Hausdorff–Besicovitch dimension is a local characteristic. For deterministic self-similar sets, the local Hausdorff–Besicovitch dimension coincides with the dimension of the set itself. For statistically homogeneous sets, however, the local Hausdorff–Besicovitch dimension may not coincide with the dimension of the whole set. The similarity transformation of metric space is the representation, g, of a space on itself whereby all distances between the points change with the same ratio, k > 0. Now, the number k is called the similarity transformation coefﬁcient. The non empty limited set E is called a self-similar set if it may be represented as the union of a limited number of two by two nonoverlapping subsets Ei ; i ¼ 1; n ðn > 1Þ, such that E is similar to E0 with coefﬁcient k. An arbitrary segment, the Sierpinski carpet and sponge are examples of self-similar sets. For fractal sets, the Hausdorff–Besicovitch dimension coincides with the self-similar dimension. Consider a set of points on the limited straight line, L. Any other set of points of limited segment L0 is self-similar (scale multiplier) KðNÞ ¼ 1=N

ð77Þ

where N is a whole number. The set of points of a rectangular area of plane L will contain any other set L0 consisting of the points of the rectangular area as a selfsimilar subset; that is, the self-similarity coefﬁcient is equal to KðNÞ ¼ 1=N 2

ð78Þ

physical properties of fractal structures

119

In general, the scale similarity coefﬁcient is KðNÞ ¼ N dn

ð79Þ

where dn is the similarity dimension. Thus, the similarity dimension is deﬁned as dn ¼ ln N= ln KðNÞ

ð80Þ

For fractal systems, the Hausdorff–Besicovitch dimension is equal to the similarity dimension, that is, df ¼ dn . We consider the triangular Sierpinski carpet as an example (Fig. 4). The iteration process means that the triangle is replaced by N ¼ 3 triangles diminished with similarity coefﬁcient K ¼ 1=2: Thus, the fractal dimension and the triangular Sierpinski carpet similarity dimension are given by df ¼ dn ¼

D.

ln 2 ln 3

ð81Þ

Multifractals

The notion of generalized (multifractal) dimension is introduced by examining heterogeneous fractal sets [22,23]. We consider the deﬁnition of multifractal dimension. Let some fractal set f .be given. We divide this set into nonoverlapping subsets An , such that the diameter ln of the set An is less than e > 0ðln < eÞ. The information (conﬁgurational) Shannon entropy of such a division is Hs ðln Þ ¼

n X

pi ln pi

ð82Þ

i¼1

where pi is the probability of An containing n points. The entropy in Eq. (82) is the measure of the information needed in order to estimate the place of the point in the ith cell. If a set f is continuous, then the probability density rðri Þ of the set f can be deﬁned as pi ðrÞ ¼ rðri ÞVi

ð83Þ

vitaly v. novikov

120

where r 2 Ai , Vi lnd e d is the volume of Ai . Then Hs ðln Þ ¼

X

pi ln pi ¼

i

¼

X i

X

pi ln rðri ÞVi ¼

X

i

pi ln ri

X

pi ðln ri þ ln Vi Þ

i

pi ln ldn ¼ hln ri d ln ln

ð84Þ

i

and Hs ðln Þ ¼ hln ri d ln ln

ð85Þ

Thus, the conﬁgurational dimension of the set f is Hs ðln Þ ln !0 lnðln Þ

ds ¼ lim

ð86Þ

or n P

pi ln pi ds ¼ lim i¼1 ln !0 lnðln Þ

ð87Þ

It follows from Eq. (87) that the information (conﬁguration) Shannon entropy, Hs ðln Þ, of a fractal set depends on the scale ln : s Hs ðln Þ ld n

ð88Þ

According to (88), the information needed to deﬁne the position of the point increases when the cell dimension, ln , approaches zero. The result obtained can be generalized if the notion of Rennie entropy is introduced, Hq [22,23] Hq ¼ ð1 qÞ1 ln

Nðl nÞ X

pqi

ð89Þ

i¼1

Hence, the Rennie dimension is dq ¼ lim

ln !0

Hq ðln Þ lnðln Þ

ð90Þ

physical properties of fractal structures

121

or ln dq ¼ ð1 qÞ1 lim

Nðl Pn Þ

pqi

i¼1

ð91Þ

lnðln Þ

ln !0

Thus, the Rennie dimension, dq , is a function of the variable q taking on values in the interval 1 < q < 1. We demonstrate that if q ! 1, then the Rennie entropy Hq is equal to the Shannon entropy Hs. In fact, if q ! 1, then Nðl nÞ X

pqi ¼

i¼1

Nðl nÞ X

ð92Þ

pi

i¼1

expanding the exponent and noting the normalization namely, Nðl nÞ X

pqi

i¼1

Nðl Pn Þ

pi we obtain

i¼1

,

i¼1

Nðl nÞ X

Nðl nÞ X

i¼1

i¼1

½pi þ ðq 1Þpi ln pi ¼ 1 þ ðq 1Þ

Nðl Pn Þ

pi ln pi

ð93Þ

Hence: ln

Nðl nÞ X

" pqi

¼ ln 1 þ ðq 1Þ

i

Nðl nÞ X

# pi ln pi ¼ ðq 1Þ

Nðl nÞ X

pi ln pi

ð94Þ

i

i¼1

that is, ðq 1Þ1 ln

Nðl nÞ X

pqi ¼

i

Nðl nÞ X

pi ln pi

ð95Þ

i

Concerning Eq. (95), it follows that if q ! 1, then the Rennie entropy Hq [Eq. (89)] coincides with the Shannon entropy Hs [Eq. (82)], and the Rennie dimension [Eq. (91)] is the information dimension [Eq. (87)], that is, ds ¼ lim dq q!1

ð96Þ

The Rennie dimension [Eq. (91)] can be rewritten as dq ¼

1 ln Zðln ; qÞ lim l !1 1q n lnð1=ln Þ

ð97Þ

vitaly v. novikov

122 where

Zðln ; qÞ ¼

Nðl nÞ X

pqi

ð98Þ

i

Zðln ; qÞ is the generalized statistical sum. If q ¼ 0, then Zðln ; qÞ ¼ 1. Now, as shown above [Eq. (87)], the Rennie dimension is equal to the information dimension. If q ¼ 0, then Zðln ; qÞ ¼ Nðln Þ, where Nðln Þ is the number of elementary geometric objects (cells) with a linear dimension ln covering the set . According to Eq. (66) f Nðln Þ ld n

ð99Þ

Thus, if q ¼ 0, then the Rennie dimension dq is the fractal dimension df of the set: df ¼ lim dq

ð100Þ

q!1

The correlation dimension dv can be deﬁned as dv ¼ lim dq

ð101Þ

q!2

Moreover, the correlation sum Cðln Þ, or the number of pairs of points the distance between which does not exceed ln , is [22,23] Cðln Þ ¼ lim

N!1 ln !2

N N X 1 X Zðl r Þ ¼ lim p2i n ij N!1 N 2 i; j¼1 i¼1

ð102Þ

ln !2

where ZðxÞ is the Heaviside unit step function; rij is the distance between points ri and rj . Thus, Cðln Þ deﬁnes the probability for two points selected at random in order to be divided by a distance less than ln : According to Eq. (98), we obtain lim ln Zðln ; qÞ ¼ ln Cðln Þ

q!2

ð103Þ

Hence, we obtain the critical index for correlation length: dv ¼ lim dq ¼ lim q!2 ln !0

ln !0

ln Cðln Þ ln l1 n

ð104Þ

physical properties of fractal structures

123

The correlation sum (integral) for a fractal set Cðln Þ depends on the scale ln as Cðln Þ ldnn ð105Þ In conclusion, we note that pi is the probability for the point to lie in the region li < e; that is, pi deﬁnes the weight (contribution) of different regions in the set measure, Md ðÞ. Thus, the choice of large q 1 contributes to putting elementary geometric objects with relatively great weights pi into the set measure, Md ðÞ. If q 1, then the contribution of cells with small weights pi is increased in Md ðÞ Lognormal Distribution Let us consider a volume V (diameter L) with dimension d. Let N particles of dimension l0 be distributed in V. We divide the whole volume V into cells with sides ln > l0 and volume ldn . We will examine only cells, occupied by at least one particle. Let the number of occupied cells k change within the limits ½ð1; 2; . . . ; Nðln Þ. This is the total number of occupied cells which depends on the cell dimension. If the medium is homogeneous, the number of particles per unit cell volume is equal. Hence f Nðln Þ ld n

ð106Þ

where df is the fractal dimension of the set formed by the particles. The number of particles in the cells is a random variable, thus the relative ﬁlling of a cell is deﬁned by a probability pk. The smaller the cell dimension ln , the smaller is the probability pk for a particle to enter the cell. For self-similar sets, the dependence pk on the cell dimensions has a power law character ð107Þ pk ðln Þ lan i where pk is the probability for a particle to be in cell number k, and ai is generally different for different k cells. For a regular (homogeneous) fractal, all exponents ai are equal to df . pi ¼

1 ldnf Nðln Þ

where Nðl1n Þ is the probability of an elementary event. Moreover, we note that here X p i ¼ c 1 e a 1 þ c 2 e a2 þ þ c n e an ¼ 1 i

ð108Þ

ð109Þ

vitaly v. novikov

124

We consider the probability distribution of ai . Let nðaÞda be the probability for ai to lie in the interval from a to a þ da; that is, nðaÞda is the relative number of k cells having the same measure pk with ak . In a multifractal, various values of a exist with a probability characterized not only by the single value a ¼ df but also by various other values, with the power law exponent f ðaÞ constituting a spectrum of fractal dimensions of the homogeneous subsets 0 of the basic set : nðaÞ ef ðaÞ

ð110Þ

So, the basic set consists of the subset of different homogeneous sets, each having its own fractal dimension f ðaÞ. In accordance with Eq. (107), we have ai ln pi = ln ln

ð111Þ

nðaÞ exp½ f ðaÞ ln e

ð112Þ

f ðaÞ ¼ df Zða a0 Þ2

ð113Þ

According to (111), we have

Now [36,37]

00

where the curvature Z ¼ f ðaÞ=2 ¼ const= ln ln ln is deﬁned by the value of the second derivative of f ðaÞ at a0 . Then h i ð114Þ nðaÞ exp constða a0 Þ2 Noting that ai ¼ ln pi = ln ln , we obtain " 2 # ln pi nðaÞ exp const þ a0 ln ln

ð115Þ

This is a lognormal distribution, it is the probability density function of the random variable pi characterizing the relative ﬁlling of the cells. We consider the asymptotic behavior of pi . pi ¼ lim ni ðln Þ=N mi =Mi

ð116Þ

where ni ðln Þ is the number of points in the cell with number i from the general set of N points belonging to the volume V, mi is the mass of ni ðln Þ points of the ith kind, and Mi is the mass of all points of the ith kind.

physical properties of fractal structures

125

Taking account of Eq. (115), we obtain "

# const ln M0 2 pðmi Þ exp 2 ln mi ln Mi ln ln

ð117Þ

where M0 lna0 . It can be shown that the probability density function of the random variable ri characterizing the dimension of the ith component is "

# const di 1 ln M0 2 pðri Þ exp ln ri di ln Mi ln2 ln

ð118Þ

where di is the fractal dimension of the ith element (component) of a heterogeneous medium. The lognormal distribution is not unlimitedly divisible. Hence, the resulting distribution for all components corresponding to the manifold of generating functions, or in other words, to the sum of lognormal distributions of the masses or dimensions of the ith components, will not be the lognormal distribution of the sum of these distributions. The lognormal distribution of the ith component of a multifractal of dimension ri obtained shows that if 1–3 components are isolated in a real multifractal, then 1–3 lognormal modes corresponding to these characteristic scales can be isolated in an experiment connected with the diffusion of radiation in a heterogeneous medium. E.

Fractal Set Constructed on a Square Lattice

By using the above-described algorithms to generate fractal sets, fractal sets constructed on square lattices have been obtained [24]. The principal or main set of bonds, n , is obtained using an iteration process, whereby at the initial step (k ¼ 0) a ﬁnite lattice is investigated in the space d ¼ 2 or d ¼ 3 with a probability p0 for the bond between the neighboring knots of the lattice to be complete and to have speciﬁc color. Bonds of like color are attributed the same properties. At the next stage of the process ðk ¼ 1; 2; . . . ; nÞ, each bond in the lattice is replaced by the lattice obtained at the previous stage (Fig. 12), The iteration process ends when the properties of the lattice no longer depend on the iteration number k. Thus, a lattice with linear dimensions l greater than the correlation length x can be obtained; that is, a lattice on which the effective macroscopic properties may be deﬁned. The set of bonds n obtained with the aid of the iteration procedure depends on the initial lattice dimension l0 , the probability p0 ; and is a self-similar fractal one.

vitaly v. novikov

126

Finite rectangular models flx ; ly ; lz g in the space of dimension d ¼ 2; 3 have also been studied, where lx ; ly ; lz are the dimensions of the lattice sides in the units of bond length (we have a constant lattice); lz for a planar lattice is zero. Formation of a connecting set (CS) is manifested by the presence in the set of complete (black) bonds connecting two opposite sides of the lattice in the direction of side lx . The fractal dimension dfk of the main set of bonds (i.e., of the frame) (Fig. 12) obtained from the iteration process for p0 ¼ 1 (all bonds colored black) can be deﬁned from the dependence of the set mass (i.e., the number of the constituent ðnÞ bonds of the frame), Mk at iteration stage number n on the linear dimension of the lattice ln : dk

ðnÞ

Mk lnf

ð119Þ

For example, for a planar square lattice with lx ¼ ly ¼ l0 , lz ¼ 0 ðnÞ

Mk ¼ ð2 l20 Þn

ð120Þ

Noting that ln ¼ ln0 , we obtain ðnÞ Mk

¼ ln

2þ lnlnl2

0

ð121Þ

Hence dfk ¼ 2 þ

ln 2 ln l0

ð122Þ

The dimension dfk of the geometric set at p0 ¼ 1 is greater than the topological dimension of space d ¼ 2. For arbitrary dimension d of Euclidean space in which the fractal set f is embedded and arbitrary form of the initial element, the mass (the number of bonds) can be deﬁned by singling out the factorial geometric coefﬁcient F: ðnÞ

Mk ¼ ðF ld0 Þn ¼ F n ldn

ð123Þ

Then Eq. (123) will read dfk ¼ d þ

ln F ln l0

ð124Þ

physical properties of fractal structures

127

Figure 12. Scheme for an iteration process on a square lattice with l0 ¼ 2: (a) p0 ¼ 1; (b) p0 ¼ 0:75 at the fourth iteration step.

It follows from Eq. (124) that the dimension of the bond set coincides with the Euclidean dimensions only in the limit of inﬁnitely large dimensions of the initial lattice, l0 : lim dfk ¼ d

l0 !1

which is obviously correct for any model.

ð125Þ

vitaly v. novikov

128

ðnÞ

If p0 6¼ 1 ð0 < p0 < 1Þ, then the mass MCS of the fractal, which is a connecting set within the scales (l0 < ln < x), depends on ln in accordance with the law: ðnÞ

MCS ldnf

ð126Þ

The density of the connecting set is ðnÞ

ðnÞ

rCS ¼

MCS

ðnÞ

Mk

df dfk

ln

ð127Þ

Now introduce Yp ¼ df dfk ¼ a

ln F ln l0

ð128Þ

where a ¼ d df . Because Yp < 0, we see that the connecting set density decreases to zero with increase of the iteration number—that is, with increase of ln : lim rCS ðln Þ ¼ 0

ln !1

ð129Þ

If p0 > pc , on scales higher than the correlation length x (ln > x), the connecting set becomes homogeneous with constant density, and Eq. (126) reads lim rCS ðln Þ ¼ r0 > 0

ln !1

ð130Þ

Thus, by giving the dimensions of the initial lattices, l0 , and the probability p0 for the bond to have a deﬁnite color, various disordered fractal sets can be obtained. F.

Cayley Tree. Ultrametric Space

The fractal dimension of the set obtained by the iteration process is df ¼

ln j ln K 1

ð131Þ

where j is the number of blocks taking part in the construction of the elementary shape of a fractal (for the Koch curve, j ¼ 4; for Cantor dust, j ¼ 2), and K is the similarity exponent showing by how much the size of the block decreases at each

physical properties of fractal structures

129

stage of the construction. If certain blocks are removed at each stage, then K 1¼)df < 1. If some block is added (Koch shape), then K 1¼)df > 1. ðK > j1 Þ The procedure of fractal set construction can be shown using the Cayley tree so that each fractal set has its own Cayley tree [25,26]. We show a Cayley tree with branch characteristic j ¼ 4. The Cayley tree is a pictorial representation of a space that is called ultrametric. Each point of the ultrametric space can be put into correspondence with an element of the fractal set; that is, the fractal set and ultrametric space are topologically equivalent sets. We remark that the main feature of an ultrametric space, as well as that of a fractal set, is its hierarchical property. The following constitutes the deﬁnition of the distance between two points in an ultrametric space. The points in an ultrametric space on a given hierarchical level are the ends of the Cayley tree branches (Fig. 13). The number of points on the nth level of the Cayley tree is equal to Nn ¼ jn . Each point on the nth level can be numbered: a1 ; a2 ; . . . ; a n ;

0 ak j 1; 1 k n

n=0

n=1

n=2

n=3

Figure 13.

Cayley tree.

ð132Þ

130

vitaly v. novikov

Thus, each point on the nth level of an ultrametric space corresponds to an n-digit number in the j-digit system of calculation (Fig. 13): fal gjn ¼ a0 ; a1 ; . . . ; al ; . . . ; an1 ; al ¼ 0; 1; . . . ; j 1

ð133Þ

These points constitute a space with ultrametric topology. The distance between two points in the ultrametric space is deﬁned by the number of steps from these points to the common limit. For example, the distance between points 00 and 03 equals 1, and the distance between points 02 and 12 equals 2 (Fig. 13). Thus, the distance between two points in the ultrametric space with coordinates given by n-digit numbers in the j-digit system of calculation only depends on which digit these numbers ﬁrst differ and does not depend on the speciﬁc values of this difference. The points of the discrete ultrametric space (Cayley tree junctions) on the nth level, namely, Nn , are divided into clusters (groups). Each cluster contains j points the distance between which is l ¼ 1 and has its progenitor on the ðn 1Þth level. The number of such clusters is Nn =j ¼ jn1 . The unity of clusters corresponds to an arbitrary distance l between points in the ultrametric space. All points are united in j subclusters with distance l 1 and having jl points. Thus, the group of clusters formed on the hierarchical level n 1 corresponds to an arbitrary distance l. If the limit transition is made when n ! 1, then the number of the points attaining level n approaches inﬁnity ðNn ! 1Þ; that is, the intervals between points xn ¼ 1=Nn become inﬁnitely small, and the ultrametric space itself becomes continuous. On the Cayley tree, the transition to a continuous ultrametric space indicates a condensation of the hierarchical levels. The distance between two points in an ultrametric space in the conventional Euclidean sense can be deﬁned as ða b Þ jn1 þ þ ða b Þ jnl þ þ 1 l l 1 rða; bÞ ¼ jra rb j ¼ þ ðan1 bn1 Þ j þ ðan bn Þ

ð134Þ

Two arbitrary points in an ultrametric space belong to any cluster characterized by the distance l n; hence the ﬁrst terms of the sum in Eq. (134) equal zero because ai ¼ bi , i ¼ 1; 2; . . . ; n l. For a continuous ultrametric space (n ! 1), knowing that j > 1, we obtain that the dominant member in the series is the member with multiplier jl, because the other members have multiplier j k, where k ¼ l 1; l 2; . . . ; 0. The values of coefﬁcients (al ¼ bl ) in Eq. (134) are limited by the number j. Representation (134) corresponds to Cayley tree division jn into n groups, each of them consisting of clusters. Each of the clusters of the group is

physical properties of fractal structures

131

characterized by the same value l of the maximum distance between the knots of the cluster. For example, the ﬁrst member in Eq. (134) describes: the contribution of those clusters with knots divided by the distance l ¼ 1. Based on the above statements, the distance rða; bÞ ﬃ jðal bl Þj j l

ð135Þ

ln rða; bÞ ﬃ l ln j

ð136Þ

or

This approximate equation means that the ultrametric space has a logarithmic metric. Thus, when constructing a fractal set, each element corresponds to a point of the ultrametric space with geometric image represented by the Cayley tree.

III.

CHAOTIC STRUCTURES A. 1.

Percolation Systems Percolation Cluster

On analyzing the ﬁltration of air through a porous medium, Broadbent and Hammersley described a novel process differing considerably from the wellknown phenomenon of diffusion [27]. Broadbent and Hammersley named such processes ‘‘percolation processes.’’ The phenomena described by the theory of percolation belong to so-called critical phenomena. These are characterized by a critical point at which the physical properties of a system dramatically change. Investigations of percolation systems are often carried out numerically on lattices, which are aggregates (sets) of junctions (sites) and bonds. Here the roles of bonds and the roles of sites are quite distinct. In the former, the transition of the set of bonds out of the unconnected domain into the connected one on increase of bond concentration p is examined, in the latter, such a transition is examined on the set of lattice sites. (We will further consider the role of bonds.) The concentration p ¼ pc at which the transition from the unconnected set of isolated clusters to the connected set, (the inﬁnite cluster), occurs is called the threshold of percolation (Table IV). If p ¼ pc þ 0, then an inﬁnite cluster exists. If p ¼ pc 0, then all clusters are isolated and ﬁnite. Applying fractal geometry to description of disordered media allows one to use the properties of scaling invariance—that is, to introduce macroscopic

vitaly v. novikov

132 (a)

Figure 14.

(b)

The percolation clusters: (a) Isolated cluster. (b) Inﬁnite cluster (schematic).

values depending on the scale of averaging. In its turn, this allows one to construct the theory of such media using the renormalization group transformation method [1,28–33], which was developed in the theory of temperature phase transitions [34]. We describe brieﬂy the main properties of a percolation cluster [35–37]. 2.

Critical Indices

As stated above, the most characteristic feature of percolation is bonding. The dimension of bonding domains (a bonding cluster)—that is, the regions in which it is possible via black bonds to go from one point of the region to another— rapidly increases with the growth of black bond concentration p. When p ¼ pc , an inﬁnite (percolation) cluster spreading over the entire lattice ﬁrst appears. The correlation length deﬁnes the connectivity of clusters. It deﬁnes the scale range within which percolation clusters behave self-similarly and, consequently, are characterized by a fractal dimension [38,39]. The correlation length x for a percolation lattice can be deﬁned as ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ+ *v u N u1 X x¼ t ðri r0 Þ2 N i¼1

ð137Þ

TABLE IV Percolation Thresholds for Lattices d

Lattice

Bond

Site

2 2 2 3 3 3 3

Square Triangular Honeycomb Simple cubic Body-centered cubic Face-centered cubic Diamond

0.5 [42} 0.33 [42,43] 0.66 [42] 0.24 [42] 0:18025 0:00015 [48] 0:119 0:002 [44] 0:388 0:05 [44]

0:590 0:010 [44] 0:500 0:005 [45] 0:70 0:01 [44] 0:320 0:004 [45] 0:243 0:010 [44] 0:208 0:0035 [45] 0:4299 0:0008 [46]

physical properties of fractal structures

133

Here N 1X ðri r0 Þ2 N i¼1

ð138Þ

is the averaged squared radius of the cluster measured from its center of gravity; ! r0 is the conditional center of gravity of the cluster ! r0 ¼

N ! X ri i¼1

N

ð139Þ

where N is the number of knots in the cluster, and ri is the coordinate of the ith bond in the cluster. The summation is over all N bonds of the cluster. Thus, according to Eq. (137) the correlation length xðpÞ is the average dimension of those clusters that contribute most to the second moment of the distribution of cluster dimensions near the percolation threshold, pc . As p ! 0 ( p ¼ p pc ) the correlation length xðpÞ ! 1 [1], namely, xðpÞ jp pc jn

ð140Þ

where n is the correlation length critical index and depends only on the dimension d of the space. Two linear dimensions, the minimum length l0 (the lattice constant) and the correlation length x, play a key role in the behavior of a percolation system. For real systems, an intermediate asymptotic region exists such that l0 < l < x where l0 is the lattice constant. In the domain x > l0 , that is in the domain of intermediate asymptotics, on the interval l all characteristics of the cluster are similar to its characteristics at the most critical point. On all scales l > x the system is homogeneous, and the self-similarity disappears. To quantitatively characterize the percolation cluster as a whole, the notion of inﬁnite cluster density is introduced, P1 ðpÞ It is the ratio of the number of the bonds belonging to the inﬁnite cluster, nk to all black bonds on the percolation lattice N: P1 ðpÞ ¼

nk N P ni i¼1

ð141Þ

vitaly v. novikov

134

l

ξ

ξ

pc Figure 15.

p

Composition dependence of the correlation length near the percolation threshold

(schematic).

Thus, P1 ðpÞ is the probability of an arbitrary bond belonging to the inﬁnite cluster. The increase of inﬁnite cluster density P1 ðpÞ near the percolation threshold pc ðp=p 1Þ is characterized by a critical index b P1 ðpÞ pb ;

p ¼ p pc > 0

ð142Þ

where b as well as v depends only on the dimension d of the space. The Typical dependence P1 ðpÞ is as shown in Fig. 16.

1

P∞

0 Figure 16.

pc

p

1

Dependence of inﬁnite cluster density on concentration.

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135

TABLE V Percolation Critical Exponents Exponent

d¼2

d¼3

n

4/3 [40, 41]

P1 ðpc pÞb

b

SðpÞ jpc pjg

g ¼ nd 2b

5=3 ½40; 41 0:15 0:03 [47] 43/18 [40, 41] 2:43 0:04 [52]

0.82 0.05 ½47 0:905 0:023 [48] 0:39 0:07 [47] 0:454 0:008 [46] 1:70 0:11 [47] 1:91 0:01 [50]

Function xðpÞ jpc pj

n

The correlation length, xðpÞ, is deﬁned by Eq. (140). Hence j p pc j ¼ jpj x1=n ;

p ! 0

ð143Þ

Here, the percolation cluster (inﬁnite cluster) density P1 ðpÞ can be represented as P1 ðpÞ ¼ xb=n

ð144Þ

The critical indices of correlation length v and percolation cluster densities b constitute the main percolation parameters. In two dimensions (d ¼ 2), these indices may be found analytically [40,41], and are, n ¼ 4=3 and b ¼ 5=36. For d ¼ 3, only numerical estimations are available: n ¼ 0:90 and b ¼ 0:40 (see Table V) The critical percolation indices depend only on the spatial dimensions and do not depend on the type or other parameters of a lattice. Another value characterizing a percolation system is the average cluster dimension SðpÞ. The critical behavior near the percolation threshold is deﬁned by the critical index g [1]: SðpÞ jpc pjg ; p < pc

ð145Þ

The density of clusters of ﬁnite dimensions, ns ðpÞ is connected to the percolation cluster density P1 ðpÞ by the equation 1 X

sns ðpÞ þ P1 ðpÞ ¼ p

ð146Þ

s¼1

The critical indices n; b; g for the percolation cluster are given in Table V.

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Figure 17. Percolation cluster density as a scaled function of its dimension, l, for three values of pðp p pc Þ [24]: 1 p ¼ 5 104 ; 2 p ¼ 1:5 103 ; 3 p ¼ 2:5 103 .

The fractal set mass, Mf (of the percolation cluster), depending on the scale x can be deﬁned as Mf xdf

ð147Þ

where df is the fractal dimension. Therefore, the percolation cluster density P1 ðpÞ is P1 ðpÞ ¼

Mf xdf d ; M

p ﬃ pc

ð148Þ

where M xd is the mass of the domain containing in the percolation cluster. From Eq. (144) and (148) we have df d ¼ b=n

ð149Þ

The percolation cluster includes those bonds forming lines, loops, dead ends and other conﬁgurations. Thus to characterize percolation cluster structure in detail, all bonds belonging to the percolation cluster are colored ‘‘red,’’ ‘‘blue’’ and so on. For example, ‘‘blue’’ bonds are a set of bonds in which current will ﬂow if the percolation cluster is placed between electrodes subject to a potential difference (Fig. 14). The set of ‘‘blue’’ bonds generates an inﬁnite cluster.

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TABLE VI Fractal Dimension of the Percolation Cluster and Its Components Fractal Dimension

d¼2

d¼3

Percolation cluster df ¼ d b=n

91/48 [39]

‘‘Blue’’ bond set

1:62 0:02 [53] 1:60 0:05 [54] 0:75 0:01 [55] 1.75 [58] 1.18 [56]

2:484 0:012 [46] 2:529 0:016 [51] 1:74 :04 [54] 1:77 0:007 [53] — 2.54 [51] 1.35 [54] 1.26 [57]

‘‘Red’’ bond set Perimeter Minimum path

The set of ‘‘red’’ bonds consists of those bonds whereby removing one bond disturbs the bonding of the percolation cluster (removing a ‘‘red’’ bond leads to an open circuit, Fig. 14), while removing a single ‘‘blue’’ bond does not lead to disturbance of inﬁnite cluster bonding. The outer boundary of the percolation cluster is the set of bonds belonging to the boundary of the percolation cluster. The fractal dimensions of these sets are shown in Table VI. 3.

Renormalization-Group Transformations

The macroscopic behavior of physical systems is determined by the microscopic behavior of these systems. Usually the microscopic ﬂuctuations are averaged, and on larger scales the averaged values satisfy the classical equations. A number of extreme (critical) situations exist when ﬂuctuations extend to macroscopic scales and exist on all intermediate scales. Temperature phase transitions and percolation processes concern such situations. In these systems a critical point exists which separates two different phase states of the system. The difﬁculties in the theoretical study of such systems are caused by the large numbers of interacting degrees of freedom, since it is necessary to have many variables in order to characterize such systems near by a critical point. For the purpose of overcoming difﬁculties in the study of critical phenomena the following method was used: sequential averaging was carried out on all scales, starting with ﬂuctuations at an atomic level and then moving step-bystep to larger scales. Such stage-by-stage modiﬁcation of scales allows one to reduce the number of degrees of freedom. This method is known as the renormalization group method. The result of the renormalization group method as applied to percolation problems consists of the following: a physical state characterized, for instance, by the parameters fY0 g evolving via a set of equations Y forms a continuous sequence of new effective equations Y ðlÞ, characterized by the new parameters

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fYðlÞg [28–33]. The new parameters fYðlÞg are functions of the initial parameters fp0 g and a scale factor l0, that is, Yk ðlÞ ¼ Y ðl0 ; fp0 gÞ

ð150Þ

In the limits of large l a trajectory of renormalization group transformation terminates in a ﬁxed point. The dimensionless correlation length x can be used as a measure of the remoteness of a percolation system from a critical point x ¼ x=l0

ð151Þ

Under renormalization group transformation the correlation length x decreases so that x

0 1 0 pi ¼ xp pi l0

ð152Þ

where p0 is the probability of making a connection (probability that connection will be black) after renormalization group transformation. This transformation has a ﬁxed point related to a percolation threshold pi ¼ pc where pi is determined from f p g ¼ Y ð f p gÞ

ð153Þ

The procedure of renormalization group transformation (153) for ﬁnite lattices is approximate as it does not consider the surface effects on a lattice properly because the connecting conﬁgurations gained after renormalization group transformation, differ from connecting conﬁgurations in a real lattice. Under renormalization group transformation some connecting conﬁgurations are lost and new connecting trajectories appear. As the initial cell l0 grows in size, boundary effects decrease [24]. To illustrate the basics of renormalization group transformation we shall consider a triangular lattice, which is featured prominently in the literature. The initial cell is connecting if all three knots are made or two of them are made, and one is vacant. In this case the probability, for the cell to be connecting, is [1] Y ð pÞ ¼ p3 þ 3p2 ð1 pÞ

ð154Þ

0

p ¼ Y ð pÞ It follows from the last equation, that three ﬁxed points: 0; 1=2; 1 exist. Two points ð0; 1Þ are unstable; however, point p ¼ 1=2ðp ¼ Y ðp Þ:Þ is stable and equal to the percolation threshold pc ¼ 1=2.

physical properties of fractal structures

139

Expansion in a Taylor series about p ¼ 1=2, neglecting ð p p Þ2 and higher yields dp0 p ¼p þ dp 0

ð p p Þ þ

ð155Þ

p¼p

or p 0 p ¼ lð p p Þ þ

ð156Þ

where dp0 l¼ dp

¼ 6p 6p2 p¼p ¼ 3=2

ð157Þ

p¼p

When p ¼ pc the correlation length tends to inﬁnity. On small scales L < x we may write x0 ¼ x and for a renormalized lattice we have x=l0 ¼ cjp0 pc jn

ð158Þ

where l0 is a lattice distance, c and n are constants being an amplitude and a critical exponent, respectively. Thus, the following equality is valid: l0 jp pc jn ¼ jp0 pc jn

ð159Þ

This relation is the basic equation governing a renormalization group in real space. According to Eq. (159) we shall obtain the critical exponent n for correlation length: n¼

ln l 0 p0 p ln ppcc

ð160Þ

Consequently, we have 1 ln l ¼ n ln l0

ð161Þ

pﬃﬃﬃ n ¼ ln 3= lnð3=2Þ ’ 1:355

ð162Þ

that is,

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If as initial cell we choose a square cell with 8 bonds [1], then Y ð pÞ ¼ p5 þ 5p4 ð1 pÞ þ 8p3 ð1 pÞ2 þ2p2 ð1 pÞ3 ; p0 ¼ Y ð p Þ

ð163Þ

It follows from Eq. (163), that p ¼ 1=2 is a nontrivial ﬁxed point and that the critical index for correlation length is n ¼ 1:43 . . .. In such calculations of n and other critical indexes it is impossible to determine the accuracy of the calculation and growth of cells of size l0 > 4 poses essential calculation difﬁculties [33]. The percolation probability of a lattice Yðp; l0 Þ with initial concentration of black connections p is calculated as the ratio of number of the number of connecting conﬁgurations to the number of all possible scatters. Typical Yðp; lÞ for a cell of size l0 ¼ 2 is presented in Fig. 18. Intersections of the bisector of coordinates axes with curves Yðp; lÞ deﬁne the percolation thresholds pc for models of size l0 (see Fig. 18). The function Yðp; lÞ converges to a step function when l0 ! 1 and the derivative dY ¼ f ðpÞ dp now converges to a Dirac delta function (f ðpÞ ! dðpÞ; if l0 ! 1).

Figure 18. The percolation lattice probability Yðp; l0 Þ for a cell of sizes l0 ¼ 2:

ð164Þ

physical properties of fractal structures

141

ξ

Figure 19.

Framework of an inﬁnite cluster according to the Scala–Shklovsky model.

By using the function Yðp; lÞ it is possible to obtain a probability density function f ðp; lÞ for the percolative conﬁgurations Z Y ð p; lÞ ¼ f ð p; lÞdp ð165Þ Thus the relation dY f ðp ; lÞ ¼ dp

¼ lp

ð166Þ

p¼p

is satisﬁed. Let us consider two models that have been used in percolation theory to describe the properties of a percolation cluster. The ﬁrst is the Scala–Shklovsky model [64]. 4.

Physical Properties

In the Scala–Shklovsky model [64] it was assumed that the structure of an inﬁnite cluster is a net with the characteristic geometric distance between knots being the percolation length x. The sites of the net are connected by single-core macrobonds with length L ðp pc Þ

ð167Þ

which can be greater than distance x between the sites of the net (Fig. 19). It was shown [65,66] that the critical index is 1 and does not depend on the dimension d of the space. The disadvantage of this model is that, according to Eq. (167), the distance x between the knots of the inﬁnite cluster net increases faster than length L of the macrobond. In the Coniglio–Sarychev–Vinogradov model, this disadvantage was removed [65].

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142

Figure 20.

Blob model of an inﬁnite cluster framework.

In the Coniglio–Sarychev–Vinogradov model, it is assumed that the inﬁnite cluster consists of blobs connected to macrobonds (Fig. 20). The blob of dimension b consists of several blobs of dimension b=2 connected by singlecore macrobonds, and so on. Therefore, the system is self-similar on any scale. It has been shown [64–67] that the average summed length L of nondoubled bonds in a fragment of the inﬁnite cluster is deﬁned as L b1=v . Whence, if b ¼ x, then L x1=n ðp pc Þ1 ; that is, the critical index ¼ 1. The relative length of a macrobond L=b b1=ðn1Þ ðp pc Þn1

ð168Þ

increases with an increase of scale b, so that this model does not have the contradictions of the Scala–Shklovsky model. We now examine the main ideas of how we deﬁne the conductivity and elasticity of fractal structures. Conductivity. Let sðlÞ be the conductivity of a fragment of the fractal structure of dimension l, where l0 < l < x. Because of the self-similarity of the structure, the ratio of conductivities on different scales l and l0 is deﬁned only by the ratio of the scales: sðlÞ l ¼ f ð169Þ 0 sðl Þ l0 For three different scales, l; l0 and l00 , the following equations are appropriate: 0 sðlÞ l sðl0 Þ l sðlÞ l ¼ f ¼ f ¼ f ; ; ð170Þ sðl0 Þ l0 sðl00 Þ sðl00 Þ l00 l00

physical properties of fractal structures

143

Hence any function f ðxÞ of the scale ratio must satisfy the equation f ðx yÞ ¼ f ðxÞ f ðyÞ

ð171Þ

It follows from Eq. (171) that f ðxÞ ¼ xg

ð172Þ

where g is an arbitrary number. Therefore, sðllÞ ¼ lg sðlÞ

ð173Þ

where l belongs to interval l > x. Now we divide the percolation cluster into d-dimensional cubes (d > 1). The conductivity of one cube with side l is designated by sðlÞ. The number of cubes per unit length in the column is equal to l1 , and the number of parallel columns is l1d . So, the conductivity of the percolation cluster is s ¼ sðlÞl2d

ð174Þ

The conductivity of a single cube, sðlÞ can be deﬁned as sðlÞ ¼ s1 l1 , (s1 is the conductivity of a ‘‘black’’ bond). Thus, we obtain s ¼ s1 l1d . Since l x, we obtain s s1 ðp pc Þnðd1Þ ;

ðp pc Þ > 0

ð175Þ

Introducing a critical index t for the conductivity s ðp pc Þt ; ðp pc Þ > 0

ð176Þ

t ¼ nðd 1Þ

ð177Þ

we obtain

Taking account of the values of n (n ¼ 1; 33 at d ¼ 2 and n ¼ 0; 8 0; 9 at d ¼ 3), we obtain t ¼ 1:3 if d ¼ 2

and

t ¼ 1:6 1:8

if d ¼ 3

The approximate estimations of the critical index of conductivity, t, so obtained, agree with the numerical evaluations (see Table VII). The critical index s for conductivity in the concentration region ðp pc Þ < 0, ð p ! pc 0Þ is deﬁned by s ðp pc Þs ;

ðp pc Þ < 0

ð178Þ

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144

TABLE VII Conductivity Critical Indexes Function

d¼2

Index

t

s ðp pc Þ ðp pc Þ > 0

t

s ðpc pÞs ðp pc Þ < 0

s

1:10 0:05 [58] 4/3 [60] 1:32 0:05 [59] 1:15 0:25 [71] 1:0 0:1 [62]

d¼3 1:6 0:1 [67] 2:00 [62] 1:95 0:1 [63] 2:46 [28] 0:7 0:05 [62] 0:9 0:01 [68] 0:7 0:05 [72]

Linear Elasticity. Here the task of deﬁning the elastic properties of a percolation system will be formulated completely if the relevant Hamiltonian is deﬁned on that set of sites and bonds with geometric parameters (numbers of sites and bonds, distances to the most remote elements, sinuosity, etc.) given statistically. The Hamiltonian describing the elastic properties of a percolation system must satisfy the following criteria: Elastic bonding must exist: At p > pc , the lattice must have ﬁnite elastic macromodulus becoming zero at p ! pc þ 0. The tensor properties of the elasticity of long chains must be generated properly. Invariance must be preserved in relation to rotations in free states of the Hamiltonian. The Born model [74], for example, satisﬁes the ﬁrst condition; however, it does not satisfy the second one because in it the longitudinal and transverse elastic constants of the linear chain of bonds (the lattice analog of a rod) decrease N 1 ; however, the rods must behave more pliably in relation to transverse shifts (the elastic constant decreases L3 ). Therefore, the Born scalar model leads to enhanced rigidity in the vicinity of pc .

j bi

i

Figure 21.

Φijk k Elastic chain of vectors.

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145

U. Cantor and I. Webman [75] have suggested a Hamiltonian describing the elasticity of continuous chains correctly, namely, H¼

N N GX Q X aij ajk dijk þ 2 ðui uj Þ2 4 i; j; k 4a i; j

ð179Þ

where dijk is the change of the angle between the bonds fðijÞ; ðjkÞg; a is the lattice constant; aij is a random variable lying in f0; 1g with probabilities ð1 pÞ and p, respectively; Q and G are local elastic constants; ðui uj Þ is the difference of the shift of sites i and j in the direction parallel to bond ði; jÞ: The rigidity of the percolation net ‘‘is supported’’ by the inﬁnite cluster system which consists of comparatively straight and single bonds connecting compact multibond domains. Thus, the elastic behavior of a percolation system can be investigated based on the behavior of chains with N vectors (bonds) fbi g (Fig. 21). The corresponding Hamiltonian is H¼

N N GX Q X 2 d2i þ 2 dbi 2 i¼1 2a i¼1

ð180Þ

In order to deﬁne the elastic constant of such a vector chain, it is necessary to deﬁne the relative change of angle di in the orientation of the bonds when ! force F is applied to the end of the chain. This relative change, di , can be found by minimizing the equation for the system energy, namely, ! 0 W ¼ H F ðRN RN Þ

ð181Þ

0

where ðRN RN Þ is the shift of the chain end from the balanced state. ! 0 The work done by the force F ðRN RN Þ can be represented as the PN summation i¼1 , and the exact expression for the change of the angles between two bonds after minimization of W reads !X N N X X F N ! ! 0 di bi þ bi dbi ; F ðRN RN Þ ¼ ð F zÞ a i¼1 i¼1 i¼1 ! ! N F zX F z a! ðRN Ri1 Þ; d ¼ F bi di ¼ bi ¼ G i¼1 G b

ð182Þ

where z is a single vector perpendicular to the plane of the chain, and Ri is the balanced position of the end of vector bi.

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146

Substituting di and bi into (180), we have H ¼ F 2 NS2? =2G F 2 aL=2Q

ð183Þ

where S2? is the square inertia of the projection of angle R onto the direction F z: S2? ¼

N N 1 X 1 X ! ! fð F zÞðRi1 RN Þg; L ¼ 2 ð F bÞ2 2 NF i¼1 aF i¼1

ð184Þ

Thus, the force constant (rigidity) of the chain is deﬁned as K ¼ G=ðNS2? Þ

ð185Þ

Thus, the elastic constant of a long chain depends not only on the length ð1=NÞ but also on its geometry, S2? . The analysis given for d ¼ 2 can also be applied to higher dimensions. As before, the deformation of the chain may be represented by a sequence of transformations, where the ith transformation includes the stretching of the bond bi and the rotation of bonds bi . . . bn about Ri¼1 : The result obtained can be used to estimate the critical index t of the macroelastic constant at p ! pc þ 0. K ðp pc Þt

ð186Þ

Dividing the percolation system into cubes with linear dimension l, the inﬁnite cluster system macroscopic elasticity can be deﬁned as K ¼ KðlÞl2d

ð187Þ

KðlÞ is the elasticity of single cube with linear dimension l. As elasticity on any scale is assumed to behave in a self-similar mode, KðlÞ can be deﬁned as KðlÞ K1 ðp pc Þt ¼ K1 lt=n

ð188Þ

(K1 is the elasticity of a bond). Moreover, according to Eq. (185), 1 ¼ l2 l1=n KðlÞ S2 b L

ð189Þ

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147

TABLE VIII Elasticity Critical Indexes d

2

t=n S=n

2.97 0.92

3 4.3 0.74

Comparing the two latter expressions for KðlÞ, we can conclude that t=n ¼ 2 þ 1=n, that is, t ¼ 2n þ 1. Using the obtained expression for n, the inﬁnite cluster system macroscopic elasticity can be deﬁned as KðlÞ K1 lð1þndÞ=n ¼ K1 ðp pc Þndþ1

ð190Þ

Thus, the volume elastic modulus of the percolation system near the percolation threshold ðp ! pc þ 0Þ may be deﬁned as K K1 ðp pc Þt ;

t ¼ nd þ 1;

p ! pc þ 0

ð191Þ

and the critical index is t ¼ nd þ 1

ð192Þ

Noting the values of n (n ¼ 1; 33 at d ¼ 2 and n ¼ 0; 8 0; 9 at d ¼ 3), we obtain t ¼ 3; 6 if d ¼ 2 and t ¼ 3:4 3:7 if d ¼ 3. The approximate estimations of the critical index of volume elasticity t obtained agree well with the numerical evaluations (see Table VIII). Comparing t and t, we see that tt ¼1þn

ð193Þ

Hence, the critical index t for elasticity is different from the critical index t for conductivity at the value of 1 þ n. Critical indices S and t, obtained numerically, are shown in Table VIII (according to Ref. 76). The critical index S for the volume elastic modulus in the concentration domain ðp pc Þ < 0 ðp ! pc 0Þ is deﬁned by K ðp pc ÞS ; B.

ðp pc Þ < 0ðp ! pc 0Þ

ð194Þ

Fractal Structure Model

Stauffer and Aharony [1] have studied chaotic fractal ensembles on square lattices where all bonds were identically colored at the initial stage and later

148

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Figure 22. Schematic for constructing a self-similar lattice via iterative growth of a square generation cell.

randomly change their color. These chaotic fractal ensembles f depend on the initial lattice size l0 and on the probability of p0 . At each stage of the growth process, each thin bond of the generating cell is replaced by a structure obtained at the previous stage. The sides marked by thin lines are assumed to be always connecting. The growth of the fractal ensemble was initiated on the two-dimensional (d ¼ 2) ﬁnite-size lattice l0 l0 ; next each bond of this lattice at the kth

physical properties of fractal structures

149

stage was replaced by a lattice generated at the preceding, (k 1)st stage. The growth was considered completed after the properties of the fractal ensemble became independent of the linear scale ln ¼ ln0 . The fractal dimension, df0 ðl0 Þ, of the principal ensemble 0 ðl0 Þ (with all bonds of the same color) generated on square lattices (lx ¼ ly ¼ l0 ) was determined, as usual, from the relationship between its mass (i.e., number of bonds) and ln , that is, [24], d0 ðl0 Þ

Mn ðl0 Þ ¼ lnf

ð195Þ

Noting that for square lattices n Mn ðl0 Þ ¼ 2l20

ð196Þ

ln 2 ln l0

ð197Þ

one obtains df0 ðl0 Þ ¼ 2 þ

It follows from Eq. (197) that fractals of dimensions 2 < df0 ðl0 Þ < 3 can be obtained by altering the size l0 of the generating (initial) square cell. For example, df0 ðl0 Þ ¼ 3 for l0 ¼ 2, while df0 ðl0 Þ ! 2 for l0 ! 1. Consider now another ensemble of structures, 0 ðl0 ; p0 Þ, with bonds that may be colored black and white with p0 as the probability of a black bond. Hence, one may distinguish between two basic states, in which the black bonds form either a bonded ensemble (connecting set) spanning the entire lattice space between two opposite faces, or via non-bonded ensemble (non connecting set) with no trajectory available to connect two opposite lattice faces. The mass of the connecting set of black bonds in the vicinity of the critical point of transition nonconnecting set!connecting set depends on the linear scale ln , Mf ðl0 ; p0 Þ lndf ðl0 ; p0 Þ

ð198Þ

where the connecting set density may be deﬁned as PCS ðl0 ; p0 Þ ¼

df ðl0 ; p0 Þdf0 ðl0 Þ M f ð l 0 ; p0 Þ ¼ ln Mn ðl0 Þ

ð199Þ

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vitaly v. novikov

The correlation length x of the connecting set is conﬁned to the range of intermediate asymptotics, which may be deﬁned as l0 << l < x

ð200Þ

In this range, the connecting set is a fractal; that is, it is geometrically similar to a percolating cluster, and its properties depend on the linear scale. Therefore, both the correlation length x and the P1 CS of the connecting set (the upper index 1 means that the limit ln ! 1 is taken) should scale with distance from the critical point (i.e., percolation threshold pc ¼ p ) as xðl0 ; p0 Þ jp0 pc jnðl0 Þ P1 CS ðl0 ; p0 Þ

bðl0 Þ

ð p0 pc Þ

ð201Þ ;

p0 > pc

ð202Þ

where p0 ! pc þ 0. Using Eqs. (201) and (202), we have the dependence of P1 CS ðl0 ; p0 Þ on the linear size of the system

P1 CS ðl0 ; p0 Þ ln

bðl0 Þ nðl0 Þ

;

when p0 ! pc þ 0

ð203Þ

Here, the critical indices for the connecting set correlation length and density are related via the fractal dimension df ðl0 Þ and bðl0 Þ; nðl0 Þ as bðl0 Þ ; nðl0 Þ df ðl0 Þ ¼ lim df ðl0 ; p0 Þ; p0 !pc þ0 bðl0 Þ ¼ lim log10 P1 CS ðl0 ; p0 Þ =log10 ½ p0 pc ;

df ðl0 Þ ¼ df0 ðl0 Þ

ð204Þ

p0 !pc þ0

vðl0 Þ ¼ lim log10 ½xðl0 ; p0 Þ=log10 jp0 pc j p0 !pc

An important characteristic of the chaotic fractal ensemble 0 ðl0 ; p0 Þ is the probability that a given conﬁguration belongs to the connecting set (i.e., the percolation probability). At the 0th (i.e., initial) stage, this probability depends on the initial density of black bonds p0 and on the size l0 of the generating cell, and it may be deﬁned as the ratio of the number of bonding conﬁgurations to the total number of possible conﬁgurations. At the ﬁrst growth stage the length of the lattice rib is l1 ¼ l20 , and the density of black bonds is p1 ¼ Yðl0 ; p0 Þ. In the

physical properties of fractal structures

151

next stages ln ¼ l0 ln1 and the probability of a bond to belong to the connecting set will depend on p0 as follows: p2 ¼ Yðl1 ; p1 Þ; ... pn ¼ Yðln1 ; pn1 Þ

ð205Þ

The unstable critical point p ¼ Yðl0 ; p Þ (pc ¼ p ) may be determined from the equality 1; p0 > pc lim pn ¼ ð206Þ n!1 0; p0 < pc In any practical (ﬁnite precision) calculations the growth trajectory of the chaotic fractal ensemble f ðl0 ; p0 Þ ends at the nth growth step (level) reaching a point indistinguishable from one of the ﬁxed points 0 or 1 of the bounding probability function Yðl0 ; p0 Þ. ðnÞ The probability PCS ðl0 ; p0 Þ that a bond belongs to the connecting set at the nth growth step is ðnÞ

PCS ðl0 ; p0 Þ ¼

n Y

Yðl0 ; pi Þ

ð207Þ

i¼0

Equation (207) shows that a complete (conducting) bond belongs to the inﬁnite cluster only when it belongs to a cluster connecting the two opposite sides of the lattice at each iteration step i ¼ 1; . . . ; n. It follows from Eqs. (206) and (207) that p0 > pc PCS ðl0 ; p0 Þ; ðnÞ lim PCS ðl0 ; p0 Þ ¼ ð208Þ n!1 0; p 0 < pc Thus, a knowledge of the function Yðl0 ; p0 Þ is crucial for determining the properties of the fractal model. For small initial lattices this function can be calculated exactly. The results for square generating cells of l0 ¼ 2; 3; 4 as well for the more general case of rectangular generating cells are given in the Appendix. Consider now fractal ensembles grown on rectangular subsets of the square lattice, lx ly ðlx 6¼ ly Þ, further referred to as rectangular generating cells. Hence, the characteristic length of the system, l0 , can be chosen in various ways. For lx > 1 ( lx ¼ 1 is trivial) the simplest and most natural choice is l0 ¼ lx . As can

vitaly v. novikov

152

A

B

A

B

Figure 23. Schematic for construction of a self-similar lattice by iterative growth of a rectangular generation cell: the meaning of the thin and thick lines is as in Fig. 22.

be seen in Fig. 23, for such a choice each iteration step increases the length of the system by the factor l0. Thus, it is natural to choose ln ¼ ln0 as the characteristic length of the system at the nth step. At the nth growth step, the mass of the ‘fractal’ ensemble so obtained will be n ðnÞ ð209Þ M0 lx ; ly ¼ 2lx ly þ lx ly

Figure 24.

The relatively dual model.

physical properties of fractal structures

153

TABLE IX The Percolation Threshold pc , Fractal Dimension of the Ensemble at p ¼ 1, df0 ðlx ; ly Þ, Mean Fractal Dimension for p ¼ pc, df , and Critical Indices b bðlx ; ly Þ, a1 , and n nðlx ; ly Þ for Various Initial Rectangular Cellsa. lx ly

pc

df0 ðlx ; ly Þ

df

lp

n

21 22 23 24 31 32 33 34 41 42 43 44

0.500 0.304 0.223 0.178 0.696 0.500 0.410 0.358 0.777 0.590 0.500 0.451

2.322 3.000 3.459 3.807 1.893 2.335 2.631 2.854 1.730 2.085 2.322 2.500

1.322 1.282 1.293 1.314 1.563 1.704 1.819 1.919 1.548 1.704 1.822 1.926

1.625 1.823 1.912 1.962 1.823 2.217 2.441 2.590 1.911 2.441 2.766 3.097

1.428 1.154 1.069 1.028 1.829 1.380 1.231 1.154 2.139 1.553 1.363 1.226

a

a1 1.000 1.718 2.166 2.493 0.330 0.631 0.811 0.935 0.182 0.381 0.500 0.574

b 1.428 1.983 2.315 2.564 0.603 0.870 0.999 1.079 0.389 0.591 0.681 0.704

ln l The lp is calculated from lp ¼ dY dp jp¼pc and n is calculated from n ¼ ln lp .

Thus, using Eq. (195) one obtains the fractal dimension df0 lx ; ly of the system as d f0

lx ; ly

ln 2lx ly þ lx ly ¼ ln lx

ð210Þ

One can easily check that 1 < df0 lx ; ly < 1, where the limits are reached for ly =lx ! 0 (yields df0 lx ; ly ¼ 1) and lx =ly ! 0 (yields df0 lx ; ly ! 1). Probability functions Yðlx ; ly ; pÞ for fractal ensembles grown on several lattices (of the generating cells lx ly where 2 lx 4; 1 ly 4) are presented in the Appendix, while calculated values ofthe percolation threshold 0 pc , fractal dimension of the ensemble at p ¼ 1, d lx ly , mean fractal dimension f at p ¼ pc df , and critical indices bðlx ; ly Þ and nðlx ; ly Þ are listed in Table IX. The index a1 in this table is calculated from 1 PCS lx ; ly ; p la n

ð211Þ

that is, a1 ¼ bðlx ; ly Þ=nðl x ;ly Þ. The results presented in Table IX were obtained in the limit p0 ! pc and df ¼ df0 lx ; ly bðlx ; ly Þ=nðlx ; ly Þ: Ending these calculations, one should note that, according to the construction procedure outlined, the lattices considered are inhomogeneous, that is, the

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154

coordination number, Z, for these lattices depends on the lattice coordinates. By introducing the average coordination number hZ i ¼ 2ðtotal number of bondsÞ=ðtotal number of sitesÞ

ð212Þ

and noting that the lattices considered are self-similar, the average coordination number can be expressed via the length of the sides of the generating cell as hZ i ¼ 2

2lx ly þ lx ly 1 lx ly þ lx ly 1

ð213Þ

It follows from Eq. (213) that for rectangular generating cells the average coordination number can vary in the range 3 hZ i 6. This range is a factor 9=2 larger than that for square generating cells (lx ¼ ly ), for which 4 hZ i 14=3

ð214Þ

Table X present results obtained for the overage coordination number hZ i and for the average number of complete bonds around a lattice site at the percolation threshold (i.e., the product of hZ i and p at the critical point) for the lattices considered. Thus, for the inhomogeneous lattices considered this product can be much different from the value 2 which has been obtained for the inﬁnite uniform square lattice [3].

TABLE X The Average Coordination Number, hZ i, and the Average Number of Complete Bonds Around a Lattice Vertex, pc hZ i, Calculated for Various Initial Rectangular Cells at the Percolation Threshold, pc lx ly

pc

21 22 23 24 31 32 33 34 41 42 43 44

1/2 0.3039 0.2227 0.1776 0.6961 1/2 0.4100 0.3580 0.7772 0.5900 1/2 0.4509

hZ i 4 14/3 5 26/5 7/2 4 17/4 22/5 10/3 34/9 4 62/15

pc hZ i 2 1.418 1.114 0.924 2.436 2 1.743 1.575 2.591 2.229 2 1.864

physical properties of fractal structures 1.

155

Properties of Finite Lattices

It follows from the results obtained above that the regularities in the statistical properties of a percolation cluster can be studied even at length intervals of the order of the lattice constant. Hence, we shall analyze in greater detail the ﬁnite scales method [78] which was implemented on the smallest models. Thus, we separate the set of all possible initial cells in two-dimensional space into classes according to the following criterion: A model belongs to class CðnÞ if the difference lx ly for the model equals n, where n 2 Z. We choose a ﬁnite-dimensional representation for the density of an inﬁnite cluster as b=n P1 a1 þ a2 gp ð l Þ 1 ¼l

ð215Þ

where the function gp ðlÞ is a correction to the scaling and l characterizes the size of the square model flx ; ly ; lz ¼ 0g in the sense that the number of bonds in the class increase as a power-law function (215) with exponent d. A computer calculation performed on small lattices ðlx < 10Þ in two dimensional space suggests that G ln P1 1 ðp ðlÞÞ = ln l is a linear function of 1= ln l (Fig. 25): b ln a1 þ a2 gp ðlÞ b B ¼ þ f ðnÞ þ þ ; ln l n n ln l

ð216Þ

G 3

0.3

2 0.2

1 0.1

0

Figure 25.

0.3

0.6

0.9

l/ln l

G ¼ ln½P1 1 ðlÞ= ln l versus 1= ln l for the model classes 1, Cð2Þ 2, Cð1Þ; and 3, Cð0Þ.

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where f ðnÞ is the difference of the coordinates of the point of intersection of the experimental straight line for a given class CðnÞ, and the G axis and the ratio b=n are singled out on the right-hand side for the purpose of generalization. Then the assumption of linearity determines the form of gp ðlÞ in Eq. (215), namely, a1 þ a2 gp ðlÞ ¼ lf ðnÞ expðBÞ

ð217Þ

During self-similar growth of the model, the similarity dimension of the statistically homogeneous fraction that now arises can be estimated right at the percolation threshold from DðlÞ ¼ d þ

ln½P1 ðp ðlÞÞ ln½F ðp ðlÞÞ þ ln l ln l

ð218Þ

which, generally speaking, takes on different values for different models. Using Eqs. (215) and (218), we obtain DðlÞ d þ

b B ln½F ðp ðlÞÞ ¼ f ðnÞ þ n ln l ln l

ð219Þ

The fractal dimension df of an inﬁnite cluster (limiting value of DðlÞ in the limit l ! 1) satisﬁes Eq. (219), which determines the asymptotic behavior of the function p ðlÞ, namely, lim

l!1

ln½p ðlÞ ¼ f ð nÞ ln l

ð220Þ

It is evident from Eq. (220) that the decrease, recorded during numerical modeling, of the nonzero quantity dðlÞ ¼ jp ðlÞ pc j (power-law decrease with exponent 1n [1]) satisﬁes Eq. (217) only if f ðnÞ ¼ 0 [hence a2 ¼ 0 in Eq. (216)]. Then the curves 1 and 3 in Fig. 24 are convex and concave, respectively, with respect to the horizontal axis, since for the classes Cð0Þ and Cð2Þ the linearity assumption leads to f ð2Þ; f ð0Þ ¼ 6 0. Thus the assumption (216) can generally be expressed as follows: There exists a value of F such that the curves G ln1 ðlÞ for classes C ðnÞ and Cð2 nÞ, where n 2 Z, are symmetric with respect to one another relative to the x axis. The x axis itself will now constitute the plot for the class Cð1Þ, which is distinguished from the other classes by the property pc lx ; ly ; 0 jCð1Þ ¼ 0:5

ð221Þ

To prove Eq. (221), we erect from the geometric centers of the interstitial squares of the model flx ; ly g, which we call the initial model, perpendiculars to

physical properties of fractal structures

157

all its bonds. We continue the perpendiculars to the periphery of the model and connect them as in Fig. 24, forming in this manner a dual square model, whose bonds are in one-two-one correspondence with the bonds of the initial model (the intersection

of

bonds in Fig. 24 indicates this correspondence). The new model lx ; ly d ly þ 1; lx 1 has the property that Yd p; lx ; ly ¼ Y p; ly þ 1; lx 1 ¼ 1 Y 1 p; lx ; ly

ð222Þ

that is, the percolation probabilities of mutually dual models are symmetric relative to the point ð0:5; 0:5Þ. A general argument in the proof of the property (221) is as follows. We form with the dual model a conﬁguration starting from a connected (unconnected) conﬁguration of the initial model according to the following principle: An unbroken bond in the initial model transforms into a broken bond corresponding to it and vice versa. Then, the resulting ‘‘symmetric’’ conﬁguration will be disconnected (connected). The formal proof is constructed for the corresponding site models which are subsets of the socalled covering lattice [80]. Since self-dual models are of class Cð1Þ, the property (221) is proved (the exact equality pc ¼ 0:5 is proved simultaneously for bonds on a square lattice). It is obvious that the inverse assertion will also be true: Any model for which pc ¼ 0:5 will belong to class Cð1Þ. The result that P1 ¼ const lb=n for class Cð1Þ is conﬁrmed because of the result b=n ¼ 0:1041 0:0013 [the average value over the data obtained using the four models with lx ¼ 3; 4; 5, and 9 [24] from Cð1Þ agrees with the exact value 5/48 [81]. Therefore small experimental models, together with models in which the number of sites is 106 or larger, are suitable for calculating the ﬁnite-dimensional scaling ratios of the critical exponents (it can be inferred that b=n is only one such ratio). 2.

Appendix. The Probability Functions.

The probability functions Yðp; lx ; ly Þ derived for various nucleating cells are presented below. Unit cell 2 1: Yðp; 1; 2Þ ¼ 2p2 ð1 pÞ3 þ 8p3 ð1 pÞ2 þ 5p4 ð1 pÞ þ p5 ; Unit cell 2 2: Yðp; 2; 2Þ ¼ 3p2 ð1 pÞ6 þ 22p3 ð1 pÞ5 þ 56p4 ð1 pÞ4 þ 54p5 ð1 pÞ3 þ 28p6 ð1 pÞ2 þ 8p7 ð1 pÞ þ p8 ;

vitaly v. novikov

158 Unit cell 2 3:

Yðp; 2; 3Þ ¼ 4p2 ð1 pÞ9 þ 42p3 ð1 pÞ8 þ 178p4 ð1 pÞ7 þ 382p5 ð1 pÞ6 þ 442p6 ð1 pÞ5 þ 328p7 ð1 pÞ4 þ 165p8 ð1 pÞ3 þ 55p9 ð1 pÞ2 þ 11p10 ð1 pÞ þ p11 ; Unit cell 2 4: Yðp; 2; 4Þ ¼ 5p2 ð1 pÞ12 þ 68p3 ð1 pÞ11 þ 398p4 ð1 pÞ10 þ 1298p5 ð1 pÞ9 þ 2575p6 ð1 pÞ8 þ 3288p7 ð1 pÞ7 þ 2977p8 ð1 pÞ6 þ 2000p9 ð1 pÞ5 þ 1001p10 ð1 pÞ4 þ 364p11 ð1 pÞ3 þ 91p12 ð1 pÞ2 þ 14p13 ð1 pÞ þ p14 ; Unit cell 2 5: Yðp; 2; 5Þ ¼ 6p2 ð1 pÞ15 þ 100p3 ð1 pÞ14 þ 743p4 ð1 pÞ13 þ 3225p5 ð1 pÞ12 þ 9036p6 ð1 pÞ11 þ 17220p7 ð1 pÞ10 þ 23402p8 ð1 pÞ9 þ 24084p9 ð1 pÞ8 þ 19416p10 ð1 pÞ7 þ 12374p11 ð1 pÞ6 þ 6188p12 ð1 pÞ5 þ 2380p13 ð1 pÞ4 þ 680p14 ð1 pÞ3 þ 136p15 ð1 pÞ2 þ 17p16 ð1 pÞ þ p17 ; Unit cell 3 1: Yðp; 3; 1Þ ¼ 2p3 ð1 pÞ5 þ 14p4 ð1 pÞ4 þ 34p5 ð1 pÞ3 þ 25p6 ð1 pÞ2 þ 8p7 ð1 pÞ þ p8 ; Unit cell 3 2: Yðp; 3; 2Þ ¼ 3p3 ð1 pÞ10 þ3 8p4 ð1 pÞ9 þ 209p5 ð1 pÞ8 þ 627p6 ð1 pÞ7 þ 1089p7 ð1 pÞ6 þ 1078p8 ð1 pÞ5 þ 677p9 ð1 pÞ4 þ 283p10 ð1 pÞ3 þ 78p11 ð1 pÞ2 þ 13p12 ð1 pÞ þ p13 ; Unit cell 3 3: Yðp; 3; 3Þ ¼ 4p3 ð1 pÞ15 þ 72p4 ð1 pÞ14 þ 594p5 ð1 pÞ13 þ 2936p6 ð1 pÞ12 þ 9582p7 ð1 pÞ11 þ 21470p8 ð1 pÞ10 þ 33494p9 ð1 pÞ9 þ 36774p10 ð1 pÞ8 þ 29642p11 ð1 pÞ7 þ 18119p12 ð1 pÞ6 þ 8514p13 ð1 pÞ5 þ 3057p14 ð1 pÞ4 þ 816p15 ð1 pÞ3 þ 153p16 ð1 pÞ2 þ 18p17 ð1 pÞ þ p18 ;

physical properties of fractal structures

159

Unit cell 3 4: Yðp;3;4Þ ¼ 5p3 ð1 pÞ20 þ 116p4 ð1 pÞ19 þ 1264p5 ð1 pÞ18 þ 8544p6 ð1 pÞ17 þ 39915p7 ð1 pÞ16 þ 135919p8 ð1 pÞ15 þ 346869p9 ð1 pÞ14 þ 672995p10 ð1 pÞ13 þ 1001865p11 ð1 pÞ12 þ 1158842p12 ð1 pÞ11 þ 1064880p13 ð1 pÞ10 þ 793300p14 ð1 pÞ9 þ 485136p15 ð1 pÞ8 þ 244390p16 ð1 pÞ7 þ 100877p17 ð1 pÞ6 þ 33646p18 ð1 pÞ5 þ 8855p19 ð1 pÞ4 þ 1771p20 ð1 pÞ3 þ 253p21 ð1 pÞ2 þ 23p22 ð1 pÞ þ p23 ; Unit cell 4 1: Yðp; 4; 1Þ ¼ 2p4 ð1 pÞ7 þ 20p5 ð1 pÞ6 þ 80p6 ð1 pÞ5 þ 152p7 ð1 pÞ4 þ 123p8 ð1 pÞ3 þ 51p9 ð1 pÞ2 þ 11p10 ð1 pÞ þ p11 ; Unit cell 4 2: Yðp; 4; 2Þ ¼ 3p4 ð1 pÞ14 þ 54p5 ð1 pÞ13 þ 445p6 ð1 pÞ12 þ 2182p7 ð1 pÞ11 þ 6984p8 ð1 pÞ10 þ 15126p9 ð1 pÞ9 þ 22288p10 ð1 pÞ8 þ 22242p11 ð1 pÞ7 þ 15628p12 ð1 pÞ6 þ 7974p13 ð1 pÞ5 þ 2988p14 ð1 pÞ4 þ 812p15 ð1 pÞ3 þ 153p16 ð1 pÞ2 þ 18p17 ð1 pÞ þ p18 ; Unit cell 4 3: Yðp; 4; 3Þ ¼ 4p4 ð1 pÞ21 þ 102p5 ð1 pÞ20 þ 1230p6 ð1 pÞ19 þ 9272p7 ð1 pÞ18 þ 48718p8 ð1 pÞ17 þ 188512p9 ð1 pÞ16 þ 553496p10 ð1 pÞ15 þ 1252416p11 ð1 pÞ14 þ 2198498p12 ð1 pÞ13 þ 3001802p13 ð1 pÞ12 þ 3204984p14 ð1 pÞ11 þ 2715264p15 ð1 pÞ10 þ 1854463p16 ð1 pÞ9 þ 1032857p17 ð1 pÞ8 þ 471428p18 ð1 pÞ7 þ 175870p19 ð1 pÞ6 þ 53028p20 ð1 pÞ5 þ 12646p21 ð1 pÞ4 þ 2300p22 ð1 pÞ3 þ 300p23 ð1 pÞ2 þ 25p24 ð1 pÞ þ p25 ;

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Unit cell 4 4: Yðp; 4; 4Þ ¼ 5p4 ð1 pÞ2 8 þ 164p5 ð1 pÞ2 7 þ 2582p6 ð1 pÞ2 6 þ 25910p7 ð1 pÞ25 þ 185667p8 ð1 pÞ24 þ 1009026p9 ð1 pÞ23 þ 4311522p10 ð1 pÞ22 þ 14818844p11 ð1 pÞ21 þ 41566143p12 ð1 pÞ20 þ 95995718p13 ð1 pÞ19 þ 183464428p14 ð1 pÞ18 þ 291036648p15 ð1 pÞ17 þ 384352578ð1 pÞ16 þ 424714914p17 ð1 pÞ15 þ 395869210p18 ð1 pÞ14 þ 314074078p19 ð1 pÞ13 þ 213777310p20 ð1 pÞ12 þ 125503512p21 ð1 pÞ11 þ 63685924p22 ð1 pÞ10 þ 27896896p23 ð1 pÞ9 þ 10497184p24 ð1 pÞ8 þ3363764p25 ð1 pÞ7 þ906060p26 ð1 pÞ6 þ 201372p27 ð1 pÞ5 þ 35960p28 ð1 pÞ4 þ 4960p29 ð1 pÞ3 þ 496p30 ð1 pÞ2 þ 32p31 ð1 pÞ þ p32

IV.

PHYSICAL PROPERTIES A.

Conductivity

Theoretical investigations of the dielectric properties of inhomogeneous media stem from works published as far back as the 1870s–1930s [82–84]. Based on these investigations, the concept of an effective medium was developed [85]. It is in essence the replacement of an inhomogeneous medium consisting of two composites with conductivities s1 and s2 by a continuous medium with an effective conductivity s. Note that such an approximation is only applicable when the wavelength of the electromagnetic wave interacting with the medium is much greater than the sizes of the inhomogeneities and the spacings between them. The effective-medium method has been sufﬁciently widely used for the description of the physical properties of inhomogeneous media [85}; however, it does not permit one to predict the behavior of the system at the metal-insulator transition near the percolation threshold [1–4]. An exact and complete description of the effective properties of a composite may be obtained if the detailed distribution of physical ﬁelds in all components of the composite is known—for example, based on the concept of a quasi-homogeneous medium [49]. The problem of deﬁning the distribution of physical ﬁelds in components of a composite with chaotic structure is rather

physical properties of fractal structures

161

difﬁcult. Hence certain assumptions are made when deﬁning the effective properties of the composite. Now we shall brieﬂy describe the main results of deﬁning the effective conductivity of the composite. The effective conductivity s and effective resistivity r are deﬁned according to the formulae hji ¼ shEi; hEi ¼ rhji

ð223Þ

where sr ¼ 1 and hji; hEi are the ﬂow current density and the corresponding electric ﬁeld, averaged over the volume V: ððð hji ¼ V 1 jðrÞ dV; hEi ¼ V 1

V ððð

EðrÞ dV

ð224Þ

V

we have locally in a composite jðrÞ ¼ sðrÞEðrÞ; EðrÞ ¼ rðrÞjðrÞ

ð225Þ

where jðrÞ, EðrÞ, sðrÞ and rðrÞ are functions of space. The constitutive relations equations, Eqs. (223–225), may also be applied to other properties–for example, dielectric permittivity and magnetic permeability. Using Eq. (223–225), the effective conductivity s of a two-component composite can be obtained via s ¼ s1 pA1 þ s2 ð1 pÞA2 ; pA1 þ ð1 pÞA2 ¼ 1

ð226Þ

where Ai ði ¼ 1; 2Þ is deﬁned by hEi ðrÞi ¼ Ai hEðrÞi; ððð 1 EðrÞ=dV hEi ðrÞi ¼ Vi

ð227Þ

Vi

where Vi is the volume occupied by the ith component (i ¼ 1; 2), and p is the volume concentration of component 1.

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162

A similar procedure can be used to deﬁne r, namely, r ¼ r1 pB1 þ r2 ð1 pÞB2 ; pB1 þ ð1 pÞB2 ¼ 1

ð228Þ

where Bi is deﬁned as

hji i ¼ Vi1

ððð

hji i ¼ Bi hji; jðrÞ dV ;

i ¼ 1; 2

ð229Þ

Vi

Unfortunately, Eq. (226) and (228) cannot be used for direct calculations of s and r because the number of unknowns—three (s, A1 , A2 in the ﬁrst case and r, B1 , B2 in the second case)—is more than the number of equations available (two in each case). Thus, it is necessary to have some additional information about the structure of the composite. We consider the simplest structure—that is, parallel layers. When the ﬂow hji is directed along the layer, we have hE1 i ¼ hE2 i ¼ hEi

ð230Þ

Therefore A1 ¼ A2 ¼ 1. From Eq. (226) we obtain sjj ¼ s1 p þ s2 ð1 pÞ

ð231Þ

where sjj is the conductivity parallel to the layers. If the ﬂow hji is directed perpendicular to the layers, that is, hj1 i ¼ hj2 i ¼ hji

ð232Þ

B1 ¼ B2 ¼ 1

ð233Þ

we have

Moreover, from Eq. (228) we obtain r? ¼ r1 p þ r2 ð1 pÞ

ð234Þ

and ﬁnally, the conductivity perpendicular to the layers s? s? ¼ ðr? Þ1

ð235Þ

physical properties of fractal structures is

p ð1 pÞ þ s? ¼ s1 s2

163

1 ð236Þ

The effective conductivity of a composite s with random structure lies between the limits s? s sjj [87]. Subtracting Eq. (231) from Eq. (236) we obtain sjj s? ¼

ðs1 s2 Þ2 pð1 pÞ s2 p þ s1 ð1 pÞ

ð237Þ

Generalizing the above result, we have s ¼ hsi K

ðs1 s2 Þ2 pð1 pÞ s2 p þ s1 ð1 pÞ

ð238Þ

where 0 K 1 is the structurally dependent coefﬁcient, and hsi ¼ sjj . Thus, a knowledge of the structural characteristics of a composite must constitute the ﬁrst step in the theoretical analysis of its effective conductivity. 1.

Maxwell Model

One of the ﬁrst models proposed in order to calculate the generalized conductivity of a composite was the Maxwell model, namely a spherical insertion (component 1) in a continuous matrix (component 2). Thus the following equation was obtained [82]: s ¼ s2 þ 2.

ps2 ðs1 s2 Þ ðs2 þ ð1 pÞðs1 s2 Þ=3Þ

ð239Þ

The Effective Medium Theory

The effective medium model has been described in [84,86]. It constitutes an isolated spherical insertion (component 1) in a continuous medium with effective (to be determined) properties. Thus, the following formula was obtained: 1 1 2 2 s0 ¼ s1 f ð x; pÞ; f ð x; pÞ ¼ a þ a þ x ; 2

1 3 1 1 p ð1 x Þ þ x ð240Þ a¼ 2 2 2 2 where x ¼ s2 =s1 .

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164 3.

Variational Approach

This method is appropriate in order to estimate the upper and lower limits of the effective conductivity of a composite. The initial estimations were [87] hs1 i1 s hsi

ð241Þ

Later they were improved [88]: hsi

pð1 pÞðs1 s2 Þ2 pð1 pÞðs1 s2 Þ2 s hsi ps2 þ ð1 pÞs1 þ s1 ps2 þ ð1 pÞs1 þ s2

ð242Þ

The formulae obtained, Eqs. (239)–(242), conform quite well to the experimental data of effective conductivity s if the conductivities of the components (s2 ; s1 ) of the composite are not more than two orders of magnitude differs (102 < s2 =s1 < 1) or at low concentrations of one of the components (e.g., p 1). If the ratio of the properties of the system is s2 =s1 < 102 , then the result yielded by Eqs. (239)–(242) and the experimental data [68,69] are essentially different. If the ratio of the properties of the system is s2 =s1 ! 0, then the results of the theory of percolation can be used to predict the effective conductivity of a composite [1–4] (Section III, A). If the properties of the system are within the range 0 < s2 =s1 < 102 , so far no suitable theory exists which can predict the effective conductivity of a heterogeneous medium. As shown below, an attempt is made to solve this problem using the ideas of the renormalization group transformation method and the theory of fractals, which is also called the geometry of chaos. 4.

Iterative Averaging Method for Conductivity

We consider inhomogeneous media with chaotic structure (Fig 26a). On Fig. 26b an illustration of how such inhomogeneous media may be split into hierarchical levels is presented. A lattice with a random distribution of parameters was chosen as an appropriate model of the chaotic structure of an inhomogeneous material Spatial microinhomogeneities (i.e., system components) were modelled by the lattice junctions, and the interjunction bonds simulated their contacts with neighbors (Fig. 27). Thus, in view of the dominant contribution of contact conditions between the components of the macroscopic properties of an medium, the general problem was again reduced to a problem of bonds. The main ensemble of bonds was derived by an iteration process in which the initial step (k ¼ 0) involved

physical properties of fractal structures (a)

165

(b) 1st step

2nd step

nth step

Figure 26. (a) A disordered cluster of particles. (b) Illustration of splitting of inhomogeneous media into hierarchical levels.

(a)

(b)

(c)

Figure 27.

A lattice junctions, with interjunction bonds simulating their contacts with neighbors.

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166

(a) 1 0.8

1

3 2

Y(p)

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

p (b) 2.5

3

f(p)

2

2 1

1.5

1 1

3

0.5 0 0

0.2

0.4

0.6

0.8

1

p Figure 28.

(a) Probability function Yk ðpÞ; (b) Derivative f ðpÞ ¼ dYðpÞ=dp.

treatment of a ﬁnite lattice in a space of dimension d ¼ 2 or d ¼ 3 and a probability p0 for a bond between neighboring lattice junctions to be unbroken (or ‘‘colored’’ with a deﬁnite color, so that bonds of the same color were assumed to have identical properties). At the next step (k ¼ 1; 2; . . . ; n), each bond of the lattice was replaced by a lattice generated at the previous step (Fig. 12) The eventual independence of the lattice properties from the iteration number n was recognized as the termination of the iteration process. Lattices with linear dimensions l (assumed to exceed by far the correlation length) generated in this way were used to calculate the effective physical properties. It is clear that the ensemble of bonds n ðl0 , p0 Þ

physical properties of fractal structures

167

derived by the iteration process is a function of both the size of the initial lattice l and the probability p. Three probability functions YðpÞ were taken to illustrate the inﬂuence of the function YðpÞ on the calculation of the effective conductivity of a composite s. These functions YðpÞ have different values of p —that is, different thresholds of percolation. This function [62] was obtained in Y1 ðpÞ ¼ p2 ½4 þ 8p 14p2 40p3 þ 16p4 þ 288p5 655p6 þ 672p7 376p8 þ 112p9 14p10

ð243Þ

According to Y1 ðpÞ ¼ p, the percolation threshold p ¼ pc1 for Y1 ðpÞ is equal to pc1 0:20846 . . ., i.e. the non-connecting sets change into connecting sets at pc1 0:20846 The value of the derivative Y1 ðpÞ at the point p ¼ pc1 is equal to f1 ðpÞjp¼pc ¼ l1 ; l1 ﬃ 1:9582

ð244Þ

We construct the probability functions YðpÞ using rectangular lattices [79] (see Appendix, Section III.): The function Y2 ðpÞ ¼ 5p2 ð1 pÞ12 þ 68p3 ð1 pÞ11 þ 398p4 ð1 pÞ10 þ 1298p5 ð1 pÞ9 þ 2575p6 ð1 pÞ8 þ 3288p7 ð1 pÞ7 þ 2977p8 ð1 pÞ6 þ 2000p9 ð1 pÞ5 þ 1001p10 ð1 pÞ4 þ 364p11 ð1 pÞ3 þ 91p12 ð1 pÞ2 þ 14p13 ð1 pÞ þ p14

ð245Þ

has an unstable ﬁxed point, that is the percolation threshold is pc2 0:178

ð246Þ

l2 ﬃ 1:9578

ð247Þ

the derivative at pc2 is

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The function (see Appendix, Section III.) Y3 ðpÞ ¼ 6p2 ð1 pÞ15 þ 100p3 ð1 pÞ14 þ 743p4 ð1 pÞ13 þ 3225p5 ð1 pÞ12 þ 9036p6 ð1 pÞ11 þ 17220p7 ð1 pÞ10 þ 23402p8 ð1 pÞ9 þ 24084p9 ð1 pÞ8 þ 19416p10 ð1 pÞ7 6

5

þ 12374p ð1 pÞ þ 6188p ð1 pÞ þ 2380p ð1 pÞ 11

12

13

ð248Þ

4

þ 680p14 ð1 pÞ3 þ 136p15 ð1 pÞ2 þ 17p16 ð1 pÞ þ p17 has an unstable ﬁxed point; that is, the percolation threshold is pc3 ¼ 0:14850

ð249Þ

l3 ﬃ 1:9839

ð250Þ

the derivative Y3 ðpÞ at pc3 is

The dependence of the functions Yk ðpÞ and their derivatives fk ðpÞðk ¼ 1; 2; 3) on the concentration p is shown in Fig 28. Now consider a two-phase system with the distribution function ð0Þ

ð0Þ

P0 ðCÞ ¼ ð1 p0 ÞdðC C2 Þ þ p0 dðC C1 Þ

ð251Þ

where dðxÞ is the Dirac function, p0 is the probability of a given local area to ð0Þ possess the property C1 (black color), and 1 p0 is the probability to possess ð0Þ the property C2 (white color). After k steps of renormalization group transformation, the density function becomes ðkÞ

ðkÞ

Pk ðCÞ ¼ ð1 pk ÞdðC C2 Þ þ pk dðC C1 Þ

ð252Þ

Here pk ¼ Yðpk1 ; lk1 Þ is the density of the connecting set of links; it is the ratio of the number of connecting sets to the total number of scatters (colors) on a square lattice. In general, the effective properties can be deﬁned according to the following scheme: Initially the properties of different conﬁgurations are deﬁned; they are then averaged; these properties are then attributed to the next stage. Deﬁning the properties of possible conﬁgurations of a set of links leads to rather cumbersome calculations. Hence we use an approximate method, meaning that we do not calculate the properties of the conﬁgurations obtained by the scatters of links on

physical properties of fractal structures

169

Figure 29. Simulation of (a, c) a connecting set and (b, d) a nonconnecting set (schematic).

a lattice. We select two kinds of link conﬁguration sets, namely, connecting sets and nonconnecting sets. A cube inside a cube cell is used as a model of connecting sets and nonconnecting sets (Fig 29); that is, the calculations of the structure of connecting sets and nonconnecting sets are formed by the cube inside a cube

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cell at each step of the iteration process: The connecting sets constitute a continuous medium from the well-conducting phase (outer cube) with an insertion of a cube from the poorly conducting phase (inner cube); the nonconnecting sets constitute a continuous medium from the poorly conducting phase (outer cube) with an insertion of a cube from the wellconducting phase (inner cube). The conductivity of the cube inside a cube cell was deﬁned in Ref. 49. The result obtained in Ref. 49 for the cell in which the cube with conductivity s2 is in the cube with conductivity s1 can be written as s¼

s1 ðc1 þ c2 Þ 2

ð253Þ

where c1 ¼

s1 þ ðs2 s1 Þð1 pÞ2=3 h i; s1 þ ðs2 s1 Þð1 pÞ2=3 1 ð1 pÞ1=3 1=3

c2 ¼

s2 þ ðs1 s2 Þð1 pÞ

h

1 ð 1 pÞ

2=3

i

ð254Þ

s2 þ ðs1 s2 Þð1 pÞ1=3

Here p is the volume concentration of a phase of conductivity s1. sðckÞ ¼

sðck1Þ ðk1Þ ðk1Þ 1 þ 2 2

ð255Þ

where ðk1Þ 1

ðk1Þ

2

sðck1Þ þ sðnk1Þ sðck1Þ ð1 pk1 Þ2=3 h i; ¼ ðk1Þ ðk1Þ ðk1Þ sc þ sn sc ð1 pk1 Þ2=3 1 ð1 pk1 Þ1=3 h i sðck1Þ þ sðck1Þ sðnk1Þ ð1 pk1 Þ1=3 1 ð1 pk1 Þ2=3 ¼ ðk1Þ ðk1Þ ðk1Þ sn þ sc sn ð1 pk1 Þ1=3

ð256Þ

where sðc0Þ ¼ s1 ; sðn0Þ ¼ s2 , and p0 ¼ p. The magnitudes of pk were determined using the formulas (243)–(248). To determine the conductivity of the nonconnecting sets snðkÞ it is necessary c; 1! 2, and to replace the indices in Eqs. (255) and (256) as follows: n ! ð1 pÞ! p: The lower indices n and c mean that the given value corresponds to: n, nonconnecting set; c, connecting set; the upper index k shows the iteration

physical properties of fractal structures

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0

Log10(s/s1)

–1

–2

–3

–4

–5 0

0.2

0.4

0.6

0.8

p

Figure 30. Comparison of the calculation of the effective conductivity of a composite using the iterative method (continuous) and using effective medium method – effective medium theory (dotted line).

step number. The iteration procedure leads ﬁnally to the effective conductivity of the composite, s: ðkÞ lim sðkÞ c ¼ lim sn ¼ s

k!1

k!1

ð257Þ

The Calculation Results. The calculations were made for a two-component medium. Calculations were executed for a two-component 3D composite with random structure. First we shall consider a comparison of the outcome for the effective conductivity calculated by means of the iterative method with the calculation using formulas (240) obtained on the basis of the effective medium theory model. Figure 30 shows a comparison between the results for effective conductivity obtained by means of the iterative method (continuous) and a calculation using the formula (240) obtained by the effective medium theory model (dashed). The ﬁgure compares the results of the calculation of the effective conductivity using the iteration method (the continuous line) to the calculation by formula (240) (the dotted line) obtained from the effective medium model. The comparison

vitaly v. novikov

172 0

Log10(s/s1)

–1

–2

–3

–4

–5 0

0.2

0.4

0.6

0.8

p Figure 31. Comparison of the calculation of the effective conductivity of a composite based on the iterative method (continuous), on the effective medium method (dotted line) and numerical modeling (dots).

(Fig. 30) shows good agreement between the two methods if s2 =s1 > 102 . If s2 =s1 102 , the calculations differ considerably in the concentration range 0:1 < p < 0:5. The difference increase as s2 =s1 ! 0: Figure 31 compares the calculation (for s2 =s1 ¼ 105 ) according to the iteration method (the continuous line), the self-coordinated ﬁeld method [Eq. (240)] (the dotted line), and numerical modeling (the dots) [68]. In the numerical modeling, the percolation threshold is pc ¼ 0:17 0:01.The function (245), for which the threshold is pc2 ﬃ 0:178, is used for the calculation. The comparison shows good agreement between the iteration method of calculation (the continuous line) and the numerical experiment (the dots). In Fig. 32, comparison of the results of calculation using the iteration method (the continuous line) with the experimental data is shown (alkali–tungsten bronzes at 300 K, s2 =s1 ¼ 103 [68,69]). The comparison shows good agreement between them. The function Y1 ðpÞ is used (pc1 0:20846). Thus, the iteration method of calculation developed here agrees well with the calculation using self-coordinated ﬁeld method if s2 =s1 ¼ 102 and to the numerical experiment if s2 =s1 ¼ 104 ðs2 =s1 ! 0Þ. This gives us hope that the iteration method of calculation can be used to estimate the generalized conductivity of composites with chaotic structure for any values of the conductivity of phases s1 and s2 and in the entire range of concentrations 0 p 1.

physical properties of fractal structures

173

0

Log10(s/s1)

–0.5 –1 –1.5 –2 –2.5 –3 0

0.2

0.4

0.6

0.8

1

p

Figure 32. Comparison of the calculation of the effective conductivity of a composite by the iterative method (continuous) and experimental data (dots).

In Fig. 33, the comparison of the calculation of the effective conductivity s with various functions YðpÞ is given. The calculation shows that the values of s closely depend on the type of function YðpÞ (that is on the values of percolation thresholds pc ) at the concentration values 0:1 p 0:5. 0

Log10(s/s1)

–1

–2

–3

3

2

–4

1

–5 0

0.2

0.4

0.6

0.8

1

p

Figure 33. Comparison of the calculation of the effective conductivity of a composite based on the iterative method with various probability functions YðpÞ : 1 Y1 ðpÞ; 2 Y2 ðpÞ; 3 Y3 ðpÞ.

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174

Chaotic fractal sets on rectangular lattices have been used to the deﬁne the effective conductivity of the composite material. The effective conductivity of the composite material is deﬁned using the fractal random structure model of a composite and the iteration method of averaging. Comparison of the calculation with experimental data is also given. B.

Frequency Dependence of Dielectric Properties

If we assume that locally the strength of the electric ﬁeld Eðr; tÞ varies periodically with frequency o as Eðr; tÞ ¼ E0 ðr; oÞ expðiotÞ

ð258Þ

then the following constitutive relation may be written [90] jðr; oÞ ¼ s ðr; oÞEðr; oÞ

ð259Þ

where jðr; oÞ is the current density and s ðr; oÞ is the complex conductivity: s ðr; oÞ ¼ sðr; oÞ þ ioeðr; oÞ

ð260Þ

For an inhomogeneous medium with a chaotic structure, the permittivity eðr; oÞ and the conductivity sðr; oÞ are random (stochastic) functions of the coordinates r. Note that from Eq. (260) we can determine the scaling expressions for the conductivity of a lattice of resistors of ﬁnite dimensions l, ﬁrst obtained in [91,92]: s ¼ s1 xt=n Gþ s1 =s2 xðtþsÞ=n ; x=l ; p > 0; p < 0; ð261Þ s ¼ s2 xs=n G s2 =s1 xðtþsÞ=n ; x=l ; where Gþ ðx; yÞ and G ðx; yÞ are functions of two variables describing the frequency and scale dependence of the conductivity above and below the percolation threshold. In recent years, great attention has been paid to the analysis of the dependence of the properties of metal-insulator composites on frequency [91–109], which is related to the difﬁculties in describing the anomalous behavior of dielectric properties in the low-frequency limit. The nature of the anomalous behavior of the frequency dependence of the dielectric properties can be clariﬁed if we consider a model medium consisting of small spherical metallic particles described by the Drude dielectric function e1 ðoÞ ¼ 1

op1 oðo þ i=t1 Þ

ð262Þ

physical properties of fractal structures

175

embedded in a matrix with a relative permittivity equal to unity (e2 ¼ 1). In Eq. (362), op1 is the plasma frequency and t1 is the relaxation time of the metallic phase. If such a medium is subjected to an external uniform electric ﬁeld E0 [Eq. (258)], then the electric ﬁeld E1 inside a spherical particle is by quasi electrostatics E1 ¼ 3E0 =ðe1 ðoÞ þ 2Þ

ð263Þ

and so the electric pﬃﬃﬃ ﬁeld E1 inside the sphere tends to inﬁnity at frequencies close to o ’ op1 = 3: At such frequencies, the applied ﬁeld resonates with the corresponding natural mode of the small metallic particle; as a result, a strong absorption appears at this frequency; that is, the imaginary part of the effective relative permittivity of the medium strongly increases in the vicinity of the frequency pﬃﬃﬃ op = 3. For inhomogeneous media—for example, a metal–insulator composite with a chaotic structure—the behavior becomes even more complex. In the majority of numerical calculations of the anomalous frequency behavior of such composites (in particular, near the percolation threshold pc ) under the action of an alternating current, lattice (discrete) models have been used, which were studied in terms of the transfer-matrix method [91,92] combined with the Frank–Lobb algorithm [93]. Numerical calculations and the theoretical analysis of the properties of composites performed in Refs. 91–109 have allowed signiﬁcant progress in the understanding of this phenomenon; however, the dielectric properties of composites with fractal structures virtually have not been considered in the literature. 1.

Iterative Averaging Method for Dielectric Properties

Each kth bond in the set of bonds n ðl0 ; p0 Þ (obtained using the iteration procedure) possesses an impedance Zk ðoÞ which consists of an ohmic resistance Rk , an inductance Lk , and a capacitance Ck in parallel so that Zk1 ðoÞ ¼ ðRk þ ioLk Þ1 þioCk

ð264Þ

In what follows, each bond will be characterized by the complex conductivity sk .noting, that the equality sk ¼ Zk1 ðoÞ must be satisﬁed.

ð265Þ

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176

Consider a two-phase system with a distribution function ð0Þ ð0Þ þ p0 d s s1 P0 ðs Þ ¼ ð1 p0 Þd s s2

ð266Þ

where dð xÞ is the Dirac delta function, p0 is the probability that a given local ð0Þ region possesses the property s1 ¼ s1, and ð1 p0 Þ is the probability that this ð0Þ region possesses the property s2 ¼ s2. After k iteration steps, the density function becomes ðkÞ ðkÞ þ pk d s s1 ð267Þ Pk ðs Þ ¼ ð1 pk Þd s s2 In what follows, we will again distinguish two types of sets of bond conﬁgurations: connecting sets and nonconnecting sets. To determine the dielectric properties of the connecting sets and nonconnecting sets, we used a cell of the cube-in-cube type (Fig. 29c, d); that is, at each step of the iteration process of the calculation of the properties, the structures of the connecting sets and nonconnecting sets were simulated by a cube-in-cube cell as follows: A connecting set comprises a continuous body of a well-conducting phase including a cube of a poorly conducting phase (Fig. 29c); a nonconnecting set comprises, a continuous body of a poorly conducting phase including a cube of a well-conducting phase (Fig. 29d). According to Eqs. (259) and (260), the effective characteristics of a medium in the quasi-stationary approximation differ from the static case only in the replacement of the conductivity s (dc conductivity) by the complex conductivity s Noting Eqs. (255) and (256), the complex conductivity of connecting sets at the kth step of the calculations was determined using the formulas sc ðkÞ ¼

sc ðk1Þ ðk1Þ ðk1Þ 1 þ 2 2

ð268Þ

where

ðk1Þ 1

ðk1Þ

2

sc ðk1Þ þ snðk1Þ sc ðk1Þ ð1 pk1 Þ2=3 h i; ¼ ðk1Þ ðk1Þ ðk1Þ sc þ sn sc ð1 pk1 Þ2=3 1 ð1 pk1 Þ1=3 h i sc ðk1Þ þ sc ðk1Þ snðk1Þ ð1 pk1 Þ1=3 1 ð1 pk1 Þ2=3 ¼ ðk1Þ ðk1Þ ðk1Þ sn þ sc sn ð1 pk1 Þ1=3 ð269Þ

physical properties of fractal structures

177

where sc ð0Þ ¼ s1 ; snð0Þ ¼ s2 , and p0 ¼ p. The magnitudes of pk were determined using formula (243). In Eqs. (268) and (269), the subscripts n and c denote that a given quantity refers to the nonconnecting set and connecting set, respectively, and the index k indicates the order number of the iteration step. To determine the complex conductivity of an nonconnecting set snðkÞ it is necessary to replace the indices in Eqs. (268) and (269) as follows: c; 1! 2, and ð1 pÞ! p: n ! Calculation Results. The calculations were performed for a two-phase (twocomponent) medium and the probability function, YðpÞ [Eq. (243)], was used in the calculations. Calculations of the dielectric properties of inhomogeneous media at various frequencies and concentrations of the phases using Eqs. (243), (268), and (269) then show that the iteration process converges; that is, lim sc ðkÞ ¼ lim snðkÞ ¼ s

k!1

k!1

ð270Þ

However, the complex local conductivity for the metallic phase with Drude dielectric function (262) was determined as 1 s1 ðoÞ ¼ s1 þ io e1 2 ð271Þ x þ g2 where x ¼ o=op ; g ¼ 1=op t1

ð272Þ

The complex local conductivity of the insulating phase was determined as s2 ðoÞ ¼ s2 þ ioe2

ð273Þ

It was assumed in the calculations that 1 ; op t1 ¼ 30; 30 s2 =s1 ¼ 102 ; t1 ¼ 1; 0:001 o=op 1:5 e1 ¼ 1; e2 ¼ 10; g ¼

ð274Þ

Figures 34 and 35 show the dependence of the effective dielectric constant e ¼ Imðs Þ=o and the effective conductivity s ¼ Reðs Þ on the concentration of the metallic phase p and the relative frequency o=op. The zeros of the effective dielectric constant e determine the plasma frequencies of the system— that is, the metal–insulator transition.

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178

It follows from the calculations (Figs. 34 and 35) that at low frequencies, a divergence arises in the effective dielectric constant and in the effective conductivity (a sharp increase in losses). This is explained by the fact that ﬁnite clusters of the metallic phase now arise in the system which are separated by thin insulating interlayers. Such structures form a hierarchical self-similar chaotic capacitance net that generates a system of resonance frequencies. (a)

0 –1000 e /e2

0.01

–2000

0.008

–3000 0.006 0 0.2

0.004

w /w p

0.4 p

0.6

0.002

0.8 1

(b)

e /e2

0 –50 –100 –150 –200

0.1 0.08 0.06 w /w p 0.04

0 0.2 0.4 p

0.6

0.02 0.8 1

Figure 34. Variation of the dielectric constant e=e2 of a metal–insulator composite as a function of the concentration of the metallic phase p and the frequency o=op.

physical properties of fractal structures

179

(c)

e /e2

2 1 0 –1 –2

0.5 0.4 0.3

0 0.2 0.4

w /w p

0.2 p

0.6 0.8 1

0.1

(d)

e /e2

1 0.8 0.6 0.4 0.2 0

1.4 1.2 0

1 0.2 0.8

0.4 p

w /w p

0.6 0.6

0.8 1

Figure 34.

(Continued ).

In addition, the frequency dependence of the effective properties is affected by the conﬁgurations of the ﬁnite clusters [110]. This may be illustrated by considering a pair of inclusions that have the form of a circle with the associated set of discrete frequencies: o21m ¼ o2p tanhðmx0 Þ; o22m ¼ o2p cothðmx0 Þ

ð275Þ

vitaly v. novikov

180 where

m ¼ 1; 2; . . . ;

r þ ðr2 4RÞ1=2 x0 ¼ ln 2R

ð276Þ

r is the spacing between the centers of the circles, and R is the radius of the circles. Thus, if such regions are formed in a composite, they create resonant circuits.

(a)

log10s

1 0 –1 –2

0.01 0.008 0.006

0 0.2

w /w p

0.004

0.4 p 0.6

0.002

0.8 1 (b)

2 1 log10s 0 –1 –2

0.1 0.08 0.06

0 0.2

0.04

0.4 p 0.6

w /w p

0.02

0.8 1

Figure 35.

Variation of the logarithm of the conductivity of a metal–insulator composite as a function of the concentration of the metallic phase p and the frequency o=op.

physical properties of fractal structures

181

(c)

2 1 log10s 0 –1 –2

0.5 0.4 0.3

0

w /w p

0.2 0.2

0.4 p

0.6 0.8 1

0.1

(d)

2 1 log10s 0 –1 –2

1. 4 1. 2 1

0 0. 2

w /w p

0. 8

0. 4 p

0. 6

0. 6

0. 8 1

Figure 35.

(Continued ).

It also follows from Eq. (275) that, at r ! 2R, we have x0 ! 0 and the frequencies (275) form a quasi-continuous spectrum [110]. It was shown in Refs. 91 and 92 that ring-shaped structures (ring clusters) generate double peaks in the frequency dependence of the conductivity. Such ring structures in the system at hand are located chaotically and hierarchically, in a self-similar way, and also lead to peaks in the conductivity. Figure 36 shows the dependence of the modulus of the ratio of the capacitance conductivity to the active conductivity h ¼ jeoj=s on p and o=op . Calculations show that the displacement current in the low-frequency region ðo=op < 1Þ behaves nonmonotonically. In the high-frequency range

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182

30 0.1

20 |∈ω |/s

10

0.08

0 0

0.06 0.2

0.04

0.4 p

0.6

w /w p

0.02

0.8 1

10 7.5 |∈ω |/s 5 2.5 0 0

0.5 0.4 0.3 0.2 0.4

w /w p

0.2 p

0.6 0.8 1

0.1

Figure 36. Variation of the ratio of the capacitance conductivity to the active conductivity h ¼ jeo=sj of a metal–insulator composite as a function of the concentration of the metallic phase p and the frequency o=op.

ðo=op > 1Þ and at concentrations of the metallic phase below the percolation threshold ðp < pc Þ, the displacement current exceeds the current through the active conductors ðh 1Þ and the surface dielectric properties become smooth. For p > pc, the current through the active conductors exceeds the displacement current ðh 1Þ.

physical properties of fractal structures

183

Now, we discuss one of the possible applications of the above model concepts to the dielectric properties of fractal systems. The optical properties of colloidal systems have not yet been explained in terms of the classical theory (e.g., in terms of the Mie theory [111–113]). In this theory, the change in the color of a solution was assumed to be due to the appearance of metallic (silver) particles of various sizes in the solution; the change in the color was attributed to the dependence of the resonance (plasma) frequency on the particle radius. However, experimental investigations show that the frequency-dependent behavior of colloidal solutions does not correlate with the statistical particle-size distribution function; that is, the role of the particle size seems to be insigniﬁcant [114]. The appearance of a longwavelength wing in the spectrum of the colloidal solution can be explained by the aggregation of particles into fractal structures. Now, a small silver particle has a frequency of plasma vibrations with a wavelength l ¼ 2pc=op ¼ 140 nm. To explain the presence of a peak at 650 nm, the classical (Lorentz) theory [111–113] requires the presence in the colloid solution of silver with a volume concentration of p ’ 0:86 (Fig. 37a), whereas the experiment yields p values that are much smaller [115], which agrees with our calculations (Fig. 37b). Thus, the shift of the peak in colloidal solutions toward the region of smaller concentrations of metal can be explained by the formation of fractal structures in these solutions. We mention some other systems that have fractal structures. For example, using sputtering regimes that correspond to the diffusional aggregation model [82], thin ﬁlms consisting of metallic fractal clusters can be obtained. Fractal structures are also characteristic of percolation clusters near the percolation threshold, as well as certain binary solutions and polymer solutions. The dielectric properties of all these systems can be predicted using the above fractal model. Conclusion. Calculations of the dependence of the conductivity and the relative permittivity of chaotic hierarchical self-similar structures of composites were performed using a fractal model in the entire range of concentrations of inhomogeneities at various frequencies of an external ﬁeld. The metal-insulator transition was shown to occur not only near the percolation threshold. It was also shown that the transition depends on the concentration of the metallic phase and the frequency of the external ﬁeld.

C.

Galvanomagnetic Properties

There has been a number of attempts [116,117] to analyze galvanomagnetic b was properties of inhomogeneous media. The effective conductivity tensor s

184

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Figure 37. Variation of the dielectric constant e=e2 of a metal–insulator colloid solution as a function of the metallic phase (silver) p: (a) calculation based on the Lorentz model; (b) calculation based on the fractal model.

introduced to evaluate the effective galvanomagnetic properties: b hEi hji ¼ s s11 s12 0 b ¼ s12 s22 0 s 0 0 s33

ð277Þ ð278Þ

physical properties of fractal structures

185

The angular brackets again mean an average over a volume V: hj i ¼

ð 1 jðrÞd3 r; V

hEi ¼

ð 1 EðrÞd3 r V

ð279Þ

where jðrÞ and EðrÞ are random functions of the coordinates. Suppose that Ohm’s law is locally satisﬁed so that b ðrÞEðrÞ j ðrÞ ¼ s s11 ðrÞ b ðrÞ ¼ s12 ðrÞ s 0

s12 ðrÞ s22 ðrÞ 0

ð280Þ 0 s33 ðrÞ 0

ð281Þ

b ðrÞ is expressed as follows: The conductivity tensor s b ij ðrÞ ¼ s b sij ðrÞ þ s b aij ðrÞ; s

ð282Þ

b sij ðrÞ is the symmetric part of the tensor (3.59) ðb b sji ðrÞÞ and where s ssij ðrÞ ¼ s b aij ðrÞ is the antisymmetric part of the tensor (3.59). ðb s saij ðrÞ ¼ b saji ðrÞÞ. In ﬁeld notation Ohm’s law in such a medium is: j þ j bðrÞ ¼ s 0 ðrÞE

ð283Þ

where bðrÞ ¼ bðr Þn - Hall parameter which is directed along the magnetic ﬁeld H ¼ Hn. The relations between the Hall coefﬁcient R, the Hall parameter b, the mobility m and the density n of carriers with charge e is R¼

b m 1 ¼ ¼ sH s en

ð284Þ

b ðrÞ according to Eq. (280) and (283) will then The conductivity tensor s read 1 bðrÞ 0 s0 ðrÞ b ðr Þ ¼ ð285Þ s b ð r Þ 1 0 2 1 þ b ðrÞ 0 0 1 þ b2 ðrÞ This tensor satisfactorily describes the conductivity of noncompensated metals and semiconductors.

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186

The ﬁrst successful attempt to get an exact solution for the effective Hall properties of a composite was made in [118,119]. Various authors acting independently of each other have given a computational method for the effective Hall properties for two-dimensional (2D) two-phase systems with statistically equivalent and isotropic allocations of the ﬁrst and second phase ðp1 ¼ p2 ¼ 0:5Þ. It was supposed, that each of the phases is characterized by two parameters: the ohmic conductivity s0 ðrÞ and the Hall factor bðrÞ. However each of properties s0 ðrÞ and bðrÞ from the conductivity tensor (285) admit of only two values: s0 ¼ s1 and b ¼ b1 in the ﬁrst phase, s0 ¼ s2 and b ¼ b2 in the second phase. The essence of ideas described in [118,119] consists in linear transformations from the old ﬁelds ðj; EÞ to new ﬁelds ðj0 ; E0 Þ such that the macroscopic properties of the new system are equivalent to those of the original system. These transformations can be applied only to a two-dimensional system, since they do not then change the laws governing a direct current: j ¼ bn E; 0

0 00

0

E ¼ dn j 00

j ¼aj þbnE ;

0

0

00

0

E ¼cE þdnj

00

ð286Þ

The transformations (286) allow one [118,119] to calculate the effective galvanomagnetic properties of a 2D inhomogeneous medium when conductivity ﬂuctuates only, and the Hall factors of the components are equal; that is, s1 6¼ s2 ; b1 ¼ b2 . If we apply complementarity—that is, in the ﬁrst phase 0 0 ðs1 ; b1 Þ we have s ¼ s2 ; b ¼ b2 , and in the second phase ðs2 ; b2 Þ 0 0 we have s ¼ s1 ; b ¼ b1 —then we shall obtain the following results for the effective Hall properties:

12 1 1 2 þ b1 h s i 1 s ¼ hsi hsi ; s s

12 1 1 2 b ¼ b1 hsi þ b1 hsi 1 s s

ð287Þ

Here the symbol hi again means volume average. The transformations (286) also allow one [118,119] to obtain results for yet another case, namely only the Hall parameter ﬂuctuates; however, the conductivities of the components are equal, that is, s1 ¼ s2 ;

b1 6¼ b2

ð288Þ

In Ref. 120 the solution for the more general case ðs1 6¼ s2 ; b1 6¼ b2 Þ has been obtained. It was given by inserting the additional coefﬁcients into the

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187

transformation (286): 0

0

j ¼ aj þ bn E ;

0

0

E ¼ cE þ dn j

ð289Þ

This transformation then allows one [120] to obtain the following result for the effective Hall properties: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s1 s2 s¼ ; 1 þ ðs1 b2 s2 b1 Þ2 = s21 þ s22 b¼s

ð290Þ

b1 þ b2 s1 þ s2

In the absence of a magnetic ﬁeld ðb1 ¼ b2 ¼ 0Þ, both Eq. (287) and Eq. (290) lead to the classical result s¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s1 s2

ð291Þ

Numerical computations for a 2D composite were made in [123]. We remark, that a novel perturbation analysis was proposed in [124]. Moreover, a composite with a columnar microstructure was studied in [125]. In Ref. 67 the behavior of Hall coefﬁcient near the percolation threshold pc in a composite containing a dielectric and metal phase was examined for b1 ¼ b2 ;

R1 s2 ¼

1 R2 s1

ð292Þ

Two concentration value regions were considered: before and after the percolation threshold. On the right of the percolation threshold ð p pc 1Þ the behavior of the Hall coefﬁcient had the following power dependence: Rð pÞ ¼ R1 ð p pc Þs1 ;

p > pc

ð293Þ

where s1 is a critical exponent of the Hall coefﬁcient. s1 ¼ 0 for a two-dimensional medium ðd ¼ 2Þ and s1 ¼ 0:9 for three-dimensions ðd ¼ 3Þ. On the left of the percolation threshold ð1 pc p Þ the behavior of Hall coefﬁcient indicated another type of power dependence: Rð pÞ ¼ R2 ðpc pÞt1 ;

pc > p

ð294Þ

where t1 is a critical exponent for the Hall coefﬁcient, which equals t1 ’ 1:1; d ¼ 2; t1 ’ 1:62; d ¼ 3 [67].

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Reference 70 provides the ﬁrst quantitative test of the random resistor network model. In Ref. 121 the authors employed the random resistor network model to determine the behavior of the low-ﬁeld Hall effect in a 3D ‘‘metal–nonmetal’’ composite near the percolation threshold. For the following power laws of effective values of ohmic conductivity s, Hall coefﬁcient R, and Hall conductivity s12, Bergman et al. 121 have obtained the critical exponents: s=s1 ð p pc Þt ; t ¼ 1:64 0:04;

R=R1 ð p p c Þ t 1 ; t1 ¼ 0:29 0:05;

. ð1Þ s12 s12 ð p pc Þt2 ; t2 ¼ 3:0 0:1

p > pc ; ð295Þ

where t2 is the critical exponent for the Hall conductivity s12. Furthermore, the following scaling assumptions have been made [122]: ð1Þ

s12 s12

ð2Þ s12

ð1Þ s12

s2

1; s1

s s1 ¼ jp pc jt Fs ðZ Þ; s2 s1

¼ jp pc jt2 Fs12 ðZ Þ; jp pc j 1;

Z¼

ð296Þ

s2 =s1 j p pc jtþs

where Fs12 and Fs are scaling functions with scaling argument Z. A similar approach was applied in Ref. 123 to describe the behavior of three-constituent ‘‘metal–insulator–superconductor’’ composite. Webman and Jortner [68] used the following formulas obtained by combining the formulas of effective medium theory with the formulas of percolation theory to calculate the effective Hall properties of a composite: s0 =s1 ¼ ð1 p=pc Þ1 ; R=R1 ¼ ð y=xÞð1 p=pc Þ2 þ ð3 p=pc Þ2 ð1 xyÞp; m=m1 ¼ yð1 p=pc Þ2 þ xð3 p=pc Þ2 ð1 p=pc Þ1 ð1 xyÞp

ð297Þ ð298Þ

Here the concentration pc is the percolation probability pc ¼ 1=3; m is Hall mobility; x ¼ s2 =s1 is the conductivities ratio; y ¼ m2 =m1 is the mobilities ratio of components. These formulas describe the behavior of effective properties satisfactorily only in the range of concentration 0:4 < p < 1. Pellegrini and Barthelemy [128] studied effective medium theory approximations for linear composite media by means of a path integral formalism. They obtained the following values of the conductivity critical exponents: s ¼ 0; t ¼ 2 in any spatial dimension d 2. Perturbation theory for a 3D composite has been described in Refs. 116, 127, and 129.

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189

From the above brief review it follows, that an adequate theoretical model of galvanomagnetic properties of composites with random structure still does not exist for the three-dimensional case. We are not aware of any papers where the inﬂuence of fractal and random structure of a 3D composite on the effective galvanomagnetic properties has been analyzed. In the following section, the detailed analysis of effective Hall properties of a 3D two-component composite will be carried out based on the fractal structure model and the iterative averaging method. 1.

Iterative Averaging Method for Hall’s Coefﬁcient

Two types of connection set are possible in a lattice, namely, a connecting set that is able to connect via black bonds two opposite sides of a lattice and a nonconnecting set that is unable to connect opposite sides. The probability Y ð pÞ of connecting set formation was calculated as a ratio of connecting set number to the number of all possible conﬁgurations. The Probability function Y ð pÞ for a three-dimensional rectangular lattice d ¼ 3; l ¼ 2 was calculated by a method similar to that of Ref. 62 and Y ð pÞ of Eq. (243). Each kth bond from the fractal random set f ðl; pÞ possesses Hall properties ðsk ; bk Þ namely ohmic conductivity and a Hall parameter. Let us consider a two-phase 3D system with the distribution function similar to [62] ð0Þ ð0Þ pðsÞ ¼ ð1 pÞd s s2 þ p d s s1 ; ð0Þ ð0Þ pðbÞ ¼ ð1 pÞd b b2 þ p d b b1

ð299Þ

where dð xÞ is the Dirac delta function; p — probability that the actual local ð0Þ ð0Þ region possesses the following Hall properties: s1 ¼ s1 ; b1 ¼ b1 ; ð1 pÞ — ð0Þ probability that the actual local region possesses other Hall properties: s2 ¼ ð0Þ s2 ; b2 ¼ b2 . After k iterative steps the distribution function will be pðsÞ ¼ p d s sðkÞ ; pðbÞ ¼ p d b bðkÞ

ð300Þ

and in the limit of large k we have the desired effective values: lim sðckÞ ¼ lim sðnkÞ ¼ s;

k!1

k!1

lim bðckÞ ¼ lim bðnkÞ ¼ b

k!1

k!1

ð301Þ

vitaly v. novikov

190 2.

Results and Discussion

For the purpose of the calculation of the effective Hall properties of 3D composite the model of a cube inside a cube (see Appendix) was applied. At each step of the iterative process evaluation of the Hall properties of the connecting and nonconnecting set structure was carried out based on a rudimentary cell of a cube inside a cube. A continuous array from the well-conducting (black) phase with a cube from poorly conducting (white) phase inserted forms the connecting set (Fig. 29c), and a continuous array from the poorly conducting (white) phase with a cube from the well-conducting (black) phase inserted forms the nonconnecting set (Fig. 29d). Hence, at the kth iterative step, if lk < x (x is the correlation length) the composite will have a self-similar random structure consisting of unit cells comprised of a cube inside a cube. The analysis of effective conductivity for a 3D composite with a random structure becomes more complicated if H 6¼ 0 because for H ¼ 0 the effective conductivity depends on only two parameters: volume concentration p and conductivity ratio x of the components ð x ¼ s2 =s1 Þ. If H 6¼ 0, however, the effective conductivity depends on four parameters: ð x; pÞ mentioned above, H and the mobility ratio y ¼ m1 =m2 . In order to describe the critical behavior of the galvanomagnetic properties quantitatively, we now deﬁne the logarithmic derivatives ws ðpÞ, wR ðpÞ of the conductivity and of Hall‘s coefﬁcient as ws ð p; x; y; HÞ ¼

log10 ½sð p þ p; x; y; HÞ log10 ½sð p; x; y; HÞ ; log10 ½p þ p pc log10 ½p pc

log10 ½Rð p þ p; x; y; HÞ log10 ½Rð p; x; y; HÞ wR ð p; x; y; HÞ ¼ log10 ½ p þ p pc log10 ½ p pc

ð302Þ

We note again that the galvanomagnetic properties near the percolation threshold are described by the following expressions: s ð pc pÞs ;

if ð pc pÞ > 0;

t

s ð p pc Þ ;

if ð p pc Þ > 0;

t1

R ð pc pÞ ; R ð p pc Þ

s1

if ð pc pÞ > 0; ;

if ð p pc Þ > 0

The critical exponents t; s and t1 ; s1 may be obtained if we ﬁnd the values of the functions ws ð p; x; y; HÞ and wR ð p; x; y; HÞ in the percolation limit ð p ! pc ;

physical properties of fractal structures

191

x ! 0; y ! 1; xy ! 1Þ; ws ð p; x; y; HÞ; tðH Þ ¼p!p lim c þ0

t1 ðH Þ ¼p!p lim wR ð p; x; y; HÞ; c þ0

x!0 y!1

x!0 y!1

ws ð p; x; y; HÞ; sðH Þ ¼p!p lim c þ0

s1 ðH Þ ¼p!p lim wR ð p; x; y; HÞ c þ0

x!0 y!1

x!0 y!1

In Figs. 38 and 39 the results of calculations for effective conductivity, Hall coefﬁcient and their logarithmic derivatives near the percolation threshold are

(a) –2

3 2.5 2 1.5 1 0.5 0 –0.5

–4 –6 –8 0.15 0.2 0.25 0.3 p

0.15 0.2 0.25 0.3 p

0

4

–5

3

χs (p)

log10 s /s1

(c)

–10 –15

1

–20

0 0.15 0.2 0.25 0.3 p

(e) 0 χs (p)

–5 –10 –15 –20 0.15 0.2 0.25 0.3 p

(d)

2

0.15 0.2 0.25 0.3 p

log10 s /s1

(b)

χs (p)

log10 s /s1

0

5 4 3 2 1 0

(f)

0.15 0.2 0.25 0.3 p

Figure 38. Dependence of the relative effective conductivity (a, c, e) and logarithmic derivative of the effective conductivity (b, d, f) on concentration near the percolation threshold: (a, b) H ! 0; (c, d) H ¼ 105 ; (e, f) H ! 1.

vitaly v. novikov

192

Figure 39. Dependence of the relative Hall coefﬁcient (a, c, e) and the logarithmic derivative of the Hall coefﬁcient (b, d, f) on concentration near the percolation threshold: (a, b) the H ! 0; (c, d) H ¼ 105 ; (e, f) H ! 1.

presented for y ¼ 1010 ; x ¼ 1010, and various values of the ﬁeld H. Using these results, we have approximately determined the critical exponents: H ! 0;

2:0 t 2:7;

0:85 s 0:9;

1:8 t1 2:2;

5

H ¼ 10 ;

2:6 t 3:0;

1:4 s 1:3;

0 t1 1:5;

H ! 1;

3:0 t 4:0;

1:4 s 1:1;

t 1 ’ s1 ’ 0

1:2 s1 2:2; 0:8 s1 2:0;

The critical exponents t and s have been thoroughly investigated in literature when H ¼ 0. The comparison between various authors 1:6 < t < 2:4;

physical properties of fractal structures

193

Figure 40.

The dependence of the Hall coefﬁcient (a) and the relative effective conductivity (b) on the concentration and on the magnetic ﬁeld when the Hall factors of components differs essentially, and their conductivities are equal ðx ¼ 1, y ¼ 1010Þ ).

0:7 < s < 0:9 (see Section III) and our results 2:0 t 2:7; 0:85 s 0:9 reveals that our results are slightly exceeded; hence, the results of our calculations in the percolation limit ð p ! pc ; x ! 0; y ! 1; xy ! 1Þ can be considered as the analysis of the qualitative dependence of the effective Hall properties. The accuracy of the effective Hall properties calculation depends on the probability function Y ð pÞ selected, Eqs. (243) and (245), and the actual structure of the inhomogeneous medium.

vitaly v. novikov

194

Note that when 1 > x > 106 ; 1 < y < 106 ð xy ¼ 1Þ, the iteration method agrees closely with various authors (Figs. 32 and 42). Figure 40 shows the results of calculation of the effective relative Hall coefﬁcient Fig. 40a and the relative effective conductivity Fig. 40b when the values of conductivity of components are equal and the mobility in the second component is very small ðm2 ¼ 1010 m1 Þ for various values of the magnetic ﬁeld H. As the magnetic ﬁeld increases, the sudden jump in dependence of the Hall coefﬁcient at p ’ 0 departs from the percolation threshold p ¼ pc (Fig. 40a). Furthermore, in the dependence of the effective conductivity on magnetic ﬁeld a minimum exists near the percolation threshold the depth of which tends to zero as H ! 1 (Fig. 40b). This is caused by the appearance of rotating currents induced by the difference in the Hall coefﬁcients of the components. Certain terms in formula (322) correspond to these rotating currents. Figure 41 shows results of the calculation of the effective relative Hall coefﬁcient for the same parameter values as Fig. 40, but with another value for the parameter y ¼ 0:1. Hence, the Hall coefﬁcient varies smoothly with concentration when H ’ 0; moreover, a jump exists near the percolation threshold when H ! 1 (Fig. 41). The dependence of the effective conductivity is roughly the same as that shown in Fig. 40. Fluctuations in the dependence of the effective Hall coefﬁcient (Fig. 41a) exist when the value of Hall coefﬁcient in second component is about R2 ’ 105 R1 if H ’ 0 at p < pc . In the range p > pc the effective Hall coefﬁcient decreases steadily to the value R1 . For H ¼ 104 the effective Hall coefﬁcient decreases steadily from R2 to R1 In the range 0 < p < pc . For p > pc the effective Hall coefﬁcient is equal to R1 and does not depend on the concentration. For the parameters given the effective conductivity is practically independent of the magnetic ﬁeld H. Discrepancies in the range p > pc are unimportant. Figure 42 shows the comparison between the calculation of the effective Hall coefﬁcient and the results of two experiments [69,130]. Fractal Properties. Figure 43 shows the dependence of the effective Hall properties of a composite on the scale (the number of iterations which according to Eq. (196) is equal to n ¼ lnlnLln 1) near the percolation threshold pc . Note that to our knowledge such a dependence is presented for the ﬁrst time. According to our calculations (Fig. 43), the dependence of the effective Hall properties of a composite on the scale can be divided into two ranges. On scales n < 5 (Fig. 43) the composite exhibits the properties of a fractal object with the characteristic power law dependence of such properties (resistivity r and Hall coefﬁcient R) on the scale: R ðLn Þa2 ; r ðLn Þa1 ;

a2 ’ 1:92; a1 ’ 3:0;

n < 5; n < 5;

H ! 0; H!0

ð303Þ

physical properties of fractal structures

195

0 –0.2 log10 R/R1

–0.4 –0.6 –0.8 –1 –6

1 0.8 0.6 0.4

–4 (a)

log10 H

–2

p

0.2 0 2 0

0 –2 log10 s/s1

–6 0 (b)

2

–4 0 –2 0. 2

–log10 H

–4

0. 4 p 0. 6 0. 8 1

–6

Figure 41. Dependence of the relative effective Hall coefﬁcient on the concentration at x ¼ 1; y ¼ 0:1 for various values of the magnetic ﬁeld H.

On scales n > 5 (Fig. 43) the Hall properties do not depend on the scale; that is, Euclidean geometry prevails. Here the composite can be described as a quasihomogeneous (‘‘gray’’) medium, whose properties correspond to effective values of properties. When n ’ 5, a transformation between a fractal and a quasihomogeneous mode of behavior of Hall properties exists. In other words, the scale nx ¼ 5 determines the correlation length x.

vitaly v. novikov

196 500 400

R

300 200 100 0 0

0.2

(a)

0.4

0.6

0.8

1

p

50 40

R

30 20 10 0 0.2 (b)

0.4

0.6

0.8

1

p

Figure 42. Comparison between calculations and experiments for the dependencies of the effective Hall coefﬁcient of the composites Bi–Cd (a) and for Nax WO3 (b) on the concentration of a phase of one of components.

Note that the accuracy of calculation by the iterative method for the Hall properties of a 3D composite with chaotic structure depends on the accuracy of the calculation of the probability function Y ð pÞ (probability that the point belongs to a connecting set) which describes the random structure of a composite). A rough approximation is the computation of the Hall properties of a connecting and a nonconnecting set based on a cube inside a cube cell by formulas from the Appendix. The results of our calculations can be considered as an analysis of the qualitative dependence of the effective Hall properties

physical properties of fractal structures

197

17 16

log10 R/R1

15 14 13 12 11 (a)

1

2

3

5

nx

6

7

n

–4 –6 –8 –10 –12 –14 –16 (b)

2

4

nx

6

8

n

Figure 43. Dependence of the relative effective Hall coefﬁcient (a) and the relative effective conductivity (b) on the scale.

(Figs. 38–43). In the future we shall calculate the probability function Y ð pÞ and the Hall properties of a connecting and nonconnecting set more accurately. Conclusion. The iterative averaging method allows us to study the Hall properties of the composite over a large range of different parameters: concentrations, conductivities, Hall coefﬁcients of components and a magnetic ﬁeld.

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198

A number of interesting results have been obtained (due to our fractal model of structure and the iterative averaging method) for the Hall properties of the composite; for example, use of a logarithmic derivative allows one to obtain critical exponents for the effective Hall coefﬁcient (Fig. 39) for various values of the magnetic ﬁeld H. When s1 ¼ s2 (Fig. 40) the effective conductivity is a constant if H ¼ 0 and tends to zero if H ! 1 near the percolation threshold. On the left of the percolation threshold ð p < pc Þ the rise in the Hall coefﬁcient is more rapid as the magnetic ﬁeld increases (Fig. 40). On the right of the percolation threshold ð p > pc Þ the Hall coefﬁcient is practically independent of the concentration p. The iterative averaging method allows one to obtain the speciﬁc dependence of the effective Hall properties on the scale (number of iteration steps). This dependence yields information about the geometry prevailing at a given scale. The transformation to the regime of Euclidean geometry (quasihomogeneous medium) occurs on a characteristic scale x where the logarithm of a property ceases to depend on the scale. Note that the scale x, as well as the dependence of the effective Hall properties, is a multiparametric dependency. Our general aims in future publications are a more accurate calculation of the probability function Y ð pÞ and a less crude approximation for the elementary cell.

3.

Appendix. Galvanomagnetic Properties of the Cube Inside a Cube Cell.

Suppose, that the magnetic ﬁeld H is directed vertically (along the axis Ox3 ) and the current h j1 i is directed horizontally along Ox1 , as shown in Fig. 44. Take the size of the outer cube as 1, and the size of the inner cube as d. The volume concentration and the size of the inner cube are now related by pﬃﬃﬃ the formula: d ¼ 3 p. Ohm’s law then becomes bE j¼s

ð304Þ

where 0

s11 s ¼ @ s12 0

s12 s22 0

1 0 0 A; s33

0

1 E1 E ¼ @ E2 A; E3

0

1 j1 j ¼ @ j2 A j3

We can express this equation as E¼b rj

ð305Þ

physical properties of fractal structures

199

Ox3

H

Ox1 Ox2 <j1>

Figure 44. The cube inside a cube model.

where

0

r11 r ¼ @ r12 0

1 0 0 A r33

r12 r22 0

ð306Þ

Now we carry out a conventional partition of the cube inside a cube cell into layered structures so that it is possible to compose the cell from them. Next we calculate the Hall properties of the cube inside a cube cell approximately by means of step-by-step averaging of the Hall properties of layered structures. Hence, the approximate evaluation of the Hall properties of the cube inside a cube cell is reduced to the evaluation of the properties of a layered medium (see Fig. 44). Galvanomagnetic Properties of Layered Structures. We shall consider three different cases of orientation for the magnetic ﬁeld H, current h j1 i and direction of layers (see Fig. 44). orientation a. The layers are parallel to the current h j1 i and perpendicular to the ﬁeld H. According to Fig. 45a the currents and ﬁelds obey the conditions D E D E9 ð1Þ ð2Þ j1 ¼ p j1 þ ð1 pÞ j1 > > > > > > D ð1Þ E D ð2Þ E > > = E1 ¼ E1 ¼ E1 D E D E ð307Þ ð1Þ ð2Þ > j2 ¼ p j2 þ ð1 pÞ j2 > > > > > > D ð1Þ E D ð2Þ E > ; E ¼ E ¼ E 2

2

2

vitaly v. novikov

200 (a)

(b)

(c)

j1

Figure 45.

Layered structure of a cube inside a cube cell.

where the angular brackets hi mean as usual the space average of a function f and the upper index indicates the material of the layer (ﬁrst or second component): ð ð D E 1 1 3 ðiÞ f ðrÞd r; f f ðrÞd3 r ð308Þ ¼ hfi ¼ V Vi V

Vi

According to the conditions (307), we shall express the longitudinal current hj1 i and the Hall current hj2 i in the ﬁrst component as follows: ð1Þ h j1 i ¼ w1 j1 ; ð309Þ ð1Þ h j2 i ¼ w2 j1 where

ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ pr11 þ ð1 pÞr11 r11 þ pr12 þ ð1 pÞr12 r12 w1 ¼ ; ð2Þ ð1Þ 2 ð2Þ ð1Þ 2 pr11 þ ð1 pÞr11 þ pr12 þ ð1 pÞr12 ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ pr11 þ ð1 pÞr11 r12 þ pr12 þ ð1 pÞr12 r11 w1 ¼ ð2Þ ð1Þ 2 ð2Þ ð1Þ 2 pr11 þ ð1 pÞr11 þ pr12 þ ð1 pÞr12

ð310Þ

It follows from the conditions (307) that the expression for the Hall ﬁeld hE2 i in case (a) is D E D E ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ hE2 i ¼ r12 j1 þ r11 j2 ¼ r12 w1 þ r11 w2 h j1 i

ð311Þ

physical properties of fractal structures

201

ðaÞ

The expression in parentheses is the off-diagonal element r12 of the tensor b r for the case (a): ðaÞ

ð1Þ

ð1Þ

r12 ¼ r12 w1 þ r11 w2

ð312Þ

It follows from the conditions (307) and from the law (309) that the longitudinal ﬁeld hE1 i in case (a) is D E D E ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ hE1 i ¼ r11 j1 r12 j2 ¼ r11 w1 r12 w2 h j1 i

ð313Þ

ðaÞ

The expression in parentheses is the diagonal element r11 of the tensor b r for case (a): ðaÞ

ð1Þ

ð1Þ

r11 ¼ r11 w1 r12 w2

ð314Þ

orientation b. Now the layers are parallel to the current h j1 i and parallel to the ﬁeld H. According to Fig. 45b the currents and ﬁelds obey the conditions D E D E ðaÞ ð2Þ hj 1 i ¼ j 1 ¼ j 1 D E D E ðaÞ ð2Þ hE1 i ¼ p E1 þ ð1 pÞ E1 D E D E ð315Þ ðaÞ ð2Þ h j 2 i ¼ p j 2 þ ð 1 pÞ j 2 D E D E ðaÞ ð2Þ hE2 i ¼ E2 ¼ E2 From now on the alphabetic upper indexes specify that the ﬁelds and currents belong to the corresponding region. As we can see in Fig. 44, the region (a) in case (b) is just a layer with the averaged properties (304) and (315). According to the conditions (315), we can express the Hall current hj2 i in layer (a) as follows: D E ðaÞ j2 ¼ w3 hj1 i ð316Þ where ð2Þ ðaÞ ð1 pÞ r12 r12 w3 ¼ ðaÞ ð2Þ p r11 þ r11

ð317Þ

vitaly v. novikov

202

It follows from conditions (315) and from law (316) that the Hall ﬁeld hE2 i in case (b) is D E D E ðaÞ ðaÞ ðaÞ ðaÞ ðaÞ ðaÞ hE2 i ¼ r12 j1 þ r11 j2 ¼ r12 þ r11 w3 hj1 i

ð318Þ ðaÞ

The expression in parentheses in Eq. (318) is the offdiagonal element r12 of the tensor b r for case (b): ðbÞ

ðaÞ

ðaÞ

r12 ¼ r12 þ r11 w3

ð319Þ

It follows from the conditions Eq. (304) that the expression for the longitudinal ﬁeld hE1 i in case (b) is D E D E ð aÞ ð2Þ hE1 i ¼ p E1 þ ð1 pÞ E1 D E D E ð aÞ ðaÞ ðaÞ ð2Þ ð2Þ ð2Þ ¼ p r11 hj1 i r12 j2 þ ð1 pÞ r11 hj1 i r12 j2

ð320Þ

According to the conditions (320) we can express the Hall currents as D

ðaÞ

E

j2 D

ð2Þ j2

E

ðbÞ

ðaÞ

¼

r12 r12

¼

r12 r12

ðaÞ

r11 ðbÞ

hj1 i; ð321Þ

ð2Þ

ð2Þ

r11

hj1 i

It follows from Eqs. (315) that hE1 i ¼ ð:::Þhj1 i and that the diagonal element ð bÞ r for case (b) is r11 of tensor b " ðbÞ r11

¼p

ðaÞ r11

ð aÞ r12

ðbÞ

ðaÞ

r12 r12 ðaÞ

r11

#

" þ ð 1 pÞ

ð2Þ r11

ð2Þ r12

ðbÞ

ð2Þ

r12 r12 ð2Þ

r11

# ð322Þ

Since r12 ¼ RH, from Eq.(322) it follows that the ohmic resistivity ð bÞ r11 H 2 . Hence the existence of a ﬁeld dependence is manifested by a difference between the Hall coefﬁcients ðR1 R2 Þ of the respective components. If this difference is not equal to zero ðR1 6¼ R2 Þ, in a given composite rotational Hall currents exist. These currents yield the contribution to the ohmic resistivity (and to the ohmic conductivity, respectively), which depends on the magnetic ﬁeld H. Hence, if there are even small inhomogeneities in the substance, then saturation ðr11 ! r1 ; H ! 1Þ does not appear for ohmic resistance with increase of the magnetic ﬁeld.

physical properties of fractal structures

203

orientation c. Here the layers are parallel to the current hj1 i and perpendicular to the ﬁeld H. According to Fig. 45c, the currents and ﬁelds satisfy the conditions D E D E ðbÞ ð2Þ hj1 i ¼ p j1 þ ð1 pÞ j1 D E D E ðbÞ ð2Þ hE1 i ¼ E1 ¼ E1 D E D E ðbÞ ð2Þ hj 2 i ¼ j 2 ¼ j 2 D E D E ðbÞ ð2Þ hE2 i ¼ p E2 þ ð1 pÞ E2

ð323Þ

According to the conditions (323), we can express the longitudinal current hj1 iin the layer ‘‘b’’ as D E ðbÞ j1 ¼ w4 hj1 i

ð324Þ

where ð2Þ

w4 ¼

r11

ðbÞ

ð2Þ

ð1 pÞr11 þ pr11

ð325Þ

It follows from the conditions (323) and from the expression (324) that the expression for the Hall ﬁeld hE2 i in case (c) is D E ðbÞ ðbÞ ð2Þ hE2 i ¼ pr12 j1 þ ð1 pÞr12 ; D E ð2Þ ðbÞ ð2Þ j1 ¼ w4 pr12 þ w4 ð1 pÞr12 hj1 i

ð326Þ

The expression in large parentheses in Eqs. (326) is the off-diagonal ðaÞ element r12 of the tensor b r for case (c)—that is, for a complete cube inside a cube cell: h i ðcÞ ðbÞ ð2Þ r12 ¼ w4 pr12 þ ð1 pÞr12 ð327Þ It again follows from the conditions (323) that the expression for the longitudinal ﬁeld hE1 i in case (c) is D E ðbÞ ðbÞ ðbÞ ðbÞ hE1 i ¼ r11 hj1 i r12 j2 ¼ r11 w4 hj1 i

ð328Þ

vitaly v. novikov

204 that is, ð cÞ

ðbÞ

ð2Þ

r11

ðbÞ

r11 ¼ r11 w4 ¼ r11

ðbÞ

ð2Þ

ð1 pÞr11 þ pr11

ð329Þ

Note that the case shown in Fig. 44 corresponds to a nonconnecting set structure, and the expressions (327) and (329) describe the average properties of a nonconnecting set structure. To obtain similar expressions for a connecting set structure, it is necessary in all the calculations of the Appendix to consistently make the change of variables: ðiÞ

ð jÞ

r11 ! r11 ;

ðiÞ

ð jÞ

r12 ! r12 ;

p ! ð1 pÞ;

i; j ¼ 1; 2

ð330Þ

Thus we shall obtain expressions similar to (327) and (329); however, they pertain to the connecting set structure. These expressions are also used for the calculation of the effective Hall properties of 3D composites. D. Elastic Properties The elastic properties of inhomogeneous media with chaotic structure can be deduced using an iterative procedure similar to that used to deﬁne effective conductivity made in the previous section. The effective elastic properties (the bulk modulus K and the shear modulus m) of the connecting set and nonconnecting set may be calculated by using standard formulas from the physics of composite materials (e.g., Hashin– Strikman formulae [133, 134]) accounting for the tensor nature of elastic properties (Fig. 46). In order to calculate the elastic properties of fractal structures according to the iterative procedure the Hashin–Strikman double-sided estimation of the elastic properties can be used [131], namely, pðK1 K2 Þ ð1 pÞðK2 K1 Þ K K1 þ ; 1 þ ð1 pÞa2 ðK1 K2 Þ 1 þ pa2 ðK2 K1 Þ pðm1 m2 Þ ð1 pÞðm2 m1 Þ m2 þ m m1 þ 1 þ ð1 pÞb2 ðm1 m2 Þ 1 þ pb2 ðm2 m1 Þ

K2 þ

ð331Þ

where ai ¼

3 ; 3Ki þ 4mi

bi ¼

6ðKi þ 2mi Þ 5Ki ð3Ki þ 4mi Þ

ð332Þ

physical properties of fractal structures (a)

(b)

A

A

Figure 46.

205

B

C

B

C

D

D

Blob models of (a) the connecting set and (b) nonconnecting set.

The Hashin–Strikman formulae were obtained by using a variational method to determine the upper Kc ; mc and the lower Kn, mn bounds of the effective elastic properties for an inhomogeneous medium [131]. The upper bound Kc ; mc corresponds to a composite structure in which spherical inclusions with elastic constants K2 , m2 are placed in a matrix of elastic constants K1 , m1; in the following, it is assumed that K1 > K2 , m1 > m2 . The lower bound Kn , mn is obtained when the components are permuted—that is, when the matrix is described by K2, m2 and the spherical inclusions are described by K1, m1 . From the ‘‘Hashin–Strikman spheres’’ (where inside a sphere of one material a sphere of the other material is placed centrally) one can form a composite as follows [131–134]: Spheres of various sizes, down to inﬁnitesimally small, are taken and the space V is densely packed by them so that vacancies do not occur. Only one condition is required: In each Hashin–Strikman sphere the volume concentrations of both the components must be the same; that is, all the Hashin– Strikman spheres must exhibit the same elastic properties. Such a composite will be further referred to as the ‘‘Hashin–Strikman composite.’’ The elastic properties of the Hashin–Strikman composite are described by the formulae that are obtained from the exactly solvable model of a single spherical inclusion of one phase in an inﬁnite matrix of the second phase and depend only on the volume concentrations and elastic properties of the constituent phases. The properties of the Hashin–Strikman composite do not depend on the scale chosen. 1.

Iterative Averaging Method for Elastic Properties

The conﬁguration elastic properties corresponding to the connecting set and nonconnecting set were calculated by applying the Hashin–Strikman formulae.

vitaly v. novikov

206

The relation connecting the modulii Kc and mc sets at step i þ 1 according to Eqs. (331) and (332) are ðiÞ

mðiþ1Þ c

¼

mðiÞ c

ðiÞ

ð1 pi ÞðKn Kc Þ

Kcðiþ1Þ ¼ KcðiÞ þ

ðiÞ

ðiÞ

ðiÞ

1 þ pi ac ðKn Kc Þ ðiÞ

þ

; ð333Þ

ðiÞ

ð1 pi Þðmn mc Þ ðiÞ

ðiÞ

ðiÞ

1 þ pi bc ðmn mc Þ

where aðiÞ c ¼

3

; ðiÞ ðiÞ 3Kc þ 4mc

bðiÞ c ¼

ðiÞ

ðiÞ

6ðKc þ 2mc Þ ðiÞ

ðiÞ

ðiÞ

5Kc ð3Kc þ 4mc Þ

ð334Þ

Kc0 ¼ K1 , m0c ¼ m1 denote the complex bulk modulus and the complex shear modulus of the ﬁrst phase of the inhomogeneous medium, and Kn0 ¼ K2 and m0n ¼ m2 - denote the complex bulk modulus and the complex shear modulus of the second phase, respectively (K1 , K2 ; m1 , m2 are elastic properties for nonhomogeneous media phases). For nonbonded conﬁgurations the viscoelastic ðiþ1Þ ðiþ1Þ and mn are described by the formulae which result from the modulii Kn following replacements c !n and pi !ð1 pi Þ. According to the iterative procedure we have ðkÞ

ðkÞ

ðkÞ mc

ðkÞ mn ;

Kc K Kn ; m

2.

lim ðkÞ lim ðkÞ k!1 Kc ¼ k!1 Kn ¼ lim ðkÞ lim ðkÞ k!1 mc ¼ k!1 mn ¼ m

K;

ð335Þ

Results of Calculation

The calculations were made for a two-component, inhomogeneous medium: K1 ; m1 -----first component;

K2 ; m2 -----second component

Apparently from the plots of log10 K (Fig. 47a) and log10 m (Fig. 47b) of the fractal ensemble versus the iteration step, number n, all these elastic properties behave like fractals before an eventual levelling off. The latter is obviously associated with the upper limit of fractal-like asymptotics, above which the elastic properties of a system are no longer m dependent on the scale—that is, on the iteration number (the loss of the self-similarity property occurs at iteration step nx ¼ logx=logl0 which deﬁnes the correlation length x at the given concentration, p).

physical properties of fractal structures 0

207

log10K

(a)

3 2 1

0

2.5

5

7.5

10

nx

n

15

0

log10 µ

(b)

3 2 1 0

2.5

5

7.5

10 nx

15

n

Figure 47. Semilogarithmic dependence of the shear elasticity modulus (a) and the bulk elasticity modulus (b) on the iteration number n for p ¼ 0:2088 (1), 0.2092 (2), and 0.2098 (3).

In order to describe the critical behavior of the inhomogeneous medium, we now deﬁne the logarithmic derivative wðpÞ of the elasticity modulus (Fig. 48) as wðpÞ ¼

log10 ½ðKðp þ pÞÞ=KðpÞ log10 ½ðp þ p pc Þ=ðp pc Þ

ð336Þ

Using the function wðpÞ we have determined the indexes t, S: for the elastic region we have t ¼ lim wðpÞ p!pc þ0

ð337Þ

for the highly elastic region we have S ¼ lim wðpÞ p!pc 0

ð338Þ

vitaly v. novikov

208

Figure 48.

The result of calculation of the function wðpÞ.

The value 3:200 0:002 obtained for the critical index t describing the singular behavior of the bulk modulus K in the vicinity of the critical point pc þ 0 is about 15% smaller than that obtained for d ¼ 3 by Sahimi and Arbabi [135]: t ¼ 3:75 0:11. We remark that for d ¼ 2, Zabolitzky et al. [136] obtained t ¼ 3:96 0:04; the results obtained in Refs. 135 and 136 are in good agreement with the relation proposed by Sahimi [137]: t ¼ t þ 2v, where t is the critical conductivity exponent of percolation networks. The values of the critical index S for the superelastic regime (p < pc ) were determined by using a very precise calculation (of a few hundred digits of accuracy) for K1 =m1 ¼ K2 =m2 ¼ 5 with the requirement that K2 =K1 ! 0 and m2 =m1 ! 0; in all cases we obtained S ¼ 0:62962 0:00002, which is in excellent agreement with the result obtained for d ¼ 3 by Sahimi and Arbabi [135,138] (S ¼ 0:65); for d ¼ 2 the most reliable estimate of 1:24 0:03 [135,138]. In the vicinity of the percolation threshold, the ratio of the bulk modulus to the shear modulus K=m tends to a constant. We found that K=m ¼ 1:33 (Fig. 49). This result is in agreement with [139], where a theoretical proof of the equality K=m ¼ 4=d in the limit p ! pc is given. The calculations of elastic properties according to the iteration procedure allow one to conclude that these calculations essentially agree with (a) the results of the percolation theory at K2 =K1 ! 0 (m2 =m1 ! 0Þ and (b) the results according to effective medium ﬁeld formulae at K2 =K1 > 102 (Fig. 50).

physical properties of fractal structures

209

5 4.5 4

5

K/ µ

3.5 3 4 2.5 3 2

2

1.5

1 0

0.2

0.4

0.6

0.8

1

p

Figure 49. Dependence on p of the ratio K=m for K1 =m1 ¼ K2 =m2 ¼ 5. The ratio K2 =K1 : 1,1010 ;

2,108 ; 3,106 ; 4,104 ; 5,101 .

As an example of comparison of theory and experiment the shear modulus dependence on volume concentration of phases for the system polybutadiene– polystyrene [140] is presented (Fig. 51). E.

Negative Poisson’s Ratio

For a long time it was believed that no isotropic material exists in nature with Poisson’s ratio less than zero [141]. Hence, various models exhibiting negative Poisson’s ratio [142–144] were merely considered as mathematical curiosities rather than bearing any relation to reality and, therefore, were regarded as having no practical importance. After foams with negative Poisson’s ratio were manufactured by Lakes [144], and expanded polytetrauoroethylene with limiting Poisson’s ratio by Evans and co-workers [145,146], increasing interest in studies of systems exhibiting such an unusual property has existed [147–155]. Various potential applications of such systems [156,157] encourage one to study new mechanisms which may lead to negative Poisson’s ratio [158,159]. In this work, we describe the results of theoretical studies of Poisson’s ratio in disordered structures composed of two phases of disparate elastic properties applying a renormalization group approach to a model of percolation on a hierarchical cubic lattice. Although this approach has been described in detail elsewhere [160], we present it brieﬂy here, for completeness. At the percolation

vitaly v. novikov

210

(a) 0

log01 K/K1

–1

–2

–3

–4

–5

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

p (b) 0

log01 µ/µ1

–1

–2

–3

–4

–5 0

0.2

0.4 p

Figure 50. Comparison of the calculation elastic properties of the according to the iteration procedure (continuous) and according to formulae effective medium ﬁeld (dotted line).

threshold, the Poisson’s ratio we obtain is in agreement with the computer simulation results and the conjecture of Arbabi and Sahimi [161]. When an isotropic body is affected from outside, it can be characterized by two parameters [141]. The ability of a body to resist volume changes is deﬁned

physical properties of fractal structures

211

1

Log10 m/m1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

p Figure 51. Dependence of shear elastic modulus on the volume concentration of block copolymers of styrene and butadiene (ﬁlled circles) [140]. The solid line represents the calculation results.

by the uniform compression modulus K; its ability to resist changes of shape by the shear modulus m. The elastic properties of a body in axial tension can be characterized using Poisson’s ratio np , which is deﬁned as the negative ratio of the transverse strain to the longitudinal strain. Poisson’s ratio np depends on K and m as follows [141]: For a three-dimensional isotropic body (d ¼ 3) np ¼

3K 2m 3 2x ¼ 6K þ 4m 6 þ 4x

ð339Þ

For a two-dimensional isotropic body (d ¼ 2) np ¼

Km 1x ¼ Kþm 1þx

ð340Þ

where x ¼ m=K. For integer d, Poisson’s ratio of a d-dimensional isotropic medium depends on the space dimension d as [162] np ¼

Kd 2m d 2x ¼ dðd 1ÞK þ 2m dðd 1Þ þ 2x

ð341Þ

vitaly v. novikov

212

Whence we see that the Poisson’s ratio can vary within the limits 1 < np <

1 d1

ð342Þ

It is also evident that Poisson’s ratio can be less than zero if x>

d 2

ð343Þ

For typical materials with d ¼ 2; 3, the condition is not fulﬁlled; that is, for typical materials np > 0. According to (314), a material with a negative Poisson’s coefﬁcient can be obtained either if it is very rigid (i.e. if its shear modulus fulﬁlls the above condition) or by forming a structure with dimension less than 2m=K (i.e., d < 2m=K) or by combining the ﬁrst and the second methods. Change of dimension of the system is impossible for continuum structures; however, it is possible in fractal structures [163]. To illustrate this idea, we will ﬁrst recall the results obtained for a planar lattice (d ¼ 2) with coordination number Z ¼ 3, bulk modulus K and shear modulus m, which are [164–169] pﬃﬃﬃ h 2 3hg ð344Þ K ¼ pﬃﬃﬃ ; m¼ h þ 6g 2 3 where h and g are parameters of the lattice, which is determined by the properties of the bonds. This results in Poisson’s ratio being given by vp ¼

Km h=g 6 ¼ ; K þ m h=g þ 18

1 np 1 3

ð345Þ

By the properties of bonds h and (h=g < 6), a Poisson’s ratio of 13 < np < 1 can be obtained. If such conﬁgurations constitute the principal contribution to the microscopic properties of the system, then this can lead to negative Poisson’s ratio. Numerical studies [167] of planar (d ¼ 2) elastic random percolation networks have shown that if their linear dimension L < 0:2x (x is the correlation length), then Poisson’s ratio for the system is negative, and if L > 0:2x, Poisson’s ratio is positive In this case, if L=x ! 1, the limiting value of Poisson’s ratio is np ¼ 0:08 0:04 and is a universal constant; that is, it does not depend on the relative values of the local elastic characteristics: If

physical properties of fractal structures

213

L=x ! 1, then np ¼ 13; if L=x ¼ 5, then np ¼ 0. Kantor and Webman [165] have proposed the following relation to describe the inﬂuence of the structure on Poisson’s ratio: m Z ¼ K 8

ð346Þ

where Z is the coordination number of the percolation lattice. If we assume that Poisson’s ratio for a d-dimensional isotropic inhomogeneous medium is given by [143] np ¼

Kd 2m db ¼ ; dðd 1ÞK þ 2m dðd 1Þ þ b

b ¼ 2x

ð347Þ

we have that if b > d, then np < 0(np ¼ 1 in the limit b=d ! 1); if b ¼ d, then np ¼ 0; if b < d, then the maximum value of np is equal to (1=d 1) (in the limit b=d ! 1). Now, if we substitute relation (346) into Eq. (347), we can write np ¼

b Z=4 ; dðd 1Þ þ Z=4

b ¼ 2x

ð348Þ

Hence it follows that the Poisson coefﬁcient at the percolation threshold is positive (np > 0) if Z < 4d, equal to zero (np ¼ 0) if Z ¼ 4d, and negative (np < 0) if Z > 4d. It is also necessary to mention that certain conﬁgurations of bonds (local regions) can possess unusual properties—in particular, a negative Poisson coefﬁcient np < 0. Thus, for example, the chain of bonds shown in Fig. 52, when stretched out, not only lengthens but also thickens. If such a conﬁguration constitutes the principal contribution to the macroscopic properties of the system, then the Poisson coefﬁcient may be negative. Thus we can conclude that by making an appropriate choice for the structure of a random medium representing the inhomogeneous medium (the coordination number Z), we can obtain a material with a negative Poisson coefﬁcient far from the percolation threshold. 1.

Results of Calculations of Poisson’s Ratio

The calculations were performed for a two-component, inhomogeneous medium.

vitaly v. novikov

214 (a)

(b)

Figure 52.

Lattice modulus with negative Poisson coefﬁcient: (a) lattice with coordination number N ¼ 3; (b) a chain of bonds.

We have for Poisson’s ratio vp for a three-dimensional isotropic system [see Eqs. (333) and (339)] after the kth iteration: ðkÞ ðkÞ ðkÞ ðkÞ Connecting set: nðkÞ c ¼ ð3Kc 2mc Þ=ð6Kc þ 4mc Þ; ðkÞ ðkÞ ðkÞ ðkÞ Nonconnecting set: nðkÞ n ¼ ð3Kn 2mn Þ=ð6Kn þ 4mn Þ ðkÞ lim nðkÞ c ¼ lim nn ¼ np

k!1

k!1

The results of calculations of the effective Poisson’s ratio np dependence on the bulk concentration of a rigid phase p at various values of a ¼ log10 ðK2 =K1 Þ are shown in Fig. 53. The calculations were made for Poisson’s ratios of the phases ranging from 0:1 to 0:4. It can be seen that at percolation threshold Poisson’s ratio of the isotropic fractal composite is np ¼ 0:2, when K2 =K1 ! 0 it is also independent of the Poisson’s ratios of the individual components of the composite. The Poisson’s ratio obtained by us near the percolation threshold is in agreement with computer simulation results and the conjecture of Arbabi and Sahimi [161]. It has been shown that an approximate theoretical treatment of percolation on a cubic lattice exactly reproduces the Poisson’s ratio obtained in computer simulation at the percolation threshold. This result may encourage one to use this approximation to describe various elastic properties of composites. It is worth noting that some critical indices have been calculated recently with a high degree of accuracy in the context of the present model. The considerations presented in this work suggest that at the percolation threshold the ratio of the shear modulus to the bulk modulus is a universal quantity, which does not depend on the elastic properties of the percolation phase. It is well known, however, that this ratio depends on the coordination number of the lattice on which the percolation takes place. Taking this into account, we conjecture that when the coordination number, Z, of the underlying lattice is more than four times larger than the dimension of the lattice, Z > 4d, Poisson’s ratio near the percolation threshold should be negative, irrespective of the value of Poisson’s ratio of the percolation phase.

physical properties of fractal structures 0.4

215

9 8 7

0.3 vp

6 5 0.2

4

3

2 0.1 1 0

–0.1 0

0.2

0.4 p

0.6

0.8

Figure 53. Dependence of the effective Poisson ratio on the volume concentration of the rigid component p for different values K2 =K1 for the Poisson ratio of the components np1 ¼ np2 ¼ 0:4 (curves above np ¼ 0:2) and np1 ¼ np2 ¼ 0:1 (curves below np ¼ 0:2).

Assuming that Eq. (333) is fulﬁlled also at the percolation threshold for random lattices with Z replaced by the average coordination number hZi i (where Zi means the coordination number of the ith lattice site), one can obtain negative Poisson’s ratios from percolation systems of particles—for example, polymer molecules, which can be due to a sufﬁciently large number of neighbors, < Z; > > 4d. Analytic studies and extensive computer simulations of various models in 2D and 3D are in progress in order to verify the above theoretical predictions. F.

Frequency Dependence of Viscoelastic Properties

Before we discuss the viscoelastic properties of nonuniform fractal structures we shall give some basic deﬁnitions from the theory of elasticity. If a body is subjected to an external force, strains appear in it, and the body itself becomes stressed. If this stress always exists during the action of the force and instantly disappears when this action ceases, then the body is ideally elastic. In this case, the relation between the stress tensor s and the strain tensor e is described by Hooke’s law [141]: s ¼ C e; e ¼ S s

ð349Þ

vitaly v. novikov

216

where C is the elastic modulus tensor and S is the compliance modulus tensor. For an isotropic medium with elastic modulus tensor C and compliance modulus tensor S we have C ¼ 3KV þ 2mD; 1 1 S¼ Vþ D 3K 2m

ð350Þ ð351Þ

where K is the volume elastic modulus, m is the shear modulus, V is the volume, and D represents deviator components of the individual tensor of the fourth rank I: I ¼ V þ D; 1 Vijkl ¼ ðdij dkl Þ; 2

1 2 Dijrl ¼ ðdik djl þ dil djk dij dkl Þ; 2 3 1; i¼k dik ¼ 0; i 6¼ k

ð352Þ

When the strain is irreversible (i.e., when a body exhibits percolation), the stress decreases rapidly and recovers again because of the displacement of structural elements. If the shape and state of the structural elements do not undergo any variations in this case, then the body is ideally viscous, and its behavior is described by the Newtonian equation [131,132] s¼Z

de dt

ð353Þ

where Z is the viscosity of the liquid. Most real bodies are viscoelastic and obey laws (349) and (353) only under certain conditions. Hence, the concept of the stress decay time or the relaxation time t is introduced to characterize the stress-strain state of real bodies. For absolutely elastic bodies, t ! 0, whereas, for ideally viscous bodies, t ! 1. Real viscous, anomalous viscous, and viscoelastic media are described in the interval 0 < t < 1. A general relation containing the law of elasticity and the law of viscosity as extreme cases can be introduced as sðtÞ ¼ K

da eðtÞ dta

ð354Þ

Hooke’s law follows from (354) for a ¼ 0, K ¼ C and Newtons law of viscosity follows for a ¼ 1; K ¼ Z:

physical properties of fractal structures

217

If the external action depends on the time, i.e. the stresses sðtÞ and strains eðtÞ depend on time, Hooke‘s law can also be introduced via the frequency domain relation [131,132] sðoÞ ¼ C ðoÞeðoÞ

ð355Þ

where sðoÞ and eðoÞ are the Fourier transforms of sðtÞ and eðtÞ respectively: 1 ð

sðoÞ ¼

1 ð iot

e sðtÞ dt; 1 1 ð

1 sðtÞ ¼ 2p

eðoÞ ¼ 1

e

iot

sðoÞ do;

1

ð356Þ

eiot eðtÞ dt

1 eðtÞ ¼ 2p

1 ð

eiot eðoÞ do

ð357Þ

1

The complex elastic modulus in the frequency domain is C ðoÞ ¼ C0 ðoÞ þ iC 00 ðoÞ

ð358Þ

where the real and imaginary parts of the modulus C ðoÞ are given by [132] 0

1 ð

C ðoÞ ¼ C1 þ o

gðtÞ sin ot dt;

00

1 ð

C ðoÞ ¼ o

0

gðtÞ cos ot dt

ð359Þ

0

Here the elastic modulus cðtÞ ¼ c1 þ gðtÞ; c1 is the asymptotic value of cðtÞ as t ! 1; gðtÞ is the response function (gðtÞ in general takes into account the history of the process) Hooke’s law for a compliance tensor S can be represented in the form eðoÞ ¼ S ðoÞsðoÞ

ð360Þ

S ðoÞ ¼ S0 ðoÞ þ iS00 ðoÞ

ð361Þ

where

00

S0 ðoÞ is the inphase (accumulation) compliance, and S ðoÞ is the quadrature (loss) compliance. One can show that the relative scattering loss of the elastic energy is related 00 only to the imaginary component S ðoÞof the elastic modulus [131,132].

218

vitaly v. novikov

We will consider below isotropic media, for which, just as Eq. (361), the concept of a complex bulk elastic modulus K ðoÞ can be introduced [131]. The complex shear modulus m and the complex viscosity Z can be written [131,132] m ðoÞ ¼ m0 ðoÞ þ im00 ðoÞ; Z ðoÞ ¼ m =io;

ð362Þ ð363Þ

Z ðoÞ ¼ Z0 ðoÞ þ iZ00 ðoÞ

ð364Þ

The relation between m0 ðoÞ; m00 ðoÞ and Z0 ðoÞ; Z00 ðoÞ is m0 ðoÞ ¼ oZ00 ðoÞ; m00 ðoÞ ¼ oZ0 ðoÞ

ð365Þ

For a medium representing a Newtonian liquid, we have m ðoÞ ¼ ioZ0 ðoÞ Viscoelastic media have been described by a variety of models involving combinations of a spring and a piston in a viscous liquid. In this (onedimensional) case, Hooke’s and Newton’s laws are [131] FH ¼ kx dx FN ¼ Z dt

ð366Þ ð367Þ

A series combination of these elements corresponds to the Maxwell model, while their parallel combination corresponds to the Kelvin–Voigt model (Fig. 54). The transition from the models to a continuous medium is performed by replacing the force F and displacements x by stresses s and strains e. The Maxwell model conforms to the series connection of these elements, and the Voigt model conforms to the parallel connection. The main disadvantage of the Maxwell model is that the static shear modulus m0 vanishes in this model, while the drawback of the Kelvin–Voigt model is that it cannot describe the stress relaxation. The Zener model [131] lacks these disadvantages. This model combines the Maxwell and Kelvin–Voigt models and describes strains closely approximating the actual physical process. The elasticity equation for the Zener model taking account of anomalous relaxation effects can be written as [131] s þ te

da s da e ¼ m e þ t s dta dta

ð368Þ

physical properties of fractal structures (a)

Figure 54.

219

(b)

Models of viscoelastic properties: (a) Maxwell model;(b) Kelvin–Voigt model.

where m0 ¼ mðoÞjo¼0 ;

ð369Þ

m1 ¼ lim mðoÞ;

ð370Þ

o!1

te =ts ¼ m0 =m1

ð371Þ

and o is the angular frequency of the impressed stimulus. d dtf aðtÞ is the Riemann– Liouville fractional differentiation operator a

ðt d a f ðtÞ 1 d ¼ ðt tÞa f ðtÞ dt dta ð1 aÞ dt

ð372Þ

c

where ðxÞ is the gamma function. By Fourier transformation of (368), we have [131] s þ ðiotÞa s ¼ 2m0 ðs þ ðiotÞa eÞ

ð373Þ

where s; e are the Fourier transforms of s; e. Now the Fourier transform of a fractional derivative is

1 ð da f ðtÞ a F expðiotÞ f ðtÞ dt ¼ ðioÞ f ðtÞ; f ðtÞ ¼ dta 1

ð374Þ

vitaly v. novikov

220 (a)

(b)

1

1

(m∞– m')/ (m∞– m0)

0.4

0.8

2

m"/(m∞– µ0)

0.3

0.6 0.4 0.2 4

0 –3

–2

–1

0 1 log10wt

1 2 3 2

3

2

3

0.2

3

0.1

4

0 –3

–2

–1

0 1 log10wt

2

3

(c) 1

1

0.8 f0

0.6 0.4

2

0.2 0

4 –3

–2

–1

3

0 1 log10t

Figure 55.

Dependence of the viscoelastic properties on log t (a) the real part of the relative shear modulus for a ¼ 0:2 (curves 1); 0:4 (2); 0:7 (3); 0:9 (4).; (b) the imaginary part of the relative shear modulus for a ¼ 0:2 (curves 4); 0:4 (3); 0:7 (2); 0:9 (1).; (c) normalized relaxation time distribution function for a ¼ 0:2 (curves 4); 0:4 (3); 0:7 (2); 0:9 (1).

It follows from this that the complex shear modulus for the standard linear body is m m0 m ðoÞ ¼ m1 1 ð375Þ 1 þ ðiote Þa Hence m1 m0 ðoÞ 1 þ ðotÞa cosðpa=2Þ ¼ ; m1 m0 1 þ ðotÞa ½2 cosðpa=2Þ þ ðotÞa m00 ðoÞ ðotÞa sinðpa=2Þ ¼ m1 m0 1 þ ðotÞa ½2 cosðpa=2Þ þ ðotÞa

ð376Þ ð377Þ

If the Fourier transform of mðoÞ is known, then the Fourier transform of the corresponding distribution f ðtÞ of relaxation times is [131] 1 1 ð378Þ f ¼ Im mðo expðipÞÞ o p

physical properties of fractal structures

221

By using Eqs. (375) and (378), we can then determine the normalized density of the distribution f0 ðtÞ of relaxation times [Eq. (377)]: f0 ðtÞ ¼

sinðpaÞ 2pfcosh½a lnðt=te Þ þ cosðapÞg

ð379Þ

where f0 ðtÞ ¼ m f ðtÞ : 1 m0 The dependence of the dispersion g2 of the relaxation time of the chaotic dynamics on the parameter a has the form 1 ð

g ¼

ln2 ðt=te Þ f0 ðtÞd ln t ¼

2

1

p2 1 a2 3 a2

ð380Þ

It has been shown in Ref. 170 that fractional derivatives can be obtained by assuming that a set of relaxation times has a fractal nature. The parameter a in Eqs. (376) and (377) is the fractal dimension of the fractal set of relaxation times and characterizes the localization (spread) of the relaxation spectrum [170]. 1.

Iterative Averaging Method for Viscoelastic Properties

According to Eq. (358), the static parameters (333), (334) can be converted into viscoelastic parameters by replacing the elastic modulii K and m by the corresponding complex modulii: K ¼ K 0 þ iK 00 ; m ¼ m0 þ i m00

ð381Þ

By using this correspondence principle for a connected set, the complex volume elastic modulus Kc and the complex shear modulus mc at the (k þ 1)th step can be written

Kcðkþ1Þ ¼ KcðkÞ þ mcðkþ1Þ ¼ mcðkÞ þ

ðkÞ

ð1 pk ÞðKn 1þ

ðkÞ

Kc

ðkÞ ðkÞ pk ac ðKn ðkÞ

ð1 pk Þðmn ðkÞ

ð382Þ

ðkÞ

mc Þ

ðkÞ

1 þ pk bc ðmn

Þ

ðkÞ Kc Þ

ð383Þ

ðkÞ

mc Þ

where

aðkÞ c ¼

3

; ðkÞ ðkÞ 3Kc þ 4mc

bðkÞ c ¼

ðkÞ

6ðKc ðkÞ

5Kc

ðkÞ

þ 2mc Þ ðkÞ

ð3Kc

ðkÞ

þ 4mc Þ

ð384Þ

vitaly v. novikov

222 ð0Þ

ð0Þ

where Kc ¼ K1 ; mc ¼ m1 are the complex volume elastic modulus and the complex shear modulus for the ﬁrst phase of the inhomogeneous medium, ð0Þ ð0Þ respectively; Kn ¼ K2 ; mn ¼ m2 are the complex volume elastic modulus and the complex shear modulus for the second phase, respectively. ðkþ1Þ ðkþ1Þ ; mn for a disconnected set are determined The elastic properties Kn from expressions that may be obtained from Eqs. (382) and (383) following the replacements c !n and pk !ð1 pk Þ. 2.

Results of Calculations for Viscoelastic Media

The calculations were made for a two-phase (two-component) inhomogeneous medium assuming that volume strains are elastic, while shear strains are viscoelastic. The ratio of local volume modulii K10 =K20 was set equal to 104 . For convenience of calculations, the local shear modulii (phase shear modulii) were written in the form 0

ð385Þ

0

¼ m2 ð1 þ iyÞ

ð386Þ

y ¼ tanðj2 Þ ¼ m002 =m02

ð387Þ

m1 ¼ m2 xð1 þ iayÞ m2 where

x ¼ m01 =m02 ¼ Z001 =Z002 a ¼ tanðj1 Þ= tanðj2 Þ

ð388Þ ð389Þ

The complex viscosity is 0

Zj ðoÞ ¼ Zj ðoÞ iZ00j ðoÞ

ð390Þ

where m0j ðoÞ ¼ oZ00j ðoÞ;

m00j ðoÞ ¼ oZ0j ðoÞ;

j ¼ 1; 2

ð391Þ

It follows from Fig. 56 (a ¼ 0:1; m001 =m01 1; m002 =m02 1) that the concentration dependence of the relaxed viscosity (o ! 0) is described by a monotonic curve and is independent of the ratio m002 =m02 . For a ¼ 0:01 and m002 =m02 ¼ 0:01 (Fig. 56b), both a local maximum and a local minimum appear near the percolation threshold, which strongly depend on the ratio m002 =m02 for a < 0:01 (Fig. 56b–d). The form of the dependence hardly changes before the percolation threshold (p < pc ) (Fig. 56b), whereas, after the percolation threshold (p > pc ),

physical properties of fractal structures 3

1

(a) 2

2.5

(b)

0.8 1

2

1

2

3 h'/h'

h'/h'2

4

1.5

0.6

2

0.4

3

1

4 0.2

0.5

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

p

0.8

0.6

0.8

1

0.6

0.8

1

p

(c)

(d) 1

1

2 3

0.5

0.6

4 0

0.4

h'/h'2

2 h'/h'2

223

3

–0.5 –1

4

0.2

–1.5 –2

0 0

0.2

0.4

0.6 p

0.8

1

0

0.2

0.4 p

Figure 56. Dependence of the logarithm of the relative effective viscosity of an inhomogeneous

fractal medium Z0 =Z02 ¼ Im½m ðoÞ=Im½m2 ðoÞ (Z02 is the viscosity of the second phase) on the concentration p of the ﬁrst phase calculated for various values of a: (a) a ¼ 0:1; (b) a ¼ 0:01; (c) a ¼ 0:001; (d) a ¼ 0:0001; (e) a ¼ 0:00001; (f) a ¼ 0:000001. The calculations were made for the ratio of the real parts of the shear modulus x ¼ 10000 and y ¼ 0:01; 10; 100; 1000.

the concave curve (Fig. 56d) becomes convex ( m001 =m01 1, m002 =m02 1). For a 0:0001 the minimum disappears and only the maximum in the vicinity of the percolation threshold remains, which also disappears for m002 =m02 ! 1 (Fig. 56d). These results show that the dependence log10 jZ0 =Z02 j on the concentration p of phases of the fractal structure becomes convex with a single maximum when ax 1—that is, when m001 m002 for x 1 (m01 m02 ). Figure 57 shows the calculated dependence of the logarithm of the relative effective viscosity Z00 =Z002 ¼ Re½m ðoÞ=Re½m2 ðoÞ on the concentration p of the ﬁrst phase demonstrating that, for a 0:001 (Fig. 57b), the relative effective

vitaly v. novikov

224 4

4 (b) 5

(a)

3

3 2 2 h" /h"

h" /h"

2

4

2 1

1

2 3 1

2 1

0

0 0

Figure 57.

0.2

0.4

0.6 p

0.8

1

0

0.2

0.4

0.6 p

0.8

1

Dependence of the logarithm of the effective relative viscosity Z00 =Z002 ¼ Re½m on the concentration p of the ﬁrst phase for the same values of x; a; and y as in

ðoÞ=Re½m2 ðoÞ Fig. 56.

viscosity Z00 =Z002 is virtually independent of a. For p < pc, the form of the dependence alters when a ! 0; however, for p > pc, it does not alter. As an example of the viscoelastic properties of an actual medium, we consider the viscoelastic properties of a charged polymeric material. Percolation properties will be exhibited by the charged polymeric composite, if the stiffness of the agglomerates of particles is greater by some orders of magnitude than the stiffness of the unperturbed polymetric compound [49]. This can occur, for example, if the polymetric compound in the vicinity of a boundary (e.g., the surface of a particle) attains the superstrong state [171–173]. The properties of agglomerates of particles (particles plus a boundary region composed of a polymetric compound) can be estimated when the region occupied by the composite is completely full [171–173]. Basing on this statement, we shall calculate the shear moduli of the charged polymeric composite. According to Ref. 172 the effective shear modulus of such a composite (real part of the modulus) at limiting doping may be estimated as m0 ¼ 2 106 [Pa] at o ¼ 0, which exceeds the shear modulus of the unperturbed polymetric compound m0p . Note that 39957:2o1:8 ¼ ½Pa 1 þ 0:40o0:8 þ 0:4183o1:6 65053o þ 0:1804o1:8 ½Pa m00p ðoÞ ¼ 1 þ 0:40o0:8 þ 0:4183o1:6 m0p ðoÞ

ð392Þ ð393Þ

In calculating of the effective shear modulus of the composite, it is assumed that the shear modulus of the particles comprising the doping compound along with its boundary region is m0f ¼ 2 106 ½Pa; m00f ¼ 0 [174] so that m f is

physical properties of fractal structures 6

225

6 5

log10 µ ¢

4

4

2

3

0

2

–2

1

–4 –4

–2

0 log10 w

2

4

Figure 58. Calculation of the frequency dependence of the effective shear (storage) modulus 0

m ðoÞ for various doping concentrations p: 1, 0.01; 2, 0.09; 3, 0.13; 4, 0.15; 5, 0.2; 6, 0.25.

independent of frequency o. The shear modulus of the polymetric compound may be determined from Eqs. (392) and (393). The calculation of the real part of the effective shear modulus m0 of a composite with fractal structure is illustrated. According to this calculation (Fig. 58) the percolation transition appears after o < 104 and at doping concentration p 0:12, i.e. for p > 0:12 in a composite with a continuous and strong skeleton composed of particles of a doping compound connected by a boundary stratum of a polymetric compound. 3.

Negative Shear Modulus

Recent studies of inhomogeneous materials containing inserts with negative stiffness indicated that such composites exhibit very interesting properties; for example, they can have much higher stiffness and higher damping coefﬁcients than the individual phases constituting them [175–178]. The analysis [175–178] of the inﬂuence of inclusions with negative shear modulus on the effective shear modulus of a composite was using the Hashin–Strikman formulae [133,134], which were obtained by assuming that the system properties do not depend on scale; that is, the system can be considered as a homogeneous medium. The idea of underlying such materials is based on the behavior of a construction which is shown on Fig. 59. This construction (a ruler deformed into the shape of the letter 00 S00 ) is in a prestressed weighted state [178], the tangential forces will not resist exterior action, and will cause bias in the direction of

vitaly v. novikov

226

Figure 59.

Buckled plastic ruler to demonstrate negative stiffness.

application of the forces. Thus, the coefﬁcient of proportionality (shear modulus) between the tangential forces and the bias will be negative. It is possible to realize such constructions if one places them in a matrix (in columns) with stable properties, for example, in a polymetric compound. Thus, it is possible to create a composite material having insertions with a negative shear modulus and a matrix–polymetric compound with a positive shear modulus. We have also studied the elastic properties of a nonuniform medium with chaotic structure in which one phase has a negative shear modulus. The analysis may be made using the fractal hierarchical structure model. The calculations were performed for a two-component, inhomogeneous medium. For simplicity, it is assumed that both phases are isotropic and that the ﬁrst phase is purely elastic whereas the second phase is elastic from the point of view of volume deformations and viscoelastic from the point of view of shear deformations. The concentration of the purely elastic phase is denoted by p. It is convenient to write the shear modulus of the second phase, m2 , in the form m2 ¼ m01 xð1 þ iyÞ

ð394Þ

y ¼ tanðj2 Þ ¼ m002 =m02 m02 =m01 ¼ x

ð395Þ

where

physical properties of fractal structures

227

and m1 ¼ m01

ð396Þ

where m01 is the (real) shear modulus of the ﬁrst phase. As mentioned earlier, fractal structures can exhibit properties different from uniform structures. To illustrate this we compare the effective shear modulus and the damping coefﬁcient of a medium with a composite material corresponding to the Hashin–Strikman formulae (referred to as the ‘‘Hashin–Strikman composite’’). In Fig. 60, the ratio of the real part of the effective shear modulus to the shear modulus of the elastic phase is shown as a function of the concentration, p, of

Figure 60. Comparison of the ratio of the effective shear modulus to the shear modulus of the elastic phase at y ¼ 0:001 as a function of the concentration of the elastic phase: (a, b) in the fractal composite and (c, d) in the Hashin–Strikman composite.

vitaly v. novikov

228

the elastic phase and the ratio x for y ¼ 103. It is assumed that the viscoelastic phase (see Figs. 60–62) has a negative shear modulus (i.e., negative real part of the complex shear modulus) and is characterized by y ¼ tan j2 ¼ 0:001 and the Poisson’s ratios of both phases (calculated from the real parts of the elastic modulii) are equal to 0:184. (The latter assumption means that the ratio of the 0 0 real parts of the shear modulii to the bulk modulus is equal to mi =Ki ¼ 0:8, where i ¼ 1; 2 denotes the phases; as a consequence, the ratio of the bulk modulus of the second phase to the bulk modulus of the ﬁrst phase is 0 0 K2 =K1 ¼ x). Apparently for the inhomogeneous fractal medium the shear modulus shows in some ranges of concentration a resonance-like behavior similar to that discussed in [175–177] whereas in the Hashin–Strikman composite there exists only one such resonance in the vicinity of the concentration p ¼ 1. To complete the illustration of the dependence of the real part of the effective 0 shear modulus m of the fractal structure composite on the ratio of the real parts of the shear modulii of the phases m02 =m01 ¼ x, the results of calculations of the 0 ratio of the modulii m =m01 for concentrations of the ﬁrst phase p ¼ 0:25 and p ¼ 0:45 are plotted against x in Fig. 61.

3 2 0.25 1 0.45 0 –1 –2 –3

0.2

0.4

0.6

0.8

1

x Figure 61. Ratio of the effective shear modulus to the shear modulus of the elastic phase for y ¼ 0:001 as a function of x in the fractal composite.

physical properties of fractal structures

229

Figure 62. Ratio of the imaginary to the real part of the effective shear modulus: (a, b) in the fractal composite and (c, d) in the Hashin–Strikman composite; the other details are the same as in Fig. 60.

According to the calculations (Figs. 60 and 61) the peaks in the dependence 0 of the real part of the effective shear modulus m of the composite of fractal structure are grouped near x 0:2 and 0:8 < x < 1. In Fig. 62 the ratio of the imaginary and real parts of the effective shear modulus (i.e., the loss tangent tan j) is shown as a function of the concentration of the elastic phase and x for y ¼ 103. The parameters of the phases are the same as in Fig. 61, again depending on x, and tan j of the fractal composite the ratio attain very large values in some ranges of concentration whereas the Hashin–Strikman composite exhibits only one large value, in the vicinity of the concentration p ¼ 1, for x in the range considered. The above comparisons show that the viscoelastic properties of the fractal composite differ qualitatively from the Hashin–Strikman one. The observed

vitaly v. novikov

230

differences can be understood by noting that the hierarchical model considered takes into account clusters of various length scales and different parameters of their ‘‘resonances’’ which are formed in the fractal composite whereas the Hashin–Strikman composite is ‘‘uniform’’ in this aspect. The nature of the peaks and the way in which they are formed depending on the effective shear modulus m , the frequency o, and the concentration p can be understood by considering a single insert with negative shear modulus 0 0 m1 ¼ m ðm ¼ xK1 Þ immersed in a medium (stabilizing matrix) of positive 0 shear modulus m2 ¼ m2 . The shear modulus m of the composite with a single inclusion can be determined from [131–133] m ¼ m2 þ

pðm1 m2 Þ 1 þ ð1 pÞb2 ðm1 m2 Þ

ð397Þ

where b2 ¼

6ðK2 þ 2m2 Þ 5m2 ð3K2 þ 4m2 Þ

ð398Þ

It follows from Eq. (397) that when m1 ¼ m0 ðm0 ¼ xK1 Þ, then m ¼ m02

pðxK1 þ m02 Þ 1 ð1 pÞb2 ðxK1 þ m02 Þ

ð399Þ

Finally, from the equation 1 ð1 pÞb2 ðxK1 þ m02 Þ ¼ 0

ð400Þ

one can determine the resonance parameters of the composite for which the peaks arise; that is, the parameters for which the external disturbance is in resonance with the inclusion parameters (being equal to its natural frequency). As the model studied constitutes a self-similar, chaotic system of clusters of various sizes, on the jth scale level, each cluster will have its own resonance parameters—that is, its own characteristic frequency, which produces the system of peaks (characteristic frequencies) in the dependence of the effective shear modulus m on the parameters of the composite. One should add that because the Hashin–Strikman composite is not very realistic and the Hashin–Strikman bounds are not particularly sharp, and since the probability function, Rðl; pÞ, used here is more or less arbitrary, the results obtained should be interpreted in a qualitative sense. We plan to study more realistic approximations in the future.

physical properties of fractal structures

231

Figure 63. Schematic for construction of a material with fractal structure.

Finally, we note that the materials with fractal structure exhibiting viscoelastic properties similar to those of the model described here can be manufactured. For example, at the ﬁrst step (the lowest size level) one produces tablets—for example, of a polymer with required inclusions. At the next step, the tablets obtained at the preceding level are used as inclusions in larger tablets. The process is continued and the hierarchy shown in Fig. 63 is obtained [160]. Conclusions. It has been shown that a hierarchical ‘‘blob’’ model when used to study viscoelastic properties of an inhomogeneous fractal medium (the fractal composite) yields results which differ qualitatively from those obtained by applying the Hashin–Strikman approximation to an inhomogeneous medium (the Hashin–Strikman composite). In particular, studies of a fractal model composed of an elastic phase (which can be regarded as a stabilizing matrix) and a viscoelastic phase with a negative shear modulus prove that the effective shear modulus and the effective loss tangent calculated exhibit much more complex behavior (more singularities) than those of the standard Hashin–Strikman model. The new singularities observed in the fractal composite are interpreted as resonances originating in (mesoscopic) clusters of various length scales which are described by different (mesoscopic) resonance parameters. Such clusters are taken into account by the hierarchical model, whereas they are neglected completely in the standard Hashin–Strikman approximation.

vitaly v. novikov

232 4.

Appendix. Fractal Model of Shear Stress Relaxation

The linear viscoelastic connection between stress (response) and velocity of strain (stimulus) can be written [132] ðt deðtÞ dt sðtÞ ¼ m1 eðtÞ þ ðm0 m1 Þ fs ðt; tÞ dt

ð401Þ

0

We shall suppose that (in order to generate the initial iteration): e ¼ e0 ZðtÞ;

de ¼ e0 dðtÞ dt

ð402Þ

where ZðtÞ is the Heaviside unit function and dðtÞ is the Dirac delta function. For a standard linear medium (the Zener model) taking into account Eq. (401) we have [131] s ¼ 2mðtÞe0

ð403Þ

t mðtÞ ¼ m1 þ ðm0 m1 Þ exp te

ð404Þ

where

where m1 ; m0 are relaxed (t ! 1) and nonrelaxed (t ¼ 0) values of the shear modulus, correspondingly; te is the relaxation time for constant deformation of the solid. Therefore the probability of f ðtÞ to change from the initial state (t ¼ 0) to a random state mðtÞ is given by the formula f0 ðtÞ ¼

m0 mðtÞ t ¼ exp m0 m1 ts

ð405Þ

In accordance with the Arrhenius formula, the relaxation time is t ¼ t0 exp½Q=kT

ð406Þ

where Q is the energy barrier between the initial and ﬁnal states, T is the temperature, k is Boltzmann’s constant, and t0 is constant. According to the fractal model and the iteration method of calculation of the shear modulus, the system will consecutively change from the initial to ﬁnal

physical properties of fractal structures

233

mð0Þ ! mð1Þ ! . . . ! mðiÞ ! mðiþ1Þ ! . . . ! mðnÞ

ð407Þ

state via

each of these changes is described by the probability of a change fi ðtÞ from the i-state to the (i þ 1) one.The function fi ðtÞ is deﬁned by an exponential relaxation time ti given by Eq. (406) and the energy barrier height Qi . According to Eq. (404), the chain of inequalities can be written tð0Þ < tð1Þ < . . . < tðiÞ < tðiþ1Þ < . . . < tðnÞ ¼ t

ð408Þ

The lower t0 and the upper t limits exist for the set of relaxation times ftðiÞ g. Therefore, in accordance with the fractal model, the set of relaxation times ftðiÞ g satisﬁes the self-similarity criterion and is bounded by the lower and upper asymptotic limits. The hierarchical chain of changes from the initial state (t ¼ 0) to the ﬁnal one (t ! 1) can be compared to a Cayley tree [25,26] (see Fig. 64). Here, the knots of the Cayley tree will correspond to static ensembles a and b which correspond to the dots in ultrametric space divided by the distance lab .

U1 n=0

l1

U2

l2

U3

n=1

n=2

l3

U4

n=3

l4

Figure 64. Schematic of a self-similar structure potential energy landscape and of the Cayley tree.

234

vitaly v. novikov

The value of lab is deﬁned by the number of steps over the levels of the Cayley tree up to the mutual knot in Fig. 64 and it yields the extent of a hierarchical link. Therefore, both the barrier height, Qab , and the relaxation time, tab , are connected with functions of the distance lab in ultrametric space, that is, Qab ¼ Qðlab Þ; tab ¼ tðlab Þ

ð409Þ

Just as in deletion of clusters the height of the barriers dividing the clusters increases and a monotonic increasing Qðlab Þ dependence is assumed. So, according to the fractal model, the collection of parallel relaxation channels is acting independently. The probability of changes between channels a and b is deﬁned by the formula fab ðtÞ ¼ exp½t=tab ; tab ¼ t0 exp½Qab =kT

ð410Þ

where Qab is the height of the energy barrier dividing the channels. The parallel action of different relaxation channels is only possible under conditions of hierarchical co-subordination of the corresponding collection of static ensembles. The hierarchical co-subordination means that the parallel net of channels of the next level having relaxation time tðiÞ does not act until channels with the given relaxation time tðiþ1Þ > tðiÞ act. Thus, the fastest processes take place ﬁrst; they correspond to surmounting the barriers of minimum height Qab . Here, static ensembles merge with each other, and the system attains a higher hierarchical level of the Cayley tree. Therefore, static ensembles a; b can combine to form clusters. Each of the clusters is characterized by the maximum height, Qab , of the barrier dividing each cluster from the others. The hierarchical co-subordination of this kind results in the deceleration of relaxation leading to the transformation of Debye’s process into more slowly decaying dependencies. Thus, according to the model of relaxation described, the dependence Qab ðlÞ and ja ðlÞ on the distance in ultrametric space can be deﬁned as Qab ðlÞ Q ln l;

ja ðlÞ ldf

ð411Þ

where df is the fractal dimension. The distribution function jðlÞ of the conditions corresponding to different points in ultrametric space reveals the degree of hierarchical linking of an inhomogeneous medium. According to Eqs. (406) and (410) and taking into consideration Eq. (411), the dependence of the relaxation time tab ðlÞ on the distance l in ultrametric

physical properties of fractal structures

235

space is tab ðlÞ ¼ t0 lQ=kT

ð412Þ

The total probability can be expressed as 1 ð

f ðtÞ ¼

jðlÞfab ðt; lÞ dl

ð413Þ

0

Taking into account that fab ðt; lÞ ¼ exp

t tab ðlÞ

ð414Þ

we have 1 ð

f ðtÞ ¼

jðlÞ exp

t dl tab ðlÞ

ð415Þ

0

which describes the relaxation due to all collections of relaxation channels, where jðlÞ is the probability of realization of a given static ensemble. Taking into account formulas (411), (412), and (415), we obtain a non-Debye relaxation law, namely, f ðtÞ tg

ð416Þ

1 df Q=kT

ð417Þ

where g¼

df is the fractal dimension of the set of relaxation times (df < 1). Thus, even simple estimations of the inﬂuence of hierarchical structure on the relaxation process yield a stress characteristic leading to anomalous relaxation of the form (416). G.

Relaxation and Diffusion Processes 1.

Non-Debye Relaxation

Anomalous (nonexponential) relaxations have long been and still are a favorite topic in the physics of inhomogeneous media [179–204]. Broadly speaking, one

vitaly v. novikov

236

may refer to three general relaxation laws encountered in experimental studies of complex systems: (i) stretched exponential [179,180,190],

b t f ðtÞ exp ; t

0 < b < 1; t > t

ð418Þ

(ii) exponential-logarithmic [181–183], h t i f ðtÞ exp B lna t

ð419Þ

(iii) algebraic decay [186], f ðtÞ

t a t

ð420Þ

where the a; b; t, and B are the appropriate ﬁtting parameters. Currently, there seems to be no quantitative microscopic theory for the cited laws [184,185,193]; moreover, sometimes even the possibility of such a theory is denied [191–193]. The main argument is that a spatial inhomogeneity (e.g., a random distribution of impurities within a matrix, or of interatomic spacings in amorphous semiconductors) will necessarily result in an extremely broad range of microscopic transition rates. Hence, a spatial disorder is expected to induce a temporal energetic disorder. Another approach to the problem of anomalous relaxations uses fractal concepts [187–189,200–203]. Here the problem is analyzed using the mathematical language of fractional derivatives [194,200–203] based on the previously mentioned Riemann–Liouville fractional differentiation operator [205–208], ðt 1 d D ½ f ðtÞ ¼ ðt tÞa f ðtÞ dt ð1 aÞ dt a

ð421Þ

c

where ðxÞ is the gamma function In spite of the reasonable success of the latter approach, use of the fractional derivative as represented by Eq. (421) renders difﬁcult the interpretation of differentiation procedures (e.g., the nonzero value of a fractional derivative of a constant), as well as their relevance to the assumed fractal ensemble. One may also note that so far fractional derivatives have been analyzed in essentially

physical properties of fractal structures

237

phenomenological terms; moreover, the evolution equations based on fractional derivatives have been constructed more by intuition (guessed), rather than obtained from ﬁrst principles. The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208–215). Apparently, Blumen et al. [189] were the ﬁrst to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200–203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like proﬁle of dielectric permittivity, etc.), and the fractal dimension of a medium. In this context, attempts to construct fractional derivatives and to clarify their relevance to the assumed fractal ensemble are believed be still relevant for the treatment of anomalous relaxations. In our previous papers [215], the analysis of the classical problem of polarization of an inhomogeneous medium permitted us to establish the relationship between anomalous relaxation and the dimension of a temporal fractal ensemble which characterizes a nonequilibrium state of a medium. Thus, the main aim of the present work is the further extension and generalization of these results in the context of a fractal model. Dielectric Relaxation. The potential of fractional derivatives in the context of anomalous relaxation will become evident, and the relationship between the exponents b and a in Eqs. (418)–(420) and the fractal dimension df will be established, in the subsequent treatment of the classical problem of polarization PðtÞ of a dielectric medium (which is, in fact, equivalent to the general problem of relaxation of the internal parameters of a nonequilibrium phase). Assume that PðtÞ contains two contributions [185], PðtÞ ¼ P0 þ P1 ðtÞ

ð422Þ

where the ﬁrst qualitity, the static polarization ðP0 ¼ w0 EÞ, varies exactly (at least, with negligibly small retardation) as the applied ﬁeld E, while the timedependent quantity, P1 ðtÞ, is retarded. Let P ¼ w1 E be the upper limit (at ﬁxed E ); then, the larger the amplitude (w1 E PðtÞ), the greater the instantaneous rate of approach P1 ðtÞ to this limit. Hence, the corresponding relaxation equation

vitaly v. novikov

238 may be written as

dP1 ðtÞ 1 ¼ ðwE P1 ðtÞÞ; dt t w ¼ w1 w0

ð423Þ

where t is the relaxation time. Thus h t i PðtÞ ¼ P0 þ P1 ðtÞ ¼ w0 þ w 1 exp E t

ð424Þ

(for a constant ﬁeld E), and PðoÞ ¼ P0 þ P1 ðoÞ ¼ ½w0 þ w=ð1 iotÞ E

ð425Þ

(for an alternating ﬁeld E ¼ E0 eiot ). Therefore, the dielectric permittivity of a medium may be deﬁned, ﬁnally in linear response, as [185] e ¼ e1 þ

e0 e1 1 þ iot

ð426Þ

where e1 ¼ lim e; e0 ¼ ejo¼0 . o!1 The next issue to concern us will be anomalous relaxation in which the ‘‘smearing out’’ of a relaxation spectrum (i.e., the deviation of complex susceptibility from its Debye form) is associated with the concept of a relaxation time distribution. As is well known, this concept implies an assembly of dipoles with a continuous distribution of relaxation times of Eq. (379). Consider Fro¨hlich’s relaxation model [221] which is based on the relaxation time distribution concept. It is usually assumed that the relaxators are homogeneously distributed along the height of potential barrier U; however, this assumption is not strictly correct. As can be seen from Fig. 65, many shallower minima may exist between two main minima; therefore, the system is involved in a continuous chain of transitions r1 ! r2 ! r3 ! . . . ! rk ! rn between adjacent minima (Fig. 65). Let Sl ðtÞ be the probability of transition from the lth minimum into the (l þ 1)th one, so that SðtÞ ¼ e where the relaxation time is

ð427Þ

l

tl ¼ t0 exp

where Ql is the barrier height.

tt

Ql kT

ð428Þ

physical properties of fractal structures

239

U

Q1

r1 r2 r3

Figure 65.

rk

rn

r

Schematic of the potential barrier landscape.

Implicit in the relaxation time distribution concept is the assumption of comparable magnitudes of barrier heights Ql (Fig. 65); hence, the characteristic times tl ¼ tlþ1 tl of transitions over the barriers will be also of comparable magnitudes. Therefore, during a time t ntl we have for n barriers with a probability 1 S(t) so that SðtÞ ¼

n Y

Sl ;

l¼1

tl Sl ¼ exp tl

ð429Þ

Assuming tl ¼ nt , it becomes clear that the dispersion of the intervals tl may be neglected in the limit of n 1; as a result, the Eq. (429) will regain its Debye form with the mean relaxation time hti deﬁned as hti1 ¼

n 1X t1 n l¼1 l

ð430Þ

Thus, the chain of transitions considered above is effectively reduced to exponential, Debye-like relaxation with the mean relaxation time hti. In other words, the concept of a relaxation time distribution implies Debye-like relaxation of a system. However, it is evident that the relaxation will become nonexponential, should a system be characterized by a complex susceptibility of, say, Cole–Cole type.

vitaly v. novikov

240

Thus, it can be concluded that the relaxation time distribution concept applies to Debye-like relaxation (even though its frequency dependence may be smeared-out), whereas it becomes inapplicable for still slower relaxation patterns. In the latter situation, the distribution of relaxation times over a selfsimilar, fractal ensemble seems a physically more reasonable assumption. As is well known, the fractality of geometrical objects implies their non-integer dimension; however, a more exact deﬁnition of the fractal concept with respect to the ensemble of relaxation times is in order. As proved by Nigmatullin [200–203], fractional derivatives in time in Newton’s equations imply that the interactions between a system and an external ﬁeld are not continuous but occur at discrete time intervals. In this context, the fractality of an ensemble of relaxation times simply means that the relaxation is not a single process with a unique relaxation time, rather it is a series of successive relaxation events with different relaxation times. Let us consider now the nonequilibrium state of a fractal-like medium assuming that this nonequilibrium state is characterized by many events such that a subsequent event is separated by a certain time interval ti from a previous event. In this case, some intervals will be eliminated from a continuous process of system evolution by a deﬁnite law. Assume that such a process is caused by a temporal fractal state of dimensionality df ; the corresponding relaxation equation can be written as Da ½P1 ðtÞ ¼

1 ðwE P1 ðtÞÞ ta

ð431Þ

and rearranged as ½1 þ ðtDÞa P1 ðtÞ ¼ wE

ð432Þ

The latter Eq. (432) can be solved using the Laplace transform [205–208], thus ½1 þ ðtsÞa P1 ðsÞ ¼

wE ; p

1 ð P1 ðsÞ ¼ est P1 ðtÞ dt

ð433Þ

0

yielding P1 ðsÞ ¼

wE 1 s 1 þ ðtsÞa

ð434Þ

physical properties of fractal structures

241

1 X 1 ðstÞa ¼ ¼ ð1Þn ðstÞaðnþ1Þ 1 þ ðtsÞa 1 þ ðstÞa n¼0

ð435Þ

since

the solution of Eq. (434) in the time domain will have the following form: aðnþ1Þ 1 X ð1Þn tt ð436Þ P1 ðtÞ ¼ wE ½aðn þ 1Þ þ 1 n¼0 where ðxÞ is the gamma function. Therefore, " aðnþ1Þ # 1 X ð1Þn tt PðtÞ ¼ P0 þ P1 ðtÞ ¼ w0 þ w E ½aðn þ 1Þ þ 1 n¼0 When a ¼ 1 in Eq. (437), Eq. (424) is recovered so that " ðnþ1Þ # 1 X ð1Þn tt PðtÞ ¼ w0 þ w E ½n þ 2 n¼0 h t i E ¼ w0 þ w 1 exp t

ð437Þ

ð438Þ

(in the derivation, the standard Eq. (439) has been used): 1 X

ðzÞn ¼ expðzÞ; ½ ð n þ 1Þ n¼0

z¼

t t

ð439Þ

Thus, the crossover from a strictly exponential to an anomalous relaxation pattern can be associated with the change of a continuous distribution of relaxation times (a ¼ 1) into a fractal-like one (0 < a ¼ df < 1). Thus, the solution of Eq. (437) in the time domain is PðtÞ ¼ w0 E 1

1 X ð1Þn n¼0

t an ! t

½an þ 1

¼ w0 E 1 Ma;1 ðzÞ

ð440Þ

where Ma;1 ðzÞ the Mittag–Lefﬂer function is Ma;g ðzÞ ¼ In our case, g ¼ 1.

1 X

zn ; ½an þ g n¼0

z¼

t a t

ð441Þ

vitaly v. novikov

242

It is convenient to use Fox functions (generalized Mellin–Barnes integras) when solving equations with fractional derivatives because Laplace and Fourier transformations for Fox functions may be expressed via Fox functions with given parameters. The connection of Mittag–Lefﬂer functions with Fox functions is as follows [216,217]:

ð0; 1Þ 1;1 z Ma;g ðzÞ ¼ H1;2 ð0; 1Þ; ð1 g; aÞ

ð442Þ

Then Eq. (440) becomes

1;1 PðtÞ ¼ w0 E 1 H1;2 z

ð0; 1Þ ð0; 1Þ; ð1 g; aÞ

ð443Þ

If a ¼ 1, then from Eq. (443) we obtain PðtÞ ¼ w0 E 1

1 X ð1Þn n¼0

t n ! t

½ n þ 1

t ¼ w0 E 1 exp t

ð444Þ

If a 6¼ 1 then according to Eq. (443) it follows that PðtÞ 1

t a t

;

t t

!1

ð445Þ

and PðtÞ

t a t

;

t t

!0

ð446Þ

Thus, the solution of an equation with fractional derivatives (431) describes relaxation in dielectrics having a power law dependence in asymptotic limits such as Eqs. (445) and (446). It follows from Eq. (447) that

PðtÞ w0 þ

w t a E ða þ 1Þ t

ð447Þ

which can be compared with Eqs. (418)–(420). For an alternating ﬁeld, Fourier transformation of Eq. (431) yields (s ¼ io)

PðioÞ w0 þ

w E ð1 þ iotÞa

ð448Þ

physical properties of fractal structures

243

and the dielectric permittivity is e ¼ e1 þ

e0 e1 1 þ ðiotÞa

ð449Þ

Equation (449) describes a frequency dependence of the Cole–Cole type. The real Re eðoÞ and imaginary Im eðoÞ parts of the total dielectric permittivity in Eq. (449) are, respectively, 2

3 h pai ð1 ZÞ 1 þ ðotÞa cos 6 7 2 ReeðoÞ ¼ e0 4Z þ 5; pa a 2a þ ðotÞ 1 þ 2 ðotÞ cos 2 2 3 h pai a ðZ 1Þ 1 þ ðotÞ sin 6 2 7 ImeðoÞ ¼ e0 4Z þ 5 pa a þ ðotÞ2a 1 þ 2ðotÞ cos 2

ð450Þ

Therefore, the dielectric loss tangent is 2 6 tan d ¼ ðZ 1Þ4

3 ðotÞ

a

7 5 pa 2a þ ðotÞ 1 þ 2 ðotÞ cos 2 a

ð451Þ

where Z ¼ ee10 . Equations (450) and (451), respectively, have been used to construct plots of the real, Re eðoÞ=e0 (Fig. 66b), and of the imaginary, Im eðoÞ=e0 (Fig. 66c), parts of the complex dielectric permittivity, as well as tan d(Fig. 66a) as a function of log ot for a medium with Z ¼ ee10 ¼ 10. As is easily veriﬁed, the relaxation spectrum pattern strongly depends on the dimension of a temporal fractal ensemble a ¼ df . Now we will try to summarize more of our results. Instead of (3.210), we consider yet another operator of fractional differentiation.

ðt

a

v þD Þ ¼ ðt Þ DaðvnÞ n n¼0 a v

1 X

v where is the binomial coefﬁcient. n

a n

ð452Þ

vitaly v. novikov

244 (a)

(b)

1 0.8 0.6 0.4 α 0.2

–2 0 log10 (wt)

2

Re

( ε(w) ε 0

(

4 3 tan(d) 2 1 0

10 7.5 5 2.5 0 –2 log10 (wt)

0

1 0.8 0.6 0.4 α 0.2 2

0

0 (c)

(

(

Im

ε(w) ε0

10 7.5 5 2.5 0

1 0.8 0.6 0.4 α 0.2

–2

0 log10 (wt)

2

0

Figure 66. Dispersion dependence of tan(d) (a), Re eðoÞ=e0 (b), Im eðoÞ=e0 (c) for different values of the parameter a.

By Eq. (452), the complex susceptibility may be written as ðta þ Da Þv ½w expðiotÞ ¼

w 0 E0 expðiotÞ tav

ð453Þ

The solution of Eq. (453) yields the standard deﬁnition of complex susceptibility, wðioÞ ¼ w1 þ

w 0 w1 v ð1 þ ðiotÞa Þ

ð454Þ

which is identical to the empirical Havriliak–Negami law [184]. Here the dielectric permittivity will be eðioÞ ¼ e1 þ

e0 e1 v ð1 þ ðioÞa Þ

ð455Þ

physical properties of fractal structures

245

with real and imaginary parts, 13 ap sin 6 C7 B 2 cos4v arctan@ A5 ap a cos þ ðotÞ 2 ; Re½eðioÞ ¼ e1 þ ðe0 e1 Þ ap 2a 1 þ ðotÞ þ2ðotÞa cos 2 0 132 ap sin 6 B C7 2 sin4v arctan@ A5 ap a þ ðotÞ cos 2 Im½eðioÞ ¼ ðe0 e1 Þ ap 2a 1 þ ðotÞ þ2ðotÞa cos 2 2

0

ð456Þ

and 2

tan d¼

0

13 ap sin 6 B C7 sin4v arctan@ ap 2 A5 a cos þ ðotÞ 2 2 0

13 ap h i sin e1 ap 6 C7 B þ cos4v arctan@ ap 2 1þ ðotÞ2aþ2ðotÞa cos A5 a 2 e0 e1 cos þ ðotÞ 2 ð457Þ

The relaxation equation for an initially polarized dielectric is ðta þ Da Þv PðtÞ ¼ 0

ð458Þ

with solution

PðtÞ ¼ P0 tav1

1 X n¼0

t an

ð1Þn

vðv þ 1Þ ðv þ n 1Þ t n! ðaðn þ vÞ þ 1Þ

ð459Þ

where P0 is the initial polarization (this solution also diverges as t ! 1). The case of a dielectric without initial polarization (when the ﬁeld is switched on at t ¼ 0) is described by Eq. (460), namely, ðta þ Da Þv PðtÞ ¼

w0 E tav

ð460Þ

vitaly v. novikov

246 U

∆U2 ∆U1 ∆U0

r Figure 67.

Schematic of the potential barrier landscape under the action of an external electrical

ﬁeld.

with solution t aðnþvÞ vðv þ 1Þ ðv þ n 1Þ t ð1Þn PðtÞ ¼ w0 E n! ð a ð n þ v Þ þ 1Þ n¼0 1 X

ð461Þ

Fractal Model. We consider in more detail the issue of relaxation times relevant to our fractal model of anomalous relaxation. As appears from the potential energy landscape for a system under the action of an external electrical ﬁeld (Fig. 67), the energy differences between minima separated by energy maxima of different levels of self-similarity diminish, the larger the number of a selfsimilarity level. In view of the standard deﬁnition of a relaxation time, t eU=kT , one may write the following chain of inequalities, t0 > t1 > t2 > . . . > ti > tiþ1

ð462Þ

It is easy to see that these inequalities meet the model requirement of the difference between relaxation times in so far as both the width and the height of the energy maxima are assumed to decrease, the larger the number of a selfsimilarity level (Fig. 65). Note, however, the existence of the upper limit for an ensemble of relaxation times; that is, the self-similar process of growing complexity of the potential energy landscape is halted at a certain level N < 1. Thus, the proposed fractal relaxation model satisﬁes the criterion of selfsimilarity; moreover, its validity is restricted by the asymptotic lower and upper

physical properties of fractal structures

247

limits. Let us now analyze the physical meaning of a self-similar potential energy landscape. Assume that a system evolves by traversing a succession of potential barriers, each next one of greater height. In this context, initial relaxation processes with short relaxation times are assumed to be followed by those with ever increasing relaxation times. Now consider a relaxing ensemble of N < 1 particles. Let this system consist of smaller subsystems (clusters), each of which, in turn, consists of still smaller subsystems (subclusters), and so on. In principle, this kind of tessellation could be repeated down to inﬁnitesimal scales; however, as mentioned above, the accepted model requires that such a self-similar increase of system complexity should halt at a certain level. In other words, the relaxation at the (n þ 1)st level would not set in until a certain fraction of particles at the previous nth level would have relaxed (here it is implicitly assumed that the enumeration of relaxation levels starts at the lower limit of self-similarity for a subcluster comprising a minimum number of relaxing particles, n ¼ 1, and attains the upper limit of self-similarity for a cluster comprising all smaller subclusters, n ¼ N, where N is the total number of hierarchical levels. Let oq be the probability of existence of each relaxation level corresponding to the kth statistical ensemble; then, the probability for a system to pass from the qth to the pth relaxation level during a time t may be deﬁned as t Sqp ðtÞ ¼ exp tqp

ð463Þ

where tqp - is a relaxation time deﬁned as tqp

Qqp ¼ t0 exp kT

ð464Þ

and Qqp is the barrier height between levels q and p. The probability for a system to attain the level n N after time t will be SðtÞ ¼

n X

oq op Sqp ðtÞ

ð465Þ

q¼1;p¼1

that is, the function SðtÞ is assumed to account for contributions from all available relaxation channels. Therefore, the relaxation process may be speciﬁed, provided the functions oq and Qqp are known. We remark that the self-similar potential energy landscape (Fig. 68) resembles a Cayley tree (Fig. 68), provided each minimum at a certain self-

vitaly v. novikov

248 U1

·

r1

n1

· · · ·

r2

n2

r3

n3

U2

U3

Figure 68.

Schematic of a self-similar potential energy landscape and of the Cayley tree.

similarity level of the former is associated with a branch of the same number on the latter [223]. The coordinates of branches on the Cayley tree comprise the ultrametric space; the metric of this space is speciﬁed by the interbranch distances which are deﬁned as the numbers of steps between the branches and a common origin (for example, the distances between branches a and b and between a and c in Fig. 6b are unity and two, respectively). Figure 68 indicates that the statistical ensembles fq; pg may merge into clusters, each of which is characterized by the maximum barrier height Qqp separating this particular cluster from its neighbor. In view of the correspondence between the ensembles fq; pg and the branches of the Cayley tree referred to above, the former may be also characterized by points fq; pg in the ultrametric space separated by the distances lqp . In this context, the barrier heights Qqp , as well as the corresponding relaxation times tqp are functions of distances lqp in the ultrametric space. Insofar as elimination of clusters from this space may be achieved by increase of the corresponding barrier heights, one may conclude that QðlÞ should be a smoothly increasing function. It follows from the above analysis that parallel contributions of various relaxation channels may be possible only via hierarchical subordination of the corresponding series of statistical ensembles. Now, the smallest statistical ensembles (subclusters) merge, and the system passes on to a higher hierarchical level of the Cayley tree (Fig. 68). After passing over barriers of higher height, Qqp , the newly created clusters merge again into larger entities (superclusters) corresponding to the next hierarchical level, and so on. It is this kind of hierarchical subordination which is believed to be the main cause of the

physical properties of fractal structures

249

critical slowing down of the relaxation process which manifests itself as the transformation of exponential, Debye-like behavior into a slower, nonexponential decay. Within the framework of phenomenological approach, consider possible patterns of temporal dependence SðtÞ at variable distribution upper bounds oðlÞ and QðlÞ. Assume that the descending tails of the probability distribution may be approximated as l ow ðlÞ exp ; l0

os ðlÞ ldf

ð466Þ

where the ﬁrst and the second functions apply to weakly hierarchical and to strongly hierarchical systems, respectively (here l0 and D are positive parameters). The reason is that the former exponential function ow ðlÞ decays at distances l l0 and, therefore, links only a limited number of hierarchical levels, while the latter, slowly changing power dependence os ðlÞ accounts, in effect, for the entire set of levels available. The landscape barrier heights QðlÞ will be approximated by three major types of functions, namely, Ql ðlÞ ¼ Q ln l;

Qp ðlÞ ¼ Qla ;

Qe ðlÞ ¼ Qel

ð467Þ

where Q is the characteristic barrier height, and a ¼ const > 0. The asymptotics at t ! 1 derived by the saddle-point method on substitution of Eq. (467) into eqs. (464) and (463), and of the result obtained using Eq. (466) and eq. (465), are shown in Table XI. It can be seen that all relaxation laws derived in this fashion are non-Debye-like, the weakest slowing down corresponding to a logarithmic growth of landscape heights in weakly hierarchical systems (i.e., the Kohlrausch–Williams–Watts stretched exponential law [179,180]). The descreasing function SðtÞ transforms into a power law as the hierarchical links become stronger and the increase of peaks on the

TABLE XI Asymptotics of the Correlator SðtÞ at t ! 1 SðtÞ

Ww Ws ðlÞ

Ql ðlÞ

b exp tt0 ;...;

1 Q : b¼ 1þ kT tg ; g ¼

df T Q

Qp ðlÞ

1=a t exp kT Q ln t0

Q kT

ln tt0

df =a

Qe ðlÞ

kT Q

ln tt0

1=l0

h idf t ln kT Q ln t0

250

vitaly v. novikov

landscape follows a power law. The alternative cases of exponential and powerlaw increases of the barrier heights in weakly and strongly hierarchical systems, respectively, would correspond to a logarithmic relaxation law, as described elsewhere [185]. Finally, a double-logarithmic slowing down (i.e., the virtual arrest) of the relaxation process is expected for strongly hierarchical systems characterized by exponential growth of barrier heights. It is instructive now to discuss the relevance of these model predictions to the structural features of inhomogeneous media. Thus we deﬁne a statistical ensemble as a set of particles in a state of constant motion. The model of hierarchical subordination was then constructed by selecting groups of ensembles from the entire statistical set available ; hence, a similar tessellation procedure should be applied to the ensemble of particles comprising an inhomogeneous medium. A rule for selecting particular clusters and subclusters in the latter should be developed. In simple terms, solid bodies may be characterized by two main features, by the pattern of their mutual arrangement (packing) of particles, and by the pattern of interparticle interactions. These features are complementary, rather than independent; nevertheless, it is the former which will be used as a criterion for partitioning the entire inhomogeneous medium system into smaller subsystems. According to current concepts, an inhomogeneous medium may be considered as a structureless body at large length scales (i.e., above the characteristic correlation length x), whereas regions of a short-range order are assumed to exist at smaller scales (below x). In this context, it is the regions of short-range order which will be identiﬁed as the primitive (1st level) clusters; a new set of primitive clusters will be deﬁned as the 2nd level clusters, and so on. Thus, the nth level cluster corresponding to the statistical ensemble of nth hierarchical level may be constructed using such a process of self-similar increase in complexity. The ‘‘blob’’ model based on these concepts has been introduced elsewhere [196]. It is now possible to establish a correspondence between the functions oðlÞ and QðlÞ and the accepted model of short-range hierarchy order. From Fig. 69, the primitive cluster comprises 7 particles, the next one at the 2-nd level 72 ¼ 49 particles, and the lth level cluster 7l particles (here l is the level number). Obviously, for a level l comprising N ¼ 7l particles, the level number may be deﬁned as l ¼ a lnðnÞ; a

1

¼ ln 7

ð468Þ

In the general case, l ln N; therefore, eq. (466) may be rewritten as ow ðNÞ N df ;

os ðNÞ ðln N Þdf

ð469Þ

physical properties of fractal structures

Figure 69.

251

Schematic of the self-similar structure of a dielectric medium.

The results obtained imply a rather small probability of large-size clusters comprising many particles for a weakly hierarchical system; the reverse is true for a strongly hierarchical system. Thus, the structural implications of the concept of strongly and weakly hierarchical systems become more transparent. The physical meaning of the function accounting for the increase of potential barrier height may be clariﬁed by considerating the microscopic kinetics of dielectric relaxation of a hierarchical structure. Assume that the initial polarization is induced in the latter, and that single particle and clusters of particles interact through dipole and multiple interactions, respectively. The relaxation processes commence after the ﬁeld is switched off at t ¼ 0. Initially, the relaxation sets in at the primitive, 1st level, insofar as the elementary dipoles can easily cross the potential barrier created by their neighbors. In contrast, for the 2nd-level clusters the barrier heights created by neighboring clusters with preferential orientation of the majority of the dipoles are so high, that no such relaxation can occur. Therefore, relaxation at the 2nd level may set in only after completion of relaxation by the majority of the dipoles at the primitive level. In other words, it is only after sufﬁcient weakening of the multipole correlations of a given cluster with its neighbors that its transition into a depolarized state becomes possible. Such self-similar processes occur in succession at the next higher level, and so on. Thus, the form of the function QðlÞ depends not only on the number of dipoles in a cluster but on the form of the multipole potential and the temperature. Finally, in view of Eq. (468), Eq. (467) can be rewritten as Ql ðNÞ ¼ Q ln½ln N ;

Qp ðNÞ ¼ Qðln N Þa ;

Qe ðNÞ ¼ QN

ð470Þ

252

vitaly v. novikov

The model considerations outlined above permit one to clarify the results presented in Table XI. For example, from the explicit deﬁnition of the Kohlrausch–Williams–Watts stretched exponent on the barrier height

Q 1 b¼ 1þ kT

ð471Þ

it can be Inferred that b ! 1 for T 0; that is, for sufﬁciently high temperatures the anomalous relaxation becomes Debye-like. Physically, this effect may be associated with the increase of the ﬂuctuation density of dipole reorientations; as a result, the relaxation of all available dipoles has already occurred at the 1st level, so that the entire chain of remaining parallel relaxation channels becomes ineffective. In like manner, it becomes easy to predict the pattern of anomalous, nonexponential relaxation at times, shorter than the relaxation time t1 at the lowest (i.e., 1-st) self-similarity level. This level may be considered as the primitive one (in a sense that it cannot be further tesselated into subclusters); hence, the relaxation should be of a classical, Debye-like type, PðtÞ et t

ð472Þ

whatever the pattern of nonclassical relaxation at longer times. Conclusions. The power of the fractional derivative technique has been demonstrated using as example the derivation of three known patterns of anomalous, non-exponential dielectric relaxation of an inhomogeneous medium in the frequency domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of microstructure of inhomogeneous media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. In this context, the regions of short-range order are identiﬁed as the primitive (1st-level) clusters; a set of primitive clusters are deﬁned as 2nd-level clusters, and so on. Different relaxation functions are derived assuming that the actual (physical) ensemble of relaxation times is conﬁned between the upper and lower limits of self-similarity. In this respect, the temporal fractal differs from a geometrical fractal (e.g., Cantor dust) for which only an upper limit (i.e., the initial segment before its subdivision) is assumed to exist. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the

physical properties of fractal structures

253

relaxation should be of a classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times. The methods and models used when describing dielectric relaxation in the previous chapter can also be used to describe diffusion of a Brownian particle. We mention that diffusion has been a focus of great attention for scientists for more than 100 years and has a vast bibliography. Recently, excellent reviews have been published [222–238]. We will only cite a few [239–251]. 2.

Anomalous Diffusion

Brownian Motion. The incessant haphazard motion of microscopic particles in a colloidal suspension is called the Brownian motion, and the particles themselves Brownian particles. The Brownian motion was ﬁrst experimentally discovered and investigated by the botanist Robert Brown (1773–1858) in 1827, who observed pollen suspended in water through a microscope. The pollen particles moved randomly, and the average value of the movements in unit time did not change for the same parameter values of the ﬂuid—for example, its temperature. Once the temperature increased, the Brownian motion became more intense. To describe the Brownian motion of a particle in a ﬂuid, Langevin assumed [250,251] that a random force, x; affecting the particle having its origin in the particles of the ﬂuid exists and that its autocorrelation function is hxðt1 Þxðt2 Þi ¼ 2kTrdðt1 t2 Þ meaning that x is Gaussian white noise. x must also obey [247,250] Wick’s theorem (Isserlis’s theorem) for the averages over the realizations of the trajectories of the particles. The equation of motion of a Brownian particle in the direction of the chosen axis, X, can then be written as mx00 þ rx0 ¼ x

ð473Þ

where m is the mass of the Brownian particle, r is the viscous drag coefﬁcient of the Brownian particle resulting from the ﬂuid. Now multipling Eq. (473) by x and taking account of the equation xx00 ¼

dðxx0 Þ x02 dt

ð474Þ

we obtain m

dðxx0 Þ mx02 þ rxx0 ¼ xx dt

ð475Þ

vitaly v. novikov

254

Averaging Eq. (475) over a large number of Brownian particles, we have m

dh xx0 i m x02 þ r h xx0 i ¼ h xxi dt

ð476Þ

Due to the haphazard motion of the Brownian particle, it may be assumed that the coordinate, x, and the force, x, are statistically independent, and the average value of their product is equal to zero: h xxi ¼ 0: Moreover, one may write assuming that the velocity distribution has reached equilibrium instantaneously 1 02 1 m x ¼ kT 2 2

ð477Þ

Then Eq. (476) becomes m

d h xx0 i þ r h xx0 i ¼ kT dt

ð478Þ

If at time t ¼ 0 the Brownian particle is at a point with coordinate x ¼ 0, then h xx0 ijt¼0 ¼ 0: Hence, the solution of Eq. (478) can be written as h xx0 i ¼

t kT 1 exp r t

ð479Þ

where the inertial relaxation time, t ¼ r=m, is small compared with the time of observation of the Brownian particle. For t >> t, which corresponds to the stationary Brownian motion (i.e., the Maxwellian distribution of velocities has been attained by the Brownian particle), we have h xx0 i ¼

kT r

ð480Þ

Using the equation 1 dðx2 Þ 2 dt

ð481Þ

d x2 kT ¼2 dt r

ð482Þ

xx0 ¼ we may transform formula (480) into

physical properties of fractal structures Integrating (482) with the initial condition mean-square displacement is 2 x ¼ 2Dt;

255

2 x jt¼0 ¼ 0 shows that the

t>0

ð483Þ

where the diffusion coefﬁcient of the Brownian particle is D¼

kT r

ð484Þ

Formula (483) was ﬁrst obtained by Albert Einstein (1879–1955) in 1905 and bears his name. Independently of Einstein, the theory of the Brownian motion was developed by Marian von Smoluchowski (1872–1917) in 1905–1906. The expression obtained by him agrees with formula (483) with a constant multiplier equal to one. Thus, it follows from formula (483) that the mean-square distance traversed by a Brownian particle is proportional to the observation time. In accordance with formula (484), the mean-square distance traversed by a Brownian particle in equal periods of time increases with temperature T and decreases with the viscosity r. The above formulae were experimentally checked by Perrin in 1908. He measured the distances traversed by Brownian particles for equal periods of time with a microscope. Based on his experiments and formulae (483) and (484), Perrin was able to deﬁne the Boltzmann constant, k, and calculated the value of Avogadro’s number NA, both closely approximating their values obtained by other methods. Diffusion (Smoluchowski) Equation for Brownian Particles. The concentration pðx; tÞ of free Brownian particles at time t is described by the diffusion equation [250] qpðx; tÞ q2 pðx; tÞ ¼D qt qx2

ð485Þ

D is the diffusion coefﬁcient. The diffusion equation (485), which is usually known as the Smoluchowski equation (a particular form of the Fokker–Planck equation), can be obtained using the equation of continuity in one dimension qpðx; tÞ qjðx; tÞ ¼ qt qx

ð486Þ

256

vitaly v. novikov

and the Fick law jðx; tÞ ¼ D

qpðx; tÞ qx

ð487Þ

If at t ¼ 0 the particle is at the origin, x ¼ 0, then the initial concentration is pðx; 0Þ ¼ dðxÞ

ð488Þ

and the solution of Eq. (485) is then 1 x2 pðx; tÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ exp ; 4Dt 2 pDt

t>0

ð489Þ

or pðx; tÞ ¼ Ct2 f ðx; Ct2 Þ 1

1

ð490Þ

where C ¼ p1ﬃﬃDﬃ, and the function

1 z 2 f ðzÞ ¼ pﬃﬃﬃ exp 2 2 p

ð491Þ

is the centered Gaussian distribution. N(0,2). According to Eq. (491), the variance of the distribution is 2 x ¼ 2Dt

ð492Þ

The diffusion equation can also be written as the forced equation qpðx; tÞ q2 pðx; tÞ ¼D þ dðxÞdðtÞ qt qx2

ð493Þ

where now of course the initial conditions are zero. This form is useful in the generalization of Eq. (485) to fractional diffusion. The investigation of the diffusion equation (485) began when Louis Bachellier (Jules Poincare’s student) wrote his thesis in 1900. It was called ‘‘The Theory of Speculations’’ and was devoted to the evolution of the stock market. Many of the most famous scientists have contributed to our knowledge of diffusion processes, amongst them Einstein, Langevin, Smoluchowski, Fokker, Planck, Levy, and others.

physical properties of fractal structures

257

The investigation of Brownian particle motion led to the development of the mathematical theory of random processes which has been widely adopted. A great contribution to the mathematical theory of Brownian motion was made by Wiener and Kolmogorov; for example, Wiener proved that the trajectory of Brownian motion is almost everywhere continuous but nowhere differentiable, Kolmogorov introduced the concept of forward and backward Fokker-Planck equations. As far as contemporary research is concerned in recent years, many works devoted to anomalous diffusion have appeared [231–242]. The main difference of anomalous diffusion from normal diffusion is that the variance of the distribution obeys: 2 1 ð494Þ H 6¼ x tH ; 2 To describe anomalous diffusion, some additional information about the diffusion process is needed, for example: speciﬁc physical models for the jump processes; the imposition of certain conditions, for example, the self-similarity condition, according to which pðx; tÞ ¼ tH Fðx; tH Þ

ð495Þ

where H is not equal to 1=2, and f ðxÞ is not a Gaussian distribution. We consider the main results which describe anomalous diffusion. Anomalous Diffusion. The standard random walk process underlying the Brownian motion assumes that the walker executes a step of ﬁxed length in a random direction at each tick of a system clock [226]. After a very large number of steps the associated random variable—namely the position of the walker a time t after he departed from the origin—will be a gaussian random variable by virtue of the central limit theorem [226] and so leads to the normal diffusion discussed above. A generalization of this process is the continuous time random walk introduced by Montroll and Weiss [231]. Such a walk is deﬁned to be passing [231] an alternation of steps and pauses with both step length and pausing (waiting) times instead of being ﬁxed in same way being governed by a step length probability distribution and a pausing time probability distribution. Thus the step length and the pausing time are random variables which are not necessarily independent although we shall assume that they are here to the diffusion limit of a large number of steps, anomalous diffusion will occur either if no average pausing time or no mean-square step length exists or if both do not exist. We consider the CTRW using the terminology of renewal theory [236].

258

vitaly v. novikov

Consider a sequence of independent identically distributed time random variables, T1 ; T2 ; . . . ; the probability that any single one of the T s satisﬁes t T t þ dt will be denoted by qðtÞdt where qðtÞ is the pdf of the time between successive steps or the pausing time density. The time intervals TðnÞ at which the nth step is taken are given by the random sum TðnÞ ¼

n X

Tj ;

Tð0Þ ¼ 0

ð496Þ

j¼1

and are called renewal (because the jump process rests) times, fTðnÞg is the set of random variables constituting the renewal process, the individual Tj are called pausing (expectation) times since Eq. (496) is a sum of independent random variables a number of relations may be simpliﬁed when expressed in terms of characteristic functions as we shall do later. Now let us suppose that in a time interval ½0; t the random walker executes n steps, and then the displacement of the walker is the random sum XðtÞ ¼

NðtÞ X

Rj

ð497Þ

j¼1

where the step lengths Rj are independent random variables each Rj having the same step length pdf pðxÞ. As an illustrative example, suppose NðtÞ is a Poisson process; then the probability that walker has moved exactly n steps after an elapsed time t is Pð N ð t Þ ¼ nÞ ¼

ðmtÞn expðmtÞ n!

ð498Þ

where m is the mean rate at which steps occur, the pdf pðxÞ of the random sum XðtÞ (i.e., the displacement of the walker at time t) is then pð x; tÞ ¼

1 X ðmtÞn n¼0

n!

expðmtÞpn ðxÞ

ð499Þ

pn ðxÞ is the pdf for the position of the random walker at the step n, p0 ðxÞ means that the walker was deﬁnitely at zero at time zero and stays there until t ¼ T1 (the time at which the ﬁrst jump occurs) p1 ðxÞ ¼ pðxÞ (the jump length pdf) for the position of the walker at step one, and pn ðxÞ ¼ pn ðxÞ (since the jump lengths are independent identically distributed random variables is the pdf for the position of the walker at step n).

physical properties of fractal structures

259

In passing we remark that Feller [218] replaces the process fX1 ¼ R1 , X2 ¼ X1 þ R2 , X3 ¼ X2 þ R3 ; . . .g with independent increments Rj by the Markov chain fX1 ; X2 ; X3 ; . . .g and calls this process pseudo-Poissonian. Subordinate Processes. The pdf of the pseudo-Poissonian process is deﬁned as pð x; tÞ ¼

1 X

Wn0 ðtÞpð x; nÞ ¼ hpð x; N ðtÞÞi

ð500Þ

n¼0

where Wn0 ðtÞ ¼

ðmtÞn expðmtÞ n!

ð501Þ

and pð x; nÞ ¼ pn ðxÞ

ð502Þ

Equation (502) is interpreted as the pdf of a process that occurs at integer times n (the operational time). Feller calls the random variable NðtÞ the randomized operational time. The operational times need not necessarily be discrete and need not be distributed according to the Poisson law, which has been only used for illustration purposes. For example, suppose pðx; tÞ is the transition probability for a Markov process fXðtÞg, and wðy; tÞ dy is the randomized operational time distribution denoted now by fTðtÞg concentrated on the positive semiaxis; then the pdf pðx; tÞ for the process fXðtÞg is 1 ð

pð x; yÞwðy; tÞdy

pð x; tÞ ¼

ð503Þ

0

We call this process a subordinate process. If fXðtÞg is a Markov process with continuous transition probabilities and fTðtÞg a process with non-negative independent increments, then fXðTðtÞÞg is also a Markov process. Thus, this process is subordinated to fXðtÞg with operational time fTðtÞg. The process fTðtÞg is called a directing (controlling) process. When the process fXðtÞg has independent increments, we again arrive at the above formula for pðx; tÞ. In particular, if fXðtÞg is Brownian motion with

vitaly v. novikov

260 transition probability

2 x pð x; tÞ ¼ ð2ptÞ1=2 exp 2t

and the directing process is the Smirnov–Le´vy process 2 t t wðy; tÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ3ﬃ exp 2y 2py

ð504Þ

ð505Þ

then the subordinate process pðx; tÞ has the Cauchy distribution t pð x; tÞ ¼ 2p

1 ð

2 x þ t2 t y2 exp dy ¼ pð t 2 þ x 2 Þ 2y

ð506Þ

0

Note that a renewal process with transition probability qðtÞ , which is not necessarily a process with independent increments, can also be chosen as the controlling process fTðtÞg. Thus, assuming that the random variables Rj and that their pausing time density qðtÞ is arbitrary in the system under examination, we obtain pðx; tÞ ¼

1 X

Wn0 ðtÞpn ðxÞ

ð507Þ

n¼0

where Wn ðtÞ ¼ PðN ðtÞ ¼ nÞ ¼ PðN ðtÞ nÞ PðN ðtÞ n þ 1Þ n

¼ Q ðtÞ Q

ðnþ1Þ

n

ð508Þ

ðtÞ ¼ Q ½1 Q

Hence [cf. Eq. 507] of Ref. 244 becomes 1 h i X Qn ðtÞ Qðnþ1Þ ðtÞ pn ð xÞ pð x; tÞ ¼

ð509Þ

n¼0

Such models of the one dimensional random walk of a particle with expectation times distributed independently according to the same pausing time law qðtÞ and independent increments (both from each other and from the expectation times) distributed with equal density pðxÞ are, as we have seen, are called Continuous-Time Random Walks. The estimation of pðx; tÞ namely the pdf associated with the position XðtÞ of the random walker at time t in the subordinate process, in the above situation is called the Montroll–Weiss problem. Its solution may be obtained by Fourier

physical properties of fractal structures

261

transformation over coordinates and Laplace transformation over time that is by using the characteristic functions: 1 ð

pðk; sÞ ¼

1 ð

dx expðikx stÞpð x; tÞ

dt 0

ð510Þ

1

where Fourier transformation over space variables is deﬁned by 1 ð

expðikxÞf ð xÞdx

F f f ð xÞg ¼

ð511Þ

1

f ðkÞ ¼ F f f ð xÞg ¼ heikx i

ð512Þ

and Laplace transformation over time variables is deﬁned by 1 ð

L f f ðt Þg ¼

expðstÞ f ðtÞdt

ð513Þ

f ðsÞ ¼ Lf f ðtÞg ¼ hest i

ð514Þ

0

Knowing by the properties of the characteristic function of independent identically distributed random variables that 1 ð

expðikxÞpn ð xÞdx ¼ ½pðkÞn

ð515Þ

1

We have like manner in terms of the Laplace transform qðsÞ of the pausing time pdf qðtÞ 1 ð

1 ð

n

expðstÞQ ðtÞdt ¼ 0

ðt

dt expðstÞ dsqn ðsÞ

0 1 ð

¼

0 1 ð

ds s

0

¼

1 s

dt expðstÞqn ðsÞ

1 ð

0

1 expðstÞqn ðtÞdt ¼ ½qðsÞn s

ð516Þ

vitaly v. novikov

262

again by the properties of the moment generating function of independent identically distribution random variables. Thus, we obtain the characteristic function of the subordinate process, namely, pðk; sÞ ¼

1 1X 1 qn ð s Þ qn ðsÞ qnþ1 ðsÞ pn ðkÞ ¼ s n¼0 s½1 qn ðsÞpn ðkÞ

ð517Þ

Hence using the Bromwich integral (complex inversion formula for the Laplace transform) and the inverse Fourier transform we have

pð x; tÞ ¼

cþi1 ð

1 ð

1

dk

2

ð2pÞ i

1

ds

1 qn ð s Þ expðikx þ stÞ s½1 qn ðsÞpn ðkÞ

ð518Þ

ci1

According to Eq. (517), we have pðk; sÞ ¼

1 qn ð s Þ s½1 qn ðsÞpn ðkÞ

ð519Þ

The estimation of the subordinate density pðx; tÞ using formula (518) demands that the form of the distributions qðtÞ and pðxÞ should be given. Their estimation is a rather complicated task in the general case. Let us consider, however, the asymptotics of qðtÞ and pðxÞ as t ! 1. In order to obtain an evolution equation describing anomalous diffusion, we will use the characteristic function of pðx; tÞ. To calculate the asymptotics of the process under study, we use the stable densities gðx; b; yÞ and gðx; a; 1Þ, 0 < b 2; a 1 as the functions pðxÞ and qðtÞ which are deﬁned below. The behavior of the characteristic function, pðk; sÞ for small values of the arguments (i.e., at large x and t), is according to Tauberian theorems in terms of the functions just mentioned [218,239] pðkÞ ¼ gðk; b; y ¼ 1Þ

ibp ibp b b sign k 1 jkj exp sign k ; ¼ exp jkj exp 2 2

k!0 ð520Þ

and qðsÞ ¼ gðis; a; 1Þ ¼ expðsa Þ 1 sa ;

s!0

ð521Þ

physical properties of fractal structures

263

In accordance with (519), we have pðk; sÞ½1 qn ðsÞpn ðkÞ ¼

1 qn ðsÞ s

ð522Þ

Taking Eqs. (520) and (521) into account, we obtain

ibp sa pðk; sÞ ¼ jkjb exp signk pðk; sÞ þ sa1 2

ð523Þ

On inverse Fourier transformation over k we have sa pð x; sÞ ¼

qb pð x; sÞ þ sa1 qxb

ð524Þ

Here it is supposed, that the Fourier transform of a fractional derivative qb pð x; sÞ qxb

ð525Þ

is (

qb pð x; sÞ F qxb

) ¼ ðikÞb pðk; sÞ

ð526Þ

so that qb pð x; sÞ 1 ¼ b qx 2p

1 ð

ðikÞb pðk; sÞ expðikxÞdk

ð527Þ

1

The product sa pð x; sÞ and the function sa1 are the Laplace transforms of the Riemann–Liouville fractional derivative ðt qa pð x; tÞ 1 q pð x; tÞ ¼ dt qta ð1 aÞ qt ðt tÞa

ð528Þ

0

and the generalized function ta dð x Þ ð1 aÞ

ð529Þ

vitaly v. novikov

264 respectively, that is,

qa pð x; tÞ L qta

¼ s pð x; sÞ; a

ta L dð x Þ ð1 aÞ

¼ sa1

ð530Þ

Hence, the distribution pðx; tÞ, as t ! 1, satisﬁes the fractional partial differential equation qa pð x; tÞ qb pð x; tÞ ta dð x Þ ¼ D þ ab qta qxb ð1 aÞ

ð531Þ

which is called the anomalous diffusion equation. Here the scale of the variables x and t is chosen in a special way, and Dab is a positive constant (the anomalous diffusion coefﬁcient). From (531) it follows that the diffusion equation for b ¼ 2 is qa pð x; tÞ q2 pð x; tÞ ta dð x Þ ¼ Dab þ a 2 qt qx ð1 aÞ

ð532Þ

Various aspects of such fractional differential equations have been studied in [205–208]. We shall now give an example of the solution of such an equation. 3.

Distribution Function of a Brownian Particle with Memory

We consider the pdf of the displacements of a Brownian particle in a process characterized by an equation like (532). The normal Fokker - Planck equation would now be (here the diffusion coefﬁcient is denoted by B) qW ð x; tÞ q2 W ð x; tÞ ¼B qt qx2

ð533Þ

If we include anomalous diffusion of the pausing times the corresponding anomalous diffusion equation is [237] qa W ð x; tÞ dð x Þ q2 W ð x; tÞ ¼ B qta ta ð1 aÞ qx2

ð534Þ

The boundary conditions for this equation are Wðx; tÞjt¼1 ¼ 0

ð535Þ

The initial condition as in the normal Brownian motion is Wðx; tÞjt¼0 ¼ dð xÞ

ð536Þ

physical properties of fractal structures

265

We apply Fourier transformation to Eq. (534). Hence 1 ð

W ðo; tÞ ¼

W ð x; tÞ expðioxÞdx

ð537Þ

1

so that qa W ðo; tÞ 1 þ Bo2 W ðo; tÞ ¼ a qta t ð1 aÞ

ð538Þ

According to the properties of fractional derivatives, from [227] the initial condition of Eq. (538) must be qa1 W ðo; tÞ jt¼0 ¼ b qta1

ð539Þ

where b is a constant. The following clear explanation of this condition can be given. The solution of Eq. (538) can be represented as the series W ðo; tÞ ¼

1 X

An ðoÞtnða1Þ

ð540Þ

n¼1

and only in this way by using the inverse Fourier transform of (539) can we obtain the initial condition for Eq. (534), namely, qa1 W ð x; tÞ jt¼0 ¼ bdðxÞ qta1

ð541Þ

The solution of Eq. (538) is, using the results of Ref. 217, W ðo; tÞ ¼ bta1 Ma;a o2 Bta þ Ma;1 o2 Bta

ð542Þ

where Ma;b ðzÞ is the generalized Mittag–Lefﬂer function [217] Ma;b ðzÞ ¼

1 X

zk ðak þ bÞ k¼0

ð543Þ

We now express Mittag–Lefﬂer function in terms of the Fox function [216] and using the formulae from Eq. (543) apply the inverse Fourier transformation

vitaly v. novikov

266

to Eq. (542). Thus, the distribution function Wðx; tÞ is # " rﬃﬃﬃ p a1 2;0 x2 a2 ; a W ð x; tÞ ¼ b t 2 H1;2 a B 4Bt 2 ð0; 1Þ 12 ; 1 " # a 2 1 ; a 1 x 2;0 2 1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ H1;2 a a 2 4Bt ð0; 1Þ 2 pBt 2;1

ð544Þ

Now since Wðx; tÞ is a pdf we have 1 ð

W ð x; tÞdx ¼ 1

ð545Þ

1

From Eq. (544) we obtain 1 ð

W ð x; tÞdx ¼ 1

1 2;0 W ð x; tÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ H1;2 2 pBta

bta1 2p þ1 ðaÞ

ð546Þ

x2 1 a2 ; a a 4Bt 2 ð0; 1Þ

1 ;1

ð547Þ

1 a2 ; a ð0; 1Þ

# 1 2;1

ð548Þ

2

Hence b ¼ 0 so that ﬁnally 1 2;0 W ð x; tÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ H1;2 2 pBta

"

x2 a 4Bt 2

For a ¼ 1, we have classical diffusion. Otherwise the dependence on time is a power law one. In order to see this consider the asymptotic behavior as t ! 1: 1 lim W ð x; tÞ pﬃﬃﬃﬃa t!1 t

ð549Þ

and as t ! 0 2 x exp a t pﬃﬃﬃﬃ lim W ð x; tÞ t!0 ta

ð550Þ

physical properties of fractal structures

267

Formula (3.328) shows that the diffusion is anomalous, that is a power low dependence on time exists. When a ¼ 1, Eq. (550) becomes Gaussian thus normal diffusion. All the above results are obtained assuming that the Maxwillian distribution of velocities is reached instantaneously by the ensemble of Brownian particles. In other words, the inertia of the particles is ignored. 4.

Inertial Effects of a Brownian Particle

We consider the Brownian motion when the assumption that the ensemble of Brownian particles is instantaneously thermalized is abandoned, in terms of simple models. Random Walks on a One-Dimensional Lattice. We consider [249, 251] stable random walks on a one-dimensional lattice. Here the particle moves at random, and the direction is deﬁned by the direction of the previous step. Each step is only carried out to the nearest neighbor. The mathematical deﬁnition of stability demands that at any time and position on the lattice of the wandering particle, two previous coordinates and the direction of the previous step be known. To describe the random walk process, we consider two probabilities, ð1Þ ð2Þ ð1Þ fpn g and fpn g, where pn is the probability to be at place j at step ð2Þ n from place j 1 at the previous step. pn is the same probability but from place j þ 1. ð2Þ pnð1Þ ¼ pð1Þ n ð j 1Þp11 þ pn ð j 1Þp12 ; ð2Þ pnð2Þ ¼ pð1Þ n ð j þ 1Þp21 þ pn ð j þ 1Þp22

ð551Þ

where p11 ¼ p22 p0 is the probability of the transition when moving in the same direction as that at the previous step, and p12 ¼ p21 q ¼ 1 p0 is the probability of the transition in the opposite direction. For the conventional random walk, p0 ¼ 1=2. Equation (551) is an example [250,251] of a persistent random walk. By correctly scaling p0 , a diffusion equation for the persistent random walk can be obtained from the recurrence relations of Eq. (551). One of the possible variants of scaling is examined when changing to continuous variables x and t. To do this, we deﬁne x and t for all j and n: x ¼ jx;

t ¼ nt

ð552Þ

In the limit t; x ! 0 lim

t;x!0

x ¼v t

ð553Þ

vitaly v. novikov

268

In order to construct partial differential equations, we can scale the probability p0 as p0 ¼ 1

t 2t

ð554Þ

where t ¼ mg is now the time between particle collisions. If t ! 0, then p0 ¼ 1=2, and we have the usual random walk process (i.e., initial effects are ignored). If t 6¼ 0, then we have a process with memory, i.e. the particle ‘‘remembers’’ its state at the previous step (the position and direction of motion). Stability embodies the fact that p0 differs but little from 1. Equations (552) and (553) are now substituted into the recurrence equations (551) yielding t ð1Þ t ð2Þ p ð x x; tÞ; p ð x x; tÞ þ 2t 2t t ð2Þ t ð1Þ p ð x þ x; tÞ pð2Þ ð x; t þ tÞ ¼ 1 p ð x þ x; tÞ þ 2t 2t pð1Þ ð x; t þ tÞ ¼

1

ð555Þ

Knowing that qpð1Þ ð x; tÞ qpð1Þ ð x; tÞ t þ x; qt qx qpð2Þ ð x; tÞ qpð2Þ ð x; tÞ pð2Þ ð x; t þ tÞ pð2Þ ð x þ x; tÞ ¼ t x qt qx

pð1Þ ð x; t þ tÞ pð1Þ ð x x; tÞ ¼

ð556Þ

and letting x and t ! 0, we obtain qpð1Þ ð x; tÞ qpð1Þ ð x; tÞ 1 ð2Þ ¼ v þ p ð x; tÞ pð1Þ ð x; tÞ ; qt qx 2t qpð2Þ ð x; tÞ qpð2Þ ð x; tÞ 1 ð1Þ ¼v þ p ð x; tÞ pð2Þ ð x; tÞ qt qx 2t

ð557Þ

From Eqs. (557) noting that the total probability density function pðx; tÞ is deﬁned as pð x; tÞ ¼ pð1Þ ð x; tÞ þ pð2Þ ð x; tÞ

ð558Þ

we obtain the diffusion equation q2 pð x; tÞ 1 qpð x; tÞ q2 pð x; tÞ ¼ D þ ; qt2 t qt qx2

D ¼ v2 t

ð559Þ

physical properties of fractal structures

269

From Eqs. (557), the diffusive particle ﬂow, jðx; tÞ, is deﬁned. jð x; tÞ ¼ vpð1Þ ð x; tÞ pð2Þ ð x; tÞ

ð560Þ

To do this, the second equation is subtracted from the ﬁrst one: q pð1Þ ð x; tÞ pð2Þ ð x; tÞ q pð1Þ ð x; tÞ þ pð2Þ ð x; tÞ ¼v qt qx 1 ð1Þ p ð x; tÞ pð2Þ ð x; tÞ t

ð561Þ

Therefore, the equation for diffusive particle ﬂow reads jð x; tÞ ¼ D

qpð x; tÞ qjð x; tÞ t ; qx qt

D ¼ v2 t ¼

kT g

ð562Þ

Equation (559) coincides with the Maxwell–Cattaneo equation [254]. Diffusion Equation with Fractional Derivatives. In normal as well as in anomalous diffusion the quantity lim

t;x!0

x t

ð563Þ

is indeterminate. However, consider the quantity

lim

t;x!0

x ¼ va ta

ð564Þ

which in general exists. In this connection, we replace t with ðtÞa in Eqs. (553) and (554); that is, we will measure time not in units t, but ðtÞa . Making the substitution

lim

t;x!0

x x ¼ v ) lim ¼ va ; 0 < a 1; t;x!0 ta t t 1 t a ) p0 ¼ 1 p0 ¼ 1 2t 2 t

ð565Þ

vitaly v. novikov

270

we arrive at fractional derivatives in the time variable. In fact, Eqs. (557) and (559) can be rewritten as 1 t a ð1Þ ð1Þ p ð x; t þ tÞ ¼ 1 p ð x x; tÞ 2 t a 1 t pð2Þ ð x x; tÞ; þ 2 t ð566Þ 1 t a ð2Þ ð2Þ p ð x; t þ tÞ ¼ 1 p ð x þ x; tÞ 2 t 1 t a ð1Þ þ p ð x þ x; tÞ 2 t Assuming that (see 5.Appendix) qa pð1Þ ð x; tÞ a t qta qpð1Þ ð x; tÞ x; þ qx qa pð2Þ ð x; tÞ a pð2Þ ð x; t þ tÞ pð2Þ ð x þ x; tÞ ¼ t qta qpð2Þ ð x; tÞ x qx pð1Þ ð x; t þ tÞ pð1Þ ð x x; tÞ ¼

ð567Þ

as x and ta ! 0. We obtain qa pð1Þ ð x; tÞ qpð1Þ ð x; tÞ 1 1 a ð2Þ ð1Þ þ ¼ v p ð x; t Þ p ð x; t Þ ; a qta qx 2 t qa pð2Þ ð x; tÞ qpð2Þ ð x; tÞ 1 1 a ð1Þ þ ¼ va p ð x; tÞ pð2Þ ð x; tÞ a qt qx 2 t

ð568Þ

Then the diffusion equation with fractional derivatives allowing for inertial effects is a a 2 q2a pð x; tÞ 1 q pð x; tÞ 2 q pð x; tÞ þ ¼ v a qt2a t qta qx2

ð569Þ

The diffusion particle ﬂow or probability current is jðx; tÞ ¼ Da

qpð x; tÞ qa jð x; tÞ ta ; qx qta

Da ¼ v2a ta

ð570Þ

physical properties of fractal structures

271

The Solution of Equation (559). We remark that inertial effects are considerably inﬂuenced by three time intervals [225]: Interval ð0; t1 Þ—ballistic mode Interval ðt1 ; t2 Þ—intermediate mode Interval ðt2 ; 1Þ—Einstein–Smoluchwski region. According to (562), the distinctive feature of the interval ðt1 ; t2 Þ is the deviation from the Fick law, that is, the Maxwell–Cattaneo law is correct rather than the Fick law (487) jðx; tÞ ¼ D

qpð x; tÞ qjð x; tÞ t qx qt

ð571Þ

combined with the continuity equation (571), this law leads to the equation for the pdf pðx; tÞ, namely, qpð x; tÞ q2 pð x; tÞ q2 pð x; tÞ þt ¼ D : qt qt2 qx2

ð572Þ

As we have already stated, Eq. (572) is called the telegraph equation.1 Applying Laplace transformation to Eq. (571), we obtain jð x; sÞ ¼ D

qpð x; sÞ stjð x; sÞ qx

ð573Þ

or jð x; sÞ ¼ DðsÞ

qpð x; sÞ qx

ð574Þ

where the diffusion coefﬁcient (memory function) is DðsÞ ¼ DkðsÞ; kðsÞ ¼

1 1 þ ts

ð575Þ

1 Editor’s Note: Severe criticisms have been concerning the use of the telegraph equation as an approximation to the conﬁguration space distribution function yielded by the exact Fokker–Planck (Klein–Kramers) equation in phase space in order to describe inertial effects in the Brownian motion see H. Risken, The Fokker–Planck Equation, Springer-Verlag, Berlin, 1984, 1989, pp. 257–261; see also Refs. [244–246] Similar considerations apply to the fractional generalization.

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Representing the diffusive ﬂow as Eq. (574) allows one to draw an analogy between the diffusion of a particle including inertial effects and the frequency dependent dielectric and viscoelastic properties. On inversion t 1 ð576Þ kðtÞ ¼ exp t t Considering Eq. (574) as the Laplace transform of a convolution and noting Eq. (576), we obtain

ðt D t t0 q jð x; tÞ ¼ jð x; 0Þ pð x; t0 Þdt0 exp ð577Þ t qx t 0

or jð x; tÞ ¼ jð x; 0Þ DkðtÞ

q pð x; tÞ qx

ð578Þ

q where kðtÞ qx pð x; tÞ is the convolution of the functions kðtÞ and ðt t t0 q dt0 pð x; t0 Þ kðtÞ rpð x; tÞ ¼ k qx t t

q qx pð x; tÞ:

ð579Þ

0

if the initial condition is pð x; 0Þ ¼ dð xÞ

ð580Þ

then, after Laplace and Fourier transformations, we obtain from (3.351) (cf Risken loc.cit.) pðo; sÞ ¼

1 þ st sð1 þ stÞ þ v2 o2

ð581Þ

Inverse transformation of Eq. (581) yields et=2t pð x; tÞ ¼ 2

(

dðhx vtÞ þ dð xi þ vtÞþ 1 ðrÞ 1 þ 4vt I0 ðrÞ þ I2tr Zðvt jxjÞ

where Zð xÞ is Heaviside’s step function pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e2 t 2 x2 r¼ 2et

) ð582Þ

ð583Þ

physical properties of fractal structures

273

I0 ðrÞ; I1 ðrÞ are modiﬁed Bessel functions of the ﬁrst kind. The Laplace transformation of the variance is L

x 2 ðt Þ

¼

q2 pðo; sÞ t2 j ¼ o¼0 qo2 sðst þ 1Þ2

ð584Þ

so we obtain the Ornstein–Uhlenbeck result2 [247]

nt o t x2 ðtÞ ¼ 2v2 t2 1 þ et t

ð585Þ

From (3.364) it follows that: If tt ! 0, then x2 ðtÞ t2 ballistic transport of particles. If tt ! 1, then x2 ðtÞ t represents the Einstein results. Let us now consider the equation for diffusive particle ﬂow with a fractional derivative (570). By Laplace transformation of equation (570), we obtain jð x; sÞ ¼ Da ðsÞ

qpð x; sÞ qx

ð586Þ

where on this occasion the s dependent diffusion coefﬁcient is Da ðsÞ ¼ Dka ðsÞ;

k a ðsÞ ¼

1 1 þ ðtsÞa

ð587Þ

Knowing that 1 X 1 ðtsÞa ¼ ¼ ð1Þn ðtsÞaðnþ1Þ 1 þ ðtsÞa 1 þ ðtsÞa n¼0

ð588Þ

in the domain of the originals ka ðsÞ reads 0

aðnþ1Þ1 1 n t a 1 ð1Þ X 1B 1 C t Ma;a ðzÞ ka ðtÞ ¼ @ A ¼ ta1 t n¼0 t ½aðn þ 1Þ

2

ð589Þ

Editor’s Note: A peculiarity of the ‘‘telegraph equation,’’ when used to approximately include inertial effects, is that it yields the correct result for the variance while yielding a poorer approximation for the distribution function than the Smoluchowski equation [see Ref. 248 and Risken (loc.cit.).]

vitaly v. novikov

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where Ma;g ðzÞ, the Mittag–Lefﬂer function is Ma;g ðzÞ ¼

1 X

zn ; ½an þ g n¼0

t a z¼ t

ð590Þ

In our case g ¼ 1: Moreover, Ma;g ðzÞ ¼

1;1 H1;2

ð0; 1Þ z ð0; 1Þ ð1 g; aÞ

ð591Þ

Then Eq. (589) can be written in the time domain as ka ðtÞ ¼ t

a1

a a 1 t ð0; 1Þ 1;1 H1;2 t t ð0; 1Þ ð1 a; aÞ

ð592Þ

if a ¼ 1, then from Eq. (592) we obtain n ! 1 t ð1Þn tt 1 X 1 kðtÞ ¼ ¼ exp t n¼0 ½n þ 1 t t

ð593Þ

Conclusions. Equation (593) brings to an end our long discussion of anomalous diffusion. Throughout its course we have seen that the characteristic feature of random inhomogeneous structures and the physical processes taking place in them is their hierarchy causing in turn anomalous behavior of their physical properties on the macrolevel, namely, anomalous large spatial ﬂuctuations of local electric and elastic ﬁelds. The fractal concept has proved to the helpful in describing such systems. In this connection our attention has been focused on using the fractal concept to make predictions concerning the physical properties of inhomogeneous media with a random structure. We remark that numerous other examples of fractal behavior than those treated here appear in the literature. We should mention resistance capacitance transmission lines [240] and fractal models for the alternating current response at a rough interface between materials of very dissimilar conductivities [241,242] and how a resistance capacitance line may be used as a semiintegrator [243]. Finally we have tried to demonstrate how rather complex phenomena may be described in uniﬁed fashion using simple fractal models and we have hopefully composed the text in such a manner that it could serve as an introduction to the subject for the beginner in the ﬁeld.

physical properties of fractal structures 5.

275

Appendix. Derivative of Fractal Functions.

In general, functions for which the total increment, h f ðxÞ ¼ f ðx þ xÞ f ðxÞ can be represented as

h f ðxÞ ¼ A½ xh þaðxÞ½ xh ;

ð594Þ

lim aðxÞ ¼ 0

x!0

ð595Þ

(i) h ¼ 1; 0: f ðxÞ belongs to the classical set of differentiable functions. (ii) h ¼ 6 1(Hoelder index): f ðxÞ belongs to the set of functions for which not the classical derivative but only the fractional derivative exists d h f ðxÞ h f ðxÞ ¼ lim x!0 ½xh dxh

ð596Þ

Wiener’s process (i.e., Brownian motion) and Kolmogorov’s turbulence (i.e., a nonsmooth vector ﬁeld) may be cited as examples of phenomena which can be described by continuous, nowhere differentiable functions (fractal functions). The displacement yðtÞ of a Brownian particle in the former (Wiener’s) process is deﬁned as jyðt þ tÞ yðtÞj ½ta

ð597Þ

whereas the singular velocity of the latter phenomenon (Kolmogorov’s turbulent ﬂow) is characterized by [219] h½vp i ½tp=3 ;

ð598Þ

where v ¼ vðx þ xÞ vðxÞ is the difference of velocities between two points separated by distance x. Assume that a function f ðxÞ is deﬁned on a fractal ensemble f , of dimension 0 < df < 1. Let the function f ðxÞ (hereafter referred to as a fractal function) be continuous through f , be self-similar at different scales, and have no tangent at any point of its trajectory. It is assumed that f ðxÞ ¼ 0 if x < 0, and jf ðxÞj < 1. Let us divide a segment ½ x; x0 in such a manner that the length of each qth fragment at the nth scale level is n xðnÞ q ¼ x ðx0 xÞ

ð599Þ

where x < 1 is the scaling factor (i.e., the index of similarity of the ensemble f , ).

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The number of dividing points of the segment ½ x; x0 at the nth step is therefore mn ¼ 1; 2; . . . ; jnþ1

ð600Þ

where j is the number of blocks (i.e., the branching index) involved in the construction of the fractal unit cell ( j ¼ 2 for Cantor’s ensemble). Let the unit scale at the nth step be xa , h

ia 1 xðnÞ ¼ ðx0 xÞa q Nn

ð601Þ

where N1 ¼ j1 ; . . .; Nn ¼ jn (that is, Nn ¼ jn determines the number of fragments at the nth scale level). This deﬁnition of the unit scale for the segment ½ x; x0 allows one to associate each point (element) of a fractal ensemble with a point of an ultrametric space which can be represented by the Cayley tree. ðnÞ ðnÞ It follows from Eq. (601) that lim xq ¼ 0; hence xq , is an inﬁnitesimal n!1 ðnÞ quantity. From now on, the increment of the function argument xq at the nth ðnÞ step will be denoted by x (that is, x ¼ xq ), while the corresponding coordinates of the dividing points will be deﬁned as xq ¼ x0 qxðnÞ q ¼ x0 qx

ð602Þ

where q ¼ 0; 1; 2; . . . ; jnþ1 . Recognition of the fractal dimension as df ¼ a further implies ðx0 xÞ x ¼ n ; 1 x

ðx0 xÞa ½xa ¼ ; Nn na 1 ¼ jn ¼ N n x

ð603Þ

Consider an increment, a f ðxÞ ¼ f ðx0 Þ f ðx0 xÞ; then the qth increment q f ðxÞ will be determined via binomial coefﬁcients with alternating signs m a f ðx0 Þ ¼

m X ½1q Cmq ð f ðx0 qxÞÞ;

m! ; ¼ q!ðm qÞ!

Cmq ð604Þ

q¼0

m¼j

nþ1

physical properties of fractal structures

277

and the function f ðxÞ in the vicinity of point x0 will be f ðxÞ ¼ ð1 a Þm f ðx0 Þ

ð605Þ

Using Eqs. (599)–(605), one can derive an analogue of the Taylor series for a function f ðxÞ f ðxÞ ¼

m X

aq ðx0 xÞaq

ð606Þ

q¼0

q

j where aq ¼ q! f ðaqÞ ðx0 Þ, and f ðaqÞ ðx0 Þ deﬁnes the fractional derivative of qth order of the fractal function f ðxÞ at the point x ¼ x0 as

qa f ðx0 Þ a q x!0 ð½x Þ

f ðaqÞ ðx0 Þ ¼ lim

ð607Þ

The coefﬁcients of the series (606) depend both on the fractional derivative of qth order of the fractal function f ðxÞ at the point x ¼ x0 and on the branching index j of the fractal ensemble for which the function f ðxÞ is speciﬁed.

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CHAPTER 8 FRACTIONAL ROTATIONAL DIFFUSION AND ANOMALOUS DIELECTRIC RELAXATION IN DIPOLE SYSTEMS WILLIAM T. COFFEY Department of Electronic and Electrical Engineering, School of Engineering, Trinity College, Dublin 2, Ireland YURI P. KALMYKOV Laboratoire de Mathe´matiques et Physique des Syste`mes, Universite´ de Perpignan, 66860 Perpignan Cedex, France SERGEY V. TITOV Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Fryazino, Moscow Region, 141190, Russian Federation

CONTENTS I. Introduction II. Microscopic Models for Dielectric Relaxation in Disordered Systems A. Continuous-Time Random Walk Model B. Fractional Diffusion Equation for the Cole–Cole Behavior C. Anomalous Dielectric Relaxation in the Context of the Debye Noninertial Rotational Diffusion Model D. Fractional Diffusion Equation for the Cole–Davidson and Havriliak–Negami Behavior E. Fundamental Solution of the Fractional Smoluchowski Equation III. Fractional Noninertial Rotational Diffusion in a Potential A. Anomalous Diffusion and Dielectric Relaxation in a Double-well Periodic Potential B. Fractional Rotational Diffusion in a Uniform DC External Field C. Fractional Rotational Diffusion in a Bistable Potential with Nonequivalent Wells

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

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1. Matrix Continued Fraction Solution 2. Bimodal Approximation IV. Inertial Effects in Anomalous Dielectric Relaxation A. Metzler and Klafter’s Form of the Fractional Klein–Kramers Equation B. Barkai and Silbey’s Form of the Fractional Klein–Kramers Equation C. Inertial Effects in Anomalous Dielectric Relaxation of Linear and Symmetrical Top Molecules 1. Rotators in Space 2. Symmetric Top Molecules D. Inertial Effects in Anomalous Dielectric Relaxation in a Periodic Potential E. Fractional Langevin Equation V. Conclusions Appendix I: Calculation of Inverse Fourier Transforms Appendix II: Exact Continued Fraction Solution for Longitudinal and Transverse Responses Appendix III: Dynamic Kerr-Effect Response: Linear Molecules Appendix IV: Ordinary Continued Fraction Solution for Spherical Top Molecules Appendix V: Kerr-Effect Response Acknowledgments References

I.

INTRODUCTION

One of the most striking features of the dielectric relaxation of disordered materials such as glass-forming liquids, amorphous polymers, and so on, is the failure of the Debye [1] theory of dielectric relaxation based on the Einstein theory of Brownian motion [2] to describe adequately the low-frequency spectrum, where the relaxation behavior may deviate considerably from the exponential (Debye) pattern and is characterized by a broad distribution of relaxation times. The relaxation process in such disordered systems is characterized by the temporally nonlocal behavior arising from the energetic disorder that produces obstacles or traps that delay the motion of the particle and introduce memory effects into the motion. Such behavior has been given the title anomalous dielectric relaxation and was ﬁrst systematically described in the pioneering article [3] by Cole and Cole in 1941 on dielectric relaxation in polar liquids. These and subsequent investigators have proposed [4,5] (see also Ref. 6) various empirical formulas describing the departure from the Debye behavior. In speciﬁc terms, the normal Debye relaxation process is characterized by a complex susceptibility wðoÞ ¼ w0 ðoÞ iw00 ðoÞ of the form wðoÞ ¼

w0 1 þ iot

ð1Þ

where w0 is the static susceptibility and t is a characteristic relaxation time known in the present context as the Debye relaxation time. Equation (1) adequately describes the low-frequency behavior of the observed complex susceptibility of many simple polar liquids.

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Equation (1) may be derived using a variety of microscopic models of the relaxation process. In the derivation of Eq. (1), Debye [1] used the theory of the Brownian motion developed by Einstein and Smoluchowski. Einstein’s theory of Brownian motion [2] is based on the notion of a discrete time walk. The walk may be described in simple schematic terms as follows. Consider a twodimensional lattice; then, in discrete time steps of length t, the random walker is assumed to jump to one of its nearest-neighbor sites, displayed, for example [7], on a square lattice with lattice constant x, the direction being random. Such a process, which is local both in space and time, can be modeled [7] in the one-dimensional analogue by the master equation Wj ðt þ tÞ ¼ ½Wj1 ðtÞ þ Wjþ1 ðtÞ=2

ð2Þ

Here the index j denotes the position of the random walker on the underlying one-dimensional lattice; j þ 1, j 1 are the adjacent lattice sites. Wj ðt þ tÞ is the probability for the random walker to be at site j at time t þ t given that it was at sites j 1 at time t. In the continuum limit t ! 0, x ! 0, expansion of Wj in a Taylor series in t and x leads to the diffusion equation qW q2 W ¼D 2 qt qx

ð3Þ

for the transition probability function Wðx; tjx0 ; t0 Þ, where x0 ¼ xðt0 Þ (i.e., at t ¼ t0 the particle was at x0 ) and the diffusion coefﬁcient D is deﬁned as D¼

ðxÞ2 x!0;t!0 2t lim

ð4Þ

Equation (3) in the presence of an external potential VðxÞ (e.g., the gravitational ﬁeld of the earth) becomes the Smoluchowski equation [8,9] qW q qW W qV ¼D þ qt qx qx kB T qx

ð5Þ

where kB T is the thermal energy. For the noninertial translational Brownian motion of a particle in space, Eq. (5) can be written as [8,9] qW W ¼ D div grad W þ grad V qt kB T

ð6Þ

Debye extended the foregoing arguments in order to establish the Smoluchowski equation [Eq. (5)] for the rotational Brownian motion of a dipolar particle about a

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diameter in suspension (planar rotation). If f is an angular coordinate and if f2 is the mean-square displacement in time t due to thermal agitation, then the underlying rotational diffusion equation is [2,8,9] qW q qW W qV ¼D þ qt qf qf kB T qf

ð7Þ

where Wðf; tjf0 ; t0 Þ is the transition probability function and D ¼ f2 =ð2tÞ is the rotational diffusion coefﬁcient, which determines the Debye relaxation time t ¼ 1=D ¼ =kB T for rotation about a ﬁxed axis ( is the viscous drag coefﬁcient of a dipole). The corresponding rotational diffusion equation for the noninertial rotational Brownian motion of a linear or spherical top molecule in space is given by [8,9] qW 1 1 2 ¼ r ðWrVÞ þ r W qt 2t kB T

ð8Þ

where r2 and r are the Laplacian and gradient on the surface of the unit sphere, Wð#; j; tj#0 ; j0 ; t0 Þ represents the transition probability density function of the orientations of the dipoles on the surface of the sphere (the orientation of the dipole moment vector l is described by the polar angle # and azimuth j), and t ¼ =2kB T is the Debye relaxation time for rotation in space. Equation (8) is the Smoluchowski equation, which is an approximate Fokker–Planck equation [7] in the space of angular coordinates for the distribution function of the orientations of the dipoles on the surface of the unit sphere when the inﬂuence of the inertia of the molecules on the relaxation process is ignored. We remark that in the context of dielectric relaxation, the rotational diffusion equation for the distribution function W (#, j, t) of the orientations of the dipoles derived by Debye [1] is simply a more general case of Eq. (3), where the tips of the unit vectors specifying the dipole orientations execute a discrete time random walk on the surface of the unit sphere. The discrete time random walk on the surface of the unit sphere then leads directly to the Debye equation [Eq. (1)]. The behavior predicted by Debye’s modiﬁcation of Einstein’s theory is substantially in accordance with the experimental evidence for simple polar liquids, supporting the hypothesis that, in such liquids, the underlying random processes are local, both in space and time. We shall term this model the ﬁrst Debye model. This model applies when one has (1) a dilute solution of dipolar molecules in a non polar liquid; (2) axially symmetric molecules; and (3) isotropy of the liquid, even on an atomic scale in the time average over a time interval small compared with the Debye relaxation time t.

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The Debye model then yields the mean dipole moment in the direction of the applied ﬁeld and the complex susceptibility wðoÞ, Eq. (1). The Smoluchowski equation on which the Debye model is based applies [10] to strong dissipative coupling to the bath so that the ﬁrst Debye model always contains the assumption that the dipolar molecule is bound so strongly to the surrounding molecules that large jumps of the dipole direction are extremely unlikely. This behavior according to Fro¨hlich [10] may be true in a number of cases, but others may exist in which the opposite (large jumps) is much more likely. A dipolar molecule will then [10] make many jumps over the potential barrier separating it from another dipole direction during the time required for an appreciable change in direction by viscous ﬂow. Clearly, such behavior holds for solids where ﬂow may be considered as entirely absent; however, it may also be expected where the viscosity is so high that ﬂow is practically negligible. In liquids, it might also happen that the motion which prevails is different for different kinds of dissolved molecules. Moreover, both large and small jump transitions may exist simultaneously. The above observations lead us to the second microscopic model considered by Debye [1] (and much extended by Fro¨hlich [10]), which is a Poisson-like process, where relaxation occurs due to the crossing by large jumps of rare members of an assembly of dipoles over an internal potential barrier in a solid due to the shuttling action of thermal agitation. This microscopic model also produces a relaxation spectrum of the form of Eq. (1); however, the overbarrier relaxation time has Arrhenius-like behavior because it depends exponentially on the height of the potential barrier. The Debye–Fro¨hlich model also constitutes a rotational Brownian motion model based on the Fokker–Planck equation, as is apparent by considering a continuous distribution of orientations [8] and a double- (multi-) well potential rather than the discrete orientation approximation treated by Debye and Fro¨hlich. It should be noted that if a continuous distribution of orientations is used, then the prefactor of the exponential in the overbarrier relaxation time depends strongly on the dissipative coupling to the heat bath and the shape of the potential, as emphasized by Kramers [11] in his famous study of the escape of particles over potential barriers due to the shuttling action of thermal agitation. Moreover, the use of the Fokker–Planck equation allows one to account for the contribution of the fast decays in the wells of the potential to the relaxation process. The Debye–Fro¨hlich model is also very useful as a picture of the solidstate-like process of reversal of the magnetization in ﬁne single-domain ferromagnetic nanoparticles possessing an internal potential barrier due to their inherent magnetocrystalline anisotropy [12–14]. In this context, taking into account the intrinsic differences between dielectric and magnetic relaxation, the model is known as the Ne´el–Brown model [12–14] of magnetic relaxation. If, on the other hand, the model is applied to the dielectric relaxation of nematic liquid crystals, it is known as the Maier–Saupe model [15].

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william t. coffey, yuri p. kalmykov and sergey v. titov

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal ﬂuctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker–Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. Returning to anomalous dielectric relaxation, it appears that a signiﬁcant amount of experimental data on disordered systems supports the following empirical expressions for dielectric loss spectra, namely, the Cole–Cole equation wðoÞ ¼

w0 ; 1 þ ðiotÞs

0 < s1

ð9Þ

w0 ; ð1 þ iotÞn

0 < n1

ð10Þ

the Cole–Davidson equation wðoÞ ¼

and the Havriliak–Negami equation wðoÞ ¼

w0 ; ð1 þ ðiotÞs Þn

0 < s 1;

0 < n1

ð11Þ

which is a combination of the Cole–Cole and Cole–Davidson equations [6]. Each of the above equations by use of the superposition principle exhibits a broad distribution of relaxation times [6,10]. In the notation of Fro¨hlich [10], we have wðoÞ ¼ w0

1 ð

f ðT 0 ÞdT 0 1 þ ioT 0

1 ð

with

0

f ðT 0 ÞdT 0 ¼ 1

ð12Þ

0

where f ðT 0 Þ is the relaxation time distribution function.This equation is the mathematical expression of the idea that the dielectric behaves as a collection of individual components each being described by a Debye equation with relaxation time T 0 . One may show that for the Debye equation [Eq. (1)] we obtain fD ðT 0 Þ ¼ dðT 0 tÞ

ð13Þ

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(dðxÞ is the Dirac delta function); thus only one relaxation mechanism is involved as is obvious by deﬁnition, while for the Cole–Cole Eq. (9) [6] fCC ðT 0 Þ ¼

sin ps pT 0 ½ðT 0 =tÞs þ ðT 0 =tÞs þ 2 cos ps

ð14Þ

for the Davidson–Cole Eq. (10) [6] fDC ðT 0 Þ ¼

8 <

sin pn ; pT 0 ðt=T 0 1Þn : 0;

ðT 0 < tÞ ðT 0 > tÞ

ð15Þ

and for the Havriliak–Negami Eq. (11) [6] sinps ðT =tÞ sin v arctan s ðT 0 =tÞ þ cos ps 0

fHN ðT 0 Þ ¼

vs

pT½ðT 0 =tÞ2s þ 2ðT 0 =tÞs cos ps þ 1v=2

ð16Þ

Thus it is apparent that the anomalous relaxation behavior may be characterized by a superposition of an inﬁnite number of Debye-like relaxation mechanisms with a distribution of relaxation times given by Eqs. (14)–(16). In the Cole–Cole, Cole–Davidson, and Havriliak–Negami relaxation mechanisms unlike the Debye equation, where the distribution function is a d function, the relaxation time distribution (ﬁrst evaluated for the Cole–Cole relaxation by Cole and Cole [3]) exhibits long-time tails typical of Le´vy probability distributions [7]. This observation has been formalized for Cole–Cole relaxation by Glo¨ckle and Nonnenmacher [16]. Returning to Eqs. (9)–(11), the Cole–Cole parameter s is a broadening parameter as the curve of w00 ðoÞ versus o broadens as s is reduced. On the other hand, the Cole–Davidson parameter n in Eqs. (10) and (11) is a skewing parameter, because in the Cole–Cole plot of w00 ðoÞ versus w0 ðoÞ the circular arc characteristic of the Debye equation is shifted toward the lowfrequency end of the spectrum [6]. An explanation of this behavior is the n-fold degeneracy induced in the Debye equation (1) by the Cole–Davidson parameter n, causing the simple pole 1=t of the Debye equation to become a branch point of order n—that is, a n-fold degenerate eigenvalue. As far as the physical mechanism underlying the Cole–Cole equation is concerned, we ﬁrst remark that Eq. (9) arises from the diffusion limit of a continuous-time random walk (CTRW) [17] (see Section II.A). In this context one should recall that the Einstein theory of the Brownian motion relies on the diffusion limit of a discrete time random walk. Here the random walker makes a jump of a ﬁxed mean-square length in a ﬁxed time, so that the only random

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variable is the direction of the walker, leading automatically by means of the central limit theorem (in the limit of a large sequence of jumps) to the Wiener process describing the Brownian motion [8]. The CTRW, on the other hand, was introduced by Montroll and Weiss [17] as a way of rendering time continuous in a random walk without necessarily appealing to the diffusion limit. In the most general case of the CTRW, the random walker may jump an arbitrary length in arbitrary time. However, the jump length and jump time random variables are not statistically independent [18–21]. In other words a given jump length is penalized by a time cost, and vice versa. A simple case of the CTRW arises when one assumes that the jump length and jump time random variables are decoupled and that the jump length variances are always ﬁnite (so that the central limit theorem applies in the limit of a large sequence of jump lengths [8]); however, the jump times may be arbitrarily long so that they obey a Le´vy distribution with its characteristic long tail [18–22]. Thus the jump length distribution ultimately becomes Gaussian with ﬁnite jump length variance, while the mean waiting time between jumps diverges on account of the underlying Le´vy waiting time distribution. Such walks, which possess a discrete hierarchy of time scales, not all of which have the same probability of occurrence, are known as fractal time random walks [19]. In the limit of a large sequence of jump times, they give rise to a fractional Fokker–Planck equation in conﬁguration space [7,18]. If this equation is now adapted to rotational Brownian motion as used by Debye [1] for the normal Fokker–Planck equation in his ﬁrst model, then the Cole–Cole equation (9) automatically follows [8,23]. Inertial effects have also been included in the model [24]. The second model of Debye or the Debye–Fro¨hlich model may also be generalized to fractional diffusion [8,25] (including inertial effects [26]). Moreover, it has been shown [25] that the Cole–Cole equation arises naturally from the solution of a fractional Fokker–Planck equation in the conﬁguration space of orientations derived from the diffusion limit of a CTRW. The broadening of the dielectric loss curve characteristic of the Cole–Cole spectrum may then be easily explained on a microscopic level by means of the relation [8,24] lp;s ¼ lp t1s

ð17Þ

between the eigenvalues lp;s and lp of the fractional and normal conﬁguration space Fokker–Planck equations, respectively. Here the relaxation behavior appears [8] as a superposition of Cole–Cole equations if the inertial effects are ignored. The fractal time random walk picture, whereby a particle is trapped in a given conﬁguration for an arbitrarily long period before executing a jump [18], immediately suggests that the Cole–Cole parameter s (here the fractal dimension of the set of waiting times between jumps) arises from the anisotropy

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of the material on a microscopic scale. Thus assumption 3 underpinning the Debye theory breaks down for Cole–Cole relaxation. The microscopic anisotropy gives rise to a distribution of microscopic potential barrier heights [23] which in turn, because the individual jump probabilities constitute a hierarchy of Poisson processes [19], give rise to a hierarchy of relaxation times not all of which have the same probability of occurrence. Such models are usually known as random activation energy models (see Ref. 20, p. 280). They appear to be consistent with the concept of a distribution of microscopic Debye-like mechanisms embodied in Eq. (12) and the Le´vy-like behavior [Ref. 6, Eqs. (3.104) and (3.105)] of the various relaxation time distributions and with the breakdown in anomalous relaxation of Einstein’s ansatz [8,18,21] that in Brownian motion the random walker executes a discrete jump of ﬁnite meansquare length in an average time t. Our purpose is to demonstrate how it is possible to describe the anomalous dielectric relaxation from microscopic models of the underlying processes. Moreover, we shall illustrate how the effects of the inertia of the molecules and an external potential arising from crystalline anisotropy or indeed any other mechanism could be included. II.

MICROSCOPIC MODELS FOR DIELECTRIC RELAXATION IN DISORDERED SYSTEMS

The Cole–Cole equation can be derived from a kinetic equation based on the concept of a continuous-time random walk—that is, a walk with a long-tailed distribution of waiting times between the elementary jumps. It is also apparent [27] that the method may be extended to both the Cole–Davidson and the Havriliak–Negami equations using an extension of an approach proposed by Nigmatullin and Ryabov [28]. There are, however, certain mathematical and conceptual difﬁculties associated with such a fractional diffusion equation approach. The ﬁrst of these is the justiﬁcation of truncation of the generalized Kramers–Moyal expansion at the second term in the space derivative in order to obtain fractional probability density diffusion equations for the orientation distribution functions. This question is easily answered in the theory of the Brownian motion because the underlying processes are Gaussian; thus it is possible to express the higher-order even moments of the distribution function in terms of powers of the second moment while the odd moments are zero. This is the crucial factor that allows one to truncate the Kramers–Moyal expansion at the second derivative in the spatial derivatives. An important corollary to the above statement is the following: Since one may express the higher-order moments in terms of powers of the second-order moment, one may also generate the hierarchy of differential recurrence relations describing the time behavior of the statistical averages by averaging the underlying Langevin equation over its

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william t. coffey, yuri p. kalmykov and sergey v. titov

realizations using Isserlis’s theorem also known as Wick’s theorem. It is not yet apparent how to do this for fractional diffusion processes due to the lack of appropriate interpretation rules (Itoˆ–Stratonovich) and the absence of an analogue of Isserlis’s theorem [8]. Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters s and n in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justiﬁcation for the fractional diffusion equation used in the explanation of the Cole–Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justiﬁcation for the neglect of spatial derivatives of higher order than the second in the Kramers–Moyal expansion. In order to accomplish this, it is ﬁrst necessary to explain the concept of the continuous-time random walk (CTRW). A.

Continuous-Time Random Walk Model

The theory of the Brownian motion, which we have described, is distinguished by a characteristic feature—namely, the concept of a collision rate—which is the inverse of the time interval between successive collision events of the Brownian particle with its surroundings; we recall the words of Einstein [2]. We introduce a time interval t in our discussion, which is to be very small compared with the observed interval of time, but, nevertheless of such a magnitude that the movements executed by a particle in two consecutive intervals of time t are to be considered as mutually independent phenomena.

This concept which is based on a random walk with a well-deﬁned characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in conﬁguration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fu¨rth), we obtain the Klein–Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clariﬁed by remarking that in the discrete time random walk, the problem [5] is always to ﬁnd the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. Einstein’s approach, ignoring inertial effects always leads to a mean-square displacement proportional to jtj. A generalization of this is of the form jtjs ,

fractional rotational diffusion

295

where the case s > 1 is referred to as superdiffusion or enhanced diffusion while s < 1 is referred to as subdiffusion. We shall now indicate how these behaviors may be interpreted in terms of CTRW. We have mentioned that the concept of a CTRW was introduced by Montroll and Weiss in 1965 [17,29] as a way to render time continuous in a random walk without an appeal to the diffusion or continuum limit. Thus Einstein’s assumption of a discrete time t in which a jump of mean-square length h2 i is executed is abandoned. Instead, a jump probability distribution function cðx; tÞ is introduced in which in general the jump length and time between jumps are coupled random variables, meaning that a jump of a certain length involves a time cost and vice versa. The jump distribution is the probability density that a random walker executes a jump from x to x þ dx in a time interval dt having remained at some site for a waiting time t. We may determine from cðx; tÞ both the jump length probability density function lðxÞ ¼

1 ð

cðx; tÞ dt

ð18Þ

0

and the waiting time probability density function wðtÞ ¼

1 ð

cðx; tÞ dx

ð19Þ

1

Here lðxÞ dx yields the probability of a jump length L in the interval x ! x þ dx, and wðtÞ dt yields the probability of a waiting time TW in the interval t ! t þ dt. Moreover, unlike in Brownian motion where the introduction of the quantities x and t deﬁnes a physical length and a physical time scale, the second moment of the jump length distribution diverges and so does the ﬁrst moment of the jump time. Thus both jump times and jump lengths exhibit chaotic behavior. Hence it is impossible to attribute underlying physical scales to such processes. This may be interpreted as the scale invariance which is typical of self-similarity and fractal behavior. The word fractal—coming from the Latin fractus, meaning broken—is used to describe dilation invariant objects which exhibit irregularities and chaotic behavior at any given scale. Speciﬁc examples are clouds and the coastline of islands. The self-similar behavior is of course constrained by certain boundaries such as in Brownian motion the selfsimilar behavior is limited to distances above the mean free path of a molecule, and so on. An important consequence of the lack of physical scales is that referring to the temporal behavior of such systems, all global characteristic

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william t. coffey, yuri p. kalmykov and sergey v. titov

times such as the mean ﬁrst passage time (i.e., the average time at which a process reaches a predetermined level for the ﬁrst time), the integral relaxation time, and so on, will diverge [8]. In analyzing the complex susceptibility in such systems, therefore, one should not use the concept of characteristic times; in constrast, the physically meaningful quantities are the frequencies of maximum absorption. The characteristic times of the normal diffusion process are merely parameters in anomalous diffusion processes. An example of divergent characteristic times is the divergence of the relaxation time of strongly interacting magnetic nanoparticle systems in the vicinity of a spin glass phase transition. Despite all these difﬁculties, it is, however, possible to analyze distribution functions having divergent moments of the kind we have mentioned. The underlying limiting distribution function is not Gaussian, it is a stable or Le´vy distribution, which is marked by the presence of long-range inverse power law tails in the distribution function which may lead to divergence of even the lowest-order moments. The tails prevent convergence to the Gaussian distribution if they pertain to a sequence of random variables, however, not the existence of a limiting distribution. It is in general very difﬁcult to treat CTRW problems when the jump time and jump length distributions are coupled. Thus we shall assume for the most part that the jump time and jump length are independent random variables so that the jump probability factorizes and we have the decoupled (separable) form cðx; tÞ ¼ wðtÞlðxÞ

ð20Þ

cðx; tÞ ¼ pðxjtÞwðtÞ

ð21Þ

cðx; tÞ ¼ pðtjxÞlðxÞ

ð22Þ

If they are coupled, we have

or

that is, a jump of a certain length involves a time cost or, on the other hand, in a given time span, the walker can only travel a maximum distance [7]. We remark that if the jump length distance is also a Le´vy process, the meansquare displacement does not exist which has led to conceptual difﬁculties in applying this process to dielectric relaxation. Using these simpliﬁcations, one can identify two specialized forms of a continuous time random walk: 1. The ﬁrst is the long rests or fractal time random walk where the mean waiting time diverges; however, the second moment of the jump length distribution remains ﬁnite. The fractal time random walk always leads to

fractional rotational diffusion

297

subdiffusion because the random walker always risks being trapped at some site for an arbitrarily long time before he can advance a distance equal to the ﬁnite variance of the jump length distribution. 2. The other uncoupled case with ﬁnite mean waiting time and divergent jump length variance is called the long-jump or Le´vy ﬂight model. A famous example of the Le´vy ﬂight model is the Weierstrass random walk. Here all steps take the same average time and the distribution of step lengths exhibits a Le´vy-type inverse power decay law for the largest jumps, which of course leads to the divergent second moment. Such a walk leads to enhanced diffusion and ultimately turbulence as the overall displacement is dominated by the largest jumps without any time cost; that is, jumps of arbitrary length all take the same time if the jump length is a Le´vy process. 3. Both divergences of the moments associated with the random walks mentioned above may be avoided by considering nonseparable CTRWs or Le´vy walks where, in contrast to Le´vy ﬂights, a Le´vy walker does not simply jump an arbitrary length in the same time [7,19] but instead has to move with a given velocity from his starting point. For the simplest case of constant velocity [7,19] we see that large jumps require a longer time than shorter ones and the overall consequence is ﬁnite mean-square displacement for all values of s. Such walks are important in the context of probability density diffusion equations for the phase-space distribution function that is the generalization of the Klein–Kramers equation to fractional diffusion. If a purely fractal time random walk is used to generalize the Klein–Kramers equation for rotational diffusion to anomalous diffusion, then a nonphysical divergence of the absorption coefﬁcient occurs at very high frequencies [8]. If, however, a Le´vy walk is used which appears to be at the root of the generalization of the Klein–Kramers equation proposed by Barkai and Silbey [30], then the undesirable divergence of the absorption coefﬁcient is removed [8]). The fact that the temporal occurrence of the motion events performed by the random walker is so broadly distributed that no characteristic waiting time exists has been exploited by a number of investigators [7,19,31] in order to generalize the various diffusion equations of Brownian dynamics to explain anomalous relaxation phenomena. The resulting diffusion equations are called fractional diffusion equations because in general they will involve fractional derivatives of the probability density with respect to the time. For example, in fractional noninertial diffusion in a potential, the diffusion Eq. (5) becomes [7,31] 2 qW q W qV s 1s q W ¼ D 0 Dt þ qt qx2 qx kB T qx

ð23Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

where s is the anomalous exponent, the fractional derivative 0 Dt1s is given by (the Riemann–Liouville deﬁnition) [7,32,33] 1s 0 Dt

¼

q s 0D qt t

ð24Þ

in terms of the convolution (recall Cauchy’s integral formula) s 0 Dt Wðx; tÞ

ðt 1 Wðx; t0 Þdt0 ¼ ðsÞ ðt t0 Þ1s

ð25Þ

0

where (z) denotes the gamma function. Equation (23) with 0 < s < 1 describes slow diffusion or subdiffusion and with 1 < s < 2 describes enhanced diffusion (s ¼ 2 deﬁnes the ballistic limit); normal diffusion occurs when s ¼ 1 [7,31]. The fractional derivative is a type of memory function with a slowly decaying power law kernel in time. Such behavior arises from random torques with an anomalous waiting time distribution—that is, from a fractal time random walk with t as the intertrapping time. The derivation of fractional diffusion equations such as Eq. (23) hinges on the observation (cf. Ref. 33, p. 118) that fractional diffusion is equivalent to a CTRW with waiting time density wðtÞ given by a generalized Mittag–Lefﬂer function (see below and also Refs. 7 and 31). The fact that wðtÞ is given by a generalized Mittag–Lefﬂer function amounts to assuming an asymptotic (longtime) power law form for the waiting time probability distribution function, namely considering slow diffusion, wðtÞ As ts t1s ;

ð0 < s < 1Þ

ð26Þ

(As is a constant). The characteristic (mean) waiting time hTW i ¼

1 ð

twðtÞ dt

ð27Þ

0

then always tends to 1 except in the limit s ! 1 (the classical Brownian motion), where wðtÞ ¼ dðt tÞ, so that hTW i ¼ t. A famous example [19] of a distribution function with a long-time tail like Eq. (26) is the Cauchy distribution wðtÞ ¼

a 1 p a2 þ t 2

ð28Þ

fractional rotational diffusion

299

with inﬁnite second moment. This distribution is just one example of a whole class of distributions which if applied to a sum of random variables do not converge to the Gaussian distribution as the number of random variables tends to inﬁnity. Thus, the central limit theorem, on which the theory of Brownian motion is based, is not obeyed because the long-time tails preclude convergence to the Gaussian distribution. Nevertheless, limiting (now called Le´vy [19], Chapter 4) distributions may exist. The divergence of the waiting time associated with the long-time-tailed nature of the waiting time probability distribution function Eq. (26) is according to Metzler and Klafter [7,31] a manifestation of the self-similar nature of the waiting time process. This has prompted many investigators [7] to use in the present context the term fractal time processes to describe anomalous relaxation. Returning to the fractional diffusion equation [Eq. (23)], that equation will now follow from Eq. (26) and CTRW theory because (Ref. 33, p. 118) the integral equation for the probability density f ðx; tÞ for a continuous-time random walker to be in a position x at time t starting from x ¼ 0 at t ¼ 0 with waiting time density given by Eq. (26) is equivalent in the diffusion limit to the fractional diffusion equation Eq. (23). We remark that postulating wðtÞ as a generalized Mittag–Lefﬂer function with long-time behavior given by Eq. (26) so that fractional diffusion may be described as a CTRW is (just as the postulate of the existence of a discrete time t, the duration of an elementary jump in the Einstein theory of the Brownian movement) equivalent to a stosszahlansatz for the Boltzmann equation. That equation must of necessity underpin the entire theory. In other words, the transition probability or ‘‘mechanism’’ of the fractional diffusion process is the CTRW. Below, we shall see in detail how the introduction of a waiting time density of the form of Eq. (26) allows one to generalize the Fokker–Planck equation of normal diffusion to fractional diffusion. We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye–Fro¨hlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye–Fro¨hlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34–36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker–Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics.

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william t. coffey, yuri p. kalmykov and sergey v. titov B.

Fractional Diffusion Equation for the Cole–Cole Behavior

We shall now demonstrate how the CTRW in the diffusion limit may be used to justify the fractional diffusion equation. We consider an assembly of permanent dipoles constrained to rotate about a ﬁxed axis (the dipole is speciﬁed by the angular coordinate f ) and a set of discrete orientations on the unit circle with ﬁxed angular spacing . We note that may not necessarily be ﬁxed; for example, if we have a Gaussian distribution of jumps, the standard deviation of serves as a ﬁxed quantity. A typical dipole may remain in a ﬁxed orientation at a given site for an arbitrary long waiting time. It may then reorient to another discrete orientation site. This is the discrete orientation model. Following the procedure suggested in [7,37] for the translational motion, we ﬁrst denote individual discrete orientations by f. . . 1; 0; 1; . . .g. A typical dipole is supposed to have orientation n ¼ 0 at time t ¼ 0. The dipole having oriented to the site n at time t is ﬁxed in that orientation that is trapped, for some random time. The random waiting times after which changes in orientation take place are denoted by fti g; i ¼ 1; 2; . . .; these times are assumed to be independent identically distributed random variables with the probability density function wðtÞ. Thus the situation is unlike that in the Einstein theory where t is ﬁxed. Note that an exponential distribution of t so that the mean waiting time is ﬁnite also leads in the diffusion limit to Einstein’s result. It follows therefore that a random walk where a mean waiting time exists and a ﬁnite jump length variance exists will always lead to Einstein’s result. (This behavior again stems from the central limit theorem.) We assume that the probability density function wðtÞ is independent of the orientation of the dipole at time t, that is, independent of n, the dipole when at orientation speciﬁed by n reorients only to its nearestneighbor sites, that is, n 1 as in Einstein’s theory, the probability of orienting from site n to n þ 1 is AðnÞ and the probability of orienting from site n to n 1 is BðnÞ, where AðnÞ and BðnÞ obey the normalization condition AðnÞ þ BðnÞ ¼ 1

ð29Þ

and are independent of the time. The probability GðtÞ that the dipole has survived in a given orientation–that is, at a given site for a time t—is ðt

1 ð

0

t

GðtÞ ¼ 1 wðtÞ dt ¼

wðtÞ dt

ð30Þ

fractional rotational diffusion

301

Using the Laplace transform of the survival probability GðtÞ at a site, we have ~ ~ ðsÞ GðsÞ ¼ s1 ½1 w

ð31Þ

Here a tilde denotes the Laplace transform, namely, ~f ðsÞ ¼

1 ð

f ðtÞest dt

ð32Þ

0

Now the waiting times are identically distributed random variables. Hence on introducing Qi ðtÞ, the probability that the dipole has changed i times in orientation ~ i ðsÞ in the time interval ð0; tÞ, we will have for the Laplace transform Q ~ wi ðsÞ ¼ s1 ½1 w ~ i ðsÞ ¼ GðsÞ~ ~ ðsÞ~ wi ðsÞ Q

ð33Þ

Following Ref. 37, let us now introduce Wðn; tÞ, which is the probability of ﬁnding the dipole in discrete orientation n at time t. Let us further introduce pi ðnÞ, which is the probability that the dipole has orientation n after i changes in orientation; then summing over all the orientation changes, we have ~ Wðn; sÞ ¼

1 X

~ ðsÞ pi ðnÞQi ðsÞ ¼ s1 ½1 w

i¼0

1 X

pi ðnÞ~ wi ðsÞ

ð34Þ

i¼0

Since only nearest neighbors are involved, now the evolution of pi ðnÞ is determined by the discrete time (i) and space (n) equation piþ1 ðnÞ ¼ Aðn 1Þpi ðn 1Þ þ Bðn þ 1Þpi ðn þ 1Þ

ð35Þ

The continuous distribution of orientations, f, is obtained by the replacement of pi ðnÞ by pi ðfÞ where pi ðfÞ df is the probability of ﬁnding the dipole after the ith jump in the angle f ! f þ df. In like manner Aðn 1Þ; Bðn þ 1Þ ! Aðf Þ; Bðf þ Þ. Thus Eq. (35) becomes piþ1 ðfÞ ¼ Aðf Þpi ðf Þ þ Bðf þ Þpi ðf þ Þ

ð36Þ

On expanding Aðf Þpi ðf Þ and Bðf þ Þpi ðf þ Þ in Taylor series, we have the following in the continuum limit with regard to terms in 2: Aðf Þpi ðf Þ ¼ AðfÞpi ðfÞ

q 2 q2 ½AðfÞpi ðfÞ þ ½AðfÞpi ðfÞ ð37Þ qf 2 qf2

Bðf þ Þpi ðf þ Þ ¼ BðfÞpi ðfÞ þ

q 2 q2 ½BðfÞpi ðfÞ þ ½BðfÞpi ðfÞ ð38Þ qf 2 qf2

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william t. coffey, yuri p. kalmykov and sergey v. titov

Moreover, by the principle of detailed balance, if the system is close to thermal equilibrium at temperature T [37], then we have AðfÞ BðfÞ ’

qVðfÞ 2kB T qf

ð39Þ

where VðfÞ is the external potential energy—for example, due to the electric ﬁeld acting on the system. Such a requirement on AðfÞ and BðfÞ guarantees that the system relaxes to the equilibrium Boltzmann distribution [37]. Equation (36) then becomes 2 q qpi ðfÞ pi ðfÞ qV piþ1 ðfÞ ¼ pi ðfÞ þ þ kB T qf 2 qf qf

ð40Þ

Now replacing n by f, we may rewrite Eq. (34) as ~ ðsÞ ~ ðsÞ 1w 1w ~ p0 ðfÞ þ Wðf; sÞ ¼ s s

1 X

pi ðfÞ~ wi ðsÞ

ð41Þ

i¼1

or according to Eq. (40) ~ ðsÞ ~ ðsÞ 1w 1w ~ dðfÞ þ Wðf;sÞ ¼ s s 1 X 2 q q pi1 ðfÞ qV ~ i ðsÞ ð42Þ pi1 ðfÞ þ pi1 ðfÞ þ þ w kB T qf 2 qf qf i¼1 We may now eliminate the summation in Eq. (42) by noting that according to Eq. (34) ~ ðsÞ 1w ~ Wðf; sÞ~ wðsÞ ¼ s

1 X

pi ðfÞ~ wiþ1 ðsÞ

ð43Þ

i¼0

on change of the summation variable to i ¼ j 1. Thus ~ ðsÞ 1w ~ dðfÞ Wðf; sÞ ¼ s ~ Wðf; sÞ qV 2 q q ~ ~ ~ ðsÞ Wðf; sÞ þ þw Wðf; sÞ þ þ kB T qf 2 qf qf

ð44Þ

fractional rotational diffusion

303

We now explicitly consider the waiting time distribution. First we reiterate that the Einstein theory of the Brownian motion relies on the central limit theorem that a sum of independent identically distributed random variables (the sum of the elementary displacements of the Brownian particle) SN ¼

N X

Xi

i¼0

! 1, provided that the ﬁrst and becomes a Gaussian distribution in the limit N second moments of Xi do not diverge. However, there are famous exceptions; for example, for the Cauchy distribution, f ðxÞ ¼

a 1 p a2 þ x2

ð45Þ

the second moment of this distribution is inﬁnite. The Cauchy distribution is just one example of a whole class of distributions which possess long inverse power law tails—for example, pðxÞ ¼

1

ð46Þ

jxj1þs

! 1, but The tails prevent [19] convergence to the Gaussian distribution for N not the existence of a limiting distribution. These distributions as we have seen are called stable distributions. If the concept of a Le´vy distribution is applied to an assembly of temporal random variables such as the fti g of the present chapter, then wðtÞ is a long-tailed probability density function with long-time asymptotic behavior [7,37], wðtÞ

sAs ; ð1 sÞt1þs

0 < s 1;

t!1

ð47Þ

The restriction to 0 < s < 1 ensures that the ﬁrst moment of the waiting-time distribution is divergent as is usual in a CTRW (corresponding to chaotic behavior of the waiting times). Moreover, in the s domain, the long-time behavior is manifested in the small s expansion [7,37] ~ ðsÞ ¼ 1 As ss þ c1 ðAs ss Þ2 þ w

ð48Þ

Thus it is obvious by the properties of the Laplace transform that the second term corresponds to the asymptotic behavior given by Eq. (47). If we now

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william t. coffey, yuri p. kalmykov and sergey v. titov

substitute Eq. (48) into Eq. (44), on multiplying across by s and simplifying, we have ~ sÞ s1 ð1 c1 As ss þ ÞdðfÞ þ ð1 c1 As ss þ ÞWðf; 2 ~ s q q ~ Wðf; sÞ qV 2 s Wðf; sÞ þ þ ¼ 0 þ ðs As þ c1 As s þ Þ qf qf kB T qf 2As ð49Þ If introducing the limiting procedure As ! 0, ! 0 with 2 !0 2As

ts ¼ lim

ð50Þ

As !0

ﬁnite, Eq. (49) becomes ~ s1 dðfÞ þ Wðf; sÞ ¼ ðtsÞs

~ Wðf; sÞ qV q q ~ Wðf; sÞ þ kB T qf qf qf

ð51Þ

or, on inversion to the time domain, becomes qWðf; tÞ ¼ t1s 0 D1s LFP Wðf; tÞ t qt

ð52Þ

1 q2 Wðf; tÞ 1 q qV Wðf; tÞ LFP Wðf; tÞ ¼ þ t kB T qf qf qf2

ð53Þ

where

is, in effect, the Fokker–Planck operator for normal rotational diffusion about a ﬁxed axis. The time t has the meaning of the intertrapping time scale which is identiﬁed with the Debye relaxation time t ¼ =kB T ( is friction coefﬁcient), is deﬁned in terms of the Riemann–Liouville fractional and the operator 0 D1s t integral deﬁnition, Eq. (24) and (25), namely, 1s 0 Dt

q ¼ 0 Ds ; qt t

s 0 Dt f ðtÞ

ðt 1 f ðt0 Þ dt0 ¼ ðsÞ ðt t0 Þ1s

ð54Þ

0

Equation (54) means [31] that Eq. (52) now contains a slowly decaying (s < 1) memory function with a power law kernel so that the process is no longer Markovian and thus depends on the history of the system. Equation (52) stems

fractional rotational diffusion

305

from a fractal-waiting-time or long-rests model, where the system is jammed in a particular conﬁguration for an arbitrary long interval. Such behavior stems in turn from assuming random forces with an anomalous waiting-time distribution. Equation (52) for fractional rotational diffusion has the same mathematical form as that for fractional translation diffusion Eq. (23) derived by Metzler and Klafter [7,31]. Thus the solution of Eq. (52) may be obtained [7,31] in similar manner using generalized Mittag–Lefﬂer or Fox functions [38]. The fractional diffusion equation [Eq. (52)] for the time evolution of the probability density function Wðf; tÞ in conﬁguration space is then the same as that previously derived [31] for a particle of one translational degree of freedom; however, rotational quantities replace translational ones (see, for example, Eq. (19) of Ref. 32). The advantage of using the CTRW formalism is that it is now possible to gain some insight into the meaning of the parameter s. It is the order of the fractional derivative in the fractional diffusion equation describing the continuum limit of a random walk with a chaotic set of waiting times (often known as a fractal-time random walk). However, a more physical and useful deﬁnition of s is as the fractal dimension of the set of waiting times which is the scaling of the waiting-time segments in the random walk with magniﬁcation. The parameter s thus measures the statistical self-similarity (or how the whole looks like its parts [19]) of the waiting time segments. In order to construct such an entity in practice, a whole discrete hierarchy of time scales is needed. For example, a fractal time Poisson process [19] with a waiting-time distribution assumes the typical form of the Le´vy stable distribution. This is explicitly discussed in Ref. 19 where a formula for s is given and is also discussed in Ref. 23 where the fractal time process is essentially generated by considering jumps over the wells of a chaotic potential barrier landscape (random activated energy modes). The microscopic picture presented in Refs. 19 and 23 appears to completely support the commonly used experimental representation of the anomalous behavior as a distribution of Debye-like relaxation mechanisms with a continuous relaxation time distribution function. C.

Anomalous Dielectric Relaxation in the Context of the Debye Model of Noninertial Rotational Diffusion

In his work on dielectric relaxation of an assembly of noninteracting dipolar molecules, Debye [1] considered two models: (a) an assembly of ﬁxed axis rotators, each having permanent dipole moment l and subjected to Brownian motion torques having their origin in the background or heat bath, and (b) the same assembly, but the restriction to ﬁxed axis rotation is removed. Nevertheless, the results in both instances are qualitatively the same, since in each case the time-dependent distribution function, if inertial effects are disregarded [39–41], depends only on a single space coordinate. In case (a) this is the azimuthal angle f while in case (b) it is the polar angle # (the colatitude). Thus in the calculations

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william t. coffey, yuri p. kalmykov and sergey v. titov

which follow which are identical to those in Chapter V of the Debye book [1] with the fractional diffusion equation in conﬁguration space (here the space of polar angles) replacing the Smoluchowski equation used by him, we shall mainly conﬁne ourselves to ﬁxed axis rotation and merely allude to the corresponding result for rotation in space. In general, an assembly of ﬁxed axis rotators qualitatively reproduces the principal features of dielectric relaxation of dipolar molecules in space while allowing considerable mathematical simpliﬁcation of the problem. Again in the spirit of the Debye calculations [1] we shall consider two separate cases of dielectric relaxation, namely (a) the response following the sudden removal of a constant ﬁeld (after effect response) and (b) the response to an alternating current (ac) ﬁeld which has been applied for a long time so that a steady state has been attained. An advantage of the two separate calculations is that they explicitly demonstrate that the concepts of linear response theory will hold (as is to be expected) for relaxation processes in fractal structures. Before proceeding, we remark that just as in the Einstein theory of the Brownian motion of which the Debye theory is a rotational version, the characteristic microscopic time scale is a time interval t1 so long that the motion of the particle at time t is independent of its motion at time t t1 , but small compared to the observation time intervals [17]. It is also supposed that during the time t1 , which is the mean of the time intervals between collision events, any external nonstochastic forces which may be applied to the system do not alter. In the fractal waiting time picture, however, the concept of collision rate does not hold and the mean of the time intervals between collision events diverges. In the ﬁxed axis rotation model of dielectric relaxation of polar molecules a typical member of the assembly is a rigid dipole of moment m rotating about a ﬁxed axis through its center. The dipole is speciﬁed by the angular coordinate f (the azimuth) so that it constitutes a system of 1 (rotational) degree of freedom. The fractional diffusion equation for the time evolution of the probability density function Wðf; tÞ in conﬁguration space is given by Eq. (52) which we write here as qWðf; tÞ q qWðf; tÞ Wðf; tÞ qVðf; tÞ ¼ ts 0 D1s þ t qt qf qf kB T qf

ð55Þ

Here Vðf; tÞ ¼ mFðtÞ cos f is the potential arising from an external applied electric ﬁeld FðtÞ. Here, just as with the translational diffusion equation treated in Ref. 7, we consider subdiffusion, 0 < s < 1 phenomena only. Here, the internal ﬁeld effects are ignored, which means that the effects of long-range torques due to the interaction between the average moments and the Maxwell ﬁelds are not taken into account. Such effects may be discounted for dilute systems in ﬁrst approximation. Thus, the results obtained here are relevant to situations where dipole–dipole interactions have been eliminated by extrapolation of data to inﬁnite dilution.

fractional rotational diffusion

307

As we have mentioned, we shall consider two classes of solution of Eq. (55): The ﬁrst is the aftereffect solution following the removal of the constant ﬁeld. In the case of the aftereffect solution of the fractional diffusion equation, a uniform ﬁeld F, having been applied to the assembly of dipoles at a time t ¼ 1 so that equilibrium conditions prevail by the time t ¼ 0, is switched off at t ¼ 0. In addition, it is supposed that the ﬁeld is weak (mF kB T); thus for t > 0, Eq. (55) becomes qWðf; tÞ q2 Wðf; tÞ ¼ ts 0 D1s t qt qf2

ð56Þ

which must be solved subject to the initial condition Wðf; 0Þ ¼ Ce

mF kB T

cos f

1 mF 1þ cos f

2p kB T

ð57Þ

where C and 2p are the normalizing constants. Just as in normal diffusion, the form of the initial condition Eq. (57) suggests that the time-dependent solution should be of the form 1 mF 1 þ gðtÞ cos f Wðf; tÞ ¼ 2p kB T

ð58Þ

Substitution of Eq. (58) into Eq. (56) then yields the following fractional differential equation for the function gðtÞ: d gðtÞ ¼ ts 0 D1s gðtÞ t dt

ð59Þ

The solution of this fractional relaxation equation [31] is gðtÞ ¼ Es ½ðt=tÞs

ð60Þ

where Es ðzÞ is the Mittag–Lefﬂer function deﬁned by Es ðzÞ ¼

1 X n¼0

zn ð1 þ s nÞ

ð61Þ

The Mittag–Lefﬂer function interpolates between the initial stretched exponential form [7,31] ðt=tÞs Es ½ðt=tÞs exp ð1 þ sÞ

ð62Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

and the long-time inverse power-law behavior Es ½ðt=tÞs ½ðt=tÞs ð1 sÞ1

ð63Þ

The Debye result for gðtÞ corresponds to s ¼ 1, namely, E1 ðt=tÞ ¼ et=t

ð64Þ

that is, the exponential function is a special case of the Mittag–Lefﬂer function [31]. The Mittag–Lefﬂer function can be expressed in terms of the Fox H-function [38] [see Eq. (A1.8) in Appendix I]. We may now calculate the mean dipole moment. The mean dipole moment due to orientation alone is given at any time t > 0 (f is a unit vector in the direction of ﬁeld F) by hl fiðtÞ ¼

2ðp

m cos fWðf; tÞ df

ð65Þ

0

so that with Eq. (58) we have hl fiðtÞ ¼

m2 F Es ½ðt=tÞs 2kB T

ð66Þ

in contrast to the Debye result embodied in Eq. (64). A practically much more important result is the behavior of the system in a periodic ﬁeld FðtÞ ¼ Feiot so that the fractional diffusion equation [Eq. (55)] for the distribution function becomes qWðf; tÞ q mF q2 Wðf; tÞ iot ¼ ts 0 D1s sin fe Wðf; tÞ þ t qt qf kB T qf2

ð67Þ

Following Debye, let us try as a solution 1 mF iot 1 þ BðoÞ cos fe Wðf; tÞ ¼ 2p kB T

ð68Þ

where BðoÞ is a constant to be determined. Substitution of Eq. (68) into Eq. (67) yields f½1 BðoÞeiot g ioBðoÞeiot ¼ ts 0 D1s t

ð69Þ

fractional rotational diffusion

309

Equation (69) may be further simpliﬁed if we recall the integration theorem of Laplace transformation as generalized to fractional calculus, [31], namely, L

f0 Dt1s f ðtÞg

¼

s1s~f ðsÞ 0 Ds t f ðtÞj t¼0; 1s~ s f ðsÞ;

0<s<1 1s < 2

ð70Þ

where ~f ðsÞ ¼ Lf f ðtÞg ¼

1 ð

est f ðtÞ dt

ð71Þ

0

(This theorem is of fundamental importance in fractional dynamics because use of it coupled with continued fraction methods [8] allows recurrence relations associated with normal diffusion to be generalized to fractional dynamics in an obvious fashion.) Regarding the ac response, if we assume that the above result may be analytically continued into the domain of the imaginaries or if we equivalently note that if D denotes the operator d=dt and GðoÞ denotes an arbitrary function of o, then [42] 1 iot eiot e GðoÞ ¼ GðoÞ D io The generalization of the above operator equation is 1 iot eiot e GðoÞ ¼ GðoÞ Dq ðioÞq

ð72Þ

where q denotes a fractional index. Equation (69), assuming Eq. (72), then simpliﬁes to BðoÞ ¼

1 1 þ ðiotÞs

ð73Þ

Thus we have in linear response 1 mF eiot cos f 1þ Wðf; tÞ ¼ 2p kB T 1 þ ðiotÞs

ð74Þ

and as before the mean moment is hl fiðtÞ ¼

m2 Feiot 1 2kB T 1 þ ðiotÞs

ð75Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

The complex susceptibility is given by the Cole–Cole equation [Eq. (9)] with the static susceptibility for ﬁxed-axis rotators w0 ¼

m2 N0 2kB T

ð76Þ

where N0 is the concentration of dipoles. Equation (9) may also be derived from the aftereffect solution Eq. (66) with the help of linear response theory. According to that theory, the complex dynamic susceptibility is given by wðoÞ ¼ w0 io

1 ð

bðtÞeiot dt

ð77Þ

0

where bðtÞ is the aftereffect solution. (In the context of the use of this theorem, we note the nonstationary nature of the stosszahlansatz or mechanism underlying the fractional dynamics.) Equation (77) with the aftereffect function bðtÞ ¼ N0

hl fiðtÞ F

given by Eq. (66) yields wðoÞ ¼ 1 io w0

1 ð

Es ½ðt=tÞs eiot dt

ð78Þ

0

Now, on noting that the Laplace transform of the Mittag–Lefﬂer function Es ½ðt=tÞs is [31] L fEs ½ðt=tÞs g ¼

1 ð

Es ½ðt=tÞs est dt ¼

1 s þ ts s1s

ð79Þ

0

Equation (9) is retrieved, demonstrating that as far as the present problem is concerned, linear response theory is obeyed in the fractional dynamics despite the nonstationary character of the mechanism underlying the fractal relaxation process. The approach we have just given may be carried over in all respects to rotation in space. Here, the space coordinate is the polar angle # (the colatitude)

fractional rotational diffusion

311

and the fractional diffusion equation assumes the form [1] [referring to the ac solution; cf. Eq. (8)] qWð#; tÞ 1 q mF qWð#; tÞ iot ¼ ts 0 D1s sin # sin #e Wð#; tÞ þ t qt 2 sin # q# kB T q# ð80Þ where the time t ¼ =ð2kB TÞ

ð81Þ

is the Debye relaxation time for rotation in space. Recalling the work of Debye, it is now apparent that Eq. (80) may be solved just as Eq. (67), which pertains to rotation about a ﬁxed axis, to yield the corresponding result for rotation in space, namely, the aftereffect solution and the ac stationary response m2 F Es ½ðt=tÞs 3kB T m2 F eiot hl fi ¼ 3kB T 1 þ ðiotÞs

hl fiðtÞ ¼

ð82Þ ð83Þ

The complex susceptibility is given again by Eq. (9) with the static susceptibility for rotators in space: w0 ¼

m2 N0 3kB T

ð84Þ

One can see that just as in normal diffusion, Eqs. (83) and (82) differ from the corresponding two-dimensional analogs [Eqs. (66) and (75)] only by a factor 2=3 and the appropriate deﬁnition of the Debye relaxation time t ¼ =ðkB TÞ and t ¼ =ð2kB TÞ, respectively. The principal result of our calculation is that the Debye theory (based on the Smoluchowski equation), when extended to fractional dynamics via a onedimensional noninertial fractional Fourier–Planck equation in conﬁguration space, can explain the Cole–Cole anomalous dielectric relaxation that appears in some complex systems and disordered materials. A further result of our calculation is that the aftereffect solution [Eq. (66)] is, with slight modiﬁcations, the moment generating function of the conﬁguration space distribution function. Hence the mean-square angular displacement of a dipole, and so on, may be easily calculated by differentiation. We must remark, however, that the fractional Debye theory can be used only at low frequencies (ot 1) just as

312

william t. coffey, yuri p. kalmykov and sergey v. titov

its normal diffusion counterpart. The most straightforward way of demonstrating the complete breakdown of the Debye theory at high frequencies is simply to calculate the dipole absorption coefﬁcient, which is proportional to ow00 ðoÞ [24]. It follows immediately from Eq. (76) that both normal and anomalous diffusion versions of the Debye theory fail at very high frequencies because neither predict a return to optical transparency at these frequencies. In normal diffusion, for example, the absorption coefﬁcient exhibits the Debye plateau phenomenon, namely, ow00 ðoÞ ! constant at o ! 1. The effect is even more pronounced in anomalous diffusion, where ow00 ðoÞ ! 1 as o ! 1. Moreover, the Gordon sum rules [43] 1 ð

ow00 ðoÞ do ¼ pN0 m2 =ð4IÞ

ð85Þ

ow00 ðoÞ do ¼ pN0 m2 =ð3IÞ

ð86Þ

0

for ﬁxed-axis rotators and 1 ð

0

for rotators in space (I is the moment of inertia of a dipole), which dictate that the dipole integral absorption remains ﬁnite, are not satisﬁed in both normal and anomalous diffusion [24]. It is signiﬁcant that the right-hand side of Eqs. (85) and (86) is determined by molecular parameters only and is independent of the temperature and the model parameters. If inertial effects are included, however, as discussed in detail by Rocard [44], Gross [39], and Sack [40] (see Refs. 41 and 43 for detailed reviews), the Gordon sum rule is satisﬁed thus the desired return to optical transparency at high frequencies is obtained. We shall show presently how one may include inertial effects in fractional relaxation processes in Section IV. In the light of previous work [45,46] in the theory of anomalous translational diffusion, we must allude to additional mechanisms that may give rise to anomalous rotational diffusion. Examples of such mechanisms are time-rescaled Brownian motion or generalized Langevin equations with Gaussian non-white noise [45,46] so that the memory function is no longer a delta function. The concept of a generalized Langevin equation with friction term given by the Riemann–Liouville deﬁnition of the fractional derivative has been used to analyze translational anomalous diffusion by Lutz [47] (see Section IV.D). Furthermore, the Cole–Davidson expression [Eq. (10)] for the complex susceptibility has also been obtained by Nigmatullin and Ryabov [28] from an appropriate generalized Langevin equation with fractional derivative memory function.

fractional rotational diffusion D.

313

Fractional Diffusion Equation for the Cole–Davidson and Havriliak–Negami Behavior

Here we demonstrate how both the Cole–Davidson and Havriliak–Negami anomalous behaviors may be embodied, just as the Cole–Cole behavior, in a fractional generalization of the Fokker–Planck equation in conﬁguration space for the ﬁrst Debye model of non-electrically interacting dipoles. In what follows, we shall use a cumulant expansion originally given by Nigmatullin and Ryabov [28], who proposed a phenomenological fractional ordinary differential equation describing the Cole–Davidson behavior. In addition, we shall demonstrate how the aftereffect function [8] following the removal of a small constant ﬁeld may be given for the Havriliak–Negami model in terms of a Fox H function [38,48] extrapolating between the stretched exponential (Kohlrausch–Williams–Watts) law at short times and the inverse long-time-tail power law at long times [7]. In Cole–Cole relaxation, the Fox H function reduces to the well-known Mittag– Lefﬂer function [7] (see Appendix I). The Debye relaxation time in this case plays the role of a time scale demarcating the transition from the stretched exponential law to the power law [20]. In order to generalize the normal Fokker–Planck equation excluding inertial effects to fractional diffusion, we ﬁrst recall the general form of that equation for normal diffusion in operator representation [49] qWðx; tÞ ¼ LFP Wðx; tÞ; qt

t > 0;

Wðx; 0Þ ¼ W0 ðxÞ

ð87Þ

where LFP is the Fokker–Planck operator. Equation (87) can equivalently be rewritten as an equation for an impulse response so that the initial condition appears as the amplitude of a forcing function dðtÞ, [50], namely, qWðx; tÞ ¼ LFP Wðx; tÞ þ dðtÞW0 ðxÞ qt

ð88Þ

where Wðx; tÞ ¼ yðtÞWðx; tÞ, yðtÞ is the Heaviside unit step function, and dðtÞ is the Dirac delta function. The fractional analog of Eq. (88), [i.e., Eq. (52)] can now be rewritten as qWðx; tÞ ¼ t1s 0 D1s LFP Wðx; tÞ þ dðtÞW0 ðxÞ t qt

ð89Þ

where the fractional derivative is given by Eq. (54). Equation (89) may be written in an equivalent form as [7,20] ts 0 Dst Wðx; tÞ ¼ tLFP Wðx; tÞ þ

ðt=tÞs W0 ðxÞ ð1 sÞ

ð90Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

The time behavior of the right-hand side of Eq. (90) indicates that the initial state W0 ðxÞ decays slowly with a long-time tail unlike the exponential decay of normal diffusion, which is an indication of the fractal time character of the process. Equations (89) and (90) are fractional analogs of the conventional Fokker–Planck equation [Eq. (88)] giving rise to the Cole–Cole anomalous behavior. Another approach to fractionalize the Fokker–Planck equation incorporating Cole–Davidson behavior can now be given by extending a hypothesis of Nigmatullin and Ryabov [28]. They noted that the ordinary ﬁrst-order differential equation describing an exponential decay d f ðtÞ þ 0 f ðtÞ ¼ 0 dt with initial condition f ðtÞjt¼0 ¼ f ð0Þ may be written as an equation for an impulse response, namely, e0 t

d 0 t e yðtÞ f ðtÞ ¼ f ð0ÞdðtÞ dt

ð91Þ

which in turn may be written as [recalling that the derivative of the step function is the Dirac delta function, qt yðtÞ ¼ dðtÞ] e0 t

d 0 t e ½yðtÞf ðtÞ þ yðtÞf ð0Þ ¼ yðtÞ0 f ð0Þ dt

ð92Þ

In order to obtain in heuristic fashion a fractional analog of Eq. (92), one may simply replace the ordinary derivative by a fractional derivative [28] so that Eq. (92) becomes e00 t Dnt fe0 t ½yðtÞf ðtÞ þ yðtÞf ð0Þg ¼ yðtÞ0 f ð0Þ

ð93Þ

In the particular application to dielectric relaxation, f ðtÞ is the aftereffect function following the removal of a constant ﬁeld [8]. The solution of Eq. (93) rendered in the frequency domain yields the Cole–Davidson equation [Eq. (10)] [28]. The approach of Nigmatullin and Ryabov [28] is, however, entirely phenomenological because no underlying kinetic equation is involved. Nevertheless, their method may also be applied to the Fokker–Planck equation [Eq. (88)] so that a kinetic equation and thus a microscopic model is involved. Indeed, we can rewrite the normal Fokker–Planck equation [Eq. (88)] as an

fractional rotational diffusion

315

equation for an impulse response in the form of the right-hand side of Eq. (91), namely, eLFP t

q LFP t ½e Wðx; tÞ ¼ dðtÞW0 ðxÞ qt

ð94Þ

Equation (94) assumes the form of Eq. (92), namely, eLFP t

q LFP t e ½Wðx; tÞ þ yðtÞW0 ðxÞ þ yðtÞLFP W0 ðxÞ ¼ 0 qt

ð95Þ

For the purpose of using a kinetic equation incorporating the Cole–Davidson mechanism according to the heuristic procedure of Nigmatullin and Ryabov [28], we may replace the partial time derivative in Eq. (95) by a fractional time derivative 0 Dnt . Thus Eq. (95) becomes [cf. Eq. (93)] tÞ þ yðtÞW0 ðxÞ þ yðtÞLFP W0 ðxÞ ¼ 0 tn1 eLFP t 0 Dnt eLFP t ½Wðx;

ð96Þ

Next we recall the cumulant operator expansion for the operator eB AeB [28], namely, e

B

½A; B; B ½A; B þ þ Ae ¼ A þ 2! 1! B

where the commutator ½A; B is deﬁned as ½A; B ¼ AB BA. This expansion allows one to represent Eq. (96) as the series of operators: t

n1

n 0 Dt

n

½0 Dt ; LFP t; LFP t ½0 Dnt ; LFP t þ yðtÞW0 þ þ ½W 1! 2!

ð97Þ

þ yðtÞLFP W0 ¼ 0 This series can be simpliﬁed by using the relationship between fractional derivatives [32]: n 0 Dt

½t f ðtÞ t 0 Dnt ½ f ðtÞ ¼ n 0 Dn1 ½ f ðtÞ t

Thus Eq. (97) reduces to tn1

n 0 Dt

nLFP 0 Dtn1 nðn 1ÞL2FP 0 Dtn2 þ yðtÞW0 þ þ ½W 1! 2!

þ yðtÞLFP W0 ¼ 0

ð98Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

which in turn assumes the shifted fractional differential operator form, namely, tÞ þ yðtÞW0 ðxÞ þ t yðtÞLFP W0 ðxÞ ¼ 0 ðt0 D1t tLFP Þn ½Wðx;

ð99Þ

Here we have recalled the binomial expansion ða þ bÞn ¼

1 X ð1Þn ðnÞn nn n a b n! n¼0

ð100Þ

where ðaÞn ¼ ðn þ aÞ=ðaÞ is a Pochhammer symbol [51]. Equation (99) represents the generalization (by heuristic reasoning) of the normal Fokker– Planck equation to anomalous diffusion governed by a Cole–Davidson relaxation mechanism. In like manner, combining the ideas embodied in the fractional diffusion Eq. (90) describing Cole–Cole relaxation and Eq. (96) describing Cole– Davidson relaxation, we may introduce the fractional kinetic equation ½ts 0 Dst tLFP n ½Wðx; tÞ þ yðtÞW0 ðxÞ þ t yðtÞLFP W0 ðxÞ ¼ 0

ð101Þ

For two particular cases n ¼ 1, 0 < s < 1 and s ¼ 1, 0 < n < 1, Eq. (101) reduces to Eqs. (90) and (99), respectively. Equation (101) represents a fractional generalization of the normal Fokker– Planck equation incorporating the Havriliak–Negami relaxation mechanism. Thus in any model described by a Fokker–Planck equation the effect of the Havriliak–Negami mechanism on the normal diffusion response may be included by solving Eq. (101). It allows one to generalize both the original Debye model and the Debye–Fro¨hlich model to include this relaxation mechanism. In the particular case of the ﬁrst Debye model, namely, an assembly of noninteracting dipoles, Eq. (101) will yield the simple Havriliak– Negami Eq. (11) as demonstrated below. E.

Fundamental Solution of the Fractional Smoluchowski Equation

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)–(11) may be obtained from our fractional generalizations of the Fokker–Planck equation in conﬁguration space (in effect, fractional Smoluchowski equations), Eq. (101), we ﬁrst consider the fractional rotational motion of a ﬁxed axis rotator [1], which for the normal diffusion is the ﬁrst Debye model (see Section II.C). The orientation of the dipole is speciﬁed by the angular coordinate f (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored.

fractional rotational diffusion

317

The normal Fokker–Planck equation for the time evolution of the probability density function Wðf; tÞ on the unit circle in conﬁguration space is then qWðf; tÞ ¼ LFP Wðf; tÞ; qt

Wðf; 0Þ ¼ W0 ðfÞ

ð102Þ

where the Fokker–Planck operator is deﬁned by Eq. (53) [see also Eq. (7)]. In the absence of external ﬁelds the Fokker–Planck operator is given by LFP ¼ t1

q2 qf2

ð103Þ

Equation (101) thus becomes tÞ þ yðtÞW0 ðfÞ þ t yðtÞLFP W0 ðfÞ ¼ 0 ½ts 0 Dst tLFP n ½Wðf;

ð104Þ

Here we wish to obtain the aftereffect solution for an assembly of ﬁxed axis rotators. Recalling that for rotation, the probability density function must be periodic in f, we expand Wðf; tÞ in the Fourier series: Wðf; tÞ ¼ yðtÞ

1 X

fp ðtÞeipf

ð105Þ

p¼1

Moreover, W must be real so that the Fourier coefﬁcient fp ðtÞ, which is the characteristic function of W must satisfy fp ðtÞ ¼ fp ðtÞ, where the asterisk denotes the complex conjugate. Substitution of Eq. (105) into Eq. (104) now yields 1 X ð1Þn ðnÞn 2n sðnnÞ sðnnÞ p t Dt ½yðtÞfp ðtÞ þ fp ð0ÞyðtÞ ¼ p2 yðtÞfp ð0Þ n! n¼0

ð106Þ Here we have noted Eq. (100) and q2 ipf e ¼ t1 p2 eipf qf2 1 X LnFP Wðf; ð1Þn fp ðtÞp2n eipf tÞ ¼ yðtÞtn LFP eipf ¼ t1

ð107Þ ð108Þ

p¼1

W0 ðfÞ ¼

1 X p¼1

fp ð0Þeipf

ð109Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

By using the integration theorem of two-sided Fourier transformation generalized to fractional calculus [32], namely, Ff0 Dat yðtÞf ðtÞg ¼ ðioÞa~f ðoÞ

ð110Þ

where ~f ðoÞ ¼

1 ð

e

iot

yðtÞf ðtÞ dt ¼

1

1 ð

eiot f ðtÞ dt

ð111Þ

0

with inverse 1 f ðtÞ ¼ 2p

1 ð

eiot~f ðoÞ dt

ð112Þ

1

we have from Eq. (106) X 1 ð1Þn ðnÞn 2n p2 ~fp ðoÞ fp ð0Þ p ðiotÞsðnnÞ ¼ fp ð0Þ io n¼0 n! io

ð113Þ

Using Eq. (100), Eq. (113) can be further rearranged as ½ fp ð0Þ io~fp ðoÞ½ p2 þ ðiotÞs n ¼ p2 fp ð0Þ

ð114Þ

Here we have also recalled that Dtnn yðtÞ ¼ Dnn1 dðtÞ. We remark that t posing the problem of the step of solution in terms of an impulse response in the time domain with the initial conditions regarded as the amplitude of the impulse has enabled us to obtain the solution in a very simple way using the generalized Fourier integration theorem [Eq. (110)]. This method, unlike the corresponding generalized Laplace transform integration theorem [32], does not involve the initial conditions. The same result may, however, be obtained using the Laplace transform if we rearrange Eq. (101) as an equation for the time-dependent part of the distribution function only so that initial conditions are not involved. Inverting Eq. (114), we have for the inverse Fourier transform over o (see Appendix I) fp ðtÞ p2ð1nÞ 1;1 2 ð1; 1Þ s ¼1 H p ðt=tÞ ðn; 1Þ; ð0; sÞ fp ð0Þ ðnÞ 1;2

ð115Þ

319

fractional rotational diffusion

1;1 where H1;2 is the Fox H function [48]. If the initial distribution function is a delta function, namely, W0 ðfÞ ¼ dðf f0 Þ, the solution (Green’s function) is given by

Wðf; tjf0 ; 0Þ 1 ð1; 1Þ 1 X p2ð1nÞ 1;1 2 s H eipðff0 Þ ; 1 p ðt=tÞ ¼ 2p p¼1 ðnÞ 1;2 ðn; 1Þ; ð0; sÞ

t 0 ð116Þ

Here we have noted that [50] dðfÞ ¼

1 1 X eipf 2p p¼1

For two particular cases n ¼ 1, 0 < s < 1 (Cole–Cole relaxation mechanism) and s ¼ 1, 0 < n < 1 (Cole–Davidson relaxation mechanism), Eq. (116) can be considerably simpliﬁed. Here Eq. (116) becomes, respectively (see Appendix I), Wðf; tjf0 ; 0Þ ¼

1 1 X Es ½p2 ðt=tÞs eipðff0 Þ ; 2p p¼1

t 0

ð117Þ

and 1 1 X p2ð1nÞ 2 gðn; p t=tÞ eipðff0 Þ ; 1 Wðf; tjf0 ; 0Þ ¼ 2p p¼1 ðnÞ

t 0 ð118Þ

where Es ðzÞ and gða; zÞ are the Mittag–Lefﬂer and incomplete Gamma functions, respectively. The Mittag–Lefﬂer function is given by Eq. (61) and the incomplete Gamma function [51] is deﬁned as ðz

gða; zÞ ¼ ta1 et dt ¼ 0

1 1X ðzÞnþa a n¼0 ða þ nÞn!

ð119Þ

For s ¼ 1, Eq. (116) yields the Green function for normal rotational diffusion of a planar rotator, namely, Wðf; tjf0 ; 0Þ ¼

1 1 X 2 eipðff0 Þp t=t ; 2p p¼1

t 0

ð120Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

which, we emphasize, is periodic in f. For comparison, the Green function Wðx; tjx0 ; 0Þ for translational diffusion of a free particle along the x axis [8] is given by 1 Wðx; tjx0 ; 0Þ ¼ 2p

1 ð

eixðxx0 Þx

2

Dt=2

dx ð121Þ

1

2 1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ eðxx0 Þ =ð4DtÞ ; 4pDt

1 < x < 1;

t 0

As one can see, the continuous x in Eq. (121) replaces the discrete p of the rotational case. In order to calculate dielectric response functions, we suppose that a uniform ﬁeld F (having been applied to the assembly of dipoles at a time t ¼ 1 so that equilibrium conditions prevail by the time t ¼ 0) is switched off at t ¼ 0. In addition, we suppose that the ﬁeld is weak (i.e., mF kB T, which is the linear response condition [8]). Thus the initial distribution function Wðf; 0Þ is the Boltzmann distribution function and is given by Eq. (57). Now one can readily obtain the corresponding aftereffect solution 1 mF 1þ f1 ðtÞ cos f Wðf; tÞ ¼ 2p kB T

ð122Þ

where the aftereffect function f1 ðtÞ is given by Eq. (115) for p ¼ 1 with f1 ð0Þ ¼ 1, namely, 1 ð1; 1Þ 1;1 ð123Þ H1;2 f1 ðtÞ ¼ 1 ðt=tÞs ðn; 1Þ; ð0; sÞ ðnÞ The function f1 ðtÞ has initially ðt tÞ stretched exponential form sn

f1 ðtÞ eðt=tÞ

=ð1þnsÞ

ð124Þ

In contrast, at long times ðt tÞ, it has the inverse power law behavior f1 ðtÞ

ðn þ 1Þðt=tÞs ðnÞð1 sÞ

ð125Þ

The behavior of f1 ðtÞ is shown in Figs. 1 and 2 for various values of s and n; the asymptotes [Eqs. (124) and (125)] are also shown in these ﬁgures for

321

fractional rotational diffusion

1"

2"

3"

100 3 2 f1(t) / f1(0)

1' 1

10−1 σ = 0.5

1,1',1": ν =0.2 2,2',2": ν =0.5 3,3',3": ν =1.0

2'

10−2

10−2

10−4

100

3'

102

t/τ Figure 1. f1 ðtÞ=f1 ð0Þ as a function of t=t for s ¼ 0:5 and different values of n [Eq. (A1.4): solid line] with the short [Eq. (124): dotted lines] and long [Eq. (125): dashed lines] time asymptotes.

comparison. For pure Cole–Davidson relaxation, where s ¼ 1 (see Fig. 2), the asymptote f1 ðtÞ

et=t ðt=tÞn1 ðnÞ

ð126Þ

must be used instead of Eq. (125) because this is the correct asymptote for the incomplete Gamma function. Thus the decay of f1 ðtÞ is essentially exponential in the limit of long times in Cole–Davidson relaxation. Thus it is apparent from Eqs. (125) and (126) that the origin of the long-time tail is the parameter s, which is ultimately due to the trapping effects inherent in the fractal time random walk—that is, the microscopic disorder. As far as the short-time behavior [Eq. (124)] is concerned, the role of n is to enhance the stretched exponential behavior for given s. Equation (122) allows us readily to evaluate the polarization PðtÞ, namely, PðtÞ ¼ N0 m

2ðp

0

cos fWðf; tÞ df ¼ w0 Ff1 ðtÞ

ð127Þ

322

william t. coffey, yuri p. kalmykov and sergey v. titov 2"

1" 10

0

3"

3

f1(t) / f1(0)

2 1 1' 10

−1

2' ν = 0.5

3'

1,1',1": σ = 0.2 2,2',2": σ = 0.5 3,3',3": σ = 1.0 10

−2

10

−4

10

−2

10

0

10

2

t /τ f1 ðtÞ=f1 ð0Þ as a function of t=t for n ¼ 0:5 and different values of s [Eq. (A1.4): solid line] with the short [Eq. (124): dotted lines] and long [Eqs. (125) (for s ¼ 0:2 and 0.5) and (126) (for s ¼ 1): dashed lines] time asymptotes.

Figure 2.

and the complex dielectric susceptibility, which is deﬁned by linear response theory as [8] wðoÞ ¼ 1 io~f1 ðioÞ w0

ð128Þ

with w0 given by Eq. (76). Thus we have the Havriliak–Negami equation [Eq. (11)]. We remark that Eq. (11) can be obtained by extracting f1 ðtÞ from the Green function, Eq. (116), and noting the linear response theory relation [8] f1 ðtÞ ¼

hcos fð0Þ cos fðtÞi0 hcos2 fð0Þi0

ð129Þ

relating the aftereffect function to the equilibrium dipole moment autocorrelation function. Here h. . .i0 means the equilibrium statistical averages over the

fractional rotational diffusion

323

equilibrium distribution function Weq ðfÞ. The autocorrelation function from Eq. (129) is deﬁned as hcos fð0Þ cos fðtÞi0 ¼

2ðp 2ðp

cos f0 cos fWeq ðfÞWðf; tjf0 ; 0Þ dfdf0

ð130Þ

0 0

As Weq ðfÞ ¼ 1=ð2pÞ for rotation in a plane, one can readily verify that Eqs. (123), (128), (129), and (130) yield Eq. (11). So far we have considered the planar rotator model. However, the above equations can readily be generalized to rotation in space. Here, the space coordinate is the polar angle # and the Fokker–Planck operator for normal rotational diffusion assumes the form [8] [cf. Eq. (80)] 1 q qVð#; tÞ qWð#; tÞ LFP Wð#; tÞ ¼ sin # Wð#; tÞ þ 2t sin # q# q# q#

ð131Þ

so that Eq. (101) now becomes tÞ þ yðtÞW0 ð#Þ þ t yðtÞLFP W0 ð#Þ ¼ 0 ½ts 0 Dst tLFP n ½Wð#;

ð132Þ

By expanding Wð#; tÞ in the Fourier series Wð#; tÞ ¼ yðtÞ

1 X

fp ðtÞPp ðcos #Þ

p¼0

(Pn ðzÞ are the Legendre polynomials [51] which now constitute the appropriate basis set), Eq. (132) may be solved to yield the corresponding results for rotation in space, namely, the aftereffect function [Eq. (123)] and the complex susceptibility [Eq. (11)], with t and w0 from Eqs. (81) and (84), respectively. Apparently as in normal diffusion, the results differ from the corresponding two-dimensional analogs only by a factor 2=3 in w0 and the appropriate deﬁnition of the Debye relaxation time. Thus we have demonstrated how the empirical Havriliak–Negami equation [Eq. (11)] can be obtained from a microscopic model, namely, the fractional Fokker–Planck equation [Eq. (101)] applied to noninteracting rotators. This model can explain the anomalous relaxation of complex dipolar systems, where the anomalous exponents s and n differ from unity (corresponding to the classical Debye theory of dielectric relaxation); that is, the relaxation process is characterized by a broad distribution of relaxation times. Hence, the empirical Havriliak–Negami equation of anomalous dielectric relaxation which has been

324

william t. coffey, yuri p. kalmykov and sergey v. titov

extensively used to analyze the experimental dielectric loss of disordered glassy systems (see, e.g., Refs. 52 and 53) may be formulated in terms of a microscopic model just as the original Debye equation. A complete understanding of the Havriliak–Negami mechanism requires an understanding of the microscopic origin of the parameters s and n. As far as s is concerned, this fractional exponent arises naturally from the diffusion limit of a fractal time walk and may be construed as arising from a chaotic set [23] of microscopic potential barrier heights; that is, s has its origin in random activation energy models [20]. Further evidence for the random activation energy model concept as the generator of the Cole–Cole mechanism is provided by the recent experimental results of Fannin and Giannitsis [54]. They successfully applied the Cole–Cole equation to the analysis of complex magnetic susceptibility data on ferroﬂuids with a distribution of particle sizes giving rise naturally to a distribution of Arrhenius-like (Ne´el) microscopic relaxation times. Here the volume of the particles appears explicitly in the argument of the experimental relaxation time. The particle size distribution then automatically leads to a hierarchy of Ne´el relaxation times. The Cole–Cole parameter s is thus a measure of the particle size distribution. Thus our understanding of the parameter s may be said to be reasonably complete. However, the same cannot be said concerning our understanding of the Cole–Davidson parameter n (although some reasons for its origin have been advanced by Nigmatulin and Ryabov [28] in their phenomenological treatment of the Cole–Davidson relaxation process). Here, unlike the Cole–Cole relaxation mechanism based on a fractal time random walk, it is not clear how the kinetic equation for the Cole–Davidson relaxation mechanism and its extension for the Havriliak–Negami relaxation may be derived from anything other than replacing the partial time derivative in the Fokker–Planck equation by a fractional partial time derivative of order n. This represents a gap in our understanding of the Cole–Davidson process. Despite these reservations, the model kinetic equations (96) and (101) are important because they allow one to incorporate the Cole–Davidson and Havriliak– Negami mechanisms into the existing microscopic theory of Debye and Cole– Cole relaxation. The kinetic equation or fractional Fokker–Planck equation [Eq. (101)] that we have proposed may be applied to a system with Havriliak–Negami behavior in the presence of an external potential. The advantage of being in possession of a kinetic equation incorporating the Havriliak–Negami mechanism then becomes apparent because it is now possible to study the effect of the anomalous behavior on fundamental parameters associated with the Brownian motion in a potential such as the Kramers escape rate. Moreover, it is possible to generalize the Debye–Fro¨hlich model of relaxation over a potential barrier to incorporate the Havriliak–Negami mechanism. We remark that in the Cole–Davidson and Havriliak–Negami mechanisms, both of which

fractional rotational diffusion

325

involve branch points in the fractional diffusion operator, the analysis of the aftereffect solution has been greatly facilitated by writing the appropriate kinetic equation as an equation for an impulse response using the properties of the unit step and delta functions. The advantage of such a formulation of the problem is that it avoids the difﬁculties associated with the inherent dependence of fractional derivatives on initial conditions. These difﬁculties are completely eliminated by considering the impulse response, which allows us to solve the problem using two-sided Fourier transforms so that the initial conditions are involved only as a forcing function. Finally we remark that in the context of stochastic resonance [8,55], Eqs. (9)–(11) for the complex susceptibility and their extensions to diffusion in a potential may be regarded as transfer functions [56] (Fourier transforms of the true impulse response of the system) whence the spectral density may be easily calculated using the Wiener–Khinchin theorem [8] so that the effect of anomalous diffusion on the stochastic resonance may be ascertained. III.

FRACTIONAL NONINERTIAL ROTATIONAL DIFFUSION IN A POTENTIAL

Anomalous rotational diffusion in a potential may be treated by using the fractional equivalent of the diffusion equation in a potential [7]. This diffusion equation allows one to include explicitly in Fro¨hlich’s model as generalized to fractional dynamics (i) the inﬂuence of the dissipative coupling to the heat bath on the Arrhenius (overbarrier) process and (ii) the inﬂuence of the fast (high-frequency) intrawell relaxation modes on the relaxation process. The fractional translational diffusion in a potential is discussed in detail in Refs. 7 and 31. Here, just as the fractional translational diffusion treated in Refs. 7 and 31, we consider fractional rotational subdiffusion ð0 < s < 1Þ phenomena only. For simplicity, we ﬁrst consider rotation about ﬁxed axis in a potential V0 ðfÞ. We suppose that a uniform ﬁeld F1 (having been applied to the assembly of dipoles at a time t ¼ 1 so that equilibrium conditions prevail by the time t ¼ 0) is switched off at t ¼ 0. In addition, we suppose that the ﬁeld is weak (i.e., mF1 kB T, which is the linear response condition). The underlying fractional rotational diffusion equation is Eq. (52) with Vðf; tÞ ¼ V0 ðfÞ mF1 ðtÞ cos f

ð133Þ

and the Fokker–Planck operator LFP is deﬁned as LFP ¼ L0FP þ Lext

ð134Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

where L0FP W ¼

1 q2 1 q qV0 W W þ t qf2 kB T qf qf

Lext W ¼

and

mF1 ðtÞ q ðsin fWÞ tkB T qf ð135Þ

The formal solution of the fractional rotational diffusion equation is obtained just as that of Eq. (135) from the Sturm–Liouville representation [7,57] Wðf; tÞ ¼ W0 ðfÞ þ

1 X

p ðfÞfp ðtÞ

ð136Þ

p¼1

with the initial (equilibrium) distribution function Wðf; 0Þ ¼ Ce½V0 ðfÞmF1 ð0Þ cos f=ðkB TÞ ¼ W0 ðfÞ½1 þ x1 ðcos f hcos fi0 Þ þ Oðx21 Þ

ð137Þ

where W0 ðfÞ ¼ eV0 =ðkB TÞ =Z is the equilibrium distribution function in the absence of the external ﬁeld F1 , which satisﬁes L0FP W0 ðfÞ ¼ 0, x1 ¼ mF1 =ðkB TÞ is the dimensionless ﬁeld parameter, Z is the partition function given by Z¼

2ðp

eV0 ðfÞ=ðkB TÞ df

ð138Þ

0

and h i0 means the equilibrium statistical average over W0 ðfÞ. Here, remembering the fact that fp must be periodic in f, the decay modes fp ðtÞ of the fractional diffusion equation obey the equation d fp ðtÞ ¼ lp;s0 D1s fp ðtÞ t dt

ð139Þ

The eigenvalues lp;s of the operator occurring in Eq. (52) 1 q 1 qV q2 þ ts qf kB T qf qf2 may also be expressed in terms of the eigenvalues lp of the Fokker–Planck operator L0FP for normal diffusion in the absence of the electric ﬁeld, namely, L0FP p ðfÞ ¼ lp p ðfÞ

ð140Þ

lp;s ¼ lp t1s

ð141Þ

so that

fractional rotational diffusion

327

The solutions of Eq. (139) are the Mittag–Lefﬂer functions Es ðzÞ [7,31], namely, fp ðtÞ ¼ Es ðlp;s ts Þ

ð142Þ

Equation (141) exempliﬁes how the eigenvalues of the normal distribution process are altered, in this case reduced, by the nonlocal character of the anomalous diffusion process. The eigenvalues of the nonlocal process are related to their Brownian counterparts by the prefactor t1s. This scaling effect is signiﬁcant in the context of escape of particles over potential barriers. There, the smallest nonvanishing eigenvalue l1 of the Fokker–Planck equation, written for the Brownian motion in a potential, yields in the high barrier limit, the Kramers escape rate [58,59]. Since we consider the anomalous diffusion analog of the overdamped Brownian motion, the scaling Eq. (141) shows that the overdamped Kramers escape rate for normal diffusion l1 eV=ðkB TÞ (V is the barrier height) is slowed by the factor t1s so that l1;s t1s l1 ¼ t1s

ð143Þ

Therefore, in the present context, the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag–Lefﬂer functions governing the highly nonexponential relaxation behavior of the system. The Sturm–Liouville representation (136) is a formal solution as a knowledge of all eigenfunctions p ðfÞ, and corresponding eigenvalues lp is required. However, this representation is very useful because it allows one to obtain a formal solution for the longitudinal complex susceptibility wðoÞ ¼ w0 ðoÞ iw00 ðoÞ. According to linear response theory [60], wðoÞ is deﬁned as wðoÞ ¼ 1 io w0

1 ð

eiot Cs ðtÞ dt

ð144Þ

0

where Cs ðtÞ ¼

hcos fiðtÞ hcos fi0 X ¼ cp Es ½tlp ðt=tÞs hcos fið0Þ hcos fi0 p

is the normalised dipole relaxation function CðtÞ, 2ðp

cp ¼

ðcos f hcos fi0 Þp ðfÞ df

0

Xð

2p

p

0

ðcos f hcos fi0 Þp ðfÞ df

ð145Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

P

p cp ¼ 1, and the angular brackets h iðtÞ denotes the statistical averages over Wðf; tÞ, namely,

hcos fiðtÞ ¼

2ðp

ð146Þ

cos fWðf; tÞ df 0

Noting the Laplace transform of the Mittag–Lefﬂer function Eq. (79), we have from Eqs. (144) and (145) wðoÞ X cp ¼ s w0 1 þ ðiotÞ =ðtlp Þ p

ð147Þ

The susceptibility from Eq. (147) may simply be evaluated in the low- ðo ! 0Þ and high- ðo ! 1Þ frequency limits. We obtain from Eq. (147) accordingly for o ! 0 and o ! 1 wðoÞ tint ðiotÞs þ

1 w0 t wðoÞ t þ w0 ðiotÞs tef

ð148Þ ð149Þ

where the parameters tint and tef are deﬁned as tint ¼

X

cp =lp

and

tef ¼ 1=

p

X

cp lp

ð150Þ

p

For normal diffusion, s ¼ 1, these parameters correspond to the integral relaxation P time tint [the area under the corresponding relaxation function C1 ðtÞ ¼ p cp elp t ] tint ¼

1 ð

C1 ðtÞ dt

ð151Þ

0

and the effective relaxation time tef [which gives precise information on the initial decay of the relaxation function C1 ðtÞ] tef ¼ 1=C_ 1 ð0Þ

ð152Þ

fractional rotational diffusion

329

It is of importance that for one-dimensional rotational Brownian motion in a potential, tint and tef deﬁned by Eq. (151) may be expressed in closed form for an arbitrary potential V0 ðfÞ Ref. 8, Chapter 4), namely, tef ¼ t

1 þ hcos 2fi0 2hcos fi20 1 hcos 2fi0

ð153Þ

and (Ref. 8, Chapter 2)

tint ¼

2p ð

t Zðhcos2 fi0 hcos fi20 Þ

0

2f 32 ð eV0 ðfÞ=kT 4 ðcos x hcos fi0 ÞeV0 ðxÞ=kT dx5 df 0

ð154Þ We note that the characteristic times tint and tef do not exist in anomalous diffusion ðs 6¼ 1Þ. This is obvious from the properties of the Mittag–Lefﬂer function [see Eqs. (62) and (63)]. However, we shall now demonstrate that the above time constants may also be used to characterize the dynamic susceptibility in anomalous relaxation. At low frequencies, ignoring the contributions of all decay modes in Eq. (147), save the slowest one, and noting Eq. (143), we will have for the complex dielectric susceptibility wðoÞ wðoÞ 1 1 ¼ ﬃ s w0 1 þ ðiotÞ =ðtl1 Þ 1 þ ðio=oc Þs

ð155Þ

where the characteristic frequency oc (here the imaginary part of wðoÞ has a maximum) may thus be deﬁned as oc ¼ t1 ðtl1 Þ1=s t1 ðt Þ1=s

ð156Þ

The characteristic time tc of the relaxation process may be deﬁned as 1=s tc ¼ o1 c ¼ tðtl1 Þ

ð157Þ

Equation (156) illustrates how anomalous relaxation inﬂuences the complex susceptibility arising from the slowest relaxation mode in the noninertial limit. Therefore, in anomalous relaxation the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag–Lefﬂer function governing the relaxation behavior of the system. We remark that the slowest decay mode, the parameters of which are determined by Eq. (156), will

330

william t. coffey, yuri p. kalmykov and sergey v. titov

be highly nonexponential in character because its behavior will be dictated by the Mittag–Lefﬂer function, Eq. (142), even though the Arrhenius character [albeit stretched according to Eq. (156)] of the activation process is preserved. Equation (155) may be used to evaluate the complex susceptibility in the low-frequency range ðo oc Þ only. In order to describe the dielectric spectra at all frequencies, we can generalize the method of Coffey et al. [8], Chapter 2, Section 2.13 developed for normal diffusion in a potential. According to this method, a knowledge of the characteristic times 1=l1 , tint , and tef allows one to give a simple analytical description of the relaxation behavior of the system in the context of a two-mode approximation. In order to accomplish this generalization for anomalous relaxation [8,25], let us now suppose that the relaxation function CðtÞ from Eq. (145), which in general comprises an inﬁnite number of Mittag–Lefﬂer functions, may be approximated by two Mittag– Lefﬂer functions only: Cs ðtÞ 1 Es ½ðt=tÞs tl1 þ ð1 1 ÞEs ½ðt=tÞs t=tW

ð158Þ

In turn, the spectrum of the longitudinal susceptibility wðoÞ may essentially be approximated by a sum of two Cole–Cole mechanisms, namely, wðoÞ 1 1 1 ¼ þ w0 1 þ ðio=oc Þs 1 þ ðio=oW Þs

ð159Þ

where the characteristic frequencies oc and oW are given by oc ¼ t1 ðtl1 Þ1=s ;

oW ¼ t1 ðt=tW Þ1=s

ð160Þ

Here, we implicitly suppose that the contribution of the high-frequency modes may be approximated by a single mode with characteristic frequency oW . The high-frequency band is due to ‘‘intrawell’’ modes corresponding to the eigenvalues lk ðk 6¼ 1Þ. These near-degenerate ‘‘intrawell’’ modes are indistinguishable in the frequency spectrum of w00 ðoÞ, appearing merely as a single high-frequency band. The low-frequency band is due to the slowest (overbarrier) relaxation mode; the characteristic frequency oc and the halfwidth of this band are determined by l1. Thus, the anomalous low-frequency behavior is dominated by the barrier crossing mode as in the normal diffusion. The parameters 1 and tW in Eqs. (159) and (160) are determined from the condition that the approximate Eq. (159) must obey the exact asymptotic Eqs. (148) and (149) yielding tint ¼ 1 =l1 þ tW ð1 1 Þ

1 and t1 ef ¼ 1 l1 þ tW ð1 1 Þ

331

fractional rotational diffusion

It follows that 1 and tW may readily be evaluated from the above equations yielding [8] tint =tef 1 l1 tint 2 þ 1=ðl1 tef Þ l1 tint 1 tW ¼ l1 1=tef

1 ¼

ð161Þ ð162Þ

Equation (159), which involves the integral relaxation time tint , the effective relaxation time tef , and the smallest nonvanishing eigenvalue l1 , correctly predicts wðoÞ both at low ðo ! 0Þ and high ðo ! 1Þ frequencies. Moreover, for a particular form of the potential V, wðoÞ may be determined in the entire frequency range 0 o < 1 as we shall presently see for a double-well periodic potential representing the internal ﬁeld due to neighboring molecules. A.

Anomalous Diffusion and Dielectric Relaxation in a Double-Well Periodic Potential

Here, we shall present both the exact and approximate solution for the anomalous dielectric relaxation of an assembly of ﬁxed axis dipoles rotating in a double-well potential: V0 ðfÞ ¼ V0 sin2 f

ð163Þ

This potential has two potential minima on the sites at f ¼ 0 and f ¼ p as well as two energy barriers located at f ¼ p=2 and f ¼ 3p=2. This model has been treated in detail for normal diffusion in Refs. 8, 61, and 62. Here we consider the fractional Fokker–Planck equation [Eq. (55)] for a ﬁxed axis rotator with dipole moment m moving in a potential [Eq. (163)]. We calculate wðoÞ by converting the solution of the fractional diffusion Eq. (55) into the calculation of successive convergents of a differential-recurrence relation just as normal diffusion [8,62]. By expanding the distribution function Wðf; tÞ in Fourier series Wðf; tÞ ¼

1 X

eipf cp ðtÞ

p¼1

we have from Eq. (55) the differential-recurrence equation: f_p ðtÞ ¼ ts 0 D1s fxV p½fp2 ðtÞ fpþ2 ðtÞ p2 fp ðtÞg; t where xV ¼ V0 =ð2kB TÞ is the barrier height parameter and fp ðtÞ ¼ Re½cp ðtÞ=ð2pÞ ¼ hcos pfiðtÞ

t 0

ð164Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

Applying the integration theorem of Laplace transformation generalized to fractional calculus, Eq. (70), we have from Eq. (164)

st~fp ðsÞ fp ð0Þ ¼ ðstÞ1s fxV p ~fp2 ðsÞ ~fpþ2 ðsÞ p2~fp ðsÞg

ð165Þ

where ~f ðsÞ denotes the Laplace transform of f ðtÞ. The exact solution of the threeterm recurrence equation [Eq. (165)] can be obtained, just as normal diffusion, in continued fraction form (see Refs. 8 and 62 for details). The initial values fp ð0Þ can be obtained as follows. At time t ¼ 0, the steady ﬁeld FðtÞ is switched off. Thus 2ðp

f2pþ1 ð0Þ ¼ hcos½ð2p þ 1Þfið0Þ ﬃ

cos½ð2p þ 1ÞfexV cos 2f ½1 þ x1 cos fdf

0 2ðp

exV cos 2f ½1 þ x1 cos fdf

0

ð166Þ where x1 ¼ mF1 =ðkB TÞ and we have supposed that the external ﬁeld is weak so that x1 1. We also have [51] ez cos 2y ¼

1 X

Im ðzÞei2my

ð167Þ

m¼1

where Im ðzÞ is the modiﬁed Bessel function of the ﬁrst kind of order m [51]. Whence, using the orthogonality property of the circular functions and noting that Im ðzÞ ¼ Im ðzÞ, we have 8 2p ð 1 x1 < X Im ðxV Þei2mf ½ei2ðpþ1Þf þ ei2ðpþ1Þf df f2pþ1 ð0Þ ¼ 4 : m¼0 0

922p 31 1 = ðX i2mf i2pf i2pf i2mf þ Im ðxV Þe ½e þe df 4 Im ðxV Þe df5 ; m¼0 m¼0 0 0 x1 Iðpþ1Þ ðxV Þ þ Ipþ1 ðxV Þ Ip ðxV Þ þ Ip ðxV Þ þ ¼ I0 ðxV Þ I0 ðxV Þ 4 x1 ½Ipþ1 ðxV Þ þ Ip ðxV Þ ð168Þ ¼ 2I0 ðxV Þ 2ðp

1 X

fractional rotational diffusion

333

On noting the initial values fp ð0Þ, we have " # p 1 X ~f1 ðsÞ tðstÞs1 ð1Þp Ipþ1 ðxV Þ þ Ip ðxV Þ Y ¼ 1þ S2kþ1 ðsÞ f1 ð0Þ ðstÞs þ 1 xV þ xV pS3 ðsÞ 2p þ 1 I1 ðxV Þ þ I0 ðxV Þ k¼1 p¼1 ð169Þ with successive convergents being calculated from the continued fraction Sp ðsÞ ¼ ¼

xV p ðstÞs þ p2 þ xV pSpþ2 ðsÞ xV p : xV 2 pðp þ 2Þ s 2 ðstÞ þ p þ xV 2 ðp þ 2Þðp þ 4Þ ðstÞs þ ðp þ 2Þ2 þ ðstÞs þ ðp þ 4Þ2 þ

~ s ðioÞ ¼ ~f1 ðioÞ=f1 ð0Þ, we may Thus, by setting s ¼ io and by noting that C calculate the complex susceptibility w (o) from Eq. (144), where the static susceptibility w0 [62] is w0 ¼

m2 N0 m2 N0 I1 ðxV Þ þ I0 ðxV Þ hcos2 fi0 ¼ 2I0 ðxV Þ kB T kB T

(N0 is the number of dipoles per unit volume). The ease of calculation of wðoÞ from Eq. (169) represents the chief advantage of the continued fraction in comparison to the Sturm–Liouville method as applied to anomalous diffusion. Now, we shall demonstrate that the characteristic times of the normal diffusion process, namely, the inverse of the smallest nonvanishing eigenvalue 1=l1 , the integral and effective relaxation times tint and tef obtained in [8,62,63], also allow us to evaluate the dielectric response of the system for anomalous diffusion using the two-mode approximation just as normal diffusion (Ref. 8, Section 2.13). Here, we can use known equations for tint , tef , and l1 for the normal diffusion in the potential Eq. (163) [8,62,63]; these equations are tint

ðp

pﬃﬃﬃﬃﬃﬃﬃﬃ te2xV 2xV sin f df ¼ exV cos 2f erf 2 4B½I1 ðxV Þ þ I0 ðxV Þ 0

I0 ðxV Þ þ I1 ðxV Þ tef ¼ t I0 ðxV Þ I1 ðxV Þ " #1 1 X p ð1Þp 2 l1 t ¼ I ðx Þ 1 e2xV p¼0 2p þ 1 pþ1=2 V

334

william t. coffey, yuri p. kalmykov and sergey v. titov

where erf(z) is the error function [51]. We recall that in the high barrier limit 2xV =8xV and tef 4xV t [8,62,63], yielding (xV 1), we have l1 1 tint tpe simple asymptotic equations for the characteristic frequencies oc and oW from Eqs. (160), namely, oc ð8xV =pÞ1=s e2xV =s =t

ð170Þ

oW ð8xV Þ1=s =t

ð171Þ

and

Thus oc and oW depend not only on the barrier height (as in normal diffusion) but also on the anomalous exponent s which substantially modiﬁes the dielectric loss spectra. The characteristic frequency oc and oW given by Eqs. (160) are shown in Fig. 3 as a function of s and xV . ^ðoÞ ¼ wðoÞ=w0 from the exact Calculations of the normalized susceptibility w continued fraction solution, Eq. (169), and the approximate bimodal approximation, Eqs. (159)–(162), are shown in Figs. 4 and 5. Here, the lowand high-frequency asymptotes, Eqs. (148) and (149), are also presented, demonstrating that the bimodal approximation obeys the exact asymptotic equations, Eqs. (148) and (149). Two bands, which appear in the dielectric loss ^00 ðoÞ at 2xV 1 (in the high barrier limit), reach a maximum at spectrum of w the characteristic frequencies oc and oW given by Eqs. (170) and (171), respectively. Apparently, the agreement between the exact continued fraction calculations and the approximate equation Eq. (159), is very good [the maximum relative deviation between the corresponding curves, which appears at o t1 , does not exceed a few (3–5) percent]. Similar (or even better) agreement exists for all values of xV and s. Such a good accuracy of the bimodal approximation is due to the fact that the inﬁnite number of highfrequency ‘‘intrawell’’ modes (these near degenerate modes are indistinguishable appearing merely as a single Cole–Cole high-frequency band in the dielectric loss spectrum) may be approximated effectively by a single mode. Thus, one may conclude that Eq. (159) accurately describes the behavior of wðoÞ for all frequencies of interest and for all values of the barrier height (xV ) and anomalous exponent (s) parameters. We remark that the bimodal approximation works extremely well both for anomalous ðs 6¼ 1Þ and normal ðs ¼ 1Þ cases (various applications for the normal diffusion in a potential are given in Ref. 8). Thus the anomalous relaxation in a double-well potential is effectively determined by the bimodal approximation, Eq. (159); the characteristic times of the normal diffusion process—namely, the inverse of the smallest nonvanishing eigenvalue, the integral, and effective relaxation times—appear as time

335

fractional rotational diffusion

0 –5 –10 log10[tw c] –15

0 1

1 2

0.8 B

3 4

0.4

0.6 s

50.2

6 4 2

0.2 0

0.4

log10[tw w]

0

1

0.6 s

3

0.8

2 B

4

1 Figure 3.

Characteristic frequencies oc and oW as functions of s and xV .

parameters. Moreover, the simple asymptotic equations, Eqs. (170) and (171), allow one to easily evaluate the characteristic frequencies of the dielectric loss spectrum in terms of the physical model parameters xV and s. Thus Fro¨hlich’s model of relaxation over a potential barrier based on the concept of normal diffusion may be generalized to anomalous diffusion in disordered energyscapes giving rise to temporally nonlocal behavior [7]. The results obtained may be regarded as a generalization of the solution for the normal Brownian motion in a

336

william t. coffey, yuri p. kalmykov and sergey v. titov

(a)

100

χ^ '(ω)

10−2

1

2

3 σ = 0.5

1: ξ V = 0.01 −4

2: ξ V = 2.5

10

3: ξ V = 5.0 10−12

−8

0

10

4

10

10

8

10 (b)

10 0 χ^''(ω)

−4

10

3

2

1

−2

10

−4

10

−12

10

−8

10

−4

10

0

ωτ

10

4

10

8

10

Figure 4.

The real (a) and imaginary (b) parts of the normalized complex susceptibility evaluated from the exact continued fraction solution [Eqs. (169): solid lines] for s ¼ 0:5 and various values of xV and compared with those calculated from the approximate Eq. (159) (asterisks). The low- (dotted lines) and high-frequency (dashed lines) asymptotes are calculated from Eqs. (148) and (149), respectively.

cosine periodic potential [8,62] to fractional dynamics (giving rise to anomalous diffusion). It should be mentioned that if one is interested only in low-frequency ðo oc Þ part of the dielectric spectrum, one may use a more simple singlemode (Cole–Cole) equation (155) [or Eq. (9) with t ¼ 1=oc ] for the normalized complex susceptibility. The characteristic frequencies oc is given by Eq. (160). The divergence of all the global characteristic times for anomalous diffusion—as deﬁned in their conventional sense (which is a natural consequence of the underlying Le´vy distribution), rendering them useless as a measure of the relaxation behavior—signiﬁes the importance of characteristic times for such processes in terms of the frequency-domain representation of

337

fractional rotational diffusion

10

0

χ^ '(ω )

3

2

ξ V = 5.0

−2

10

1: σ = 1.00 2: σ = 0.75

−4

10

3: σ = 0.50 10

χ^ ''( ω )

(a)

1

10

0

10

−2

10

−4

−12

10

−8

10

3

10

−12

Figure 5.

10

−8

−4

2

10

10

0

10

4

ωτ

8

(b)

1

−4

10

10

0

10

4

10

8

The same as in Fig. 4 for xV ¼ 5 and various values of s.

such processes. As shown in this section, the anomalous diffusion process in the periodic potential will have two associated characteristic times, namely, the inverse of the overbarrier frequency oc and the inverse of the well frequency o W . Moreover, we have the novel and surprising result that the anomalous dielectric relaxation at any frequency is governed by the characteristic times l1 1 , tint , and tef of the normal diffusion processes. Thus, the difﬁculties associated with divergent global characteristic times in anomalous diffusion may be avoided. In conclusion, we remark that throughout this discussion, we have conﬁned ourselves to an interpretation of Cole–Cole behavior. Phenomenological fractional diffusion equations in the time domain describing other types of anomalous relaxation such as Cole–Davidson or Havriliak–Negami behavior can be treated in a similar manner. As further applications of the approach developed, we consider in Sections III.C and III.D two problems of fractional rotational diffusion in space, namely,

338

william t. coffey, yuri p. kalmykov and sergey v. titov

(i) the fractional rotational diffusion in a dc electric ﬁeld and (ii) the fractional rotational diffusion in a uniaxial potential with a superimposed dc electric ﬁeld (i.e., a double-well potential with nonequivalent wells). B.

Fractional Rotational Diffusion in a Uniform DC External Field

The application of a strong direct current (dc) electric ﬁeld F0 to a polar ﬂuid comprised of dipolar molecules results in a transition from free thermal rotation of the molecules to partial orientation with hindered rotation. This change in the character of the molecular motion under the inﬂuence of the ﬁeld has a marked effect on the dielectric properties of the ﬂuid insofar as dispersion and absorption of electromagnetic waves will be observed at the characteristic frequencies of rotation of the molecule in the ﬁeld F0 . A similar effect arises in magnetic relaxation of ferroﬂuids subjected to a strong dc magnetic ﬁeld H0 . The similarity of the problems of dielectric relaxation of a polar ﬂuid and magnetic relaxation of a ferroﬂuid is not surprising because, from a physical point of view, the rotational Brownian motion of magnetic particles (magnetic dipoles) in a constant magnetic ﬁeld H0 is similar to that of polar molecules (electric dipoles) in a constant electric ﬁeld F0 . Orientational relaxation of Brownian particles in the context of the normal rotational diffusion in the presence of a dc ﬁeld has been treated in detail, for example, in Refs. 8 and 64–71. In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external ﬁeld F1 applied parallel and perpendicular to the bias ﬁeld F0 may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker–Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the inﬁnite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. Let us suppose that the uniform dc ﬁeld F0 is directed along the Z axis of the laboratory coordinate system and that a small probing ﬁeld F1 , having been applied to the assembly of dipoles in the distant past ðt ¼ 1Þ so that

fractional rotational diffusion

339

equilibrium conditions are attained by the time t ¼ 0, is switched off at t ¼ 0. In addition it is supposed that the ﬁeld F1 is weak (i.e., mF1 kB T, which is the linear response condition; m is the permanent dipole moment of a molecule). Here, the appropriate fractional rotational diffusion equation [8] is qW ¼ t1s 0 D1s LFP W t qt

ð172Þ

where LFP is the Fokker–Planck operator for normal noninertial rotational diffusion deﬁned by Eq. (8), namely, 1 1 r2 W þ r ðWrVÞ 2t kT 1 1 q qW W qV 1 q qW W qV ¼ sin # þ þ þ 2 2t sin # q# q# kB T q# sin # qj qj kB T qj

LFP W ¼

ð173Þ # and j are the polar and azimuthal angles, respectively, and V is the orientational potential energy of the particle in the ﬁeld. The characteristic relaxation time t for the normal diffusion is given by Eq. (81). For the longitudinal and transverse responses, one has Vk ð#; tÞ ¼ mF0 cos # mF1 ðtÞ cos #

ð174Þ

V? ð#; j; tÞ ¼ mF0 cos # mF1 ðtÞ cos j sin #

ð175Þ

and

respectively. According to linear response theory [8], the longitudinal and transverse components of the complex dielectric susceptibility wg ðoÞ ¼ w0g ðoÞ iw00g ðoÞ, ðg ¼ k; ?Þ are deﬁned as wg ðoÞ ¼ 1 io wg

1 ð

eiot Cg ðtÞ dt

ð176Þ

hcos #iðtÞ hcos #i0 hcos #ið0Þ hcos #i0

ð177Þ

0

where Ck ðtÞ ¼

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william t. coffey, yuri p. kalmykov and sergey v. titov

and C? ðtÞ ¼

hcos j sin #iðtÞ hcos j sin #ið0Þ

ð178Þ

are the normalized relaxation functions, wk ¼

m2 N0 ð1 þ x2 coth2 xÞ and kB T

w? ¼

m2 N0 ðcoth x 1=xÞ kB Tx

are the components of the static susceptibility tensor, x ¼ mF0 =ðkB TÞ is the dimensionless ﬁeld parameter, h. . .iðtÞ denotes the statistical averages over the assembly of rotators in the presence of a small probing ac electric ﬁeld, and h. . .i0 means the equilibrium statistical averages. The formal step-off transient solution of Eq. (172) for t 0 is obtained from the Sturm–Liouville representation Wð#; j; tÞ ¼ W0 ð#Þ þ

1 X

gp ð#; jÞEs ðlgp;s ts Þ

ð179Þ

p¼1

with the initial (equilibrium) distribution function

Wð#; j; 0Þ ¼ CeVg ð#;j;0Þ=kT ¼ W0 ð#Þ 1 þ x1 ðug hug i0 Þ þ Oðx21 Þ

ð180Þ

where W0 ð#Þ ¼ ex cos # =Z is the equilibrium distribution function, Z is the partition function given by

Z¼

ð1

exx dx ¼ 2 sinh x=x

ð181Þ

1

and the angular brackets h. . .i0 mean the equilibrium statistical average over W0. The Sturm–Liouville representation is very useful as it allows one readily to obtain a solution for the relaxation functions Cg ðtÞ. According to Eqs. (176)–(179), Cg ðtÞ ¼

X p

cgp Es ðlgp;s ts Þ

cgp wg ðoÞ X ¼ p 1 þ ðiotÞs =ðtlg Þ wg p

ð182Þ ð183Þ

341

fractional rotational diffusion

P (where p cp ¼ 1). In the low- ðo ! 0Þ and high- ðo ! 1Þ frequency limits, the susceptibility tensor components may readily be evaluated in terms of the integral relaxation time tint and the effective relaxation time tgef . We have from Eq. (183) wg ðoÞ tgint

1 ðiotÞs þ w0g ð0Þ t

ð184Þ

wg ðoÞ t þ 0 wg ð0Þ ðiotÞs tgef

ð185Þ

for o ! 0, and

for o ! 1, where tgint ¼

X

cg =lgp p p

and

tgef ¼ 1=

X

ð186Þ

cg lg p p p

The relaxation times tgint and tgef were obtained in the context of the normal rotational diffusion model in Refs. 8, 65, and 67 and are given by (in our notation) k tint

tx ¼ 2 ð1 þ x coth2 xÞ sinh x

ð1 1

exz ½z coth x þ exð1þzÞ ð1 þ coth xÞ2 dz 1 z2 ð187Þ

k tef

t

¼

x xLðxÞ 2 LðxÞ

ð188Þ

for the longitudinal response and t? int

¼

t

1 X

xI3=2 ðxÞ n¼1

ð1Þ

nþ1

n ð2n þ 1ÞInþ1=2 ðxÞ Y ~S? ð0Þj s¼1 k nðn þ 1Þ k¼1

t? ef ¼ 2t

LðxÞ x LðxÞ

! ð189Þ ð190Þ

for the transverse response, where LðxÞ ¼ coth x 1=x is known as the Langevin function, Inþ1=2 ðxÞ are the spherical Bessel functions [51], and the continued fraction ~ S? k ð0Þjs¼1 is deﬁned in Appendix II, Eq. (A2.9). The behavior of the relaxation times tgint and tgef is very similar: They are very close to each other and decrease with increasing x (see Fig. 6).

342

william t. coffey, yuri p. kalmykov and sergey v. titov 1.0

1: τint / τ : τef / τ ⊥

2: τint / τ ⊥

Relaxation Times

: τef / τ

0.5

2 1

0.0 0

5

ξ

10

15

Integral (tgint : solid lines) and effective (tgef : ﬁlled circles and asterisks) relaxation times vs. x for normal rotational diffusion in a dc bias ﬁeld. Equations (187)–(190) have been used in the calculation.

Figure 6.

According to Eq. (147), each of the inﬁnite number of relaxation modes (corresponding to the eigenvalues lgk ) gives a contribution to the spectra wg ðoÞ. However, as we shall see, these near-degenerate individual modes are indistinguishable in the frequency spectrum of wg ðoÞ appearing merely as a single band. This is due to the fact that in contrast to the rotational diffusion in a double-well potential treated above, for the single-well potential under consideration, the overbarrier relaxation mode is not involved in the relaxation process. Thus noting that tgint tgef for all values of x (see Fig. 6), the spectrum of wg ðoÞ may be approximated by the Cole–Cole equation wg ðoÞ 1

0 wg ð0Þ 1 þ ðio=ogc Þs

ð191Þ

ogc ¼ t1 ðt=tgef Þ1=s

ð192Þ

where

is the characteristic frequency at which the loss spectrum a00g ðoÞ attains its maximum. Noting the low-temperature behavior of the effective relaxation

343

fractional rotational diffusion

5 4 3 log [twc] 2 10 1

0

0.2 2 0.4 s

4 0.6

6 0.8

x

8 1 10

Frequency okc as a function of x and s.

Figure 7. k

times, namely, tef t=x and t? ef 2t=x at x 1 [8,67], one can readily obtain from Eqs. (188), (190), and (192) okc t1 x1=s

1=s 1 and o? at x 1 c t ð2xÞ

ð193Þ

The frequencies okc and o? c as functions of x and s are plotted in Figs. 7 and 8. In the time domain, the single-mode approximation, equation Eq. (191), is equivalent to assuming that the relaxation function Cg ðtÞ as determined by the exact equation, Eq. (182) (which in general comprises an inﬁnite number of Mittag–Lefﬂer functions), may be approximated by one Mittag–Lefﬂer function only, namely " Cg ðtÞ Es

t t s g tef t

# ð194Þ

Equation (191) correctly predicts wg ðoÞ both at low ðo ! 0Þ and high ðo ! 1Þ frequencies; moreover, wg ðoÞ may be determined in the entire frequency range as one shall presently see. In order to estimate the accuracy of the approximate Eq. (191), the longitudinal and transverse components of the complex susceptibility are evaluated by converting the problem of solving the

344

william t. coffey, yuri p. kalmykov and sergey v. titov

4 3 log10 [twc] 2

1

0

0.2 2 0.4 s

4 0.6

6 0.8

x

8 1 10

Figure 8.

Frequency o? c as a function of x and s.

fractional diffusion Eq. (172) with V given by Eq. (174) into the calculation of successive convergents of a differential-recurrence relation just as normal diffusion [8,67]. Just as a double-well potential, the exact solutions for the longitudinal and transverse responses are given in terms of ordinary continued fractions in Appendix II. The results of the calculation of the normalized ½m2 N0 =ðkB TÞ ¼ 1 loss spectra w00k ðoÞ and w00? ðoÞ from these exact continued fraction solutions and the approximate Eqs. (191) are shown in Figs. 9–12; here, the low- and high-frequency asymptotes, Eqs. (148) and (149), are also presented. Apparently as x increases, the spectra shift to higher frequencies in accordance with Eq. (193); simultaneously, the half-width of the spectra increases with decreasing s. Furthermore, the agreement between the exact continued fraction calculations and the approximate Eq. (191) is good [the maximum relative deviation between the corresponding curves does not exceed a few (3–5) percent]. Similar (or even better) agreement exists for all values of x. The accuracy of the single-mode approximation is because for F1 ¼ 0, the potentials (174) and (175) are single-well potentials so that the long-lived mode due to overbarrier relaxation (as in multiwell potentials with two or more metastable states) [63] does not exist. Thus the inﬁnite number of high-frequency ‘‘intrawell’’ modes (these near-degenerate modes are indistinguishable appearing merely as a single high-frequency band in the

345

fractional rotational diffusion

10

−1

σ = 0.5

1 - ξ = 0.01 2- ξ =5 3 - ξ = 10

χ||''(ω)

1

10

2

−3

3

10

−5 −4

−2

10

10

10

0

ωτ

10

2

10

4

10

6

Figure 9. Dielectric loss spectra w00k ðoÞ evaluated from the exact continued fraction solution [Eqs. (176) and (A2.3): solid lines] for s ¼ 0:5 and various values of x, and compared with those calculated from the approximate Eq. (191) (asterisks). The low- (dotted lines) and high-frequency (dashed lines) asymptotes are calculated from Eqs. (184), (187), and (185), (188), respectively.

ξ = 5.0

10−1

χ ''( ω ) ||

1 - σ = 1.00 2 - σ = 0.75 3 - σ = 0.50 10−3

3 2 1

10−5 −4

−2

10

0

10

4

10

Figure 10.

The same as in Fig. 9 for x = 5 and various values of s.

ωτ

10

2

10

10

6

346

william t. coffey, yuri p. kalmykov and sergey v. titov

10

σ = 0.5 1 - ξ = 0.01 2- ξ =5 3 - ξ = 10

−1

χ⊥''(ω)

1 10

2

−2

3

10

−3

10

−4

10

−2

10

0

ωτ

10

2

10

4

10

6

Figure 11. Dielectric loss spectra w00? ðoÞ evaluated from the exact continued fraction solution [Eqs. (176) and (A2.8): solid lines] for s ¼ 0:5 and various values of x, and compared with those calculated from the approximate Eq. (191) (asterisks). The low- (dotted lines) and high-frequency (dashed lines) asymptotes are calculated from Eqs. (184), (189), and (185), (190), respectively.

χ⊥''(ω )

ξ = 5.0 10

−1

10

−2

1 - σ = 1.00 2 - σ = 0.75 3 - σ = 0.50

3 2 10

−3

10

1

−4

Figure 12.

10

−2

10

0

ωτ

10

2

10

4

The same as in Fig. 11 for x ¼ 5 and various values of s.

10

6

fractional rotational diffusion

347

dielectric loss spectrum) may be effectively approximated by a single mode. Thus one may conclude that Eq. (191) accurately describes the behavior of wk ðoÞ and w? ðoÞ for all frequencies of interest and for all values of the ﬁeld strength (x) and anomalous exponent (s) parameters. Hence the generalized Debye model can explain the anomalous relaxation of complex dipolar systems where the anomalous exponent s differs from unity (corresponding to the classical Debye theory of dielectric relaxation); that is, the relaxation process is characterized by a broad distribution of relaxation times. In particular, the theory may be applied to dilute suspensions of ﬁne magnetic particles (ferroﬂuids) by a simple change of notation. Experiments on the magnetization induced by a weak ac ﬁeld superimposed on a strong dc magnetic ﬁeld may be realized in practice in a ferroﬂuid because a large value of x can be achieved with a moderate constant magnetic ﬁeld due to the large value of the magnetic dipole moment m (104–105 Bohr magnetons) of singledomain particles. As observed by Fannin and Giannitsis [54,73], with increasing x, both the magnetic loss spectra and the relaxation times for ferroﬂuids in a strong dc magnetic ﬁeld decrease compared with those in the isotropic case. The anomalous relaxation behavior naturally appears in ferroﬂuids due to the broad distribution of particle volume v (for ﬁne particles, the magnetic moment and the Debye relaxation time strongly depend on v) [74]. The results obtained may be regarded as a generalization of the solution for the normal Brownian motion in a dc bias ﬁeld [8,67] to fractional dynamics (giving rise to anomalous diffusion). We remark that the single-mode approximation also works extremely well for normal diffusion ðs ¼ 1Þ in a dc external ﬁeld. C.

Fractional Rotational Diffusion in a Bistable Potential with Nonequivalent Wells

In this section, we shall demonstrate how to evaluate the linear response of an assembly of noninteracting polar Brownian particles in a uniaxial potential Kðn eÞ2 (K is the anisotropy constant, e ¼ l=m, and n is the unit vector in the direction of the easy axis) with a superimposed strong dc electric ﬁeld F0 . In order to retain axial symmetry, we suppose that the ﬁeld F0 and axis of the uniaxial anisotropy potential n are directed along the Z axis of the laboratory coordinate system. The total potential is an asymmetric double-well potential, which can be written in a dimensionless form as V0 ð#Þ=ðkB TÞ ¼ xV cos2 # x cos #

ð195Þ

where xV ¼ K=ðkB TÞand x ¼ mF0 =ðkB TÞ are the barrier height and bias ﬁeld parameters, respectively. The potential (195) is a bistable potential with two nonequivalent wells (see Fig. 13); for xV ¼ 0, it reduces to that considered in the

348

william t. coffey, yuri p. kalmykov and sergey v. titov ϑ

0

π

0

β [V0 (ϑ )−V0 (ϑ max )]

1

2 1

2 −5 3

−10

Figure 13.

4

ξV = 2

1: ξ = 0 2: ξ = ξV 3: ξ = 2 ξV 4: ξ = 3 ξV

Potential V0 ð#Þ=kB T ¼ xV cos2 # x cos # for xV ¼ 2 and various values of x.

previous section. This model has been treated in Refs. 74–80 in the context of the normal diffusion. A very interesting feature of the asymmetric double-well potential is that at values of the ﬁeld which are much less than that required to destroy the double-well structure of the potential, the integral relaxation time (which is the area under the curve of the decay function of the electric polarization) switches from the exponential (Arrhenius-like) behavior characteristic of an overbarrier relaxation process to simple algebraic behavior characteristic of the relaxation process in a well [79]. This constant ﬁeld effect, which is due to the depletion for high-bias ﬁelds of the population of the shallower well of the potential, has been described in detail in Ref. 8, Section 1.20. In terms of the complex susceptibility, the effect of the bias ﬁeld is then to destroy the overbarrier contribution to the susceptibility so that only the highfrequency contribution due to the intrawell modes remains. The dielectric response in the context of anomalous rotational diffusion in the potential Eq. (195) may be calculated 81 in the same manner as normal rotational diffusion (Ref. 8, Chapter 8). Here we present (following [81]) both exact and approximate solutions of fractional Fokker–Planck equation for this particular bistable asymmetric potential. The exact solution of the problem reduces to the solution of inﬁnite hierarchies of differential-recurrence relations for the corresponding relaxation functions. The longitudinal and transverse components of the

fractional rotational diffusion

349

susceptibility tensor may be calculated exactly by matrix continued fractions from the Laplace transform of these relaxation functions using linear response theory. We shall also demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times, allow one to evaluate the dielectric response for anomalous diffusion in the context of the two-mode approximation. It will be shown how this procedure yields a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The longitudinal component of the susceptibility tensor in general exhibits a very broad band low-frequency peak arising from the overbarrier relaxation, superimposed on which is a weaker high-frequency wing. However, if the ﬁeld F0 is sufﬁciently strong, the lowfrequency peak as explained earlier can be made to disappear due to depletion of the dipoles in the shallowest well of the potential. We suppose that a small probing ﬁeld F1 , having been applied to the assembly of dipoles in the distant past ðt ¼ 1Þ so that equilibrium conditions have been attained at time t ¼ 0, is switched off at t ¼ 0. Our starting point is the fractional Smoluchowski equation (172) for the evolution of the probability density function Wð#; j; tÞ for normal diffusion of dipole moment orientations on the unit sphere in conﬁguration space (# and j are the polar and azimuthal angles of the dipole, respectively), where the Fokker– Planck operator LFP for normal rotational diffusion in Eq. (8) is given by LFP ¼ L0FP þ Lext , where L0FP W ¼

1 1 r2 W þ r ðWrV0 Þ ; 2t kB T

Lext W ¼

1 r ðWrVg Þ 2tkB T ð196Þ

and the potential energy Vg of a dipole due to the perturbing ﬁeld F1 is Vg ð#; j; tÞ ¼ mF1 ðtÞug ð#; jÞ

ð197Þ

Here g ¼ k for the longitudinal response and g ¼ ? for the transverse response with uk ¼ cos # and u? ¼ cos j sin #. Our objective is to ascertain how anomalous diffusion modiﬁes the dielectric relaxation in a bistable potential with two nonequivalent wells, Eq. (195). The formal step-off transient solution of Eq. (172) for t 0 is obtained from the Sturm–Liouville representation, Eq. (179), with the initial (equilibrium) distribution function Wð#; j; 0Þ ¼ Ce½V0 ð#ÞþVg ð#;j;0Þ=ðkB TÞ ¼ W0 ð#Þ½1 þ x1 ðug hug i0 Þ þ Oðx21 Þ ð198Þ

350

william t. coffey, yuri p. kalmykov and sergey v. titov

where W0 ð#Þ ¼ eV0 =ðkB TÞ =Z is the equilibrium distribution function in the absence of the external ﬁeld E1 , which satisﬁes L0FP W0 ð#Þ ¼ 0, Z is the partition function given by [8] Z¼

ð1

2

exxþxV x dx ¼

1

rﬃﬃﬃﬃﬃﬃﬃﬃ o pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ p xV h2 n e erfi½ xV ð1 þ hÞ þ erfi½ xV ð1 hÞ 4xV ð199Þ

Ðx 2 erfiðxÞ ¼ p2ﬃﬃp et dt is the error function of imaginary argument, h ¼ x=ð2xV Þ is 0

the reduced ﬁeld (or asymmetry) parameter, and x1 ¼ mF1 =ðkB TÞ. The longitudinal and transverse normalized relaxation functions Cg ðtÞ and components of the complex dielectric susceptibility wg ðoÞ are given by Cg ðtÞ ¼

hug iðtÞ hug i0 X g ¼ c E ½lgp tðt=tÞs p p s hug ið0Þ hug i0 cgp wg ðoÞ X ¼ p 1 þ ðiotÞs =ðtlg Þ wg p

ð200Þ ð201Þ

where lgp;s are expressed in terms of the eigenvalues lgp of the operator L0FP for the normal diffusion, namely, L0FP gp ð#; jÞ ¼ lgp gp ð#; jÞ so that lgp;s ¼ lgp t1s , and wk ¼

m2 N0 m2 N0 ðhcos2 #i0 hcos #i20 Þ; w? ¼ hcos2 j sin2 #i0 kB T kB T

are the components of the static susceptibility tensor. 1.

Matrix Continued Fraction Solution

As we have already mentioned, it is difﬁcult to evaluate dielectric parameters from Eq. (201) because a knowledge of all the eigenvalues lgk and corresponding amplitudes cgk is required. A more simple (from the computational point of view) solution can be given in terms of matrix continued fractions. The general transient response solution of Eq. (172) for t 0, one can be sought in the form Wð#; j; tÞ ¼ W0 ð#Þ þ

1 X l 1 X ð2l þ 1Þðl mÞ! g cl;m ðtÞeimj Pm l ðcos #Þ 4p l¼0 m¼l ðl þ mÞ!

ð202Þ

fractional rotational diffusion

351

where the Pm l ðxÞ are the associated Legendre functions and the equilibrium distribution function may be expanded in the Legendre polynomials Pl ðxÞ, namely, W0 ð#Þ ¼

1 1 X ð2l þ 1ÞGl Pl ðcos #Þ 4p l¼0

ð203Þ

and Gl ¼ hPl ðcos #Þi0 . By substituting Eq. (202) into Eq. (172), we have the fractional differential recurrence relations for the longitudinal and transverse k relaxation functions fl ðtÞ and fl? ðtÞ, namely, d k xV lðl þ 1Þ lðl þ 1Þ k s 1s f ðtÞ ¼ t 0 Dt fl ðtÞ dt l ð2l 1Þð2l þ 3Þ 2 x xV ðl 1Þ k k k þ ½f ðtÞ flþ1 ðtÞ þ f ðtÞ 2ð2l þ 1Þ l1 ð2l 1Þð2l þ 1Þ l2 xV ðl þ 2Þ k flþ2 ðtÞ ð204Þ ð2l þ 1Þð2l þ 3Þ d ? lðl þ 1Þ 3 lðl þ 1Þ ? s 1s f ðtÞ ¼ t 0 Dt fl ðtÞ xV dt l ð2l 1Þð2l þ 3Þ 2

x ? ? þ ðtÞ l2 flþ1 ðtÞ ðl þ 1Þ2 fl1 2ð2l þ 1Þ xV lðl þ 1Þ2 xV l2 ðl þ 1Þ ? ? flþ2 fl2 ðtÞ ðtÞ þ ð2l þ 1Þð2l þ 3Þ ð2l 1Þð2l þ 1Þ ð205Þ where

k fl ðtÞ

and fl? ðtÞ are deﬁned by k

k

fl ðtÞ ¼ cl;0 ðtÞ ¼ hPl ðcos #ÞiðtÞ hPl i0

ð206Þ

fl? ðtÞ

ð207Þ

¼

Re½c? l;1 ðtÞ

¼

hcos jP1l ðcos #ÞiðtÞ

so that the relaxation function Cg ðtÞ describing the decay of the electric polarization is Cg ðtÞ ¼ f1g ðtÞ=f1g ð0Þ. We now present the solution of Eqs. (204) and (205) in terms of matrix continued fractions. The advantage of posing the problem in this way is that exact formulae in terms of such continued fractions may be written for the Laplace transform of the aftereffect function, the relaxation time, and the complex susceptibility. The starting point of the calculation is Eqs. (204) and (205) written as the matrix differential recurrence relation g g g gþ g ½Qg C_ gl ðtÞ ¼ ts 0 D1s t l Cl1 ðtÞ þ Ql Cl ðtÞ þ Ql Clþ1 ðtÞ

ð208Þ

352

william t. coffey, yuri p. kalmykov and sergey v. titov

Here the vectors Cgl ðtÞ are deﬁned as

0

Cgl ðtÞ ¼ @

g f2l1 ðtÞ

f2lg ðtÞ

1 A

ð209Þ

g where the 2 2 matrices Qg l ; Ql in Eq. (208) are

0

1 4xV ðl 1Þð2l 1Þl xð2l 1Þl B ð4l 1Þð4l 3Þ C 4l 1 B C k C Ql ¼ B B C @ 2xV ð2l 1Þð2l þ 1Þl A 0 ð4l þ 1Þð4l 1Þ 1 0 2xV xlð2l 1Þ 1 lð2l 1Þ C B 4l 1 ð4l 3Þð4l þ 1Þ C B k C B Ql ¼ B C A @ xlð2l þ 1Þ 2xV lð2l þ 1Þ 1 4l þ 1 ð4l þ 3Þð4l 1Þ 1 0 2xV ð2l þ 1Þð2l 1Þl 0 C B ð4l 1Þð4l þ 1Þ C B kþ C B Ql ¼ B C @ xlð2l þ 1Þ 4xV ðl þ 1Þð2l þ 1Þl A 4l þ 1 ð4l þ 1Þð4l þ 3Þ 0 1 4xV l2 ð2l 1Þ 2xl2 B ð4l 1Þð4l 3Þ C 4l 1 B C B C Q? ¼ l B C @ 2xV lð2l þ 1Þ2 A 0 ð4l 1Þð4l þ 1Þ 1 0 2lð2l 1Þ 3 xð2l 1Þ2 C B xV ð4l 3Þð4l þ 1Þ lð2l 1Þ 2ð4l 1Þ C B ? C B Ql ¼ B C 2 A @ xð2l þ 1Þ 2lð2l þ 1Þ 3 lð2l þ 1Þ xV ð4l 1Þð4l þ 3Þ 2ð4l þ 1Þ 1 0 2xV lð2l 1Þ2 0 C B ð4l 1Þð4l þ 1Þ C B ?þ C Ql ¼ B C B @ 2xl2 4xV l 2 ð2l þ 1Þ A ð4l þ 1Þð4l þ 3Þ 4l þ 1

353

fractional rotational diffusion

The three-term matrix recurrence relation, Eq. (208), may now be solved for the ~ g ðoÞ in terms of matrix continued fractions to yield [8] Fourier–Laplace transform C 1 " ! # 1 l X Y ~ g ðoÞ ¼ tðiotÞs1 Dg ðoÞ Cg ð0Þþ Qgþ Dg ðoÞ Cg ð0Þ ð210Þ C 1

1

1

k1

l¼2

k

l

k¼2

where Dgk ðoÞ is the 2 2 matrix continued fraction deﬁned as g g 1 Dgk ðoÞ ¼ ðiotÞs I Qgk Qgþ k Dkþ1 ðoÞQkþ1

ð211Þ

the tilde denotes the one-sided Fourier transform, namely, ~ g ðoÞ C l

¼

1 ð

eiot Cgl ðtÞ dt

0

and the initial value vectors Cgl ð0Þ ¼

g ð0Þ f2l1 f2lg ð0Þ

ð212Þ

can be also evaluated in terms of continued fractions. The initial value vectors Cgl ð0Þ in Eq. (212) may be determined using matrix continued fractions because the initial values flg ð0Þ are expressed just as in normal diffusion in terms of the equilibrium averages (Ref. 8, Chapter 8)

k fl ð0Þ ¼ x1 hcos #Plþ1 i0 hcos #i0 hPl i0 lþ1 l hPlþ1 i0 þ hPl1 i0 hP1 i0 hPl i0 ¼ x1 2l þ 1 2l þ 1 lðl þ 1Þ ½hPl1 i0 hPlþ1 i0 fl? ð0Þ ¼ x1 hcos j sin #Pl i0 ¼ x1 2ð2l þ 1Þ and, in turn, the equilibrium quantities hPl i0 satisfy the set of equations (Ref. 8, Chapter 8) k hP2l3 i0 k hP2l1 i0 kþ hP2lþ1 i0 Ql þ Ql þ Ql ¼0 ð213Þ hP2l2 i0 hP2l i0 hP2lþ2 i0 The solution of Eq. (213) is then given by

hP2l1 ðcos #Þi0 hP2l ðcos #Þi0

k

k

k

¼ Sl ð0ÞSl1 ð0Þ . . . S1 ð0Þ

0 1

ð214Þ

354

william t. coffey, yuri p. kalmykov and sergey v. titov k

k

k

k

where Sl ð0Þ ¼ Dl ð0ÞQl and Dl ð0Þ is given by Eq. (211) at s ¼ 0. Thus the initial conditions Cgl ð0Þ using matrix continued fractions are 1 0 1 2l 2l 1 i hP 1 0 6 0 B hP2l1 i0 hP2l3 i0 k 4l 1 C C B @ A þ Cl ð0Þ ¼ x1 6 4l 1 A hP i 4 @ 2l hP2l2 i0 2l 0 0 0 hP1 i0 4l þ 1 0 1 3 0 0 hP i 0 2lþ1 A 5 þ @ 2l þ 1 0 hP2lþ2 i0 4l þ 1 82 13 0 0 1 2l > > hP i <6 0 0 1 0 7 B 4l 1 C @ 2l þ 1 ASk ð0Þ þ B C7Sk ð0Þ ¼ x1 6 lþ1 5 l 4 A @ 2l 0 > > : hP1 i0 4l þ 1 4l þ 1 0 19 2l 1 = 0 0 k k A ð0Þ . . . S ð0Þ S þ@ 4l 1 1 l1 ; 1 0 0 2

0

and 0 1 1 lð2l 1Þ lð2l 1Þ hP i B 0 4l 1 C 2l3 0 C þB 4l 1 A @ A lð2l þ 1Þ hP2l2 i0 0 0 4l þ 1 3 0 1 0 0 hP2l1 i0 hP2lþ1 i0 7 A 7 þ @ lð2l þ 1Þ 0 hP2l i0 hP2lþ2 i0 5 4l þ 1 82 0 13 0 1 lð2l 1Þ > > 0 0 0 <6 B 7 4l 1 C @ lð2l þ 1Þ ASk ð0Þ þ B C7Sk ð0Þ ¼ x1 6 lþ1 4 @ A 5 l > 0 lð2l þ 1Þ > : 0 4l þ 1 4l þ 1 0 19 lð2l 1Þ = 0 0 k k @ A S ð0Þ . . . S1 ð0Þ þ 4l 1 ; l1 1 0 0

2 0 6 6@ 0 C? l ð0Þ ¼ x1 4 0

Equations (210) and (211) constitute the exact solution of our problem formulated in terms of matrix continued fractions. Having determined the ~ g ðioÞ ¼ ~f g ðioÞ=f g ð0Þ, one may Laplace transform ~flg ðoÞ and noting that C 1 1 calculate the susceptibility wg ðoÞ from Eq. (201).

355

fractional rotational diffusion 2.

Bimodal Approximation

The matrix continued fraction method we have just described yields the exact solutions for the complex susceptibility for all values of the thermal and anisotropy energies. Consequently, that method is an indispensable tool in estimating the accuracy of approximate solutions for typical parameters of the system. For example, in normal diffusion, the Kramers escape rate provides a close approximation to the smallest nonvanishing eigenvalue, the inverse of which provides an approximation to the longest relaxation time in a system where barrier crossing is involved as well as the relaxation time for bistable potentials with equivalent wells ½8. The matrix continued fraction method then allows one to determine the range of system parameters (e.g., barrier height, friction, etc.) in which the approximate solution for the longest relaxation time provided by the Kramers method is valid. Nevertheless, in practical applications such as the analysis of experimental results, the matrix continued fraction method is of very limited use since the dependence of the susceptibility tensor on the model parameters (anisotropy constants) is not obvious from this method. Thus it is desirable to try to obtain simple approximate formulas describing the dynamical behavior. This has been accomplished in previous sections for similar models in the context of the two-mode approximation. For the model under consideration, the corresponding longitudinal complex susceptibility wk ðoÞ can be effectively described then by the sum of two Cole–Cole spectra, namely, wk ðoÞ 1 1 1 ¼ þ k s k wk 1 þ ðio=oc Þ 1 þ ðio=oW Þs

ð215Þ

k

where the characteristic frequencies okc and oW are given by k

k

okc ¼ t1 ðtl1 Þ1=s ;

oW ¼ t1 ðt=tW Þ1=s

ð216Þ

k

and l1 is the smallest nonvanishing eigenvalue of the Fokker–Planck operator L0FP for the normal rotational diffusion deﬁned by Eq. (196). The parameters 1 and tW are deﬁned in terms of the characteristic times of the normal diffusion k k (the integral relaxation time tint , the effective relaxation time tef , and the inverse k of the smallest non-vanishing eigenvalue 1=l1 ) k

1 ¼

k

k k

tint =tef 1 k k

k k

l1 tint 2 þ 1=ðl1 tef Þ

;

tW ¼

l1 tint 1 k

k

l1 1=tef

ð217Þ

356

william t. coffey, yuri p. kalmykov and sergey v. titov

Furthermore, for the transverse response, noting that ? ? t? int ﬃ tef ﬃ l1

ð218Þ

for all values of the model parameters, the spectrum of w? ðoÞ may be approximated by the single Cole–Cole equation w? ðoÞ 1 ¼ s w? 1 þ ðio=o? cÞ

ð219Þ

where the characteristic frequency o? is given by 1 ? 1=s o? c ¼ t ðt=tef Þ

ð220Þ

In the foregoing equations tgint , tgef , and lg1 are calculated as follows. The k effective relaxation times tef and tgef are given in terms of equilibrium averages hP1 i0 and hP2 i0 only (Ref. 8, Chapter 8): 2hP2 i0 þ 1 3hP1 i20 1 hP2 i0 1 hP2 i0 t? ef ¼ t 1 þ hP2 i0 =2 k

tef ¼ t

ð221Þ ð222Þ

where exV sinh x x xV Z 2xV 3exV x 3x2 3 1 coshx sinhx þ 2 hP2 i0 ¼ 2xV 4x 2 2xV Z 8xV V

hP1 i0 ¼

ð223Þ ð224Þ

the partition function Z is given by Eq. (199). The integral relaxation time tgint can be evaluated numerically from the continued fraction solution ~ g ð0Þ=Cg ð0Þ at s ¼ 1. Moreover, the longitudinal integral relaxation tgint ¼ C k time tint is given in an exact integral form [8] as k tint

ð1

2

ðz

32

eV0 ðzÞ=kT 4 ðz0 hcos #i0 ÞeV0 ðz Þ=kT dz05 ¼ dz 1 z2 Z hcos2 #i0 hcos #i20 1 1 2t

0

ð225Þ

fractional rotational diffusion

357

k

where z ¼ cos #. The smallest eigenvalue l1 can also be evaluated numerically as described in detail in Ref. 8, Chapter 8. In the low-temperature limit, 2xV þ x 1, 1 ; 1 1 , and tW in Eq. (159) may be evaluated from the simple equations [79] wk 1

4ð1 h2 Þ ½ð1 þ hÞe2xV h þ ð1 hÞe2xV h 2

wk ð1 1 Þ ½2xV ð1 þ hÞ2 5 þ h 1 tW t 2xV ð1 þ hÞ 1þh

ð226Þ ð227Þ ð228Þ

k

Moreover, l1 can be approximated by Brown’s formula ½76 k

l1 ¼

3=2

2 2 xV ð1 h2 Þ ½ð1 þ hÞexV ð1þhÞ þ ð1 hÞexV ð1hÞ 1=2 tp

ð229Þ

For small values of xV , this formula is inadequate. However, we can then use an k approximate equation for the effective relaxation time tef because for k k xV 1 l1 1=tef [8]. Also, for small values of x, one can use the exact Taylor series expansion ([8], Section 8.3.2) 2 48 2 32 3 15552 4 k xV xV þ x l1 t ¼ 1 x V þ 875 21875 58953125 V 5 1 1 686 2 2 1 4 þ x x x x þ þ 10 875 V 84375 V 7000 k

k

k

ð230Þ

Selected numerical values of tint , tef , 1=l1 , tW , and t? ef are presented in Table I of Ref. [81]. Equations (228) and (229) allow one to readily estimate the asymptotic k behavior of the characteristic frequencies okc and oW in Eq. (160). k The frequencies okc and oW as functions of xV (i.e., the inverse dimensionless temperature) at a ﬁxed value of h ¼ 0:2 are plotted in Figs. 14 and 15 for various values of s. As one can see in these ﬁgures, the fractional exponent k s strongly inﬂuences the temperature dependence of okc and oW . The results of the calculation of the real w0g ðoÞ and imaginary w00g ðoÞ parts of susceptibility from both the exact continued fraction solutions and the approximate Eqs. (159) and (219) for typical values of the model parameters are shown in Figs. 16–19. The agreement between the exact continued fraction calculations and the approximate Eqs. (159) and (219) is very good [the maximum relative deviation between the corresponding curves does not exceed a few (3–5) percent]. Similar

358

william t. coffey, yuri p. kalmykov and sergey v. titov

10

0

1: σ = 1.0 2: σ = 0.8 3: σ = 0.5 4: σ = 0.2

h = 0.2 −1

10

−2

τωc

||

10

1

10

−3

2

4

10

−4

10

−5

3 (a)

0

5

10

ξV

15

(b)

10 0

4 10 −1

τωc

||

3

σ = 0.8

10 −2

10

1: h = 0.0 2: h = 0.2 3: h = 0.4 4: h = 0.6

−3

0

1

5

2

10

ξV k

15

okc [solid lines: Eq. (160) with exact values of l1 ] versus xV (inverse temperature) for (a) h ¼ 0:2 and various values of s and (b) s ¼ 0:8 and various values of h. Dashed lines: Eq. (160) k with l1 from Eq. (229).

Figure 14.

359

fractional rotational diffusion 10

6

10

4

10

2

1: σ = 1.0 2: σ = 0.8 3: σ = 0.5 4: σ = 0.2

τωW

||

4

3 2 1 h = 0.2

10

0

0

5

10 ξV

15

20

Figure 15. okW versus xV for h ¼ 0:2 and various values of s [solid lines: Eqs. (216) and (217) k

k

k

with exact values of l1 , tint , and tef ].

(or even better) agreement is obtained for all values of x and s. Thus one may conclude that Eqs. (159) and (219) accurately describe the behavior of wg ðoÞ for all frequencies of interest and for all values of the ﬁeld strength (x), anisotropy constants (xV ), and anomalous exponent (s) parameters. We remark that h ¼ 0 corresponds to the symmetric bistable potential while h ¼ 1 is the value of h at which the bistable structure of the potential—that is, the nucleation ﬁeld— disappears. Furthermore, for normal diffusion in the asymmetric bistable potential given by Eq. (195), it may be shown that the shallower of the two potential wells becomes completely depopulated at h 0:17 so that the relaxation is no longer dominated by the low-frequency peak generated by the overbarrier process ½79. All that effectively remains is the high-frequency absorption peak due to the well process. The behavior is reminiscent of a phase transition controlled by the asymmetry parameter h. With a simple change of notation the present theory may be used to set the Gilroy and Philips model ½82 of structural relaxation processes in amorphous materials and Dyre and Olsen’s minimal model for beta relaxation in viscous liquids ½83 in the framework of the general theory of stochastic processes. Moreover, the formulation of the theory in terms of kinetic equations as the

χ|| '(ω)

360

william t. coffey, yuri p. kalmykov and sergey v. titov 10 0

1

10

−1

2

10

−2

σ = 0.5

ξV = 10

1: ξ = 0 2: ξ = 2 3: ξ = 5 4: ξ = 10

3 10

−3

10

−4

4

10

10

−1

10

−3

10

−5

−1 0

10

−6

−2

10 ωτ

10

2

10

6

1

χ||'' (ω)

2

ξV = 10 σ = 0.5

4 10

1: ξ = 0 2: ξ = 2 3: ξ = 5 4: ξ = 10

3

−7

10

−1 0

−6

10

−2

10 ωτ

10

2

6

10

w0k ðoÞ and w00k ðoÞ versus ot evaluated from the exact matrix continued fraction solution (solid lines) for s ¼ 0:5, xV ¼ 10 and various values of x and compared with those calculated from the approximate Eq. (215) (ﬁlled circles) and with the low-(dotted lines) and highfrequency (dashed lines) asymptotes Eqs. (184) and (185), respectively.

Figure 16.

361

fractional rotational diffusion

σ = 0.5 −1

10

ξ=2

1 2

χ||' (ω)

3 4

−3

10

1: ξV = 1 2: ξV = 5 3: ξV = 10 4: ξV = 15

−5

10

10

10

−1 2

10

−8

10

4

3

−4

ωτ

10

0

2

10

4

10 8

1: ξV = 1

1

2: ξV = 5

−2

3: ξV = 10

χ ''|| (ω )

4: ξV = 15

10

−4

σ = 0.5 ξ=2

10

−6

10 −12

Figure 17.

10 −8

10 −4

ωτ

10 0

10 4

108

The same as in Fig. 16 for s ¼ 0:5, x = 2 and various values of xV .

diffusion limit of, in general, fractal time random walks allows anomalous diffusion effects to be incorporated into both of these models. In the context of dielectric relaxation, we remark that the area of applicability of these results is restricted to the low-frequency range, as deﬁned

362

william t. coffey, yuri p. kalmykov and sergey v. titov 10−1 1

σ = 0.5 ξV = 10

2 3

χ⊥' (ω)

10

−2

10−3

1: ξ = 0 2: ξ = 7 3: ξ = 15 10−4 10−4

10−2

10 0

ωτ

10 2

10 4

10 6

10 2

10 4

10 6

10 −1 σ = 0.5 ξV = 10

10

−2

χ ''⊥ (ω)

1 10 −3

3 2 1: ξ = 0 2: ξ = 7 3: ξ = 15

10 −4

10 −5

10−4

10−2

10 0

ωτ

w0? ðoÞ and w00? ðoÞ versus ot evaluated from the exact matrix continued fraction solution (solid lines) for s ¼ 0:5, xV ¼ 10 and various values of x and compared with those calculated from the approximate Eq. (219) (ﬁlled circles) and with the low- (dotted lines) and highfrequency (dashed lines) asymptotes Eqs. (184) and (185), respectively.

Figure 18.

363

fractional rotational diffusion 10

0

σ = 0.5 ξ=2

1 10

2

−1

χ⊥' (ω)

3 10

−2

1: ξV = 1 10

−3

10

−4

2: ξV = 5 3: ξV = 10

10

10

−4

10

−2

10

0

ωτ

10

2

10

σ = 0.5 ξ=2

−1

4

10

6

1: ξV = 1 2: ξV = 5 3: ξV = 10

1

−2

2

χ''⊥ (ω)

10

3

10

−3

10

−4 −4

10

Figure 19.

−2

10

0

10

ωτ

10

2

4

10

6

10

The same as in Fig. 18 for s ¼ 0:5, x ¼ 2 and various values of xV .

by the inequality otg 1 ðg ¼ k; ?Þ, because the theory does not include the effects of molecular inertia. A consistent treatment of inertial effects must be carried out using the appropriate inertial kinetic equation for the evolution of the probability density function in phase space ½8.

364

william t. coffey, yuri p. kalmykov and sergey v. titov IV.

INERTIAL EFFECTS IN ANOMALOUS DIELECTRIC RELAXATION

We shall now discuss the role played by inertial effects in anomalous relaxation on the basis of various generalizations of the inertial Fokker–Planck equation for the motion of a Brownian particle in a potential (or Klein–Kramers equation) to anomalous diffusion. In order to give a physically meaningful description of the anomalous dielectric behavior at high frequencies, inertial effects must be included just as in normal relaxation ½43. As already mentioned in Section II.C, the omission of inertial effects in the relaxation process gives rise, in the context of dielectric relaxation, to the phenomenon of inﬁnite dielectric absorption at high frequencies ½43. Inclusion of them on the other hand yields the return to transparency at high frequencies (in the far-infrared region). In practice, models of the inertial rotational Brownian motion ½43 are frequently used in studying orientational relaxation in liquids in order to compare spectra obtained by various probe techniques such as dielectric relaxation, the dynamic Kerr effect, infrared absorption, Raman scattering, and so on, with the corresponding theoretical spectra ½8; 43; 84. In the context of Brownian motion, inertial effects may be included either by averaging the inertial Langevin equation or by constructing a kinetic equation for the evolution of the probability density function in phase space. The resulting differential recurrence relation may then be solved by various methods to yield the complex susceptibility ½8. Coffey and others ½8; 24 have recently shown how inertial effects may be included in the fractional dynamics of an assembly of noninteracting rotators (rotating either in a plane or in space). They have also demonstrated that this generalized model can reproduce nonexponential (Cole–Cole type) anomalous dielectric relaxation behavior at low frequencies, and how the unphysical high-frequency behavior of the absorption coefﬁcient due to the neglect of inertia may be removed in fractional relaxation just as in inertia corrected Debye relaxation. The inclusion of inertial effects in the fractional dynamics just as in the conventional Brownian dynamics thus ensures the desired return to optical transparency at high frequencies. Moreover, Gordon’s sum rule ½24 for the integral dipolar absorption is satisﬁed. The starting point of the solution outlined in Refs. 8 and 24 is a generalization of the Klein– Kramers equation to fractional dynamics. The various generalizations of the Klein–Kramers equation which have been proposed have been fully described by Metzler [85], and so we shall merely list them here. Moreover, we shall select the particular generalization of the Klein–Kramers equation which provides a physically acceptable description of the far-infrared absorption in dipolar ﬂuids.

365

fractional rotational diffusion A.

Metzler and Klafter’s Form of the Fractional Klein–Kramers Equation

Metzler and Klafter ½86 and Metzler ½85 have proposed a fractional Klein– Kramers equation (FKKE), which, according to them, corresponds to a multiple trapping picture where the tagged particle executes translational Brownian motion in accordance with the Langevin equation: m€xðtÞ ¼ mb_xðtÞ

q V½xðtÞ; t þ FðtÞ qx

ð231Þ

where FðtÞ is the usual white noise driving force. The particle then gets successively immobilized in traps whose mean distance apart is ¼ ðkB T=mÞt

ð232Þ

where t is the mean time between successive trapping events. The time spans spent in the traps are governed by the waiting-time probability density function wðtÞ Aa t1a ;

0
ð233Þ

which corresponds, as before, to a divergence of the characteristic waiting time due to ½31 ‘‘the relatively frequent occurrence of long waiting times.’’ The process is thus ½85 a separable fractal time random walk with ﬁnite jump length variance and thus describes subdiffusion. The divergence of the average waiting time combined with the separability of the process is very signiﬁcant if the walk is used to model dielectric relaxation because it leads to inﬁnite integral absorption. We illustrate this by referring to the motion of a ﬁxed axis rotator, the normal Brownian rotation of which is described by the Langevin equation (see Ref. 8, Chapters 4 and 10): € ¼ IbfðtÞ _ q V½fðtÞ; t þ lðtÞ I fðtÞ qf

ð234Þ

The procedure based on a generalized Chapman–Kolmogorov equation in phase space proposed by Metzler and Klafter ½7; 85; 86 then leads, assuming the diffusion limit, to the following generalization of the Klein–Kramers

366

william t. coffey, yuri p. kalmykov and sergey v. titov

equation: qW qW 1 qV qW q _ k B T q2 W 1a 1a _ ¼ 0 Dt t þ þb ðfWÞ þ f qt qf I qf qf_ I qf_ 2 qf_

ð235Þ

_ tÞ and the Debye relaxation time t is here identiﬁed with t . where W ¼ Wðf; f; According to Metzler ½85, the entire Klein–Kramers operator in the square brackets acts nonlocally in time; that is, drift friction and diffusion terms are under the time convolution and are thus affected by the memory. In the present application, the ensuing nonlocal behavior causes divergence of the absorption coefﬁcient as we shall presently illustrate. Thus, a model based on a FKKE of the form of Eq. (235) cannot produce the desired return to transparency at high frequencies. Before illustrating this behavior however, we remark that according to Metzler and Klafter ½7; 86 and Metzler ½85, Eq. (235) reduces in a manner analogous to the reduction of the Klein–Kramers equation, to the Smoluchowski equation in the high friction limit. In other words, Eq. (235) simpliﬁes, in the high friction limit, to the fractional Fokker–Planck equation ½Eq. (55) for the evolution of W in conﬁguration space f only, if the replacement s ! a is made. Equation (235) may also be separated, just as the normal Klein–Kramers equation, in the time and space variables in the form _ fÞFðtÞ W ¼ ðf;

ð236Þ

This procedure yields the time behavior corresponding to a given phase-space eigenfunction p;n of the normal diffusion solution as a Mittag–Lefﬂer function instead of the exponential decay characteristic of the Klein–Kramers equation just as in Eq. (142). Moreover, the scaling laws for the eigenvalues of the Fokker– Planck operator described in Section III will also apply. In addition, one stationary solution will be the Maxwell–Boltzmann distribution, while the other will pertain to a steady diffusion current over a barrier ½7. According to Metzler and Klafter ½7; 86, the second solution leads to a generalization of the Kramers theory of escape of particles over potential barriers to anomalous diffusion including inertial effects and thus energy-controlled diffusion effects. Assuming a solution of Eq. (235) in the form of Eq. (236) corresponds to posing the solution as a Sturm–Liouville problem. However, the problem may be also be cast into the form of a solution to a differential recurrence relation, and thus as a calculation of successive convergents of a continued fraction in the frequency domain. This is accomplished, just as in normal diffusion [8], by means of a Fourier expansion in the Hermite polynomials in the angular velocity and the circular functions to yield, inter alia, the Laplace transform of the complex susceptibility as an inﬁnite continued fraction in the frequency domain, as we now describe.

367

fractional rotational diffusion

In Eq. (235), we assume, in the present context, that a weak uniform electric ﬁeld F applied along the initial line is suddenly switched off at t ¼ 0, so that linear response theory may be used to describe the ensuing response. Now, the fractional derivative in (235) acts only on functions of the time. Hence, we may seek a solution of Eq. (235), just as in normal Brownian motion, in the form of the Fourier–Hermite series _ tÞ ¼ Wðf; f;

1 X 1 Z Z2 f_ 2 X 1 _ iqf c ðtÞ Hn ðZfÞe e n n! n;q 2 2p3=2 n¼0 q¼1

where the Hn are the Hermite polynomials ½51 and Z ¼ linearized initial distribution is _ 0Þ ¼ Wðf; f;

ð237Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ I=ð2kB TÞ. The

Z Z2 f_ 2 mF cos f þ e 1 þ kB T 2p3=2

ð238Þ

Straightforward manipulation of the recurrence relations of the Hn ½51 d Hnþ1 ðxÞ ¼ 2nHn1 ðxÞ; dx

Hnþ1 ðxÞ ¼ 2xHn ðxÞ 2nHn1 ðxÞ

(essentially as in Ref. 8, Chapter 10) then leads by the orthogonality of the Hn , to a differential-recurrence relation for the Fourier coefﬁcients cn;q ðtÞ, namely, iq n p ﬃﬃﬃﬃﬃ c c_ n;q ðtÞ þ ta 0 D1a ðtÞ þ 2nc ðtÞ þ ðtÞ ¼0 ½c nþ1;q n1;q n;q t g 2g

ð239Þ

where g ¼ ð2Z2 b2 Þ1 is a dimensionless inertial (Sack’s) parameter ½40. It is obvious from Eq. (239) how all recurrence relations associated with the Brownian motion may be generalized to fractional dynamics governed by Eq. (235). Since we consider the linear dielectric response, this may be written for the only case of interest, namely, q ¼ 1 (q ¼ 1 follows by symmetry) as a recurrence relation for cn;1 ðtÞ. The Laplace transform of this set, noting the generalized integration theorem of Laplace transformation, is pﬃﬃﬃﬃﬃﬃﬃﬃ ½gðtsÞa þ n~cn;1 ðsÞ þ i g=2½~cnþ1;1 ðsÞ þ 2n~cn1;1 ðsÞ ¼ dn;0 c0;1 ð0ÞgtðtsÞa1 ð240Þ where di; j is Kronecker’s delta and n 0. In writing Eq. (240), we have noted that on account of the initial condition, Eq. (238), and the orthogonality property of the Hn , all the cn;1 ð0Þ will vanish with the exception n ¼ 0.

368

william t. coffey, yuri p. kalmykov and sergey v. titov

Equation (240) is a three-term algebraic recurrence relation for the ~cn;1 ðsÞ, which may be solved exactly for ~c0;1 ðsÞ using standard continued fraction methods as in the normal Brownian motion (see Ref. 8, Chapter 2, Section 2.7.3): ~c0;1 ðsÞ ¼

gtðstÞa1 c0;1 ð0Þ pﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ~1 ðsÞ gðstÞa þ i g=2S

tðstÞa1 c0;1 ð0Þ 1

ðstÞa þ

1 þ gðstÞa þ

2g 2 þ gðstÞa þ ð241Þ

where successive convergents are calculated from the continued fraction ~ Sn ðsÞ ¼

pﬃﬃﬃﬃﬃ in 2g pﬃﬃﬃﬃﬃﬃﬃﬃ gðstÞa þ n þ i g=2~Snþ1 ðsÞ

ð242Þ

^ðoÞ may be given by linear response The normalized complex susceptibility w theory ^ðoÞ ¼ w

~c0;1 ðioÞ wðoÞ ¼ 1 io w0 c0;1 ð0Þ

ð243Þ

We remark in passing that ~c0;1 ðsÞ will also yield the Laplace transform of the characteristic function of the conﬁguration space probability density function. Equations (241)–(243) then lead to the generalisation of the Gross–Sack result ½39; 40 for a ﬁxed axis rotator to fractal time relaxation governed by Eq. (235), namely, ^ðoÞ ¼ 1 w

gðiotÞa g

gðiotÞa þ 1 þ gðiotÞa þ

ð244Þ

2g 2 þ gðiotÞa þ

Equation (244), in turn, can be expressed in terms of the conﬂuent hypergeometric (Kummer) function Mða; b; zÞ ½51, namely, ^ðoÞ ¼ 1 w

ðitoÞa Mð1; 1 þ g½1 þ ðitoÞa ; gÞ 1 þ ðitoÞa

ð245Þ

fractional rotational diffusion

369

Equation (245) can be readily derived by comparing Eq. (244) with the continued fraction Mða; b; zÞ ¼ ðb 1ÞMða 1; b 1; zÞ b 1 z þ

1 az ða þ 1Þz bzþ b þ 1 z þ

ð246Þ

where a ¼ 1, z ¼ g, and b ¼ 1 þ g½1 þ ðitoÞa , by noting that Mð0; b 1; zÞ ¼ 1 [51]. The continued fraction (246) can be obtained from the known recurrence relation [51] bð1 b þ zÞMða; b; zÞ þ bðb 1ÞMða 1; b 1; zÞ ¼ azMða þ 1; b þ 1; zÞ We remark that Mð1; 1 þ b; zÞ ¼ bzb ez gðb; zÞ, where gðb; zÞ is the incomplete gamma function ½51 deﬁned by Eq. (119). For s ¼ 1, Eq. (245) can be reduced to Sack’s result (Eq. (3.19c) of Ref. ½40). Successive convergents of Eq. (244) may be calculated as follows. The ﬁrst convergent is the generalization to fractional dynamics of the Debye result, the Cole–Cole equation ½Eq. (9), namely, ^ðoÞ ¼ w

1 1 þ ðiotÞa

ð247Þ

1

ð248Þ

The next convergent, namely, ^ðoÞ ¼ w

a

1 þ ðiotÞ þ gðiotÞ2a

represents a generalization of the Rocard result for normal inertia corrected rotational diffusion ½44 which takes the form ^ðoÞ ¼ w

1 1 þ iot gðotÞ2

ð249Þ

For rotation about a ﬁxed axis ½8. Moreover, the effect of the Brownian (intertrapping) dynamics appears explicitly through the inertial parameter g. Thus, one expects intertrapping effects to manifest themselves at high frequencies. We also remark that Eq. (248), unlike the Rocard equation ½Eq. (249), does not obey the Gordon sum rules ½see Eq. (85), which will have an important bearing on our subsequent discussion.

370

william t. coffey, yuri p. kalmykov and sergey v. titov

Equation (239) may also be used to calculate the angular velocity correlation function (AVCF) in the fractional dynamics. From Eq. (239) with n ¼ 1 and q ¼ 0, we have c1;0 ðtÞ ¼ 0 c_ 1;0 ðtÞ þ ðta =gÞ0 D1a t

ð250Þ

_ On choosing a sharp set of initial values of fð0Þ, say f_ 0, Eq. (250) has a solution in the time domain c1;0 ðtÞ ¼ f_ 0 Ea ½ðt=tÞa =g

ð251Þ

where, as usual, Ea ðzÞ is the Mittag–Lefﬂer function. The equilibrium AVCF _ fðtÞi _ then follows by multiplying Eq. (251) by f_ 0 ¼ f_ 0 ð0Þ and co ðtÞ ¼ hfð0Þ averaging over a Maxwell distribution of f_ 0 , thus co ðtÞ ¼ ðkB T=IÞEa ½ðt=tÞa =g

ð252Þ

Equation (252) represents the generalization of the conventional result based on the Ornstein–Uhlenbeck (inertia-corrected Einstein) theory of the Brownian motion ½87 to fractional dynamics. By way of illustration, we show in Fig. 20 1.0

Normalized AVCF

α = 0.7 α = 1.0

0.5

3 2

3'

2' 1 1' 0.0 0.0

0.1

0.2

t

0.3

0.4

0.5

Figure 20. Normalized angular velocity correlation function (AVCF) co ðtÞ=co ð0Þ for a ¼ 0:7 and a ¼ 1 and g ¼ 0:1 (1 and 10 ), 0.5 (2 and 20 ), 0.9 (3 and 30 ). Note the pronounced increase in the long time tail as g is increased.

371

fractional rotational diffusion

the variation of the AVCF with inertial parameter g, for a fractional index a ¼ 0:7. The long-time tail ðt tÞ due to the asymptotic ta -like dependence of the Mittag–Lefﬂer function is apparent as is the stretched exponential behavior at short times ðt tÞ. Despite the success in predicting reasonable behavior of the AVCF, and the recovery of the Debye result in the limit of small inertial effects, the theory based on Eq. (235) and the exact continued fraction solution, Eq. (244), or any of its convergents, utterly fails to explain the return to transparency at high frequencies as is illustrated in Fig. 21. Here, we show the absorption coefﬁcient ow00 ðoÞ for a ¼ 1 (normal diffusion) and a ¼ 0:5. It is apparent that just as in the Debye theory, Eq. (245) predicts nonphysical behavior of the absorption coefﬁcient because it does not predict the return to transparency at high frequencies. The effect of including inertia based on the model of Eq. (235) is simply to produce an enhanced Debye plateau for all a < 1; thus the Gordon sum rule for the dipole integral absorption of one-dimensional rotators, Eq. (85), is violated. The unphysical behavior of the present model appears to be a consequence of allowing the fractional operator, or memory function, to operate on the Liouville terms in the FKKE, so that the form of the underlying Boltzmann equation is not preserved. 1.5

ωχ''(ω)/χ0

1.0 α = 0.5, γ = 10 α = 0.5, γ = 0

0.5

α = 1.0, γ = 0 α = 1.0, γ =10

0.0

−1

0

1 log10(ωτ)

2

3

Figure 21. ow00 ðoÞ=w0 versus log10(ot) for a ¼ 0:5 and g ¼ 10 (solid line), a ¼ 1 and g ¼ 10 (dot-dashed line with the Debye plateau), a ¼ 0:5 and g ¼ 0 (dotted line), and a ¼ 1 and g ¼ 0 (dashed line).

372

william t. coffey, yuri p. kalmykov and sergey v. titov

Another way of regarding the divergence of the integrated absorption is to recall that in Brownian motion, the characteristic time associated with the farinfrared absorption for a ﬁxed Debye time is essentially the angular velocity correlation time as expressed through Sack’s parameter g. In the separable fractal time random walk, however, the angular velocity correlation time no longer dictates any particular time scale for the process. Such behavior results from the fact that both angular velocity and conﬁguration variables are subject to the same separable fractal time random walk, since a fractional derivative appears in the Liouville term. As is usual in separable fractal time random walks, no global characteristic time scales exist. Hence, the problem must be considered in the frequency domain. Thus the only meaningful time scale is that provided (at low frequencies) by the inverse of the characteristic frequency at which wðoÞ has a maximum. No very short time scale exists because the far-infrared absorption is effectively inﬁnite. Thus it is impossible to use the inverse of the maximum far-infrared absorption frequency to deﬁne a very short time scale. The lack of such a short time scale is synonymous with inﬁnite integrated absorption. Faced with these difﬁculties, we shall presently illustrate that if a generalization of the Klein–Kramers equation, ﬁrst proposed by Barkai and Silbey ½30, where the fractional derivatives do not act on the Liouville terms, is used, then the desired return to transparency at high frequencies is achieved. Moreover, the Gordon sum rule, Eq. (85), is satisﬁed. In conclusion of this subsection, we remark that the divergence of the integral absorption is not unusual in models that incorporate inertial effects. For example, in the wellknown Van Vleck–Weisskopf model [88], the divergence results from the stosszahlansatz used by them, just as in the present problem. B.

Barkai and Silbey’s Form of the Fractional Klein–Kramers Equation

We have seen from the particular form of the fractional Klein–Kramers equation ½Eq. (235) that the effect of allowing the entire Klein–Kramers operator to act nonlocally in time is to cause inﬁnite dielectric absorption just as the noninertial case. Thus, Eq. (235) provides a physically unacceptable picture of the highfrequency dielectric behavior. The root of this difﬁculty apparently being that in writing Eq. (235), the convective derivative or Liouville term, in the underlying Klein–Kramers equation, is operated upon by the fractional derivative. Hence, the convective derivative is under the time convolution and thus is affected by the memory, thus exhibiting nonlocal behavior or ‘‘Le´vy sneaking,’’ a term coined by Metzler ½85. Thus, the form of the Boltzmann equation for the single-particle distribution function is not preserved, meaning that now both angular velocity and conﬁguration variables are governed by the same separable fractal time random walk. Such a walk has ﬁnite jump length variance and fractal waiting

fractional rotational diffusion

373

time distribution. Thus, the angular velocity correlation time as we have already stated can no longer dictate the time scale of the fast high-frequency process giving rise to the far-infrared absorption, ultimately resulting in divergence of the integrated absorption. The diverging integrated absorption resulting from a separable fractal time walk and to a lesser extent the inﬁnite variance associated with Le´vy ﬂights suggests that, in general, separable CTRWs may not be useful as a description of inertial effects in dielectric relaxation. In the FKKE proposed by Barkai and Silbey ½30 in contrast, the memory or fractional derivative term is supposed to act only on the dissipative part of the normal Klein–Kramers operator, so that Barkai and Silbey’s equation [where, however, rotational quantities (angle f, moment of inertia I, etc.) replace translational ones (position x, mass m, etc.)] is of the form, again referring to ﬁxed axis rotators ½cf. Eq. (235) qW qW mF sin f qW q _ k B T q2 W 1a 1a _ ¼ f þ þ 0 Dt t b ðfWÞ þ qt qf I I qf_ 2 qf_ qf_

ð253Þ

This equation according to Barkai and Silbey is given for 0 < a < 1, which pertains to subdiffusion in velocity space. By solving this equation using the continued fraction method, we shall demonstrate that the subdiffusion in velocity space gives rise to enhanced diffusion in conﬁguration space. Although Eq. (253) has hitherto been regarded as valid for subdiffusion only, we shall demonstrate that if the equation is also regarded as describing enhanced diffusion in velocity space, then the enhanced diffusion in velocity space gives rise to subdiffusion in conﬁguration space. We remark that if we set a ¼ 1 in Eq. (253), then the velocity f_ has the form of the Ornstein–Uhlenbeck process ½87. Thus, in anomalous relaxation described by Eq. (253), the velocities acquire a fractional character and we are dealing with the fractional Ornstein–Uhlenbeck process in the velocity variables. In order to justify a diffusion equation of the form of Eq. (253), Barkai and Silbey ½30 consider a ‘‘Brownian’’ test particle that moves freely in one dimension and that collides elastically at random times with particles of the heat bath which are assumed to move much more rapidly than the test particle. The times between collision events are assumed to be independent, identically distributed, random variables, implying that the number of collisions in a time interval ð0; tÞ is a renewal process. This is reasonable, according to Barkai and Silbey, when the bath particles thermalize rapidly and when the motion of the test particle is slow. Thus, the time intervals between collision events ti are described by a probability density wðtÞ that is independent of the mechanical state of the test particle. Hence, the process is characterized by free motion with constant velocity for a time t1 , then an elastic collision with a bath particle, then a free motion evolution for a time t2 , then again a collision. The most

374

william t. coffey, yuri p. kalmykov and sergey v. titov

important assumption of the model is that wðtÞ decays as a power law for long times, namely, wðtÞ t1a ;

0
ð254Þ

The above stochastic collision model then leads to a generalization, Eq. (253), of the Fokker–Planck equation for the evolution of the phase distribution function for mechanical particles, where the velocities acquire a fractional character ½30, rather than both the displacements and the velocities as in Eq. (235). In the present context, all these comments apply, of course, to rotational Brownian motion. An equation that resembles Eq. (253), and indeed, coincides with it when the external potential is zero, has been proposed by Metzler and Sokolov ½89, in order to extend the equation [of which Eq. (235) is an example] originally proposed by Metzler and Klafter, to include enhanced diffusion in conﬁguration space. We should also remark that according to Metzler, the Barkai and Silbey equation arises from a Le´vy random walk—that is, a walk governed by a jump probability density function in which the step lengths are coupled to the waiting times, so that a long jump is penalized by a large time cost. In the particular Le´vy walk corresponding to Eq. (253), according to Metzler (who analyzed the problem using a generalized Chapman–Kolmogorov equation), the length of the step associated with a jump is curtailed by utilizing a jump probability density function of the form (v0 is the sharp velocity in the free motion of the test particle) cðx; tÞ ¼ pðxjtÞ ¼ dðx v0 tÞwðtÞ=2

ð255Þ

Thus the explicit spatiotemporal coupling characteristic of the Le´vy walk is retained. The coupling ensures ﬁnite jump length variance in contrast to a Le´vy ﬂight, which arises from a separable random walk with ﬁnite mean jump time. The behavior is thus quite different from the simple one-dimensional CTRW model considered in Section II.A, where, in order to derive the one-dimensional fractional Fokker–Planck equation for diffusion in conﬁguration space, the jump length and waiting-time random variables are assumed independent. However, we shall presently see from the exact continued fraction solution of the Barkai and Silbey equation (termed by Metzler the Le´vy rambling model) that in the high friction limit the result for the complex susceptibility is the same, namely, Eq. (247), as predicted by the fractal time random walk model. This is of course true of the normal Klein–Kramers equation. However, Eq. (253) is not separable. _ ¼ 0 can only be obtained by supposing Indeed, the stationary solution with W that both convective and dissipative terms each vanish separately. Nevertheless, Eq. (253) can also be solved by continued fraction methods as we shall

375

fractional rotational diffusion

demonstrate. The fact that Eq. (253) does not separate in the space and time variables appears to be entirely consistent with the Le´vy walk picture where a jump length involves a time cost and vice versa. This concludes our abbreviated discussion of the advantages and drawbacks of the various fractional diffusion equations, which have been proposed in order to generalize the Klein–Kramers equation. We shall now justify our assertion that the Barkai–Silbey equation leads to a physically acceptable description of the absorption coefﬁcient at high frequencies. Just as in Section IV.A, we may seek a solution of Eq. (253) in the form of the Fourier series, Eq. (237). The linearized initial (at t ¼ 0) distribution function is given by Eq. (238). Straightforward manipulation of the recurrence relations of the Hermite polynomials Hn again leads to a differential-recurrence relation for the Fourier coefﬁcients cn;q ðtÞ as ½cf. Eq. (239) iq n cn;q ðtÞ ¼ 0 t_cn;q ðtÞ þ pﬃﬃﬃﬃﬃ ½cnþ1;q ðtÞ þ 2ncn1;q ðtÞ þ t1a 0 D1a t g 2g

ð256Þ

Here, as above, g is the Sack inertial parameter. Noting the initial condition, Eq. (238), all the cn;1 ð0Þ in Eq. (256) will vanish with the exception n ¼ 0. On using the integration theorem of Laplace transformation as generalized to fractional calculus, we have from Eq. (256) the three-term recurrence relation ½cf. Eq. (240) for the only case of interest q ¼ 1 (since the linear dielectric response is all that is considered):

sﬃﬃﬃﬃﬃ n 1 1a ~cn;1 ðsÞ þ i ½~cnþ1;1 ðsÞ þ 2n~cn1;1 ðsÞ ¼ tc0;1 ð0Þdn;0 ð257Þ ts þ ðtsÞ g 2g

ðn > 1Þ. Equation (257) is a three-term algebraic recurrence relation for the ~cn;1 ðsÞ, which may be solved exactly for ~c0;1 ðsÞ as in the previous section, namely, ~c0;1 ðsÞ ¼

tc0;1 ð0Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1=2g~ S1 ðsÞ st þ

st þ i

tc0;1 ð0Þ 1 st þ ðstÞ1a =g þ

2g st þ 2ðstÞ1a =g þ ð258Þ

where successive convergents are calculated from the continued fraction ~ Sn ðsÞ ¼

pﬃﬃﬃﬃﬃ in 2g pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ st þ nðstÞ1a =g þ i 1=2g~Snþ1 ðsÞ

ð259Þ

376

william t. coffey, yuri p. kalmykov and sergey v. titov

^ðoÞ ¼ w ^0 ðoÞ i^ The normalized complex susceptibility w w00 ðoÞ is, as usual, given by linear response theory ½Eq. (243). On combining Eqs. (243) and (258), we have after simple algebra the generalization of the Gross–Sack result ½39; 40 for a ﬁxed-axis rotator to fractional relaxation governed by the Barkai and Silbey equation ½Eq. (253), namely ½24 ½cf. Eq. (244), ^ðoÞ ¼ 1 w

QðiotÞs

ð260Þ

Q

QðiotÞs þ

2Q

1 þ QðiotÞs þ

2 þ QðiotÞs þ

3Q 3 þ

where s ¼ 2 a and Q ¼ gðiotÞ2ða1Þ . Just as Eq. (244), Eq. (260) can be expressed in terms of the conﬂuent hypergeometric function Mða; b; zÞ [cf. Eq. (245)], namely, ^ðoÞ ¼ 1 w

ðitoÞs Mð1; 1 þ Q½1 þ ðitoÞs ; QÞ 1 þ ðitoÞs

ð261Þ

Equation (261) can be readily derived, by comparing Eq. (260) with the continued fraction Eq. (246), where now a ¼ 1;

z ¼ Q;

and b ¼ 1 þ Q½1 þ ðitoÞs

In the high damping limit ðg 1Þ, Eq. (261) can be simpliﬁed, yielding the generalization of the Rocard equation to fractional dynamics governed by the Barkai and Silbey equation [Eq. (253)], namely, ^ðoÞ ¼ w

1 s

1 þ ðiotÞ gðotÞ2

ð262Þ

We remark that Eq. (262), unlike the form of the Rocard equation of the Le´vy sneaking model, Eq. (248), has an inertial term similar to the Rocard equation for normal diffusion, Eq. (249). This has an important bearing on the high-frequency behavior because return to transparency can now be achieved, as we shall demonstrate presently. The exact solution, Eq. (260), also has satisfactory high-frequency behavior. We further remark that, on neglecting inertial effects (g ! 0), Eq. (261) yields the Cole–Cole formula [Eq. (9)]—that is, the result predicted by the noninertial fractional Fokker– Planck equation. ^00 ðoÞ versus oZ for various values of a and g Dielectric loss spectra w are shown in Figs. 22–24. It is apparent that the spectral parameters (the characteristic frequency, the half-width, the shape) strongly depend on

377

fractional rotational diffusion 10 1

1,1' : α = 0.25 2,2' : α = 0.5 3,3' : α = 1.0 4,4' : α = 1.5

γ = 0.02

4

10−1 4'

^

χ''

2 1 3 −3

10

3'

1'

2' 10−6

10−4

10−2

ηω

100

10 2

Figure 22. Dielectric loss spectra w^00 ðoÞ for g ¼ 0:02 and various values of a: 0.25 (curves 1 and

10 ), 0.5 (2, 20 ), 1 (3, 30 ), and 1.5 (4, 40 ). Solid lines (1, 2, 3, and 4): Eq. (261); crosses (10 , 20 , 30 , and 40 ): Eq. (9) with s ¼ 2 a; ﬁlled circles: Eq. (263).

1,1': γ = 2 × 10−8

10 1 2,2'

1,1'

3,3'

2,2': γ = 2 × 10−6

4

3,3': γ = 2 × 10−4 10

−1

χ^ ''

5

4,4': γ = 2 × 10−2 5,5': γ = 2

10 −3

10 −5

10

5'

4'

−6

10

−4

α = 0.5 −2

10 ηω

10

0

10

2

Figure 23. w^00 ðoÞ for a ¼ 0:5 and various values of g: 2 108 (curves 1 and 10 ), 2 106

(2, 20 ), 2 102 (3, 30 ), 2 102 (4, 40 ), and 2 (5, 50 ). Solid lines (1, 2, 3, 4, and 5): Eq. (261); crosses (10 , 20 , 30 , 40 , and 50 ): Eq. (9) with s ¼ 2 a; ﬁlled circles: Eq. (263).

378

william t. coffey, yuri p. kalmykov and sergey v. titov 10 0

α = 1.5

5 1

2

3

4 5'

10 −1

χ^ ''

4'

10

1,1': γ = 2 × 10−8

−2

3'

2,2': γ = 2 × 10−6 2'

3,3': γ = 2 × 10−4

10 −3

10−6

4,4': γ = 2 × 10−2 5,5': γ = 2 10−4 Figure 24.

10−2 ηω

1'

100

102

The same as in Fig. 23 for a ¼ 1:5.

both the anomalous exponent a (which pertains to the velocity space), and the ^00 ðoÞ is entirely inertial parameter g. Moreover, the high-frequency behavior of w determined by the inertia of the system. The nonphysical a dependence of ^00 ðoÞ from Eq. (235) at high frequencies is absent. It is apparent, just as in w Brownian dynamics, that inertial effects produce a much more rapid fall-off of ^00 ðoÞ at high frequencies. Furthermore, the Gordon sum rule for the integral w absorption, Eq. (85), is satisﬁed. It is apparent from Figs. 22 and 24, assuming that the Barkai–Silbey equation [Eq. (253)] also describes subdiffusion in conﬁguration space (i.e., s < 1 or 1 < a < 2), that Eq. (261) also provides a physically acceptable description of the dielectric loss spectrum. In the high ^00 ðoÞ may be approximated damping limit ðg 1Þ, the low-frequency part of w by the Cole–Cole equation [Eq. (9)]. When s > 1 (enhanced diffusion in ^00 ðoÞ is similar (see Figs. conﬁguration space), the low-frequency behavior of w ^00FR ðoÞ in the free rotation limit ð ¼ 0Þ, 13 and 14) to that of the dielectric loss w which is given by [8] ^00FR ðoÞ ¼ w

pﬃﬃﬃ 2 2 pZoeZ o

ð263Þ

_ fðtÞi, _ In order to calculate the equilibrium AVCF co ðtÞ ¼ hfð0Þ one can simply use the same method as in Section IV.A. Since the dynamics of the

379

fractional rotational diffusion 1.0 γ = 1.0

1 - α = 0.5 2 - α = 1.0 3 - α = 1.5

0.5 AVCF

1 2 0.0 3

0

2

4

t /τ

6

8

10

Figure 25. Normalized angular velocity correlation function co ðtÞ=co ð0Þ for g ¼ 1 and various values of a: 0.5 (curve 1), 1.0 (curve 2), and 1.5 (curve 3).

Barkai–Silbey model are also governed by Eq. (250), we have the same result, Eq. (252), as the Metzler–Klafter model, namely, co ðtÞ ¼ ðkB T=IÞEa ½ðt=tÞa =g

ð264Þ

This is simply the result of Barkai and Silbey [30] for the translational velocity correlation function h_xð0Þ_xðtÞi, where the translational quantities are replaced by rotational ones. For a > 1, the AVCF exhibits oscillations (see Fig. 25), which is consistent with the large excess absorption occurring at high frequencies. We remark that both the Barkai–Silbey and Metzler–Klafter generalizations of the Klein–Kramers equation yield identical results for the AVCF in the absence of the external potential due to the decoupling of the velocity and phase space. It thus appears, unlike the fractional kinetic equation of Section IV.A, namely Eq. (235), that the Barkai–Silbey [30] kinetic equation, Eq. (253), can provide a physically acceptable description of the high-frequency dielectric absorption behavior of an assembly of ﬁxed axis rotators. The explanation of this appears to be the fact that in the equation proposed by Barkai and Silbey, the form of the Boltzmann equation, for the single-particle distribution function, is preserved; that is, the memory function of which the fractional derivative is an example does not affect the Liouville terms in the kinetic equation. Exactly the same conclusions apply to an assembly of rotators, which may rotate in space.

380

william t. coffey, yuri p. kalmykov and sergey v. titov C.

Inertial Effects in Anomalous Dielectric Relaxation of Linear and Symmetrical Top Molecules

Although the ﬁxed-axis rotator model considered above reproduces the principal features of dielectric relaxation of an ensemble of dipolar molecules and allows one considerable mathematical simpliﬁcation of the problem, this model may only be used for the qualitative evaluation of dielectric susceptibility only [8]. The quantitative theory of dielectric relaxation requires an analysis of molecular reorientations in three dimensions. Here we shall generalize the results given above and demonstrate how the analogous FKKE pertaining to rotation in space may also be solved to yield the complex dielectric susceptibility in terms of continued fractions, thus extending the results of Sack [40] (originally given for normal rotational diffusion in space including inertial effects) to fractional dynamics. The after effect solution for the dynamic Kerr effect [59] may be treated in analogous fashion and is also presented. 1.

Rotators in Space

We consider the rotational motion of a thin rod, or rotator, representing the linear polar molecule, which is subjected to an external electric ﬁeld F [8,41]. We assume that the ﬁeld F is parallel to the Z axis of the laboratory coordinate system OXYZ. In the molecular coordinate system oxyz rigidly connected to the rotator, the components of the angular velocity x of the rotator and of the torques K produced by the ﬁeld F are [41] _ j_ sin #; j_ cos #Þ; x ¼ ðox ; oy ; oz Þ ¼ ð#;

K ¼ ðmF sin #; 0; 0Þ

where #ðtÞ and jðtÞ are the polar and azimuthal angles, respectively, and l is the dipole moment of a rotator. In order to describe the fractional rotational diffusion, we use the FKKE for the evolution of the probability density function W in conﬁguration angular-velocity space for linear molecules in the same form as for ﬁxed-axis rotators—that is, the form of the FKKE suggested by Barkai and Silbey [30] for one-dimensional translational Brownian motion. For rotators in space, the FKKE becomes qW qW qW qW mF qW þ ox þ oy cot # oy sin # ox ¼ 0 D1a LFP W t qt qy qox qoy I qox ð265Þ where 1a LFP W ¼ 0 Dt1a b 0 Dt

q kB T qW q kB T qW ox W þ oy W þ þ qox I qox qoy I qoy ð266Þ

381

fractional rotational diffusion

is the fractional Fokker–Planck operator, b ¼ =I, is the viscous damping coefﬁcient of a dipole, I is the moment of inertia of the rotator about the axis of rotation, ¼ t1a , t is the intertrapping time scale that we identify with the Debye relaxation time for linear molecules, and a is the exponent characterizing the anomalous diffusion process. For a ¼ 1, the fractional Fokker–Planck operator of Eq. (266) reduces to that corresponding to normal inertia corrected rotational diffusion considered by Sack [40] and McConnell [41]. Just as for a ¼ 1, Eq. (265) is independent of the azimuthal angle j and the z-component of the angular velocity oz so that one may now ignore the dependence of W on j and oz . Let us suppose that the uniform ﬁeld F, having been applied to the assembly of dipoles at a time t ¼ 1 so that equilibrium conditions prevail by the time t ¼ 0, is switched off at t ¼ 0. In addition, it is supposed that the ﬁeld is weak (i.e., mF kB T, which is the linear response condition). We seek a solution of the FKKE, Eq. (265), for the case F ¼ 0 at t > 0 by using the method of separation of variables in the form of the series Wð#; ox ; oy ; tÞ ¼ Z2 eZ

2

ðo2x þo2y Þ

1 X l X 1 X

l;m m m al;m n cn ðtÞsn ðox ; oy ÞPl ðcos #Þ

l¼0 m¼0 n¼0

ð267Þ where m m cl;m n ðtÞ ¼ hsn ðox ; oy ÞPl ðcos #Þi

ð268Þ

are the associated Legendre functions of order l ðm lÞ [51], Pm l ðzÞpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z ¼ I=ð2kB TÞ, and the angular brackets h i denote ensemble averages over the distribution function W, namely, hAiðtÞ ¼

1 ð

1 ð 2p ð

A ð#; ox ; oy Þ Wð#; ox ; oy ; tÞ sin #d#dox doy

ð269Þ

1 1 0

The orthogonal functions sm n ðox ; oy Þ are given in terms of ﬁnite series of products of Hermite polynomials Hn [51] in the components of the angular velocity as s2mþM ðox ; oy Þ ¼ n

n X r2mþM ðn; qÞ q¼0

q!ðn qÞ!

H2n2qþM ðZox ÞH2q ðZoy Þ;

M ¼ 0; 1

ð270Þ

where the coefﬁcients r2mþM ðn; qÞ can be determined from the following

382

william t. coffey, yuri p. kalmykov and sergey v. titov

recurrence relations:

1 r2m ðn;qÞ ¼ n q þ 2

2q þ 1 2q þ 1 r2m1 ðn;q þ 1Þ 1 r2m1 ðn;qÞ þ ðn qÞ 2m 1 2m 1

r2mþ1 ðn;qÞ ¼ ð1 þ q=mÞr2m ðn;qÞ ðq=mÞr2m ðn;q 1Þ with r0 ðn; qÞ ¼ r1 ðn; qÞ ¼ 1 (e.g., r2 ¼ n 2q, r3 ¼ n 4q, r4 ¼ nðn 1Þ 8qðn qÞ, etc.). The normalizing constants am n are obtained by means of the orthogonality properties of the Pm l ðcos #Þ and Hn ðzÞ, namely [51], ðp

0

m Pm l ðcos #ÞPl0 ðcos #Þ sin #d# ¼

0 1 ð

Hn ðxÞHn0 ðxÞex dx ¼ 2

2ðl þ mÞ! dl;l0 dm;m0 ð2l þ 1Þðl mÞ!

pﬃﬃﬃ n p2 n!dn;n0

ð271Þ

ð272Þ

1

so that 1 ðal;m ¼ n Þ

ðl þ mÞ!bm n pð2l þ 1Þðl mÞ!24n1

where b0n ¼ 1, b1n ¼ 1=½2ðn þ 1Þ, b2n ¼ 1=n2 , . . . . Our separation of the variables procedure based on Eq. (267) differs from that introduced by Sack [40] and allows us to solve the FKKE in a simpler manner. Substituting Eq. (267) into Eq. (265), taking the inner product, and utilizing the orthogonal properties and known recurrence relations [51] for the associated Legendre functions Pm l ðcos #Þ and the Hermite polynomials Hn ðzÞ then yields the inﬁnite hierarchy of differential recurrence relations for the cl;m n ðtÞ governing the orientational relaxation of the system, namely, d l;0 1 l;1 l;1 c ðtÞ ¼ 2nZ0 D1a t1a bcl;0 t n ðtÞ þ cn ðtÞ þ 2cn1 ðtÞ dt n 2 d l;2 ðtÞ ¼ ð2n þ 1ÞZ0 Dt1a t1a bcl;1 Z cl;1 n ðtÞ þ cn ðtÞ dt n 1 1 l;0 l;0 c ðtÞ lðl þ 1Þðn þ 1Þ c ðtÞ þ ðtÞ þ cl;2 n 4 nþ1 4 nþ1 d ðl þ 2Þðl 1Þn l;1 1a 1a cn ðtÞ t bcl;2 Z cl;2 n ðtÞ ¼ 2nZ0 Dt n ðtÞ dt 4 1 l;3 l;3 ðl þ 2Þðl 1Þðn þ 1Þcl;1 n1 ðtÞ þ cn1 ðtÞ þ cn ðtÞ 4 Z

and so on.

ð273Þ

ð274Þ

ð275Þ

383

fractional rotational diffusion

In dielectric relaxation l ¼ 1 so that by taking the Laplace transform of Eqs. (273)–(275) over the time variables and noting the generalized integral theorem for Laplace transforms, we then have a system of algebraic recurrence relations for the Laplace transform of c1;m n ðtÞ (m = 0, 1) governing the dielectric response, namely, 1;0 c1;1 c1;1 ½Zs þ 2nx~c1;0 n1 ðsÞ ~ n ðsÞ 2~ n ðsÞ=2 ¼ dn;0 Zc0 ð0Þ

½Zs þ ð2n þ

1Þx~c1;1 n ðsÞ

þ 2ðn þ

1Þ½~c1;0 n ðsÞ

þ ~c1;0 nþ1 ðsÞ=4

ð276Þ ¼0

ð277Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Here x ¼ b0 ðtsÞ1a =2, b0 ¼ bZ ¼ = 2IkB T is the inertial effects parameter 0 (large b corresponds to small inertial effects and vice versa), and we have noted 1;0 that all the c1;0 n ð0Þ vanish with the exception n ¼ 0, namely, c0 ð0Þ ¼ x=3, where x ¼ mF=ðkB TÞ. The last equality follows from the linearized initial (at t ¼ 0) distribution function, which has the Maxwell–Boltzmann form Wð#; ox ; oy ; 0Þ ¼

1 2 Z2 ðo2x þo2x Þ x2 Z e 1 þ x cos # þ cos2 # þ oðx2 Þ ð278Þ 2p 2

The solution of Eqs. (276) and (277) can obtained as a scalar continued fraction as follows. First of all, Eqs. (276) and (277) can be rearranged to yield 1;0 þ 1;0 ðZs qn Þ~c1;0 cnþ1 ðsÞ q c1;0 n ðsÞ qn ~ n~ n1 ðsÞ ¼ dn;0 Zc0 ð0Þ

ð279Þ

where qn ¼

n ðn þ 1Þ 2nx ð2n 1Þx þ Zs ð2n þ 1Þx þ Zs qþ n ¼

ðn þ 1Þ=4 ð2n þ 1Þx þ Zs

q n ¼

4n ð2n 1Þx þ Zs

Equation (279) can now be solved using the continued fraction ~c1;0 0 ðsÞ ¼

Zs q0

Zc01;0 ð0Þ qþ 0 q1 Zs q1

ð280Þ qþ 1 q2

. Zs q2 . .

384

william t. coffey, yuri p. kalmykov and sergey v. titov

Equation (280) can be further rearranged to yield ~c1;0 n ðsÞ c1;0 0 ð0Þ

Z

¼

ð281Þ

1

Zs þ

1

x þ Zs þ

2

2x þ Zs þ

2

3x þ Zs þ 4x þ Zs þ

3 5x þ Zs þ

g Having determined ~c1;0 0 ðsÞ ¼ hP1 ðcos #ÞiðsÞ, one can calculate the normalized ^ðoÞ which is given by linear response theory as [7] complex susceptibility w ^ðoÞ ¼ wðoÞ=w0 ¼ w ^0 ðoÞ i^ w w00 ðoÞ ¼ 1 io

~c1;0 ðioÞ ~c1;0 0 ð0Þ

ð282Þ

1;0 The static susceptibility w0 is given by Eq. (84). Here, the quantity ~c1;0 0 ðioÞ=c0 ð0Þ coincides with the one-sided Fourier transform of the normalized dipole autocorrelation function C1 ðtÞ ¼ hcos #ð0Þ cos #ðtÞi0 , namely,

~c1;0 0 ðioÞ c1;0 0 ð0Þ

¼

1 ð

C1 ðtÞeiot dt

ð283Þ

0

The zero on the angular braces denotes that the ensemble average is taken in the absence of the ﬁeld. The approach developed may also be extended to treat all the other averages hPn ðcos #ÞiðtÞ characterizing orientational relaxation in ﬂuids [43]. In particular, the evaluation of the average of the second-order Legendre polynomial hP2 ðcos #ÞiðtÞ (e.g., this quantity describes the dynamic Kerr effect [8]) is given in Appendix III. The inﬁnite continued fraction, Eq. (281), is very convenient for the purpose of calculations so that the complex dielectric susceptibility, Eq. (282), can be readily evaluated for all values of the model parameters Z, b0 , and a. For a ¼ 1, the anomalous rotational diffusion solution, Eq. (281), coincides with that of Sack [40] for normal rotational diffusion. Moreover, in a few particular cases, Eqs. (281) and (282) can be considerably simpliﬁed. In the free rotation limit ð ¼ 0Þ, which corresponds to the continued fraction [Eq. (281)] evaluated at x ¼ 0, that fraction can be expressed (just as for normal rotational diffusion [40]) in terms of the exponential integral function E1 ðzÞ [51] so that the normalized complex susceptibility is ^ðoÞ ¼ 1 þ Z2 o2 eZ w

2

o2

E1 ðZ2 o2 Þ

ð284Þ

385

fractional rotational diffusion

Furthermore, just as in the one-degree-of-freedom ﬁxed-axis rotation model, in the high damping limit ðb0 1Þ, Eq. (282) can be simpliﬁed yielding the generalization to fractional dynamics of the Rocard [44] equation, namely, ^ðoÞ ¼ w

1

ð285Þ

1 þ ðiotÞs ðoZÞ2

where s ¼ 2 a. On neglecting inertial effects (Z ! 0), Eq. (285) gives Eq. (9). ^00 ðoÞ and absorption o^ Dielectric loss w w00 ðoÞ spectra for various values of a 0 ^0 ðoÞ] is w00 ðoÞ versus w and b are shown in Figs. 26–29. The Cole–Cole plot [^ presented in Fig. 20. It is apparent that the half-width and the shape of dielectric spectra strongly depend on both a (which in the present context pertains to anomalous diffusion in velocity space) and b0 (which characterizes the effects of molecular inertia). In the high damping limit ðb0 1Þ and for a > 1 corresponding to s < 1 (subdiffusion in conﬁguration space), the low-frequency part ^00 ðoÞ may be approximated by the modiﬁed Debye equation, Eq. (9). On the of w ^00 ðoÞ is entirely determined by the other hand, the high-frequency behavior of w 0 inertia of system. For a given value of b , the inertial effects become more pronounced when a ! 2 (see Fig. 29). Just as in Brownian dynamics, it is ^00 ðoÞ at high apparent that inertial effects produce a much more rapid fall-off of w frequencies. As before, the fractional needle model satisﬁes the Gordon sum rule for the dipole integral absorption of rotators in space, Eq. (86). 1

10

β' =5

1,1' : α = 0.5 2,2' : α = 1.0 3,3' : α = 1.5

3

10−1 χ^''(ω)

3' 2

10−3

2'

1

1' −5

10

10−6

10−4

10−2 ηω

100

102

Figure 26. Dielectric loss spectra w^00 ðoÞ for b0 ¼ 5 and various values of a: a ¼ 0:5 (curves 1 and 10 ), 1 (curves 2 and 20 ), and 1.5 (curves 3 and 30 ). Solid lines (1, 2, and 3): Eqs. (281) and (282); crosses (10 , 20 , and 30 ): Eq. (9) with s ¼ 2 a.

386

william t. coffey, yuri p. kalmykov and sergey v. titov 10 2

1,1': β' = 5000 2,2': β' = 500 3,3': β' = 50 4,4': β' = 5 5,5': β' = 0.5

10 0 −2

ηωχ^''(ω)

10

α = 0.5

5'

10−4

3,3'

4'

2,2'

4 5

1,1'

10−6 10−8 10−10 −6

10

−4

−2

10

10

0

10

ηω

10

2

Dielectric absorption spectra o^ w00 ðoÞ for a ¼ 0:5 and various values of b0 : 5000 0 0 (curves 1 and 1 ), 500 (curves 2 and 2 ), 50 (curves 3 and 30 ), 5 (curves 4 and 40 ), and 0.5 (curves 5 and 50 ). Solid lines (1, 2, 3, 4, and 5): Eqs. (281) and (282); crosses (10 , 20 , 30 , 40 , and 50 ): Eq. (9) with s ¼ 2 a.

Figure 27.

4'

5'

α = 1.5

1,1': β ' = 5000 2,2': β ' = 500 3,3': β ' = 50 4,4': β ' = 5 5,5': β ' = 0.5

−1

ηωχ^''(ω)

10

3' 2' 1'

1 −3

10

5

−4

10

−2

10 Figure 28.

0

ηω

10

The same as in Fig. 27 for a ¼ 1:5.

4

3

2

2

10

387

fractional rotational diffusion

0.5

1 - α = 1.00 2 - α = 1.25 3 - α = 1.50 4 - α = 1.60

β ' = 50 1

−Im[χ^(ω)]

2

3 4

0.0 0

1

Re[χ^ (ω)]

Figure 29. Cole–Cole plots for b0 ¼ 50 and various values of a: a ¼ 1 (curve 1), 1.25 (curve 2), 1.5 (curve 3), and 1.6 (curve 4). Solid lines (1, 2, 3, and 4): Eqs. (281) and (282); symbols: Eq. (9) with s ¼ 2 a.

The behavior of the dielectric spectra for the two-rotational-degreeof-freedom (needle) model is similar but not identical to that for ﬁxed-axis rotators (one-rotational-degree-of-freedom model). Here, the two- and onerotational-degree-of-freedom models (fractional or normal) can predict dielectric parameters, which may considerably differ from each other. The differences in the results predicted by these two models are summarized in Table I. It is apparent that the model of rotational Brownian motion of a ﬁxed-axis rotator treated in Section IV.B only qualitatively reproduces the principal features (return to optical transparency, etc.) of dielectric relaxation of dipolar molecules in space; for example, the dielectric relaxation time obtained in the context of these models differs by a factor 2. TABLE I. Comparison of the Results for Fixed-Axis Rotators and Rotators in Space

Characteristic relaxation time Static susceptibility Generalized Rocard equation Gordon’s sum rule

Fixed-Axis Rotators

Rotators in Space

¼ =ðkB TÞ ¼ 0 0 ¼ 2 N0 =ð2kB TÞ ð!Þ ^ ¼ 1þði!Þ12ð!Þ2

¼ =ð2kB TÞ ¼ 0 =2 0 ¼ 2 N0 =ð3kB TÞ ð!Þ ^ ¼ 1þði!Þ1 ð!Þ2

1 Ð 0

Dielectric loss at ¼ 0 (free rotation limit)

!00 ð!Þd! ¼ N4I0

^00FR ð!Þ ¼

2

pﬃﬃﬃ 2 2 !e !

1 Ð 0

!00 ð!Þd! ¼ N3I0

2

^00FR ð!Þ ¼ 2 !2 e

!

2 2

388

william t. coffey, yuri p. kalmykov and sergey v. titov

We remark that the advantage of using the continued fraction method is that solutions for the complex susceptibility may be easily obtained, to any desired degree of accuracy by elementary algebraic manipulation without using special functions. We further remark that the continued fraction solutions that we have given, with a few elementary modiﬁcations, also yield the Laplace transform of the characteristic function of the conﬁguration space distribution function including inertial effects. Thus all desired statistical averages such as the mean-square angular displacement, and so on, may be simply calculated by differentiation. 2.

Symmetric Top Molecules

The approach for rigid rotators proposed above can be extended [90] to the orientational relaxation of an assembly of dipolar nonpolarizable symmetrical top molecules undergoing fractional diffusion in space (treated originally by McConnell [41], Morita [91], and Coffey et al. [8,92] for normal diffusion). The rotational Brownian motion of a symmetric top molecule in the molecular coordinate system oxyz rigidly connected to the top is characterized by the angular velocity o and the angular momentum M deﬁned as [41] _ j_ sin #; c_ þ j_ cos #Þ and o ¼ ðox ; oy ; oz Þ ¼ ð#;

M ¼ ðIox ; Ioy ; Iz oz Þ

where I and Iz are the moments of inertia about the axis of symmetry and about an axis perpendicular to that axis, respectively, and #, j, and c are the Euler angles (# is the angle between the axis of symmetry of the molecule and the Z axis of the laboratory coordinate system, j is the azimuthal angle, and c is the angle characterizing rotation about the axis of symmetry). In order to describe the fractional rotational diffusion, we use the FKKE for the evolution of the probability density function W in conﬁguration–angular-velocity space for symmetrical top molecules in the same form as for linear molecules. For symmetric top molecules, the FKKE becomes (in the absence of external ﬁelds) [90] qW qW Iz qW qW þ ox þ oy cot # oz ox oy qt q# qox qoy I q kB T qW ¼ t1a 0 Dt1a b ox W þ qox I qox q kB T qW q kB T qW þb oy W þ oz W þ þ bz qoy I qoy qoz Iz qoz

ð286Þ

where b ¼ =I, bz ¼ z =Iz , and z are the viscous damping coefﬁcients, and t is again an intertrapping time scale that we identify with the Debye relaxation time for normal diffusion of symmetrical top molecules given by Eq. (81). For a ¼ 1,

389

fractional rotational diffusion

Eq. (4) reduces to the corresponding normal Fokker–Planck equation for inertiacorrected rotational diffusion considered, for example, in Ref. 91. Let us suppose that the uniform electric ﬁeld F (having been applied to the assembly of polar nonpolarizable symmetric top molecules at a time t ¼ 1 so that equilibrium conditions prevail by the time t ¼ 0) is switched off at t ¼ 0. In addition, it is supposed that the ﬁeld is weak (i.e., mF kB T, which is the linear response condition). For t > 0, the evolution of W satisﬁes Eq. (265). Just as a ¼ 1, Eq. (4) is independent of the angles j and c so that for the problem in question one may ignore the dependence of W on j and c. Thus, we seek a solution of Eq. (265) by using the method of separation of variables in the form of the series Wð#; ox ; oy ; oz ; tÞ ¼ Zz Z2 eZ

1 X l X 1 X 1 X

2

ðo2x þo2y ÞZ2z o2z jmj

l;m m al;m n;k bn;k ðtÞsn;k ðox ; oy ; oz ÞPl ðcos #Þ

ð287Þ

jmj

ð288Þ

l¼0 m¼l n¼0 k¼0

where m bl;m n;k ðtÞ ¼ hPl ðcos #Þsn;k ðox ; oy ; oz Þi jmj

Pl ðcos #Þ are the associated Legendre functions [51], and the functions sm n;k ðox ; oy ; oz Þ ðl; n; k ¼ 0; 1; 2; . . . ; l m lÞ are expressed as ﬁnite series of products of Hermite polynomials Hn ðzÞ in the components ox ; oy , and oz of the angular velocity, namely, ðox ; oy ; oz Þ ¼ Hk ðZz oz Þ s2mM n;k

n X r2mM ðn; qÞ q¼0

q!ðn qÞ!

H2n2qþMem ðZox ÞH2qþem ðZoy Þ

ð289Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Here Z ¼ I=ð2kB TÞ, Zz ¼ Iz =ð2kB TÞ, em ¼ 0 for m 0, em ¼ 1 for m < 0, M ¼ 0 or 1, and the coefﬁcients r2mþM ðn; qÞ of the ﬁnite series are determined by the recurrence relations [92] 1 2q þ 1 2q þ 1 r2p1 ðn;q þ 1Þ r2p ðn;qÞ ¼ n q þ 1 r2p1 ðn;qÞþðn qÞ 2 2p 1 2p 1 q q r2pþ1 ðn;qÞ ¼ 1 þ r2p ðn;qÞ r2p ðn;q 1Þ p p 2qþ 2 2q þ 3 rð2p1Þ ðn;q þ 1Þ r2p ðn;qÞ ¼ ðn qÞ 1 rð2p1Þ ðn;qÞ þ 2p 1 2p 1 2q þ 1 q 1 rð2pþ1Þ ðn;qÞ ¼ 1 þ 2 r2p ðn;qÞ r2p ðn;q 1Þ 2p 2p n q þ 1

390

william t. coffey, yuri p. kalmykov and sergey v. titov

with r0 ðn; qÞ ¼ r1 ðn; qÞ ¼ 1 and p 0. The above recurrence relations and the orthogonality of the Hermite polynomials ensure that the functions sm n;k ðox ; oy ; oz Þ are also orthogonal, namely, 1 ð

1 ð

1 ð

0

m Z ðox þoy ÞZz oz sm dox doy doz n;k ðox ; oy ; oz Þsn0 ;k0 ðox ; oy ; oz Þe 2

2

2

2

2

1 1 1

dn;n0 dm;m0 dk;k0 and that they form a complete set in angular velocity space. The angular brackets h i denote ensemble averages over the distribution function W, namely, hAiðtÞ ¼

1 ð

1 ð

1 ð ðp

A ð#; ox ; oy ; oz Þ Wð#; ox ; oy ; oz ; tÞ sin #d#dox doy doz

1 1 1 0

ð290Þ The normalizing constants al;m n;k are obtained by means of the orthogonality properties of the Pm l ðcos #Þ and Hn ðzÞ, Eqs. (271) and (272), so that Þ1 ¼ ðal;2mM n;k

n p3=2 22nþkþMþ1 k!ðl þ mÞ! X r2mM ðn; qÞ 2 ð2l þ 1Þðl mÞ! q!ðn qÞ! q¼0

ð291Þ

ð2n 2q þ M em Þ!ð2q þ em Þ! Substituting Eq. (267) into Eq. (265), taking the inner product, and utilizing the orthogonality properties and known recurrence relations [51] for the associated Legendre functions and the Hermite polynomials then yields the inﬁnite hierarchy of differential recurrence relations for the bl;m n;k ðtÞ governing the orientational relaxation of the system, namely, d l;0 1 l;1 l;1 bn;k ¼ ½2nb0 þ kb0z =Bt1a 0 D1a bl;0 ð292Þ t n;k þ bn;k þ 2bn1;k dt 2 d 1 l;2 0 0 1a 1a l;1 bn;k þ bl;2 Z bl;1 0 Dt n;k ¼ ½ð2n þ 1Þb þ kbz =Bt n;k þ bnþ1;k dt 4 pﬃﬃﬃ 1 l;1 1 l;0 l;0 ð1 d1;1 Þlðl þ 1Þðn þ 1Þ bn;k þ bnþ1;k B bn;kþ1 þ kbl;1 n;k1 ð293Þ 4 2 d 1 l;3 0 0 1a 1a l;2 t bn;k þ bl;3 Z bl;2 n;k ¼ ½2nb þ kbz =B0 Dt n1;k þ bn;k dt 4 h pﬃﬃﬃ 1 l;2 n l;1 i l;1 ðl þ 2Þðl 1Þ ðn þ 1Þbn1;k þ bn;k 2 B bn;kþ1 þ kbl;2 ð294Þ n;k1 4 2

Z

fractional rotational diffusion

391

and so on. Here b0 ¼ Z=I, b0z ¼ Zz =I, and B ¼ Iz =I. For linear molecules (B ¼ 0, b0z =B ! 0), Eqs. (292)–(293) reduce to Eqs. (273)–(275). In dielectric relaxation, l ¼ 1 so that once again by taking the Laplace transform of Eqs. (292)–(294) over the time variables and noting the generalized integral theorem for Laplace transforms, we then have a system of algebraic recurrence relations for the Laplace transform of b1;m n;k ðtÞ (m ¼ 0,1) [so that ðtÞ hcos yiðtÞ] governing the dielectric response. These equations can be b1;0 0;0 written as a system of algebraic recurrence relations in the frequency domain using Laplace transformation, namely, 1 ~1;1 1;0 ~1;1 b1;0 ½Zs þ 2nrðsÞ þ krz ðsÞ=B~ n;k ¼ Zb0;0 ð0Þdnþk;0 þ bn;k þ 2bn1;k 2 1 ~1;0 ~1;0 b1;1 ½Zs þ ð2n þ 1ÞrðsÞ þ krz ðsÞ=B~ n;k ¼ ðn þ 1Þbnþ1;k 2ðn þ 1Þbn;k 2 1 pﬃﬃﬃ ~1;1 Bðbn;kþ1 þ 2k~b1;1 n;k1 Þ 2

1 pﬃﬃﬃ ~1;1 ~b1;1 b b1;1 ¼ þ 2k ½Zs þ ð2n þ 1ÞrðsÞ þ krz ðsÞ=B~ B n;k n;kþ1 n;k1 2

ð295Þ

ð296Þ ð297Þ

Here rðsÞ ¼ b0 ðtsÞ1a , rz ðsÞ ¼ b0z ðtsÞ1a , and we have noted that all the b1;0 n;k ð0Þ vanish with the exception of n þ k ¼ 0, namely, b1;0 0;0 ð0Þ ¼ x=3, where x ¼ mF=ðkB TÞ. This initial condition follows from the linearized initial (at t ¼ 0) distribution function, which has the Maxwell–Boltzmann form Wð#; ox ; oy ; 0Þ ¼

Zz Z2 Z2 ðo2x þo2x Þ Z2z o2z x2 2 cos e 1 þ x cos # þ # þ oðx2 Þ 2p3=2 2 ð298Þ

In order to solve the hierarchy of recurrence equations [Eqs. (295)–(297)], we ~ n ðsÞ comprising three subvectors: introduce a supercolumn vector C 1 0 1;m ~b ðsÞ 0 1 n;0 B ~b1;m ðsÞ C ~c1;0 n1 ðsÞ B n1;1 C B 1;m ~ n ðsÞ ¼ @ ~c1;1 ðsÞ C C B ~ ð299Þ ðsÞ ¼ ; c C A .. n C B n1 1;1 A @ . ~cn1 ðsÞ ~b1;m ðsÞ 0;n The subvector ~c1;m n ðsÞ has the dimension n þ 1. The three index recurrence equations [Eqs. (295)–(297)] for ~ b1;m n;k ðsÞ can then be transformed into the matrix three-term differential-recurrence equation ~ ~ n ðsÞ Qþ C ~ ½ZsI3n Qn ðsÞC n nþ1 ðsÞ Qn Cn1 ðsÞ ¼ dn;1 ZC1 ð0Þ

ð300Þ

392

william t. coffey, yuri p. kalmykov and sergey v. titov

where 0

q n

0

B Q n ¼ @0 0

C p n A

0 p n

0

0

0

0

B þ Qþ n ¼ @ qn

0

0

0

0

q0n ðsÞ

In =2

B Qn ðsÞ ¼ @ 2ðrn In Þ 1 B B0 B B. . qn ¼ 2 B B. B B0 @ 0 0 qþ n

0 1 .. . . . . 0 0

B 0 1B B ¼ B. 2 B .. @ 0

0

0 q1n ðsÞ 1 0 C 0C C .. C .C C C 1C A 0 nðn1Þ 0

n1 .. .

0 .. . . ..

0

1

0

0 B B1 B pﬃﬃﬃB . B .. p ¼ B n B B B0 @ 0 0 0 pﬃﬃﬃ B .. BB B. þ pn ¼ B 2 B0 @

C 0 A

0

n

1

0

q1n ðsÞ

0

0

1

C pþ n A

pþ n

0

1

0

0

C 0C C .. C .C A 0

0

0

.. .

0 .. .

0 .. .

n 2

0

1 .. .

0 . .. . ..

0

1

0

0

0 n1 1 0 .. C .C C C 0C A 1

1

nðnþ1Þ

1 C C C C C C C C A

nðn1Þ

nðnþ1Þ

393

fractional rotational diffusion 0 B B B rn ¼ B B @ 0 B B B B qM n ðsÞ ¼ B B B @

n1

0

0

0 .. .

n2 .. .

0

0

1

C 0C C .. C .. . .C A 0

nn

1

M ðsÞ fn1;0

0

0

0

M ðsÞ fn2;1

0

.. .

.. .

..

.. .

0

0

M f0;n1 ðsÞ

.

C C C C C C C A nn

M Here fn;k ðsÞ ¼ ð2n þ MÞrðsÞ krz ðsÞ=B and In is the unit matrix of dimension n n. The initial value vector C1 ð0Þ is

0

1 x=3 C1 ð0Þ ¼ @ 0 A 0 ~ 1 ðsÞ is then given by the and Cn ð0Þ ¼ 0 for all n 2. The exact solution for C matrix continued fraction [8] ~ 1 ðsÞ ¼ Z C

I3 ZsI3 Q1 ðsÞ Qþ 1

I6 I9

ZsI6 Q2 ðsÞ Qþ 2

..

Q 3

Q 2

C1 ð0Þ

ZsI9 Q3 ðsÞ . ð301Þ where the fraction lines denote matrix inversion. Having determined the spectrum ~ b1;0 0;0 ðioÞ from Eq. (301), one can also evaluate the normalized complex susceptibility from linear response theory as ^ðoÞ ¼ w

~ b1;0 wðoÞ 0;0 ðioÞ ¼ 1 io 1;0 w0 b0;0 ð0Þ

ð302Þ

where the static susceptibility w0 is given by Eq. (84). For linear and spherical top molecules, the solution can be considerably simpliﬁed and presented as ordinary continued fractions. For linear molecules

394

william t. coffey, yuri p. kalmykov and sergey v. titov

ðIz ¼ 0Þ, the solution has already been obtained above. For spherical tops (b0 ¼ b0z and B ¼ 1), the corresponding solution is (see Appendix IV) ~ b1;0 0;0 ðsÞ b1;0 0;0 ð0Þ

Z

¼ sZ þ

1 1 þ sZ þ rðsÞ þ 4½sZ þ 2rðsÞ

ð3 1=2Þ=2 2 sZ þ 2rðsÞ þ 1 ð5 1=3Þ=2 þ sZ þ 3rðsÞ þ 6½sZ þ 4rðsÞ sZ þ 4rðsÞ þ

ð303Þ For a ¼ 1, rðsÞ ¼ b0 and Eq. (303) coincides with that of Sack [40] for normal rotational diffusion with a corrected misprint [8]. In the high damping limit ðb0 1Þ, Eq. (303) can be simpliﬁed, yielding the generalization of the Rocard equation to fractional dynamics, Eq. (285). The results of numerical calculations indicate that the matrix continued fraction solution [Eqs. (301) and (282)] and the ordinary continued fraction solution for linear molecules, Eq. (281), and for spherical tops, Eq. (303), yield the same results. The approach we have developed may also be extended to treat all the other averages hPn ðcos #ÞiðtÞ characterizing orientational relaxation in ﬂuids [8,43], in particular, to evaluate the average of the second-order Legendre polynomial hP2 ðcos #ÞiðtÞ (this quantity describes the dynamic Kerr effect), which is given in Appendix V. The inﬁnite matrix continued fraction, Eq. (301), is easily computed so that the complex dielectric susceptibility, Eq. (302), can be readily evaluated for ^00 ðoÞ typical values of the model parameters Z, b0 , b0z , B, and a. Dielectric loss w spectra for various values of a, b0 , b0z , and B are shown in Figs. 30–34. It is apparent that the half-width and the shape of the dielectric spectra strongly depend on a (here pertaining to anomalous diffusion in velocity space), b0 , b0z (which characterize the effects of damping and molecular inertia), and B (which accounts for the shape of the molecule). For high damping, b0 , b0z 1, ^00 ðoÞ may be approximated by the Cole–Cole the low-frequency part of w equation, Eq. (9) (see Figs. 30–34). On the other hand, the high-frequency ^00 ðoÞ is entirely determined by the inertia of system. Just as in behavior of w normal Brownian dynamics, it is apparent that inertial effects produce a much ^00 ðoÞ at high frequencies. Indeed, one can again show more rapid fall-off of w that our fractional model satisﬁes the Gordon sum rule for the dipole integral absorption, Eq. (86). We remark that all the results of this section are obtained by using the Barkai–Silbey [30] fractional form of the Klein–Kramers equation for the evolution of the probability distribution function in phase space. In that equation, the fractional derivative, or memory term, acts only on the right-hand side—that is, on the diffusion or dissipative term. Thus, the form of the Liouville operator, or convective derivative, is preserved [cf. the right-hand side

395

fractional rotational diffusion 10

1

1,1' : α = 0.5 2,2' : α = 1.0 3,3' : α = 1.5 b' = b'z = 5

−1

B=1

3'

χ '' (ω )

10

3

2'

^

2 1

1'

−3

10

−5

10

10−6

10−4

10−2

ηω

100

102

Figure 30. Dielectric loss spectra w^00 ðoÞ for b0 ¼ b0z ¼ 5 and various values of a: a ¼ 0:5

(curves 1 and 10 ), a ¼ 1 (curves 2 and 20 ), and a ¼ 1:5 (curves 3 and 30 ). Solid lines (1, 2, and 3): Eqs. (301) and (302); asterisks (10 , 20 , and 30 ): Eq. (9) with s ¼ 2 a.

0,4

1

α = 1.5 β ' = β z' = 5

2 3

χ^''(ω )

0,3

1 : B = 0.1 2:B=1 3:B=2

0,2

0,1

0,0 10

−3

10

−2

−1

10 ηω

10

0

10

1

Figure 31. Dielectric loss spectra w^00 ðoÞ for a ¼ 1:5, b0 ¼ b0z ¼ 5 and various values of B: B ¼ 0:1 (curve 1), B ¼ 1 (curve 2), B ¼ 2 (curve 3); circles: Eq. (285); asterisks: Eq. (9) with s ¼ 2 a.

396

william t. coffey, yuri p. kalmykov and sergey v. titov 1

α = 0.5 β' = βz' = 5

1,2

2

1 : B = 0.1 2:B=1 3:B=2

3

χ^"(ω)

0,8

0,4

0,0 0,01

0,1 ηω

1

Figure 32. Dielectric loss spectra w^00 ðoÞ for a ¼ 0:5, b0 ¼ b0z ¼ 5 and various values of B: B ¼ 0:1 (curve 1), B ¼ 1 (curve 2), B ¼ 2 (curve 3); circles: Eq. (285); asterisks: Eq. (9) with s ¼ 2 a. 10

0

1 2 1'

−1

10

3 2'

^

χ''(ω)

4 3'

−2

10

5

1,1' : β ' =1 2,2' : β ' =10 2 3,3' : β ' =10 3 4,4' : β ' =10 4 5,5' : β ' =10

−3

10

α = 1.5 B = 1.0

−4

10

−6

10

−4

10

−2

10 ηω

4' 5'

10

0

2

10

Figure 33. Dielectric loss spectra w^00 ðoÞ for a ¼ 1:5, B ¼ 1 and various values of b0 ¼ b0z : b0 ¼ 1 (curve 1), b0 ¼ 10 (curve 2), B ¼ 100 (curve 3), b0 ¼ 103 (curve 4), and b0 ¼ 104 (curve 5). asterisks: Eq. (9) with s ¼ 2 a.

397

fractional rotational diffusion 10 1

B = 1, α = 0.5 10 0

4,4'

5,5'

3,3'

2,2'

1,1'

−1

χ^''(ω)

10

1,1': β ' = 1 2,2': β ' = 10 2 3,3': β ' = 10 3 4,4': β ' = 10 4 5,5': β ' = 10

−2

10

−3

10

−4

10

−6

10

−4

10

−2

10 ηω

0

10

10

2

Figure 34. Dielectric loss spectra w^00 ðoÞ for a ¼ 0:5, B ¼ 1 and various values of b0 ¼ b0z :

b0 ¼ 1 (curve 1), b0 ¼ 10 (curve 2), b0 ¼ 100 (curve 3), b0 ¼ 103 (curve 4) and b0 ¼ 104 (curve 5). asterisks: Eq. (9) with s ¼ 2 a.

of Eq. (286)]. Thus, Eq. (286) has the conventional form of a Boltzmann equation for the single-particle distribution function. The preservation of the Liouville operator is equivalent to stating that the Newtonian form of the equations of motion underlying the Klein–Kramers equation is preserved. Thus, the high-frequency behavior is entirely controlled by the inertia of the system and does not depend on the anomalous exponent. Consequently, the fundamental sum rule, Eq. (86), for the dipole integral absorption of single axis rotators is satisﬁed, ensuring a return to transparency at high frequencies as demanded on physical grounds. As far as comparison with experimental data is concerned, the fractional Klein–Kramers model under discussion may be suitable for the explanation of dielectric relaxation of dilute solution of polar molecules (such as CHCl3, CH3Cl, etc.) in nonpolar glassy solvents (such as decalin at low temperatures; see, e.g., Ref. 93). Here, in contrast to the normal diffusion, the model can explain qualitatively the inertia-corrected anomalous (Cole–Cole-like) dielectric relaxation behavior of such solutions at low frequencies. However, one would expect that the model is not applicable at high frequencies (in the far-infrared region), where the librational character of the rotational motion must be taken

398

william t. coffey, yuri p. kalmykov and sergey v. titov

into account. The failure of the fractional Klein–Kramers model for nonhindered (free) rotation to account for the high-frequency (Poley) absorption even though it explains the return to transparency at high frequencies is to be expected in view of the assumption made in the theory that all electrical interactions between dipoles may be neglected. D.

Inertial Effects in Anomalous Dielectric Relaxation in a Periodic Potential

It is the purpose of this section to include [26] the effect of an internal ﬁeld potential (and thus dielectric relaxation due to barrier crossing by dipoles) in the fractional inertia-corrected Brownian dynamics model considered above. In the noninertial limit, the model has been treated in Section III.A. The model can be considered as a generalization of the model for the normal Brownian motion in a cosine periodic potential to fractional dynamics (giving rise to anomalous diffusion) and also represents a generalization of Fro¨hlich’s model of relaxation over a potential barrier. The ﬁrst succesful attempts to calculate the complex susceptibility including inertial effects and a potential arising from the external ﬁeld for normal rotational diffusion in a cosine periodic potential were made by Risken and Vollmer [94] and Reid [95], who gave numerical results in a limited number of specialized cases. Only very recently, however, has it become possible to treat the calculation of the Fourier coefﬁcients in a systematic way for the conventional Brownian motion. The difﬁculty arises because when inertial effects are included, the two recurring numbers n and q always give rise to a multivariable recurrence relation. Matrix continued fractions are therefore an ideal way of solving such recurrence relations. This has been accomplished in Refs. 96 and 97, where it has been shown that the linear and nonlinear response of an assembly of ﬁxed-axis rotators in the presence of a strong spatially uniform external ﬁeld (that is a cos y potential) may be systematically solved using the matrix continued fraction method. Here we generalize the results [96,97] by including the effect of an internal ﬁeld potential (and so dielectric relaxation due to barrier crossing by dipoles) in the fractional diffusion. As in [26], our approach is based on the FKKE for the translational Brownian motion in a potential proposed by Barkai and Silbey [30]. The solution of the rotational analog of this FKKE is accomplished using the matrix continued fraction method. We illustrate by considering one of the simplest microscopic models of dielectric relaxation, namely: an assembly of rigid dipoles each of moment m each rotating about a ﬁxed axis through its center. A dipole has moment of inertia I and is speciﬁed by the angular coordinate f. The internal ﬁeld due to molecular interactions is represented by an N-fold cosine potential. V0 ðfÞ ¼ V0 cos Nf

ð304Þ

fractional rotational diffusion

399

We suppose that a uniform ﬁeld F (having been applied to the assembly of dipoles at a time t ¼ 1 so that equilibrium conditions prevail by the time t ¼ 0) is switched off at t ¼ 0. In addition, we suppose that the ﬁeld is weak (i.e., mF kB T, which is the linear response condition). For t 0 and t ! 1, the distribution functions are linearized Boltzmann distributions, namely, _

Wt0

2

eðZfÞ þxV cos Nf ð1 þ x cosðf ÞÞ 2p ð

_

2

eðZfÞ þxV cos Nf ð1 þ x cosðf ÞÞ df

ð305Þ

0

_ ¼ W0 ðf; fÞ½1 þ x cosðf Þ xhcosðf Þi0 and _ 2 þx cos Nf _ ¼ Z 1 eðZfÞ V Wt!1 W0 ðf; fÞ

ð306Þ

respectively. Here Z is the partition function, is the angle between F and the z axis in the plane zx, x¼

mF ; kB T

xV ¼

V0 kB T

ð307Þ

_ and h i0 means the equilibrium statistical averages over W0 ðf; fÞ. Our goal is to evaluate the transient relaxation of the electric polarization deﬁned as PF ðtÞ ¼ mN0 ½hcosðf ÞiðtÞ hcosðf Þi0 ¼ cos Pk ðtÞ þ sin P? ðtÞ ð308Þ where Pk ðtÞ ¼ mN0 ½hcos fiðtÞ hcos fi0

ð307Þ

P? ðtÞ ¼ mN0 ½hsin fiðtÞ hsin fi0

ð310Þ

and

are the longitudinal and transverse components of the polarization. According to linear response theory [8,60], the decay of the longitudinal and transverse components of the polarization of a system of noninteracting planar dipoles, when a small uniform external ﬁeld E is switched off at time t ¼ 0, is Pk ðtÞ ¼ cos ECk ðtÞ

ð311Þ

400

william t. coffey, yuri p. kalmykov and sergey v. titov

and P? ðtÞ ¼ sin EC? ðtÞ

ð312Þ

where [23,83] Ck ðtÞ ¼

m2 N0 ½hcos fð0Þ cos fðtÞi0 hcos fð0Þi20 kB T

ð313Þ

m2 N0 ½hsin fð0Þ sin fðtÞi0 hsin fð0Þi20 kB T

ð314Þ

and C? ðtÞ ¼

are the longitudinal and transverse relaxation functions. The longitudinal wk ðoÞ and transverse w? ðoÞ components of the complex susceptibility tensor are deﬁned by Eq. (176). By supposing that the local conﬁguration potential is uniformly distributed in a plane, we may deﬁne the averaged susceptibility wðoÞ as wðoÞ ¼ ½wk ðoÞ þ w? ðoÞ=2

ð315Þ

which yields after elementary manipulation of Eqs. (313) and (314) wðoÞ ¼ 1 io w0

1 ð

hcos fðtÞi0 eiot dt

ð316Þ

0 2

m N0 . where fðtÞ ¼ fðtÞ fð0Þ and w0 ¼ 2k BT The starting point in our calculation of wðoÞ from Eq. (316) is the FKKE for _ tÞ in the phase space ðf; fÞ _ in Barkai the probability density function Wðf; f; and Silbey’s form [30] for the one-dimensional translational Brownian motion of a particle, where, however, rotational quantities (angle f, moment of inertia I, etc.) replace translational ones (position x, mass m, etc.) so that for t > 0 [cf. Eq. (253)]

qW _ qW qW þf ðNV0 sin Nf þ mF sin fÞ qt qf Iqf_ q _ kB T q2 W 1a t b ð f WÞ þ ¼ 0 D1a t I qf_ 2 qf_

ð317Þ

Here b ¼ =I, is the damping coefﬁcient of a dipole, and t is the intertrapping time scale that we identify with the Debye relaxation time.

fractional rotational diffusion

401

We seek a solution of Eq. (317), for the step-off transient response (F ¼ 0 at t > 0) by using the method of separation of variables in the form of the Fourier series, Eq. (237). Just as in Section IV.A, for the statistical moments (correlation i½qfðtÞfð0Þ _ i0 , we have the recurrence relation functions) cn;q ðtÞ ¼ hHn ðZfðtÞÞe that is given by iq inNxV ½cnþ1;q ðtÞ þ 2ncn1;q ðtÞ þ ½cn1;qþN ðtÞ cn1;qN ðtÞ 2 2 ¼ 0 D1a t1a nb0 cn;q ðtÞ t

Z_cn;q ðtÞ þ

ð318Þ 0

where b ¼ bZ. On using the integration theorem of Laplace transformation generalized to fractional calculus, we have from Eq. (318) ½2Zs þ ng02a ðZsÞ1a ~cn;q ðsÞ þ iq½~cnþ1;q ðsÞ þ 2n~cn1;q ðsÞ ð319Þ þ inNxV ½~cn1;qþN ðsÞ ~cn1;qN ðsÞ ¼ 2Zdn;0 c0;q ð0Þ p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Here g0 ¼ t=Z ¼ 2=ðIkB TÞ is the inertial effects parameter (so that large g0 pﬃﬃﬃﬃﬃﬃﬃﬃ characterizes small inertial effects and vice versa, g0 ¼ 2=g, where g ¼ IkB T= 2 is the inertial parameter used by Sack [40]) and cn;q ð0Þ ¼ 0 for n 1 because hHn i0 ¼ 0 for n 1 for the equilibrium Maxwell–Boltzmann distribution [8]. We remark that the calculation of the longitudinal and transverse components of the complex susceptibility tensor differs only in the term c0;q ð0Þ, which must be evaluated at ¼ 0 and ¼ p=2, respectively. The calculation of the averaged susceptibility from Eqs. (315) and (316) can be carried out formally by solving Eq. (319). The complex susceptibility is then given by wðoÞ ¼ 1 io ~c0;1 ðioÞ w0

ð320Þ

As we have already mentioned, we choose as an example of an internal ﬁeld potential a double-well potential ðN ¼ 2Þ that will allow us to treat overbarrier relaxation (for N ¼ 1 corresponding to a uniform electric ﬁeld this process does not exist). In order to solve Eq. (319), we shall use matrix continued fractions [8,26]. This is accomplished as follows. We introduce the column vectors 1 0 .. 1 0 .. . C B . C B ~cn1;2 ðioÞ C B C B B ~c0;2 ðioÞ C C B ~cn1;1 ðioÞ C B C B B ~c0;1 ðioÞ C ~ ~ 1 ðoÞ ¼ B C C B ðn 2Þ ð321Þ C B ~c0;1 ðioÞ C and Cn ðoÞ ¼ B ~cn1;0 ðioÞ C C B B ~cn1;1 ðioÞ C C B ~c0;2 ðioÞ C B A @ B ~cn1;2 ðioÞ C A @ .. .. . .

402

william t. coffey, yuri p. kalmykov and sergey v. titov

Now, from the recurrence Eq. (319) we have the matrix recurrence equations ~ ~ n ðoÞ Qþ C ~ ð2iZo Qn ðoÞÞC n nþ1 ðoÞ Qn Cn1 ðoÞ ¼ 2Zdn;1 C1 ð0Þ

where

0 B B B B B B B B B 1 B B C1 ð0Þ ¼ I0 ðxV Þ B B B B B B B B B @

.. . I3 ðxV Þ 0 I2 ðxV Þ 0 I1 ðxV Þ I0 ðxV Þ 0 I1 ðxV Þ 0 I2 ðxV Þ .. .

ð322Þ

1 C C C C C C C C C C C C C C C C C C C C A

ð323Þ

and the matrices Qn ðoÞ, Qþ n , and Qn are deﬁned by

ð324Þ

..

.

Qn ðoÞ ¼ g02a ðiZoÞ1a ðn 1ÞI 1 0 .. .. .. .. .. .. . . . . C B . . C B B 2 0 0 0 0 C C B B 0 1 0 0 0 C C B C B Qþ ¼ i C B 0 0 0 0 0 n C B B 0 0 0 1 0 C C B C B B 0 0 0 0 2 C A @ .. .. .. .. . . .. . . . . . . 0 .. . .. .. .. .. .. .. . . .. . . . . B . B 0 xV 0 B xV 0 2 B B x 1 0 xV 0 0 B V B Qn ¼ 2iðn 1ÞB 0 0 xV 0 xV 0 B B 1 0 0 0 xV 0 B B B 0 x 0 0 0 2 V @ .. .. .. .. .. .. .. . . . . . . . ..

.

..

.

ð325Þ

.

1

..

.. . 0 0 0 .. .

C C C C C C C 0 C C xV C C C C x 0 V A .. . . .. . .. ð326Þ

403

fractional rotational diffusion

and I is the unit matrix of inﬁnite dimension. The exceptions are the matrices Qþ 1 and Q 2 , which are given by

.. . 2 0 xV 0 0 .. .

.. . 0 1 0 xV 0 .. .

.. .. . . 0 0 xV 0 0 xV 1 0 0 2 .. .. . .

1

.

.. . 0 0 0 2 .. .

.. . 0 0 1 0 .. .

.. . 0 0 0 0 .. .

.. .

C C C C C C C C A

.. ... . 0 0 0 0 0 0 xV 0 0 xV .. .. . .

ð327Þ

1

.

.. .. . . xV 0 0 xV 0 0 0 0 0 0 .. .. . .

.. . 0 1 0 0 .. .

..

.

..

B B B B B B Q ¼ 2i 2 B B B B @

..

.

0

.. . 2 0 0 0 .. .

..

.

..

B B B B B Qþ ¼ i 1 B B B @

..

.

0

.. .

C C C C C C ð328Þ C C C C A

Here we have taken into account the initial conditions for c0;q ð0Þ, namely, 2ðp

c0;q ð0Þ ¼ heiðq1Þf i0 ¼

eiðq1Þf exV cos Nf df

0 2p ð

¼ dq1mN

Im ðxV Þ I0 ðxV Þ

ð329Þ

exV cos Nf df

0

where the In are the modiﬁed Bessel functions of the ﬁrst kind of order n [51]. In order to prove Eq. (329), we use the relation [51] eiðq1Þf exV cos Nf ¼

1 X

Im ðxV ÞeiðmNqþ1Þf

ð330Þ

m¼1

By invoking the general method for solving the matrix recurrence Eq. (322) [8], ~ 1 ðoÞ in terms of a matrix continued we have the exact solution for the spectrum C fraction, namely, ~ 1 ðioÞ ¼ 2Z1 ðioÞC1 ð0Þ C

ð331Þ

404

william t. coffey, yuri p. kalmykov and sergey v. titov

where the matrix continued fraction n ðioÞ is deﬁned by 1 n ðioÞ ¼ ½2iZoI Qn Qþ n nþ1 ðioÞQnþ1

ð332Þ

The exact matrix continued fraction solution [Eq. (331)] we have obtained is very convenient for the purpose of computation. As far as practical calculation of the inﬁnite matrix continued fraction [Eq. (332)] is concerned, we approximate it by some matrix continued fraction of ﬁnite order (by setting þ Q n ; Qn ¼ 0 at some n ¼ N). Simultaneously, we restrict the dimensions of the þ matrices Q n , Qn , and Qn to some ﬁnite number M. Both of the numbers N and M depend on the barrier height xV and damping g0 parameters and must be chosen by taking into account the desired degree of accuracy of the calculation (with decreasing g0 and increasing xV both N and M must be increased). Having ~ 1 ðoÞ from Eqs. (331) and (332), we may evaluate the complex calculated C dielectric susceptibility wðoÞ from Eq. (320) for all values of the model parameters Z, g0 , xV , and a. The real, w0 ðoÞ, and imaginary, w00 ðoÞ, parts of the complex susceptibility for various values of a (which in the present context pertains to anomalous diffusion in velocity space), xV , (which is the barrier height parameter), and g0 (which characterizes the effects of molecular inertia; g0 ! 0 and g0 ! 1 characterize large and small inertial effects, respectively) are shown in Figs. 26–32 [the calculations were carried out for w0 ¼ 1]. For xV ¼ 0, the calculation demonstrates that the matrix continued fraction algorithm yields the same results as the exact analytic solution for this particular case, Eq. (261). The shape of the dielectric spectra strongly depends on the anomalous exponent a (Fig. 35), xV (Fig. 36–38), and g0 (Fig. 39–41). In general, three bands may appear in the dielectric loss w00 ðoÞ spectra, and the corresponding dispersion regions are visible in the spectra of w0 ðoÞ. One anomalous relaxation band dominates the low-frequency part of the spectra and is due to the slow overbarrier relaxation of the dipoles in the double-well cosine potential as identiﬁed by Fro¨hlich [10]. The characteristic frequency oR of this lowfrequency band strongly depends on the barrier height xV and the friction parameter g0 as well as on the anomalous exponent a. Regarding the barrier height dependence the frequency oR decreases exponentially as the barrier height xV is raised. This behavior occurs because the probability of escape of a dipole from one well to another over the potential barrier exponentially decreases with xV (cf. Figs. 36–38). As far as the dependence of the lowfrequency part of the spectrum for small inertial effects ðg0 > 10Þ is concerned, the frequency oR decreases as g0 increases as is apparent by inspection of curves 3–6 in Figs. 39–41. For large inertial effects ðg0 < 0:1Þ the frequency oR decreases with decreasing g0 for given values of xV for the enhanced diffusion in conﬁguration space ða < 1Þ and for normal diffusion a ¼ 1 (cf. curves 1–3 in

405

fractional rotational diffusion

−1

10

3

−χ''(ηω)

1 1 - α = 0.5 2 - α = 1.0 3 - α = 1.5

−3

10

2

ξV = 3

γ ' = 10 −5

10

10−8

10−6

10−4

10−2

100

10−2

100

ηω

1.0

χ'(ηω )

3

2

1

0.5

1 - α = 0.5 2 - α = 1.0 3 - α = 1.5 0.0 γ ' = 10

10−8

ξV = 3

10−6

10−4 ηω

Figure 35. Real and imaginary parts of the complex susceptibility wðZoÞ versus normalized frequency Zo for g0 ¼ 10 and xV ¼ 3 and various values of the fractional parameter a.

406

william t. coffey, yuri p. kalmykov and sergey v. titov 101

γ ' = 0.4

α = 1.5

−χ''(ηω)

4

2

3

1

10−1

1 - ξV = 0.1 2 - ξV = 1.0

10−3

3 - ξV = 3.0 4 - ξV = 5.0 −9

10

−7

10

−5

−3

10

ηω

−1

10

10

1

10

1.0

3

4

1

2

0.5

χ'(ηω)

1 - ξV = 0.1 2 - ξV = 1.0 3 - ξV = 3.0 4 - ξV = 5.0

0.0

γ ' = 0.4

10

−9

10

−7

α = 1.5

10

−5

ηω

10

−3

10

−1

10

1

Real and imaginary parts of the complex susceptibility wðZoÞ versus Zo for a ¼ 1:5 and g0 ¼ 0:4 and various values of the barrier height parameter xV .

Figure 36.

407

fractional rotational diffusion 1 - ξV = 0.1

10

4

3 - ξV = 3.0

2

3

1

4 - ξV = 5.0

−1

10 −χ''(ηω)

α = 1.0

γ ' = 0.4

2 - ξV = 1.0

0

−2

10

−3

10

−4

10

−7

10

−5

10

−3

−1

10 ηω

1

10

10

1.0

χ'(ηω)

4

0.5

3

2

1

1 - ξV = 0.1 2 - ξV = 1.0 3 - ξV = 3.0 4 - ξV = 5.0

0.0

α = 1.0 −7

10

−5

10

Figure 37.

−3

10 ηω

γ ' = 0.4 −1

10

1

10

The same as in Fig. 36 for a ¼ 1.

Figs. 40 and 41); for the subdiffusion ða > 1Þ, however, this frequency does not exhibit such behavior. We remark that a very high-frequency band is visible in all the ﬁgures. This band is due to the fast inertial librations of the dipoles in the potential wells. This band corresponds to the THz (far-infrared) range of frequencies and is usually associated with the Poley absorption [24]. For xV 1, the characteristic

408

william t. coffey, yuri p. kalmykov and sergey v. titov 10

0

10

−1

10

−2

10

−3

α = 0.5

3

4

−χ''(ηω)

γ ' = 0.4

1

2

1 - ξV = 0.1 2 - ξV = 1.0 3 - ξV = 3.0

10

4 - ξV = 5.0

−4 −9

−7

10

−5

10

10

−3

ηω

10

−1

1

10

1.0

10

1 2

0.5

3

χ'(ηω )

4

0.0

1 - ξV = 0.1 2 - ξV = 1.0 3 - ξV = 3.0 α = 0.5

4 - ξV = 5.0 −0.5

10

−9

10

−7

Figure 38.

γ ' = 0.4

10

−5

ηω

10

−3

The same as in Fig. 36 for a ¼ 0:5.

10

−1

10

1

409

fractional rotational diffusion 10

0

α = 1.5 −1

10

ξV = 3

6 5

−χ''(ηω)

4 3

−2

10

2

123456-

1 −3

10

γ ' = 0.01 γ ' = 0.1 γ'=1 γ ' = 10 γ ' = 100 γ ' = 1000

−4

10

10−9

10−7

10−5

10−3

ηω

1.0

10−1

101

10−1

101

1 3 5

2

4

6 χ '(ηω )

0.5 1 - γ ' = 0.01 2 - γ ' = 0.1 0.0

3- γ '=1 4 - γ ' = 10 5 - γ ' = 100 6 - γ ' = 1000

10−9

10−7

α = 1.5

10−5 ηω

ξV = 3

10−3

Figure 39. Real and imaginary parts of the complex susceptibility wðZoÞ versus Zo for a ¼ 1:5 and xV ¼ 3 and various values of the friction parameter g0.

410

william t. coffey, yuri p. kalmykov and sergey v. titov

10−1

4

5 4

5

6 −χ''(ηω)

6 1

2

10−3

123456-

10−5 10−7

3

2

3 γ ' = 0.01 γ ' = 0.1 γ'=1 γ ' = 10 γ ' = 100 γ ' = 1000

10−5

1 α =1

10−3

ξV = 3 10−1

ηω

α =1

101

ξV = 3

1.0

5

6

1

4 2

3

χ'(ηω)

0.5

0.0

–0.5 −7 10

123456-

γ ' = 0.01 γ ' = 0.1 γ'=1 γ ' = 10 γ ' = 100 γ ' = 1000

−5

10

Figure 40.

−3

10 ηω

−1

10

1

10

The same as in Fig. 39 for a ¼ 1.

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ frequency of librations oL increases as xV =I (this frequency is weakly dependent on a). As far as the behavior as a function of g0 is concerned, the amplitude of the high-frequency band decreases progressively with increasing g0 for small inertial effects g0 1, as one would intuitively expect. On the other

411

fractional rotational diffusion 5

6

100

4 2

10−2

3

−χ''(ηω)

1

10−4

1 - γ ' = 0.01 4 - γ ' = 10 2 - γ ' = 0.1 5 - γ ' = 100 6 - γ ' = 1000 3- γ'=1

10−6

10−8

10−6

α = 0.5

10−4 ηω

ξV = 3

10−2

100

1.5 α = 0.5

ξV = 3

χ'(ηω)

1.0

2

3

0.5

6

5

123456-

γ ' = 0.01 γ ' = 0.1 γ'=1 γ ' = 10 γ ' = 100 γ ' = 1000

4

1 0.0

–0.5 10−8

10−6 Figure 41.

10−4 ηω

10−2

The same as in Fig. 39 for a ¼ 0:5.

100

412

william t. coffey, yuri p. kalmykov and sergey v. titov

hand, for large inertial effects g0 1, a ﬁne structure appears in the highfrequency part of the spectra (due to resonances at high harmonic frequencies of the almost free motion in the (anharmonic) cosine potential) again in accordance with intuition. We further remark that the high-frequency ^00 ðoÞ is entirely determined by the inertia of system. ðo oL Þ behavior of w Moreover, just as in the normal Brownian dynamics, the inertial effects ^00 ðoÞ at high frequencies. It can be shown that the produce a rapid fall-off of w present fractional model satisﬁes the Gordon sum rule for the dipole integral absorption of rotators in a plane [see Eq. (85)]. For a ¼ 1, the anomalous rotational diffusion solution coincides with that for normal rotational diffusion. Finally, it is apparent that between the low-frequency and very highfrequency bands, at some values of model parameters, a third band exists in the dielectric loss spectra (see, e.g., Fig. 30). This band is due to the high-frequency relaxation modes of the dipoles in the potential wells (without crossing the potential barrier) which will always exist in the spectra even in the noninertial limit (see Section III.B). Such relaxation modes are generally termed the intrawell modes. The characteristic frequency of this band depends on the barrier height xV and the anomalous exponent a. In Fig. 42, a comparison of experimental data for 10% v/v solution of a probe molecule CH2Cl2 in glassy decalin at 110 K [98] with the theoretical dielectric loss spectrum e00 ðoÞ ðe0 e1 Þw00 ðoÞ=w0 calculated from Eqs. (320) and (331) is shown. The reduced moment of inertia Ir used in the calculation is deﬁned by Ir1 ¼ Ib1 þ Ic1, where Ib and Ic are the principal moments of inertia about molecular axes perpendicular to the principal axis a along which the dipole moment vector is directed. For the CH2Cl2 molecule, Ir ¼ 0:24 1038 g cm2 [98]. The use of the reduced moment Ir allows one to obtain the correct value for the dipolar integral absorption for twodimensional models. The phenomenological model parameters xV , g0 , and a were adjusted by using the best ﬁt of experimental data. It is known that in order to describe the low-frequency dielectric relaxation in such organic glasses, one must consider anomalous diffusion and relaxation [98]. The highfrequency Poley absorption is also observed in molecular glasses in the farinfrared region (e.g., Ref. 98). Figure 42 indicates that our generalized Fro¨hlich model explains qualitatively the main features of the whole broadband (0-THz) dielectric loss spectrum of the CH2Cl2/decalin solution in contrast to the normal diffusion in a periodic potential (curve 2), which is incapable of explaining the anomalous dielectric relaxation behavior at low frequencies. One can also see in Fig. 33 that the low-frequency part of the loss ^00 ðoÞ, which may be approximated by the Cole–Cole equation [Eq. spectrum w (9)] with s ¼ 2 a and t ¼ 1=oR , is also explained by the generalized Fro¨hlich model.

413

fractional rotational diffusion 100

ε''

10−2

3

10−4

1 2 10−6

10−8 10

1

10

3

10

5

7

10 f [Hz]

10

9

10

11

10

13

Figure 42. Broad-band dielectric loss spectrum of 10% v/v solution of probe molecule CH2Cl2 in glassy decalin at 110 K. Filled circles are the experimental data [98]. Curve 1 is the best ﬁt for the anomalous diffusion in the double-well cosine potential (a ¼ 1:5, xV ¼ 8, and g ¼ 0:003); curve 2 is the best ﬁt for the normal diffusion (a ¼ 1, xV ¼ 7, and g ¼ 0:001) in the double-well cosine potential. Dashed line (curve 3) is the Cole–Cole equation [Eq. (9)] with s ¼ 2 a.

We remark that all the above results are obtained by using the Barkai– Silbey [30] fractional form of the Klein–Kramers equation for the evolution of the probability distribution function in phase space. In that equation, the fractional derivative, or memory term, acts only on the right-hand side—that is, on the diffusion or dissipative term. Hence the form of the Liouville operator, or convective derivative, is preserved [cf. the right-hand side of Eq. (317)] so that Eq. (317) has the conventional form of a Boltzmann equation for the single-particle distribution function. We reiterate that the preservation of the Liouville operator is equivalent to stating that the Newtonian form of the equations of motion underlying the Klein–Kramers equation is preserved. Thus the high-frequency behavior is entirely controlled by the inertia of the system and does not depend on the anomalous exponent. Consequently, the fundamental sum rule, Eq. (85), for the dipole integral absorption of singleaxis rotators is satisﬁed, ensuring a return to transparency at high frequencies

414

william t. coffey, yuri p. kalmykov and sergey v. titov

as demanded on physical grounds. We also remark that a general characteristic of the systems we have treated is that they are nonlocal both in space and time and thus give rise to anomalous diffusion. The generalized Fro¨hlich model we have outlined incorporates both resonance and relaxation behavior and thus may simultaneously explain both the anomalous relaxation (low-frequency) and far infrared absorption spectra of complex dipolar systems. E.

Fractional Langevin Equation

In previous sections, we have treated anomalous relaxation in the context of the fractional Fokker–Planck equation. As far as the Langevin equation treatment of anomalous relaxation is concerned, we proceed ﬁrst by noting that Lutz [47] has introduced the following fractional Langevin equation for the translational Brownian motion in a potential V: m

d vðtÞ þ mga 0 Dt1a vðtÞ þ qx V½xðtÞ; t ¼ lðtÞ dt

ð333Þ

where vðtÞ ¼ x_ ðtÞ is the velocity of the particle, m is the mass of the particle, ga is vðtÞ and lðtÞ are, respectively, the the friction coefﬁcient, and mga 0 D1a t generalized frictional and random forces with the properties lðtÞ ¼ 0;

lðt0 ÞlðtÞ ¼

mkB Tga jt t0 ja2 ða 1Þ

ð334Þ

(the parameter a corresponds to 2 a as used in Ref. 47). The overbar means the statistical average over an ensemble of particles starting at the instant t with the same sharp values of the velocity and the position. The fractional derivative 0 D1a t in Eq. (333) has the form of a memory function so that Eq. (333) may be regarded as a generalized Langevin equation (a didactic account of the generalized Langevin equation is given by Mazo [9]): ðt d m vðtÞ þ Ka ðt t0 Þvðt0 Þ dt0 þ qx V½xðtÞ; t ¼ lðtÞ dt

ð335Þ

0

The memory function Ka ðtÞ is given (in accordance with the ﬂuctuation dissipation theorem) by Ka ðtÞ ¼

1 lð0ÞlðtÞ kB T

ð336Þ

fractional rotational diffusion

415

Lutz [47] also supposed that the random force lðtÞ is Gaussian. Equation (335) may also describe non-Gaussian processes. However, in that case, the higherorder moments lðt1 Þlðt2 Þ . . . lðtn Þ may not be expressed in terms of lðtÞ and lðt0 ÞlðtÞ. The formal exact solution of Eq. (335) for a free Brownian particle ðV ¼ 0Þ may readily be obtained using Laplace transforms [47]. We have ~vðsÞ ¼

~ lðsÞ vð0Þ 1 þ s þ s1a ga m s þ s1a ga

ð337Þ

so that, noting the Laplace transform of the Mittag–Lefﬂer function, Eq. (79), we obtain

vðtÞ ¼ vð0ÞEa ðt ga Þ þ m a

1

ðt

Ea ððt t0 Þa ga Þlðt0 Þ dt0

ð338Þ

0

and ðt

xðtÞ ¼ xð0Þ þ vðt0 Þ dt0 ¼ xð0Þ þ vð0Þ t Ea;2 ðga ta Þ 0

þ m1

ðt ðt00

ð339Þ Ea ððt t0 Þa ga Þlðt0 Þ dt0 dt00

0 0

Here we have noted that ðt

Ea ðga t0a Þ dt0 ¼ t Ea;2 ðga ta Þ

0

where Ea;b ðzÞ is the generalized Mittag–Lefﬂer function deﬁned by [89] Ea;b ðzÞ ¼ In particular, Ea;1 ðzÞ ¼ Ea ðzÞ.

1 X

zk ; ðb þ kaÞ k¼0

a; b > 0

ð340Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

One can also obtain the ﬁrst-order statistical moments. We have for the mean displacement [47] xðtÞ ¼ xð0Þ þ vð0Þt Ea;2 ðga ta Þ

ð341Þ

and for the ﬁrst moment of the velocity vðtÞ ¼ vð0ÞEa;1 ðga ta Þ

ð342Þ

Moreover, we have from Eq. (342) the equilibrium velocity correlation function cv ðtÞ [cf. Eq. (264) for the angular velocity correlation function] cv ðtÞ ¼ hvð0ÞvðtÞi0 ¼ ðkB T=mÞEa;1 ðga ta Þ

ð343Þ

since for the Maxwell–Boltzmann distribution hv2 ð0Þi0 ¼ kB T=m. Noting that x2 ðtÞ is given by ðt

x ðtÞ ¼ x ð0Þ þ 2 xðt0 Þvðt0 Þ dt0 2

2

ð344Þ

0

we may obtain from Eqs. (338) and (339) [30] 2kB T hx2 ðtÞi0 hx ð0Þi0 ¼ m 2

¼2

ðt ðt0

Ea ðga ðt0 t00 Þa Þ dt00 dt0

0 0 ðt

kB T 2kB T 2 t Ea;3 ðga ta Þ ðt t0 ÞEa ðga ðt t0 Þa Þ dt0 ¼ m m 0

ð345Þ Lutz also compared his results with those predicted by the fractional Klein– Kramers equation for the probability density function f ðx; v; tÞ in phase space for the inertia-corrected one-dimensional translational Brownian motion in a potential V of Barkai and Silbey [30], which in the present context is qf qf 1 qV qf q k B T q2 f 1a þv ¼ 0 D t ga ðvf Þ þ qt qx m qx qv qv m qv2

ð346Þ

Here x and v ¼ x_ are the position and the velocity of the particle, respectively. Lutz showed that Eqs. (343) and (345) can be obtained in the context of both

fractional rotational diffusion

417

(Langevin and Fokker–Planck) methods. However, the two methods apparently predict different equations for the second moment of the velocity, namely, 2 v2 ðtÞ ¼ ðv2 ð0Þ kB T=mÞEa;1 ðga ta Þ þ kB T=m

ð347Þ

by the Langevin method and v2 ðtÞ ¼ ðv2 ð0Þ kB T=mÞEa;1 ð2ga ta Þ þ kB T=m

ð348Þ

by the Fokker–Planck method. Likewise, each approach apparently predicts different results for all higher-order moments (e.g., vn ðtÞ for n 2). Thus, it has been concluded [47] that the fractional equations, Eqs. (346) and (333), describe fundamentally different stochastic processes, although they share striking common features. The above results are indicative of a wider problem, which will be encountered in all attempts to calculate higher-order statistical moments from the generalized Langevin equation. Namely, the fact that a knowledge of the ﬁrst two moments of the random force is insufﬁcient to calculate higher moments. In other words, the advantage conferred by Isserlis’s theorem (see Ref. 8, Chapter 1, Section 1.3) in the calculation of statistical moments for Markovian Gaussian processes is entirely lost when memory effects are taken into account. This is particularly important in the context of the averaging procedure for the construction of differential-recurrence equations from the Langevin equation, which we have used throughout the book, because it is no longer apparent how the general term of the hierarchy of the averages may be calculated. Thus, at the present time, it is not clear how our procedure may be extended to Langevin equations with a memory term and thus extended to fractional Langevin equations. Similar arguments will of course apply to fractional Fokker–Planck equations such as the Barkai–Silbey or Metzler–Klafter equations, since an analog of the Isserlis theorem is needed in order to justify truncation of the fractural Kramers– Moyal expansion. The general nature of the problems that are encountered in identifying a generalized Langevin equation with a Fokker–Planck equation possessing a memory kernel have been succinctly discussed by Mazo in Chapters 10 and 11 of Ref. 9. For example, taking as a dynamical variable the momentum p of a particle and retaining his notation, we have the generalized Langevin equation ðt dp ¼ Kðt t0 Þpðt0 Þ dt0 þ FðtÞ dt 0

ð349Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

He demonstrates that this equation cannot, in general, be identiﬁed with the Fokker–Planck equation [ f ¼ f ðx; p; tÞ] ðt qf p qf q p 0 q þ ¼ Kðt t Þ þ f ðx; p; t0 Þ dt0 qt m qx qp qp mkB T

ð350Þ

0

In view of these difﬁculties, it appears then that the best course to adopt at the present time is merely to regard the right-hand side of a fractional Klein– Kramers equation as a stosszahlansatz for the Boltzmann equation of which, for example, the Barkai–Silbey equation is simply a special case. We remark that there is nothing unusual about this hypothesis, because there are many examples of collision operators (kinetic models) in statistical mechanics, where it is possible to have a well-behaved right-hand side of the Boltzmann equation, where no corresponding Langevin equation exists, because one cannot separate the stochastic forces into systematic and random parts. Indeed, the classical theory of the Brownian motion is rather particular insofar as a Newtonian-like equation of motion (e.g., the Langevin equation) underlies the dynamical process. There exist many examples of collision operators where a dynamical (Langevin) equation is not deﬁned—for example, the Van Vleck–Weisskopf model [88], the Bhatnagar–Gross–Krook model [39,100], and so on. These kinetic models yield physically acceptable results for the observed variables (such as the complex susceptibility). Therefore, at the present stage of development, it appears to us that the best way forward is to regard kinetic equations such as the Barkai–Silbey equation, as belonging to that particular class of kinetic (collision) models which takes into account long-memory effects. Collision models are, in general, described by a Boltzmann equation such as [39,40,88,100] qf qf qv qf df þv þ ¼ qt qx qx qv dt

ð351Þ

where the right-hand side represents the disturbance of the streaming motion of the distribution function due to collisions. In particular, Sack [40] and Gross [39] (considering rotation about a ﬁxed axis) have shown, by means of an expansion of the distribution function in Fourier series in the angular variable f, have shown how differential-recurrence relations for four distinct collision mechanisms may be obtained. These, in turn, may be solved [25,49] using continued fraction methods to yield the complex susceptibility. We may summarize by stating that as far as progress using the generalized Langevin equation is concerned, the main problem is the lack of a stochastic integral formalism (analogous to the Wiener integrals for normal diffusion), which

fractional rotational diffusion

419

would allow one to calculate the statistical averages needed for the construction of the hierarchy of differential-recurrence equations from the Langevin equation. V.

CONCLUSIONS

By constructing the appropriate probability density diffusion equations, we have demonstrated how conventional Brownian motion solutions for dielectric relaxation may be generalized to fractional dynamics, thereby providing one with a reasonably well-grounded framework for treating the rotational diffusion in disordered fractal systems. Such systems are, in general, governed by a joint probability distribution for two random variables, which are, in general, vectorvalued. The two random variables in question are the waiting time and the jump length. Thus the situation is radically different from that in a discrete-time random walk where the random walker makes a step of ﬁxed mean-square length in a discrete time t so that considering a one-dimensional random walk, the only variable is the direction of the walker. Instead, in the continuous-time random walk, both the jump length and the time duration between steps are random variables. In general, it is very difﬁcult to treat such walks if these two random variables are not independent. Thus apart from the sections in which inertial effects are considered, we have for the most part concentrated on random walks where one has a Le´vy distribution of waiting times and in general a Gaussian jump length distribution so that the variance of the jump length is ﬁnite. The Le´vy distribution of jump times has a convincing physical origin insofar as it is easy to visualize such a chaotic distribution of waiting times as resulting from random activation energies that naturally arise from a chaotic distribution of potential barrier heights that is microscopic disorder. In particular, in the diffusion limit of such fractal time random walks, we have shown how probability density diffusion equations may be constructed in substantially the same manner as those pertaining to normal diffusion. The advantage of such a formulation is that one may just as in normal diffusion easily introduce an external potential. Thus, because of the simple scaling relation that exists between the eigenvalues of the fractional probability density diffusion equation and those of the normal diffusion equation, one may predict the effect of anomalous diffusion on important parameters of the relaxing system such as the Kramers escape rate, and so on. Moreover, just as in normal diffusion, one may derive simple formulas for the complex polarizability, and so on. These predict accurately the effect of anomalous diffusion on the interwell (overbarrier) and intrawell relaxation processes. The overall conclusion is that one may analyze systems governed by such fractal time random walks (of which the Cole–Cole relaxation is the most important example) in a manner almost as simple as the Debye relaxation, which arises from normal diffusion. However, one should remark that in order to obtain diffusion equations for fractal time random walks

420

william t. coffey, yuri p. kalmykov and sergey v. titov

yielding the Cole–Cole relaxation, it is necessary to truncate the fractional analog of the Kramers–Moyal expansion for normal diffusion. Such a truncation may be rigorously justiﬁed for the normal diffusion since Isserlis’s theorem allows one to express all moments of the transition probability [101] in terms of powers of the second moment. Thus one may rigorously justify for normal diffusion the truncation of the Kramers–Moyal expansion at the second moment. One cannot do this at present for the diffusion limit of fractal time random walks because of the lack of a fractional diffusion equivalent of Isserlis’s theorem. In other words, one requires the generalization of Isserlis’s theorem (which pertains to Gaussian white noise) to fractal time Le´vy processes. One of the most important consequences of this is that it is impossible at present to establish a correspondence between the diffusion equation for the fractal time random walk and a Langevin equation. Without an analog of Isserlis’s theorem, one cannot by averaging the appropriate Langevin equation over its realizations generate the same hierarchy of differential recurrence relations for the relaxation functions as that which arises from the fractional diffusion equation. In spite of these difﬁculties, however, the fractal time random walk for the most part provides a reasonable theoretical model for the Cole–Cole process. The Cole–Davidson and Havriliak–Negami processes are, however, more difﬁcult to justify because the fractional diffusion equations governing these processes are essentially generated by a purely mathematical device involving the replacement of ordinary differential operators by fractional ones as detailed in the text. The merit of such an approach is that because of the existence of a kinetic equation, albeit generated by a purely mathematical transformation, one may again incorporate a potential into systems governed by these relaxation processes. Thus one may predict the relaxation behavior in a manner similar to that used for the fractal time random walk. However, unlike in the Cole–Cole process, it is not readily apparent from the underlying physics of the problem how one may justify on physical grounds the purely mathematical replacement of the normal diffusion operator by a fractional diffusion operator in order to generate kinetic equations. We have summarized the present state of the theory when the inertial effects are ignored. Initially, in studying these we considered a diffusion equation in conﬁguration angular velocity space in which the fractional diffusion operator acts on both the Liouville and dissipative terms. Thus the resulting diffusion equation no longer has the form of a Boltzmann equation. Now allowing the fractional diffusion operator to act on the Liouville term destroys the Hamiltonian character of the noiseless motion so that Hamilton’s canonical equations no longer apply. The result of this is that at high frequencies where the relaxation behavior is controlled by the inertia of the system a nonacceptable divergence of the absorption coefﬁcient is obtained. This behavior suggests that the fractional dynamics should be included in the dissipative term only. In this way the form of the Boltzmann equation is retained. Moreover, the noiseless

fractional rotational diffusion

421

motion is still governed by Hamilton’s equations. Thus the absorption coefﬁcient no longer diverges at very high frequencies. We remark that allowing the fractional derivatives to act only on the dissipative term, causes the resulting diffusion equation (ﬁrst given by Barkai and Silbey [30]) to be nonseparable in the space and time variables. This appears to be consistent with a coupled Le´vy walk picture where any given jump length involves a time cost and vice versa. Although such a diffusion equation fully incorporates inertial effects and produces physically meaning results insofar as the Cole–Cole relaxation behavior is reproduced at low frequencies and the absorption coefﬁcient returns to zero at high frequencies, much work still remain to be done in order to provide a rigorous justiﬁcation for such inertial kinetic equations. Moreover, the question still remains how such equations may arise from generalized Langevin equations. APPENDIX I: CALCULATION OF INVERSE FOURIER TRANSFORMS For simplicity, we calculate fp ðtÞ for p ¼ 1. Commencing with Eq. (114) for p ¼ 1, we have ð ~f1 ðioÞ 1 1 1 ¼ ¼ eio t f1 ðtÞ dt f1 ð0Þ io ioð1 þ ðiotÞs Þn f1 ð0Þ 1

ðA1:1Þ

0

The inverse Laplace transformation yields f1 ðtÞ 1 ¼ f1 ð0Þ 2pi

gþi1 ð

du ut 1 e 1 u ð1 þ ðutÞs Þn

ðA1:2Þ

gi1

The part inside the brackets in Eq. (A1.2) can be rearranged as follows: 1 1 n ¼ ð1 þ ðutÞs Þ ð1 þ ðutÞs Þn ðutÞsn Using Eq. (100), we have (Ref. 51, Eqs. (6.1.22) and (15.1.1)) ð1 þ ðutÞs Þn ¼ 1 nðutÞs þ ¼

1 X ðnÞ n¼0

n!

n

nðn þ 1Þ nðn þ 1Þðn þ 2Þ ðutÞ2s ðutÞ3s þ 2! 3!

ððutÞs Þn

ðA1:3Þ

422

william t. coffey, yuri p. kalmykov and sergey v. titov

This equation is then substituted back into Eq. (A1.2) to get f1 ðtÞ 1 ¼ f1 ð0Þ 2pi

" # 1 X du ut ðnÞn sn s n e 1 ðutÞ ððutÞ Þ u n! n¼0

gþi1 ð gi1

¼1t

sn

gþi1 ð 1 X ðnÞn ðts Þn 1 du ut snsn e u 2pi u n! n¼0 gi1

¼ 1 ðt=tÞsn

1 X ðnÞ n¼0

ððt=tÞs Þn n! ð1 þ sn þ snÞ n

ðA1:4Þ

Finally f1 ðtÞ ~ s; 1 þ sn; ðt=tÞs Þ ¼ 1 ðt=tÞsn fðn; f1 ð0Þ

ðA1:5Þ

~ is a generalization of a Wright function [99] containing an extra where f Pochhammer symbol. Equation (A1.5) can be also expressed in terms of the Fox 1;1 , [102], namely, H function H1;2 f1 ðtÞ 1 ð1; 1Þ 1;1 H1;2 ¼1 ðt=tÞs ðn; 1Þ; ð0; sÞ f1 ð0Þ ðnÞ

ðA1:6Þ

For p 6¼ 1, the corresponding equation is 1 fp ðtÞ p2 ðt=tÞsn X ð1Þn ðn þ nÞðp2 ðt=tÞs Þn ¼1 fp ð0Þ ðnÞ n¼0 ð1 þ sn þ snÞn! ð1; 1Þ p2ð1nÞ 1;1 2 H1;2 p ðt=tÞs ¼1 ðnÞ ðn; 1Þ; ð0; sÞ

ðA1:7Þ

For n ¼ 1 and s ¼ 1, the function fp ðtÞ from Eq. (A1.7) reduces, respectively, to X 1 ð0; 1Þ fp ðtÞ ðp2 ðt=tÞs Þn 1;1 s 2 ¼ H1;2 p ðt=tÞ ¼ fp ð0Þ ð1 þ nsÞ ð0; 1Þ; ð0; sÞ n¼0 s

¼ Es ðp ðt=tÞ Þ 2

ðA1:8Þ

423

fractional rotational diffusion and

fp ðtÞ p2ð1nÞ 1;1 2 ð1; 1Þ ¼1 H1;2 p t=t fp ð0Þ ðnÞ ðn; 1Þ; ð0; 1Þ nX n 1 2 p ðt=tÞ ð1Þ ðn þ nÞðp2 ðt=tÞÞn p2ð1nÞ ¼1 gðn; p2 t=tÞ ¼1 ðnÞ n¼0 ð1 þ n þ nÞn! ðnÞ ðA1:9Þ the Mittag–Lefﬂer function Es ðzÞ and the incomplete Gamma function gða; zÞ are deﬁned accordingly by Eqs. (61) and (119); for p ¼ 1, the corresponding equations have been obtained in Ref. 102. Here we recalled that the Fox H m;n can be expressed as a series [48] function Hp;q m Q ðbj ðbi þ kÞBj =Bi Þ X m 1 X j¼1; j6¼i ða1 ; A1 Þ; . . . ðap ; Ap Þ m;n ¼ Hp;q z q Q ðb1 ; B1 Þ; . . . ðbq ; Bq Þ i¼1 k¼0 ð1 bj þ ðbi þ kÞBj =Bi Þ j¼1þm n Q

ð1 aj þ ðbi þ kÞAj =Bi Þ

j¼1 p Q

ðaj ðbi þ kÞAj =Bi Þ

ð1Þk zðbi þkÞ=Bi k!Bi

j¼1þn

if ¼

q P

Bj

j¼1

p P

Aj 0 and Bk ðbj þ lÞ 6¼ Bj ðbk þ sÞ; j 6¼ k; j; k ¼ 1; 2; . . . ; m;

j¼1

l; s ¼ 1; 2; . . .; and n Q ð1 aj ð1 ai þ kÞAj =Ai Þ X m 1 X j¼1; j6¼i ða1 ; A1 Þ; . . . ðap ; Ap Þ m;n Hp;q z ¼ p Q ðb1 ; B1 Þ; . . . ðbq ; Bq Þ i¼1 k¼0 ðaj þ ð1 ai þ kÞAj =Ai Þ j¼1þn m Q

ðbj þ ð1 ai þ kÞBj =Ai Þ

j¼1 q Q

ð1 bj ð1 ai þ kÞBj =Ai Þ

ð1Þk zð1ai þkÞ=Ai k!Ai

j¼1þm

if ¼

q P j¼1

Bj

p P

Aj 0 and Ak ð1 aj þ lÞ 6¼ Aj ð1 ak þ sÞ; j 6¼ k; j; k ¼

j¼1

1; 2; . . . ; n; l; s ¼ 1; 2; . . .; Ai ; Bj > 0; i ¼ 1; 2; . . . ; p; j ¼ 1; 2; . . . ; q. Nearly

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william t. coffey, yuri p. kalmykov and sergey v. titov

all special functions appearing in applied mathematics can be expressed in terms of the Fox H function [48]. Solutions of Eq. (90), which governs the Cole–Cole mechanism, have been given in Refs. 22 and 101. Here we give details of the aftereffect solution for an assembly of ﬁxed axis rotators. Thus we expand the probability density function Wðf; tÞ in the Fourier series, Eq. (105). By substituting Eq. (105) into Eq. (90), applying the Fourier transformation and using Eq. (110), we have ½ðioÞs þ p2 =ts ~fp ðoÞ ¼ ðioÞs1 fp ð0Þ or s1 ~fp ðoÞ ¼ tðiotÞ fp ð0Þ ðiotÞs þ p2

ðA1:10Þ

Noting that the Laplace transform of the Mittag–Lefﬂer function is given by Eq. (79), one has Eq. (A1.8) [3,4]. The Green function is given by Eq. (117). In like manner for the Cole–Davidson mechanism, substitution of Eq. (105) into Eq. (99) yields 1 X ð1Þn ðn nÞ 2n nn nn p t 0 Dt ½yðtÞfp ðtÞ þ fp ð0ÞyðtÞ ¼ yðtÞp2 fp ð0Þ ðn þ 1ÞðnÞ n¼0

ðA1:11Þ yðtÞ ¼ Dnn1 dðtÞ and using Eq. (110), we have Recalling that Dnn t t X 1 ð1Þn ðn nÞ 2n p2 ~fp ðoÞ fp ð0Þ p ðiotÞnn ¼ fp ð0Þ io n¼0 ðn þ 1ÞðnÞ io or ½fp ð0Þ io~fp ðoÞðp2 þ iotÞn ¼ p2 fp ð0Þ Using the known relation 1 ð

gðn; ctÞeiot dt ¼

0

one can readily obtain Eq. (A1.9).

ðnÞ ioð1 þ io=cÞn

ðA1:12Þ

425

fractional rotational diffusion

APPENDIX II: EXACT CONTINUED FRACTION SOLUTION FOR LONGITUDINAL AND TRANSVERSE RESPONSES The complex susceptibility components wg ðoÞ can be evaluated from Eq. (147) by calculation of the eigenvalues lgk for normal rotational diffusion (see Section III.C). However, wg ðoÞ may be much more effectively calculated by using the continued fraction method (see Ref. 103 for detail). Let us ﬁrst evaluate the longitudinal response. By expanding the distribution function Wð#; tÞ in a Fourier series (here W is independent of j) Wð#; tÞ ¼ W0 ð#Þ þ

1 X

ðn þ 1=2ÞPn ðcos #Þ fn ðtÞ

n¼0

one has from Eq. (172) a differential-recurrence equation just as for normal diffusion [8]: þ f_n ðtÞ ¼ ts 0 D1s ½q t n fn1 ðtÞ þ qn fn ðtÞ þ qn fnþ1 ðtÞ

ðA2:1Þ

where W0 ð#Þ ¼ expðx cos #Þ=Z is the equilibrium distribution function, the Pn ðzÞ are the Legendre polynomials [51], and fn ðtÞ ¼ hPn iðtÞ hPn i0 are the þ relaxation functions, so that Ck ðtÞ ¼ f1 ðtÞ=f1 ð0Þ and qn ; q n ; qn are deﬁned as qn ¼

nðn þ 1Þ ; 2

q n ¼

xnðn þ 1Þ ; 2ð2n þ 1Þ

qþ n ¼

xnðn þ 1Þ 2ð2n þ 1Þ

Applying the integration theorem of one-sided Fourier transformation generalized to fractional calculus, we have from Eq. (A2.1) þ~ ~ ~ iot~fn ðioÞ fn ð0Þ ¼ ðiotÞ1s ½q n fn1 ðioÞ þ qn fn ðioÞ þ qn fnþ1 ðioÞ

ðA2:2Þ

where ~f ðsÞ denotes the Laplace transform, deﬁned by Eq. (71). The three-term recurrence Eq. (A2.2) can be solved exactly for the Fourier– Laplace transform ~f1 ðioÞ in terms of ordinary continued fractions to yield s1 X 1 n ~ 2n þ 1 Y ~Sk ðioÞ ~ k ðioÞ ¼ f1 ðioÞ ¼ 2tðiotÞ C ð1Þnþ1 fn ð0Þ f1 ð0Þ nðn þ 1Þ k¼1 k x f1 ð0Þ n¼1

where the continued fraction Skn ðsÞ is deﬁned by the recurrence equation k

s 1 þ Skn ðioÞ ¼ q n ½ðiotÞ qn qn Snþ1 ðioÞ 1 x 2ðiotÞs x ~k 1þ Snþ1 ðioÞ þ ¼ 2n þ 1 nðn þ 1Þ 2n þ 1

ðA2:3Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

The initial values fn ð0Þ are evaluated just as normal diffusion [8,67]:

fn ð0Þ ¼ x1 hP1 Pn i0 hP1 i0 hPn i0 nþ1 n ¼ x1 hPnþ1 i0 þ hPn1 i0 hP1 i0 hPn i0 2n þ 1 2n þ 1

ðA2:4Þ

where hPn i0 ¼

Inþ1=2 ðxÞ I1=2 ðxÞ

ðA2:5Þ

x1 ¼ mF1 =ðkB TÞ and Ip ðzÞ is the modiﬁed Bessel function of the ﬁrst kind [51]. Here we have used the relation ð2n þ 1ÞP1 Pn ¼ ðn þ 1ÞPnþ1 þ nPn1 [51]. In particular, "

# 2 2 I5=2 ðxÞ 1 I3=2 ðxÞ 1 þ 2 f1 ð0Þ ¼ x1 ¼ x1 1 þ 2 coth2 x 3 I1=2 ðxÞ 3 I1=2 ðxÞ x The appropriate differential-recurrence equation for the transverse relaxation functions gn ðtÞ ¼ hcos jP1n ðcos #ÞiðtÞ

ðA2:6Þ

so that C? ðtÞ ¼ g1 ðtÞ=g1 ð0Þ can be obtained from Eq. (172) with V given by Eq. (175) just as for normal diffusion [8]. By expanding the distribution function W in a Fourier series (here W is dependent on j) Wð#; j; tÞ ¼ W0 ð#Þ þ

1 X l 1 X ð2l þ 1Þðl mÞ! cl;m ðtÞeimj Pm l ðcos #Þ 4p l¼0 m¼l ðl þ mÞ!

we obtain

d þ gn ðtÞ ¼ ts 0 Dt1s q n gn1 ðtÞ þ qn gn ðtÞ þ qn gnþ1 ðtÞ dt

ðA2:7Þ

where Pm n ðzÞ is the associated Legendre function [51], gn ðtÞ ¼ Re½cn;1 ðtÞ, and þ ; q qn ; q n n are deﬁned as qn ¼

nðn þ 1Þ ; 2

q n ¼

xðn þ 1Þ2 ; 2ð2n þ 1Þ

qþ n ¼

xn2 2ð2n þ 1Þ

427

fractional rotational diffusion

Just as the longitudinal response, Eq. (A2.7) can be solved exactly for the Fourier–Laplace transform ~ g1 ðioÞ in terms of ordinary continued fractions to yield s1 X 1

g1 ðioÞ 2tðiotÞ ~ ? ðioÞ ¼ ~ ¼ C g1 ð0Þ xg1 ð0Þ

ð1Þnþ1

n¼1

2n þ 1 n2 ðn

þ 1Þ

2

gn ð0Þ

n Y

S? k ðioÞ

k¼1

ðA2:8Þ where the continued fraction Skn ðsÞ is deﬁned by the following recurrence equation: q n ? ðiotÞ qn qþ n Snþ1 ðioÞ 1 s xðn þ 1Þ 2ðiotÞ xn ~S? ðioÞ þ1þ ¼ nð2n þ 1Þ nðn þ 1Þ ðn þ 1Þð2n þ 1Þ nþ1

S? n ðioÞ ¼

s

ðA2:9Þ

and the initial values gn ð0Þ are given by gn ð0Þ ¼ x1

nðn þ 1Þ nðn þ 1Þ Inþ1=2 ðxÞ hPn1 i0 hPnþ1 i0 ¼ x1 I1=2 ðxÞ 2ð2n þ 1Þ 2x

Equations (A2.3) and (A2.8) are the exact solutions of the problem. They allow one to calculate the longitudinal and transverse components of the complex susceptibility from Eqs. (176)–(178). APPENDIX III. DYNAMIC KERR-EFFECT RESPONSE: LINEAR MOLECULES The physical quantity of interest from an experimental point of view and which is appropriate to Kerr effect relaxation is the electric birefringence function K(t) deﬁned by [69] KðtÞ ¼ b2

2pN0 0 ðak a0? ÞhP2 ðcos #ÞiðtÞ n

where N0 denotes the number of molecules per unit volume, a0k and a0? are the components of the optical polarizability due to the electric ﬁeld (optical frequency) of the light beam passing through the liquid medium, and n is the mean refractive index. The coefﬁcient b2 depends on the particle depolarization factors and the dielectric susceptibility of the medium.

428

william t. coffey, yuri p. kalmykov and sergey v. titov

In the transient (step-off) Kerr-effect response, it is also possible to obtain from Eqs. (273)–(275) for l ¼ 2 the system of recurrence equations for the Laplace transforms of the corresponding relaxation functions c2;m n ðtÞ ðm ¼ 0; 1; 2Þ pertaining to that response, namely, 1 2;1 2;0 c ðsÞ 2~c2;1 ½Zs þ 2nx~c2;0 n1 ðsÞ ¼ dn;0 Zc0 ð0Þ n ðsÞ ~ 2 n ½sZ þ ð2n þ 1Þx~c2;1 n ðsÞ þ

ðA3:1Þ

3ðn þ 1Þ 2;0 ~cnþ1 ðsÞ þ 6ðn þ 1Þ~c2;0 n ðsÞ 2

1 2;2 ~c2;2 c ðsÞ ¼ 0 n ðsÞ ~ 4 nþ1

ðA3:2Þ

c2;1 c2;1 ½sZ þ 2nx~c2;2 n ðsÞ þ n~ n ðsÞ þ 4ðn þ 1Þ~ n1 ðsÞ ¼ 0

ðA3:3Þ

Here, we have taken into account that all the c2;m n ð0Þ vanish with the exception 2 n ¼ 0 and m ¼ 0, namely, c2;0 0 ð0Þ ¼ x =15. This follows from the initial Maxwell–Boltzmann distribution, Eq. (278). Just as in the dielectric response, the solution of Eqs. (A3.1)–(A3.3) for g yÞiðsÞ can be obtained in terms of an inﬁnite continued ~c2;0 ðsÞ ¼ hP2 ðcos 0

fraction, namely, ~c2;0 0 ðsÞ c2;0 0 ð0Þ

Z

¼ Zs þ

3 5 Zs þ x þ Zs þ 2x

b0 Zs þ a1

b1 Zs þ a2

b2

. Zs þ a3 . . ðA3:4Þ

where an ¼ ð2n þ 1Þx þ

4n þ 3 4n þ 5 þ 2nx þ Zs 2ðn þ 1Þx þ Zs

and bn ¼

16ðn þ 1Þðn þ 2Þ ½2ðn þ 1Þx þ Zs2

429

fractional rotational diffusion

The scalar recurrence Eqs. (A3.1)–(A3.3) may also be recast in the form of the matrix three-term recurrence relation ~ ~ n ðsÞ Qþ C ~ ðZs Qn ÞC n nþ1 ðsÞ Qn Cn1 ðsÞ ¼ d1;n ZC1 ð0Þ

ðA3:5Þ

~ 0 ðsÞ ¼ 0, where C 0

~c2;0 n1 ðsÞ

1

B C ~ n ðsÞ ¼ B ~c2;1 ðsÞ C C @ n1 A ~c2;2 n1 ðsÞ 0 B C1 ð0Þ ¼ B @

c2;0 0 ð0Þ 0 0

1

ðA3:6Þ

C C A 0

0

B B Q n ¼ @0 0 0

2

0

C 0C A

0 4n 0

1 ðA3:7Þ

0 0

0

1

B C B C Qþ n ¼ @ 3n=2 0 1=4 A 0 0 0 0 1 2xðn 1Þ 1=2 0 B C C Qn ¼ B 6n xð2n 1Þ 1 @ A 0 ðn 1Þ 2xðn 1Þ

ðA3:8Þ

ðA3:9Þ

The solution of Eq. (A3.5) is given by ~ 1 ðsÞ ¼ Z C

I ZsI Q1

Qþ 1

I ZsI Q2 Qþ 2

I ..

Q 3

Q 2

C1 ð0Þ

ZsI Q3 . ðA3:10Þ þ where the column vector C1 ð0Þ and the matrices Q n ; Qn ; and Qn are deﬁned by Eqs. (A3.6)–(A3.9).

430

william t. coffey, yuri p. kalmykov and sergey v. titov

APPENDIX IV: ORDINARY CONTINUED FRACTION SOLUTION FOR SPHERICAL TOP MOLECULES A solution of the system of moment equations (295)–(297) can be obtained in terms of ordinary continued fractions for spherical top molecules (b0 ¼ b0z , and B ¼ 1) as follows. On substituting Eqs. (295) and (297) into Eq. (296) and introducing a new quantity an;k deﬁned as an;k ¼ ~b1;1 n;2k =k!, we obtain Zs þ ð2ðn þ kÞ þ 1ÞrðsÞ þ þ

2ðn þ kÞ þ 3 2½Zs þ ð2ðn þ kÞ þ 2ÞrðsÞ

2ðn þ kÞ þ 2 an;k 2½Zs þ 2ðn þ kÞrðsÞ

2 nþ1 anþ1;k ¼ b1;0 ð0Þdnþk;0 s 0;0 4½Zs þ ð2ðn þ kÞ þ 2ÞrðsÞ

kþ1 an;kþ1 4½Zs þ ð2ðn þ kÞ þ 2ÞrðsÞ

4ðn þ 1Þ 4k 2 an1;k an;k1 Zs þ 2ðn þ kÞrðsÞ Zs þ 2ðn þ kÞrðsÞ

ðA4:1Þ

Furthermore, for the series Sm ¼

m X

ami;i

ðA4:2Þ

i¼0

where the summation is taken over the elements an;k with n þ k ¼ m, e.g., S0 ¼ a0;0 , S1 ¼ a1;0 þ a0;1 , S2 ¼ a2;0 þ a1;1 þ a0;2 , etc.), we have from Eqs. (A4.1) and (A4.2)

2m þ 3 2m þ 2 þ Zs þ ð2m þ 1ÞrðsÞ þ Sm 2½Zs þ 2ðm þ 1ÞrðsÞ 2½Zs þ 2mrðsÞ 2 mþ1 4m þ 6 Smþ1 Sm1 ¼ b1;0 ð0Þdm;0 s 0;0 4½Zs þ 2ðm þ 1ÞrðsÞ Zs þ 2mrðsÞ ðA4:3Þ

or 2 1;0 ðZs qm ÞSm qþ m Smþ1 qm Sm1 ¼ b0;0 ð0Þdm;0 s

ðA4:4Þ

fractional rotational diffusion

431

where qm ¼ ð2m þ 1ÞrðsÞ

2m þ 3 mþ1 2½Zs þ 2ðm þ 1ÞrðsÞ Zs þ 2mrðsÞ

mþ1 4½Zs þ 2ðm þ 1ÞrðsÞ 4m þ 6 q m ¼ Zs þ 2mrðsÞ qþ m ¼

ðA4:5Þ ðA4:6Þ ðA4:7Þ

The continued fraction solution of the three-term recurrence equation, Eq. (A4.4), is S0 ¼

2 Zs

Zs q0

Zb1;0 0;0 ð0Þ qþ 0 q1 Zs q1

ðA4:8Þ qþ 1 q2

. Zs q2 . .

~1;0 is then given by The quantity b 0;0 ~ b1;0 0;0 ¼

b1;0 1 0;0 ð0Þ þ S0 s 2Zs

ðA4:9Þ

Taking into account Eqs. (A4.4)–(A4.9) and noting the equality 1 1 2n þ 3 2n þ 3 ðn þ 1Þð2n þ 5Þ 1 1 2 nþ2 þ ¼ 2 n þ 2 n þ 2 2A 2A 2ðn þ 2ÞA þB Aþ A B we obtain ~ b1;0 0;0 ðsÞ b1;0 0;0 ð0Þ

¼

Z sZ þ Z2

ðA4:10Þ

where the inﬁnite continued fraction Z2 is determined by the recurrence relation Zn ¼

n=2 1 nðn þ 3Þ þ sZ þ rðsÞðn 1Þ þ ðn þ 2Þ½sZ þ nrðsÞ 2ðn þ 2Þ½sZ þ nrðsÞ þ Znþ2 ðA4:11Þ

Equations (A4.10) and (A4.11) yield Eq. (303).

432

william t. coffey, yuri p. kalmykov and sergey v. titov APPENDIX V: KERR-EFFECT RESPONSE

One may also readily derive differential-recurrence equations for the statistical moments involving the associated Legendre functions of order 2 (l ¼ 2) pertaining 2;0 to the dynamic Kerr effect, namely, b2;m n;k ðtÞ [so that b0;0 ðtÞ ¼ hP2 ðcos #ÞiðtÞ]. These equations can be written as a system of algebraic recurrence relations in the frequency domain using Laplace transformation, namely, 1 ~2;1 2;0 ~2;1 ½Zs þ 2nrðsÞ þ krz ðsÞ=B~ b2;0 n;k ¼ Zc0;0 ð0Þdnþk;0 þ bn;k þ 2bn1;k 2 ½Zs þ ð2n þ 1ÞrðsÞ þ krz ðsÞ=B~ b2;1

ðA5:1Þ

n;k

3 ~2;0 1 ~2;2 ~2;2 b2;0 ¼ ðn þ 1Þ~ nþ1;k 6ðn þ 1Þbn;k þ bnþ1;k þ bn;k 2 4 1 pﬃﬃﬃ ~2;1 ~2;1 Þ Bðbn;kþ1 þ 2kb n;k1 2 ½Zs þ ð2n þ 1ÞrðsÞ þ krz ðsÞ=B~ b2;1

ðA5:2Þ

n;k

1 2;2 1 pﬃﬃﬃ ~2;1 bnþ1;k þ ~ Bðbn;kþ1 þ 2k~ ¼ ~ b2;2 b2;1 n;k þ n;k1 Þ 4 2 ~2;2 ½Zs þ 2nrðsÞ þ krz ðsÞ=Bb n;k pﬃﬃﬃ 2;2 2;1 2;1 ¼ n~ bn;k 4ðn þ 1Þ~ bn1;k Bð~ bn;kþ1 þ 2k~b2;2 n;k1 Þ 2;2 ½Zs þ 2nrðsÞ þ krz ðsÞ=B~ bn;k

~2;1 ¼ n~ b2;1 n;k 4ðn þ 1Þbn1;k þ

ðA5:3Þ

ðA5:4Þ

pﬃﬃﬃ 2;2 Bð~ bn;kþ1 þ 2k~b2;2 n;k1 Þ

ðA5:5Þ

Then the hierarchy of equations for ~ b2;m n;k ðsÞ Eqs. (A5.1)–(A5.5) can be transformed into the matrix three-term differential-recurrence equation ~ ~ n ðsÞ Qþ C ~ ½ZsI5n Qn ðsÞC n nþ1 ðsÞ Qn Cn1 ðsÞ ¼ dn;1 ZC1 ð0Þ

ðA5:6Þ

~ ðsÞ is comprised of the ﬁve subvectors where the supercolumn vector C n 0

~c2;0 n1 ðsÞ

1

C B 2;1 B ~cn1 ðsÞ C C B C ~ n ðsÞ ¼ B C C; B ~c2;1 ðsÞ C B n1 C B 2;2 @ ~cn1 ðsÞ A ~c2;2 n1 ðsÞ

0

~b2;m ðsÞ n;0

1

C B 2;m C B ~b B n1;1 ðsÞ C 2;m C B ~cn ðsÞ ¼ B C .. C B . A @ ~b2;m ðsÞ 0;n

ðA5:7Þ

433

fractional rotational diffusion þ and the supermatrices Q n , Qn , and Qn are given by 1 0 0 q 0 0 0 n C B 0 0 C B 0 0 p n C B B 0 0 0 C Q n ¼ B 0 pn C C B 0 0 2p @ 0 v n n A

0 0

0

B þ B 3qn B B 0 Qþ ¼ n B B @ 0 0 0

v n

0

q0n ðsÞ

2p n

0

0

0

0

0 pþ n

pþ n 0

vþ n 0

0 0

0 0

0 2pþ n

In =2

0

0

C C C C C þC 2pn A 0 0 vþ n

0

B 0 In B 6ðrn In Þ q1n ðsÞ B 1 B Qn ðsÞ ¼ B 0 0 qn ðsÞ 0 B 0 0 rn 0 qn ðsÞ @ 0 0 0 rn Here the submatrices p n, submatrices v n are 0 n 0 B0 n 1 B. .. .. B. v . . n ¼ 4B . @0 0 0

0

1

1

0

C C C C C C 0 A q0n ðsÞ 0 In

M q n , rn , and qn (s) are deﬁned above, and the

1 0 0C .. C .C C 2A

0

0

1 B 0 1B vþ n ¼ @ .. 4 . 0

;

0 1 .. . 0

0 0 . .. . .. 1

nðn1Þ

1 0 0C .. C .A 0

nðnþ1Þ

~ 1 ðsÞ is then given by the The exact solution, for the Laplace transform C matrix continued fraction, namely, ~ 1 ðsÞ ¼ Z C

I5 ZsI5 Q1 Qþ 1

I10 ZsI10 Q2 Qþ 2

I15 ..

Q 3

Q 2

C1 ð0Þ

ZsI15 Q3 . ðA5:8Þ

434

william t. coffey, yuri p. kalmykov and sergey v. titov

with initial conditions 1 x2 =15 B 0 C C B C C1 ð0Þ ¼ B B 0 C @ 0 A 0 0

and

Cn ð0Þ ¼ 0

for all n 2

As an example the results of numerical calculations for linear ðIz ¼ 0Þ molecules based on the above matrix continued fraction solution have been compared with that of Ref. 67 presented in terms of ordinary continued fractions. The numerical calculations show that both matrix and ordinary continued fraction solutions yield the same results. We remark that evaluation of the Kerr effect response in the context of the fractional noninertial rotational diffusion model has been carried out by De´jardin and Jadzyn [104]. Acknowledgments The support of this work by INTAS (project 01–2341) and HEA Ireland (Programme for Research in Third Level Institutions, Nanomaterials Initiative) is gratefully acknowledged. WTC thanks the Queen’s University of Belfast for the award of a Distinguished Visitor Fellowship for the period 2002–2005. The Trinity College Dublin Trust is thanked for ﬁnancial support.

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83. J. C. Dyre and N. B. Olsen, Phys. Rev. Lett. 91, 155703 (2003). 84. A. I. Burstein and S. I. Temkin, Spectroscopy of Molecular Rotation in Gases and Liquids, Cambridge University Press, Cambridge, 1994. 85. R. Metzler, Phys. Rev. E 62, 6233 (2000). 86. R. Metzler and J. Klafter, J. Phys. Chem. B 104, 3851 (2000). 87. G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930). 88. J. H. Van Vleck and V. F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945). 89. R. Metzler and I. M. Sokolov, Europhys. Lett. 58, 482 (2001). 90. S. V. Titov, Y. P. Kalmykov, and W. T. Coffey, Phys. Rev. E 69, 031114 (2004). 91. A. Morita, J. Chem. Phys. 76, 3198 (1982). 92. W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, J. Phys A: Math. Gen. 36, 4947 (2003). 93. C. J. Reid and M. W. Evans, J. Chem. Soc., Faraday Trans. 2 75, 1369 (1979). 94. H. Risken and H. D. Vollmer, Mol. Phys. 46, 55 (1982). 95. C. J. Reid, Mol. Phys. 49, 331 (1983). 96. W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, J. Chem. Phys. 115, 9895 (2001). 97. W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, J. Chem. Phys. 120, 9199 (2004). 98. C. J. Reid and M. W. Evans, J. Chem. Soc., Faraday Trans. 2 75, 1369 (1979). 99. A. Erde´lyi, ed., Higher Transcendental Functions, Bateman Manuscript Project, Vol. 3, McGraw-Hill, New York, 1953. 100. P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511 (1954). 101. W. T. Coffey, D. S. F. Crothers, D. Holland, and S. V. Titov, J. Mol. Liquids 114, 165 (2004). 102. R. Hilfer, Phys. Rev. E 65, 061510 (2002). 103. Y. P. Kalmykov, Phys. Rev. E 70, 051106 (2004). 104. J. L. De´jardin and J. Jadzyn, J. Chem. Phys. 122, 074502 (2005).

CHAPTER 9 FUNDAMENTALS OF LE´VY FLIGHT PROCESSES ALEKSEI V. CHECHKIN and VSEVOLOD Y. GONCHAR Institute for Theoretical Physics, National Science Center, Kharkov Institute for Physics and Technology, Kharkov 61108, Ukraine JOSEPH KLAFTER School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel RALF METZLER NORDITA—Nordic Institute for Theoretical Physics, DK-2100 Copenhagen Ø, Denmark

CONTENTS I. Introduction II. Definition and Basic Properties of Le´vy Flights A. The Langevin Equation with Le´vy Noise B. Fractional Fokker–Planck Equation 1. Rescaling of the Dynamical Equations C. Starting Equations in Fourier Space III. Confinement and Multimodality A. The Stationary Quartic Cauchy Oscillator B. Power-Law Asymptotics of Stationary Solutions for c 2, and Finite Variance for c > 2 C. Proof of Nonunimodality of Stationary Solution for c > 2 D. Formal Solution of Equation (38) E. Existence of a Bifurcation Time 1. Trimodal Transient State at c > 4 2. Phase Diagrams for n-Modal States F. Consequences IV. First Passage and Arrival Time Problems for Le´vy Flights A. First Arrival Time B. Sparre Anderson Universality Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

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C. Inconsistency of Method of Images V. Barrier Crossing of a Le´vy Flight A. Starting Equations B. Brownian Motion C. Numerical Solution D. Analytical Approximation for the Cauchy Case E. Discussion VI. Dissipative Nonlinearity A. Nonlinear Friction Term B. Dynamical Equation with Le´vy Noise and Dissipative Nonlinearity C. Asymptotic Behavior D. Numerical Solution of Quadratic and Quartic Nonlinearity E. Central Part of PðV; tÞ F. Discussion VII. Summary Acknowledgements References VIII. Appendix. Numerical Solution Methods A. Numerical Solution of the Fractional Fokker–Planck Equation [Eq. (38)] via the Gru¨nwald–Letnikov Method B. Numerical Solution of the Langevin Equation [Eq. (25)]

I.

INTRODUCTION

Random processes in the physical and related sciences have a long-standing history. Beginning with the description of the haphazard motion of dust particles seen against the sunlight in a dark hallway in the astonishing work of Titus Lucretius Carus [1], followed by Jan Ingenhousz’s record of jittery motion of charcoal on an alcohol surface [2] and Robert Brown’s account of zigzag motion of pollen particles [3], made quantitative by Adolf Fick’s introduction of the diffusion equation as a model for spatial spreading of epidemic diseases [6], and culminating with Albert Einstein’s theoretical description [4] and Jean Perrin’s experiments tracing the motion of small particles of putty [5], the idea of an effective stochastic motion of a particle in a surrounding heat bath has been a triumph of the statistical approach to complex systems. This is even more true in the present Einstein year celebrating 100 years after his groundbreaking work providing our present understanding of Brownian motion. In Fig. 1, we display a collection of typical trajectories collected by Perrin. Classical Brownian motion of a particle is distinguished by the linear growth of the mean-square displacement of its position coordinate x [9–11],1 hx2 ðtÞi ’ Dt 1

Editor’s note. The inertia of the particle is ignored.

ð1Þ

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Figure 1. Random walk traces recorded by Perrin [5]: Three trajectories obtained by tracing a small grain of putty at intervals of 30sec. Using Einstein’s relation between the macroscopic gas constant and the diffusion constant, Perrin found a quite accurate result for Avogadro’s number. Reﬁned results were successively obtained by Westgren and Kappler [7,8].

and the Gaussian form

1 x2 Pðx; tÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 4Dt 4pDt

ð2Þ

of its probability density function (PDF) Pðx; tÞ to ﬁnd the particle at position x at time t. This PDF satisﬁes the diffusion equation q q2 Pðx; tÞ ¼ D 2 Pðx; tÞ qt qx

ð3Þ

for natural boundary conditions Pðjxj ! 1; tÞ ¼ 0 and d function initial condition ÐPðx; 0Þ ¼ dðxÞ. If the particle moves in an external potential x VðxÞ ¼ Fðx0 Þdx0 , the force FðxÞ it experiences enters additively into the diffusion equation, and the diffusion equation [Eq. (3)] is the particular term of the Fokker–Planck equation known as the Smoluchwski equation [11, 12] q Pðx; tÞ ¼ qt

q V 0 ðxÞ q2 þ D 2 Pðx; tÞ qx mZ qx

ð4Þ

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where m is the mass of the particle and Z the friction constant arising from existing from exchange of energy with the surrounding heat bath. This Fokker– Planck equation is a versatile instrument for the description of a stochastic process in external ﬁelds [12]. Requiring that the stationary solution deﬁned by qPðx; tÞ=qt ¼ 0 is the equilibrium distribution, V 0 ðxÞ ! V 0 ðxÞ Pst ðxÞ ¼ N exp ð5Þ ¼ N exp DmZ kB T where N is the normalization constant and kB T the thermal energy, one obtains the Einstein–Stokes relation D¼

kB T mZ

ð6Þ

for the diffusion constant. The second important relation connected with the Fokker–Planck equation [Eq. (4)] is the linear response 1 hx2 ðtÞiF¼0 hxðtÞiF0 ¼ F0 2 kB T

ð7Þ

between the ﬁrst moment (drift) in presence of a constant force F0 and the variance in absence of that force, sometimes referred to as the second Einstein relation. The Fokker–Planck equation can be obtained phenomenologically following Fick’s approach by combining the continuity equation with the constitutive equation for the probability current j, q q Pðx; tÞ ¼ jðx; tÞ; qt qx

jðx; tÞ ¼ D

q Pðx; tÞ qx

Alternatively, that equation follows from the master equation [11]2 ðn o q Wðxjx0 ÞPðx0 ; tÞ Wðx0 ; xÞPðx; tÞ dx0 Pðx; tÞ ¼ qt

ð8Þ

ð9Þ

by Taylor expansion of the transition probabilities W under speciﬁc conditions. The master equation is thus a balance equation for the ‘‘state’’ Pðx; tÞ, and as such is a representation of a Pearson random walk: The transition probabilities quantify jumps from position x0 to x and vice versa [11]. Finally, the Fokker–Planck equation emerges from the Langevin equation [13] (ignoring inertial effects)3: dxðtÞ FðxÞ ¼ þ ðtÞ dt mZ 2 3

The differential form of the Chapman–Kolmogorov equation [11]. That is, we consider the overdamped case.

ð10Þ

fundamentals of le´vy ﬂight processes

443

relating the velocity of a particle to the external force, plus an erratic, timeﬂuctuating force ðtÞ. This random force ðtÞ is supposed to represent the many small impacts on the particle by its surroundings (or heat bath); It constitutes a measure of our ignorance about the microscopic details of the ‘‘bath’’ to which the particle is coupled. On the typical scale of measurements, the Langevin description is, however, very successful. The random force ðtÞ is assumed independent of x, and it ﬂuctuates very rapidly in comparison to the variations of xðtÞ. We quantify this by writing ðtÞ ¼ 0;

ðtÞðt0 Þ ¼ dðt t0 Þ

ð11Þ

where noise strength and overbars denotes bath particle averages. d denotes the Dirac-delta function, ðtÞ is Gaussian, white noise which obeys Isserlis’s (Wick’s) theorem [13]. We will see below the differences which occur when the noise is no longer Gaussian. Gaussian diffusion is by no means ubiquitous, despite the appeal of the central limit theorem. Indeed, many systems exhibit deviations from the linear time dependence of Eq. (1). Often, a nonlinear scaling of the form [14–16] hx2 ðtÞi ’ Dta

ð12Þ

is observed, where the generalized diffusion coefﬁcient now has the dimension cm2 =seca . One distinguishes subdiffusion (0 < a < 1) and sub-ballistic, enhanced diffusion (1 < a < 2). Subdiffusive phenomena include charge carrier transport in amorphous semiconductors [17], tracer diffusion in catchments [18], or the motion of inclusions in the cytoskeleton [19], just to name a few.4 In general, subdiffusion corresponds to situations where the normal diffusion is slowed down by trapping events [21–25]. Conversely, sub-ballistic, enhanced diffusion can stem from advection among random directional motions [26,27], from trapping of a wave-like process [28], or in Knudsen diffusion [29,30,31], among others. Trapping processes in the language of continuous time random walk theory are characterized by a waiting time drawn from a waiting time distribution cðtÞ, exhibiting a long tail, cðtÞ ’ t1b , where 0 < b < 1 [14,21,32]. Now no characteristic waiting time exists; and while this process endures longer and longer, waiting times may be drawn from this cðtÞ. The nonexistence of a characteristic waiting time alters the Markovian character of normal diffusion, giving rise to slowly decaying memory effects (‘semi-Markov’ character). Among other consequences, this causes the aging effects. From a probability theory point of view, such behavior corresponds to the limiting distribution of a sum of positive, independent identically distributed random 4

An extensive overview can be found in Ref. [20].

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variables with a diverging ﬁrst moment, enforcing by the generalized central limit theorem a one-sided Le´vy stable density with characteristic function [14,33,34] 1 ð

cðuÞ ¼ LfcðtÞg

b

eut cðtÞ dt ¼ eðt=tÞ ;

0 < b < 1:

ð13Þ

0

The above relation is valid also for b ¼ 1. Indeed, in that limit, we have cðuÞ ¼ et=t , whence cðtÞ ¼ dðt tÞ. This sharp distribution of the waiting time is but one possible deﬁnition of a Markovian process. In the remainder of this review, we solely focus on processes with b ¼ 1. Apart from trapping, there also exist situations where, as far as ensemble average hi is concerned, the mean square displacement does not exist. This corresponds to a jump length distribution lðxÞ emerging from an Le´vy stable density for independent identically distributed random variables of the symmetric jump length x, whose second moment diverges. The characteristic function of this Le´vy stable density is [14,33,34] 1 ð

lðxÞeikx dx ¼ expðsa jkja Þ

lðkÞ ¼ FflðxÞg

ð14Þ

1

for 0 < a 2. For a ¼ 2, one immediately recovers a Gaussian jump length distribution with ﬁnite variance s2 . Figure 2 describes the data points for the

Figure 2. The starting point of each step from Fig. 1 is shifted to the origin. This illustrates the continuum approach of the jump length distribution if only a large number of jumps is considered [5].

fundamentals of le´vy ﬂight processes

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Figure 3. Comparison of the trajectories of a Gaussian (left) and a Le´vy (right) process, the latter with index a ¼ 1:5. While both trajectories are statistically self-similar, the Le´vy walk trajectory possesses a fractal dimension, characterizing the island structure of clusters of smaller steps, connected by a long step. Both walks are drawn for the same number of steps (7000).

jump lengths collected by Perrin, which were then ﬁtted to a Gaussian. Asymptotically for 0 < a < 2, relation (14) implies the long-tailed form lðxÞ ’ jxj1a

ð15Þ

During the random walk governed by lðxÞ with 0 < a < 2, longer and longer jump lengths occur, leading to a characteristic trajectory with fractal dimension a. Thus, processes with an underlying Le´vy stable jump length distribution are called Le´vy ﬂights [35,36]. A comparison between the trajectory of a Gaussian and a Le´vy ﬂight process is shown in Fig. 3, for the same number of steps. A distinct feature of the Le´vy ﬂight is the hierarchical clustering of the trajectory. Le´vy-ﬂight processes have been assigned to spreading of biological species [37–39], related to the high efﬁciency of a Le´vy ﬂight as a search mechanism for exactly this exchange of long jumps and local exploration [40], in contrast to the locally oversampling (in one or two dimensions) of a Gaussian process. A number of trajectories monitored for the motion of spider monkeys are displayed in Fig. 4, along with the power-law motion length distribution for individual monkeys and the entire group [41]. Le´vy ﬂights have been also used to model groundwater ﬂow [42], which exhibits Le´vy stable features, these are implicated in plasma processes [43] and other turbulent phenomena, among many others, see, for instance [16,20]. It is worthwhile noting that a diverging kinetic energy has been reported for an ion in an optical lattice [44]. Le´vy ﬂights are the central topic of this review. For a homogeneous environment the central relation of continuous time random walk theory is given by [14,45] Pðk; uÞ ¼

1 cðuÞ 1 u 1 cðuÞlðkÞ

ð16Þ

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Figure 4. Daily trajectories of adult female (a,b) and male (c) spider monkeys. In panel d, a zoom into the square of c is shown [41]. On the right, the step length distribution is demonstrated to approximately follow power-law statistics with exponent ag corresponding to Le´vy motion.

that is, the Fourier–Laplace transform of the propagator, which immediately produces in the limit ks ! 0 and ut ! 0 (i.e., long distance and long time limit, in comparison to s and t) the characteristic function Pðk; tÞ ¼ exp Djkja t

ð17Þ

with diffusion coefﬁcient D ¼ sa =t with dimensions cma =sec. That is, the PDF Pðx; tÞ of such a Le´vy ﬂight process is a Le´vy stable density. In particular, it decays like Pðx; tÞ ’ Dt=jxj1þa . Although the variance of Le´vy ﬂights diverges, one can obtain by means of rescaling of fractional moments a relation that is formally equivalent to expression (1), namely [46] hjxj i2= ’ Dt2=a

ð18Þ

where 0 < < a for convergence. This scaling relation indicates that Le´vy ﬂights are indeed move superdiffusively. We note here that instead of the decoupled jump length and waiting time distributions used in this continuous time random walk description of Le´vy ﬂights, one can introduce a coupling between lðxÞ and cðtÞ, such that long jumps invoke a higher time cost than short jumps. Such a coupling therefore introduces a ﬁnite ‘‘velocity,’’ leading to the name Le´vy walks, compare [45,47,48]. These are non-Markovian processes, which we shall not consider any further.

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Continuous time random walk processes with decoupled lðxÞ and cðtÞ can be rephrased in terms of a generalized master equation [49]. This is also true for a general external force FðxÞ, where we obtain a relation of the type q Pðx; tÞ ¼ qt

1 ð

dx 1

0

ðt

dt0 Kðx; x0 ; t t0 ÞPðx0 ; t0 Þ

ð19Þ

0

The kernel K determines the jump length dependence of the starting position x0 , as well as the waiting time. Only in the spatially homogeneous case, is Kðx;0 x; t t0 Þ ¼ Kðx x0 ; t t0 Þ [50,51]. In continuous time random walk language, one needs to replace lðxÞ by ðx; x0 Þ [52]. A convenient way to formulate a dynamical equation for a Le´vy ﬂight in an external potential is the space-fractional Fokker–Planck equation. Let us quickly review how this is established from the continuous time random walk. We will see below, how that equation also emerges from the alternative Langevin picture with Le´vy stable noise. Consider a homogeneous diffusion process, obeying relation (16). In the limit k ! 0 and u ! 0, we have lðkÞ 1 sa jkja and cðuÞ 1 ut, whence [52–55] uPðk; uÞ 1 ¼ Djkja Pðk; uÞ

ð20Þ

From the differentiation theorem of Laplace transform, L f_ ðtÞ ¼ uPðuÞ Pðt ¼ 0Þ, we infer that the left-hand side in ðx; tÞ space corresponds to qPðx; tÞ=qt, with initial condition Pðx; 0Þ ¼ dðxÞ. Similarly in the Gaussian limit a ¼ 2, the right-hand side is Dq2 Pðx; tÞ=qx2 , so that we recover the standard diffusion equation. For general a, the right-hand side deﬁnes a fractional differential operator in the Riesz–Weyl sense (see below) and we ﬁnd the fractional diffusion equation [52–56] q qa Pðx; tÞ ¼ D Pðx; tÞ qt qjxja

ð21Þ

where we interpret Ffqa gðxÞ=qjxja g ¼ jkja gðkÞ. The drift exerted by the external force FðxÞ should enter additively (as proved in Ref. 52), and we ﬁnally obtain the fractional Fokker–Planck equation for Le´vy ﬂight processes, [52,54–56] q Pðx; tÞ ¼ qt

q V 0 ðxÞ qa þD Pðx; tÞ qx mZ qjxja

ð22Þ

The fractional Fokker–Planck equation (22) which ignores inertial effects can be solved exactly for an harmonic potential (Ornstein–Uhlenbeck process),

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giving rise to the restoring Hookean force FðxÞ ¼ mo2 x. In the space of wavenumbers k, the solution is [57] i ZDjkja h ao2 t=Z Pðk; tÞ ¼ exp 1 e ð23Þ ao2 which is a Le´vy stable density with the same stable index a, but time-dependent width ZD=ðaoÞ ½1 expðao2 t=ZÞ. In particular, the stationary solution Pst ðxÞ ¼ F

1

ZDjkja exp ao2

ZD ao2 jxj1þa

ð24Þ

leads to an inﬁnite variance. Thus, although the harmonic potential introduces a linear restoring force, the process never leaves the basin of attraction of the Le´vy stable density with index a, imposed by the external noise. In particular, due to the diverging variance, the Einstein–Stokes relation and the linear response found for standard diffusion,5 no longer hold. After addressing the Langevin and fractional Fokker–Planck formulations of Le´vy ﬂight processes in some more detail, we will show that in the presence of steeper than harmonic external potentials, the situation changes drastically: The forced Le´vy process no longer leads to an Le´vy stable density but instead to a multimodal PDF with steeper asymptotics than any Le´vy stable density. Mutimodality of the PDF and a converging variance are just one result, which one would not expect at ﬁrst glance. We will show that Le´vy ﬂights in the presence of non-natural boundary conditions are incompatible with the method of images, leading to subtleties in the ﬁrst passage and ﬁrst arrival behaviour. Moreover, we will demonstrate how a driving Le´vy noise alters the standard Kramers barrier crossing problem, thereby preserving the exponential decay of the survival probability. Finally, we address the long-standing question of whether or not a Le´vy ﬂight with a diverging variance (or diverging kinetic energy) exhibits pathological behavior. As we will show, within a proper framework, nonlinear dissipative effects will cause a truncation of the Le´vy stable nature; however, within a ﬁnite experimental window, Le´vy ﬂights are a meaningful approximation to real systems. These questions touch on the most fundamental properties of a stochastic process, and the question of the thermodynamic interpretation of processes that leave the basin of attraction of standard Gaussian processes. Le´vy ﬂights, despite having been studied for many decades, still leave numerous open questions. In the following we explore the new physics of Le´vy ﬂight processes and demonstrate their subtle and the intruguing nature. 5

And, in generalized form, also for subdiffusive processes [58].

fundamentals of le´vy ﬂight processes II.

449

DEFINITION AND BASIC PROPERTIES OF LE´VY FLIGHTS

In this section, we formulate the dynamical description of Le´vy ﬂights using both a stochastic differential (Langevin) equation and the deterministic fractional Fokker–Planck equation. For the latter, we also discuss the corresponding form in the domain of wavenumbers, which is a convenient form for certain analytical manipulations in later sections. A.

The Langevin Equation with Le´vy Noise

Our starting point in the stochastic description is the overdamped Langevin equation [54,59]6 dx FðxÞ ¼ þ a ðtÞ ð25Þ dt mZ where F ¼ dV=dx is an external force with potential VðxÞ, which we choose to be VðxÞ ¼

ajxjc c

ð26Þ

with amplitude a > 0 and exponent c 2 (for reasons that become clear below); as before, m is the particle mass, Z the friction coefﬁcient, and a ðtÞ represents a stationary white Le´vy noise with Le´vy index a (1 a 2). By white Le´vy noise a ðtÞ we mean that the process tþt ð

LðtÞ ¼

a ðtÞ dt

ð27Þ

t

that is, the time integral over an increment t, is an a-stable process with stationary independent increments. Restricting ourselves to symmetric Le´vy stable distributions, this implies a characteristic function of the form pa ðk; tÞ ¼ expðDjkja tÞ

ð28Þ

The constant D in this description constitutes the intensity of the external noise. In Fig. 5 we show realizations of white Le´vy noises for various values of a. The sharply pronounced ‘spikes’, due to the long-tailed nature of the Le´vy stable distribution, are distinctly apparent in comparison to the Gaussian case a ¼ 2. 6

A more formal way of writing this Langevin equation is xðt þ dtÞ xðtÞ ¼

1 dVðxÞ dt þ D1=a a ðdtÞ mZ dx

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Figure 5. Examples of white Le´vy noise with Le´vy index a ¼ 2; 1:7; 1:3; 1:0. The outliers are increasingly more pronounced the smaller the Le´vy index a becomes. Note the different scales on the ordinates.

B.

Fractional Fokker–Planck Equation

The Langevin equation [Eq. (25)] still deﬁnes a Markov process, and it is therefore fairly straightforward to show that the corresponding ﬂuctuation-averaged (deterministic) description is given in terms of the space-fractional Fokker–Planck equation (22) [54,60]. In what follows, we solve it with d-initial condition Pðx; 0Þ ¼ dðxÞ

ð29Þ

The space-fractional derivative qa =qjxja occurring in the fractional Fokker– Planck equation (22) is called the Riesz fractional derivative. We have already seen that it is implicitly deﬁned by a q Pðx; tÞ F ð30Þ ¼ jkja Pðk; tÞ: qjxja The Riesz fractional derivative is deﬁned explicitly, via the Weyl fractional operator ( Da Pðx;tÞþDa Pðx;tÞ da Pðx; tÞ ; a 6¼ 1 þ 2 cosðpa=2Þ ¼ ð31Þ d djxja dx HPðx; tÞ; a¼1

fundamentals of le´vy ﬂight processes

451

where we use the following abbreviations: ðDaþ PÞðx; tÞ

1 d2 ¼ ð2 aÞ dx2

ðx

Pðx; tÞ dx

1

ðx xÞa1

ð32Þ

and ðDa PÞðx; tÞ

1 d2 ¼ ð2 aÞ dx2

1 ð

x

Pðx; tÞ dx ðx xÞa1

ð33Þ

for the left and right Riemann–Liouville derivatives (1 a < 2), respectively, and [61] 1 ð 1 Pðx; tÞ dx ð34Þ ðHPÞðx; tÞ ¼ p xx 1

is the Hilbert transform. Note that the integral is to be interpreted as the Cauchy principal value. The deﬁnitions of qa =qjxja demonstrate the strongly nonlocal property of the space-fractional Fokker–Planck equation. 1.

Rescaling of the Dynamical Equations

Passing to dimensionless variables x0 ¼ x=x0 ;

t0 ¼ t=t0

ð35Þ

with x0 ¼

mDZ 1=ðc2þaÞ ; a

t0 ¼

xa0 D

ð36Þ

the initial equations take the form (we omit primes below) dx dV ¼ þ a ðtÞ dt dx

ð37Þ

instead of the Langevin equation (25), and qPðx; tÞ q dV qa Pðx; tÞ ¼ Pðx; tÞ þ qt qx dx qjxja

ð38Þ

instead of the fractional Fokker–Planck equation (22); also, VðxÞ ¼ instead of Eq. (26).

jxjc c

ð39Þ

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Starting Equations in Fourier Space

For the PDF Pðx; tÞ and its Fourier image Pðk; tÞ ¼ FfPðx; tÞg, we use the notation Pðx; tÞ Pðk; tÞ;

ð40Þ

where the symbol denotes a Fourier transform pair. Since [62] Da Pðx; tÞ ðikÞa Pðk; tÞ

ð41Þ

HPðx; tÞ isignðkÞPðk; tÞ

ð42Þ

qa Pðx; tÞ jkja Pðk; tÞ qjxja

ð43Þ

and

we obtain

for all a. The transformed fractional Fokker–Planck equation [Eq. (38)] for the characteristic function then follows immediately: qPðk; tÞ þ jkja Pðk; tÞ ¼ V k Pðk; tÞ qt

ð44Þ

with the initial condition Pðk; t ¼ 0Þ ¼ 1

ð45Þ

Pðk ¼ 0; tÞ ¼ 1

ð46Þ

and the normalization

The external potential VðxÞ becomes the linear differential operator in k, 1 ð

V k Pðx; tÞ ¼

e 1

ikx

q dV Pðx; tÞ dx qx dx

1 ð

eikx signðxÞjxjc1 Pðx; tÞ dx

¼ ik

ð47Þ

1

Next, by using the following inverse transforms ð ixÞa PðxÞ Da PðkÞ

ð48Þ

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453

iðsignðxÞPðxÞ HPðkÞ

ð49Þ

and

we obtain the explicit expression for the external potential operator, (

c1 k Pðk; tÞ; c 6¼ 3; 5; 7; . . . Dþ Dc1 V k Pðk; tÞ ¼ 2 cosðpc=2Þ m d2m ð1Þ k dk2m HPðk; tÞ; c ¼ 3; 5; 7; . . .

ð50Þ

Note that for the even potential exponents c ¼ 2m þ 2 , m ¼ 0; 1; 2; . . . , we ﬁnd the simpliﬁed expression V k ¼ ð1Þmþ1 k

q2mþ1 qk2mþ1

ð51Þ

in terms of conventional derivatives in k. We see that the force term can be written in terms of fractional derivatives in k-space, and therefore it is not straightforward to calculate even the stationary solution of the fractional Fokker–Planck equation [Eq. (38)] in the general case c 2 = N. In particular, in this latter case, the nonlocal equation [Eq. (38)] in x-space translates into a nonlocal equation in k-space, where the nonlocality shifts from the diffusion to the drift term. III.

CONFINEMENT AND MULTIMODALITY

In the preceding section, we discussed some elementary properties of the spacefractional Fokker–Planck equation for Le´vy ﬂights; in particular, we highlighted in the domain of wave numbers k the spatially nonlocal character of Eq. (38), and its counterpart (44). For the particular case of the external harmonic potential corresponding to Eq. (26) with c ¼ 2, we found that the PDF does not leave the basin of attraction imposed by the external noise a ðtÞ—that is, its stable index a. In this section, we determine the analytical solution of the fractional Fokker– Planck equation for general c 2. We start with the exactly solvable stationary quartic Cauchy oscillator, to demonstrate directly the occurring steep asymptotics and the bimodality, that we will then investigate in the general case. The ﬁndings collected in this section were ﬁrst reported in Refs. 60, 63 and 64. A.

The Stationary Quartic Cauchy Oscillator

Let us ﬁrst consider a stationary quartic potential with c ¼ 4 for the Cauchy– Le´vy ﬂight with a ¼ 1 that is, the solution of the equation d 3 d x Pst ðxÞ þ Pst ðxÞ ¼ 0 dx djxj

ð52Þ

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aleksei v. chechkin et al.

or d 3 Pst ðkÞ ¼ signðkÞjkjPst ðkÞ dk3

ð53Þ

in the k domain. Its solution is pﬃﬃﬃ 3jkj p 2 jkj cos Pst ðkÞ ¼ pﬃﬃﬃ exp 2 2 6 3

ð54Þ

whose inverse Fourier transform results in the simple analytical form 1 : Pst ðxÞ ¼ pð1 x2 þ x4 Þ

ð55Þ

We observe surprisingly that the variance hx2 i ¼ 1

ð56Þ

of the solution (55) is ﬁnite, due to the long-tailed asymptotics Pst ðxÞ x4 . In addition, aspﬃﬃshown in Fig. 6, this solution has two global maxima at ﬃ xmax ¼ 1= 2 along with the local minimum at the origin (that is the position of the initial condition). These two distinct properties of Le´vy ﬂights are a central theme of the remainder of this section.

0.45 0.4 0.35

fst(x)

0.3 0.25 0.2 0.15 0.1 0.05 0 –4

–3

–2

–1

0 x

1

2

3

4

Figure 6. Stationary PDF (55)pof ﬃﬃﬃﬃﬃﬃﬃthe ﬃ Cauchy-Le´vy ﬂight in a quartic (c ¼ 4) potential. Two global maxima exist at xmax ¼ 1=2, and a local minimum at the origin also exists.

fundamentals of le´vy ﬂight processes B.

455

Power-Law Asymptotics of Stationary Solutions for c 2, and Finite Variance for c > 2

We now derive the power-law asymptotics of the stationary PDF Pst ðxÞ for external potentials of the form (39) with general c 2. Thus, we note that as x ! þ1, it is reasonable to assume Da Pst Daþ Pst

ð57Þ

since the region of integration for the right-side Riemann–Liouville derivative Da Pst ðxÞ, ðx; 1Þ, is much smaller than the region of integration for the left-side derivative Daþ Pst ðxÞ, ð1; xÞ, in which the major portion of Pst ðxÞ is located. Thus, at large x we get for the stationary state, ðx d dV 1 d2 Pst ðxÞ dx Pst ðxÞ ﬃ0 2 dx dx 2 cosðpa=2Þ dx ðx xÞa1

ð58Þ

1

This relation corresponds to the approximate equality x

c1

1 d Pst ðxÞ ﬃ 2 cosðpa=2Þ dx

ðx

Pst ðxÞ dx

1

ðx xÞa1

ð59Þ

We are seeking asymptotic behaviors of Pst ðxÞ in the form PðxÞ C1 =xm (x ! þ1, m > 0Þ. After integration of relation (59), we ﬁnd 2C1 cosðpa=2Þð2 aÞ mþc x ﬃ m þ c

ðx 1

Pst ðxÞ dx ðx xÞa1

ð60Þ

The integral on the right-hand side can be approximated by 1 xa1

ðx Pst ðxÞ dx ﬃ 1

1 xa1

1 ð

Pst ðxÞ dx ¼ 1

1 xa1

ð61Þ

Thus, we may identify the powers of x and the prefactor, so that m¼aþc1

ð62Þ

and C1 ¼

sinðpa=2ÞðaÞ p

ð63Þ

456

aleksei v. chechkin et al.

By symmetry of the PDF we therefore recover the general asymptotic form Pst ðxÞ

sinðpa=2ÞðaÞ ; pjxjm

x ! þ1

ð64Þ

for all c 2. This result is remarkable, for several reasons: (i) despite the approximations involved, the asymptotic form (64) for arbitrary c 2 corresponds exactly to previously obtained forms, such as the exact analytical result for the harmonic Le´vy ﬂight (linear Le´vy oscillator), c ¼ 2 reported in Ref. 57; the result for the quartic Le´vy oscillator with c ¼ 4 discussed in Ref. 60 and 64; and the case of even power-law exponents c ¼ 2m þ 2 (m 2 N0 ) given in Ref. 60. It is also supported by the calculation in Ref. 65. (ii) The prefactor C1 is independent of the potential exponent c; in this sense, C1 is universal. (iii) For each value a of the Le´vy index a critical value ccr ¼ 4 a

ð65Þ

exists such that at c < ccr the variance hx2 i is inﬁnite, whereas at c > ccr the variance is ﬁnite. (iv) We have found a fairly simple method for constructing stationary solutions for large x in the form of inverse power series. The qualitative consequence of the steep power-law asymptotics can be visualized by direct integration of the Langevin equation for white Le´vy noise, the latter being portrayed in Fig. 5. Typical results for the sample paths under the inﬂuence of an external potential (39) with increasing superharmonicity are shown in Fig. 7 in comparison to the Brownian case (i.e., white Gaussian noise). For growing exponent c, the long excursions typical of homogeneous Le´vy ﬂights are increasingly suppressed. For all cases shown, however, the qualitative behavior of the noise under the inﬂuence of the external potential is different from the Brownian noise even in this case of strong conﬁnement. In the same ﬁgure, we also show the curvature of the external potential. Additional investigations have shown that the maximum curvature is always very close to the positions of the two maxima, leading us to conjecture that they are in fact identical. C.

Proof of Nonunimodality of Stationary Solution for c > 2

In this subsection we demonstrate that the stationary solution of the kinetic equation (38) has a nonunimodal shape. For this purpose, we use an

fundamentals of le´vy ﬂight processes

457

Figure 7. Left column: The potential energy functions V ¼ xc =c, (solid lines) and their curvatures (dotted lines) for different values of c: c ¼ 2 (linear oscillator), and c ¼ 4; 6; 8 (strongly non-linear oscillators). Middle column: Typical sample paths of Brownian oscillators, a ¼ 2, with the potential energy functions shown on the left. Right column: Typical sample paths of Le´vy oscillators, a ¼ 1. On increasing m the potential walls become steeper, and the ﬂights become shorter; in this sense, they are conﬁned.

alternative expression for the fractional Riesz derivative (compare, e.g., Ref. 62), d a PðxÞ sinðap=2Þ ð1 þ aÞ p djxja 1 ð Pðx þ xÞ 2PðxÞ þ Pðx xÞ dx x1þa 0

ð66Þ

458

aleksei v. chechkin et al.

valid for 0 < a < 2. In the stationary state (qP=qt ¼ 0), we have from Eq. (38) da P ðxÞ d st sgnðxÞjxjc1 Pst ðxÞ þ ¼0 dx djxja Thus, it follows that at c > 2 (strict inequality) da Pst ðxÞ ¼0 djxja x¼0

ð67Þ

ð68Þ

or, from deﬁnition (66) and noting that Pst ðxÞ is an even function, 1 ð

dx

Pst ðxÞ Pst ð0Þ ¼0 x1þa

ð69Þ

0

we can immediately obtain a proof of the nonunimodality of Pst , from the latter relation, which we produce in two steps: 1. If we assume that the stationary PDF Pst ðxÞ is unimodal, then due to the symmetry x ! x, it necessarily has one global maximum at x ¼ 0. Here the integrand in equation (69) must be negative, and therefore contradicts equation (69). Therefore, Pst ðxÞ is nonunimodal. 2. We can in addition exclude Pð0Þ ¼ 0, as now the integrand will be positive, which again contradicts Eq. (69). Since PðxÞ ! 0 at x ! 1, based on statements 1 and 2, one may conclude that the simplest situation is such that x0 > 0 exists with the property 1 ð

dx

PðxÞ Pð0Þ <0 x1þa

ð70Þ

dx

PðxÞ Pð0Þ >0 x1þa

ð71Þ

x0

and xð0

0

that is, the condition for a two-hump stationary PDF for all c > 2. At intermediate times, however, we will show that a trimodal state may also exist. If such bimodality occurs, it results from a bifurcation at a critical time t12 [64] when evolution commences (as usually assumed) from the delta function at the origin. A typical result is shown in Fig. 8, for the quartic case c ¼ 4 and

fundamentals of le´vy ﬂight processes

459

Figure 8. Time evolution of the Le´vy ﬂight-PDF in the presence of the superharmonic external potential [Eq. (26)] with c ¼ 4 (quartic Le´vy oscillator) and Le´vy index a ¼ 1:2, obtained from the numerical solution of the fractional Fokker–Planck equation, using the Gru¨nwald–Letnikov representation of the fractional Riesz derivative (full line). The initial condition is a d-function at the origin. The dashed lines indicate the corresponding Boltzmann distribution. The transition from one to two maxima is clearly seen. This picture of the time evolution is typical for 2 < c 4 (see below).

Le´vy index a ¼ 1:2: from an initial d-peak, eventually a bimodal distribution emerges. D.

Formal Solution of Equation (38)

Returning to the general case, we rewrite Eq. (44) in the equivalent integral form, ðt Pðk; tÞ ¼ pa ðk; tÞ þ dt pa ðk; t tÞV k Pðk; tÞ 0

ð72Þ

460

aleksei v. chechkin et al.

where pa ðk; tÞ ¼ expðjkja tÞ

ð73Þ

is the characteristic function of a free (homogeneous) Le´vy ﬂight. This relation follows from equation (44) by formally treating it as a nonhomogeneous linear ﬁrst-order differential equation, where V k plays the role of the nonhomogeneity. Then, Eq. (44) is obtained by variation of parameters. [Differentiate Eq. (72) to return to Eq. (44).] Equation (72) can be solved formally by iteration: Let f ð0Þ ðk; tÞ ¼ pa ðk; tÞ

ð74Þ

then f

ð1Þ

ðt

ðk; tÞ ¼ pa ðk; tÞ þ dtpa ðk; t tÞV k f ð0Þ ðk; tÞ

ð75Þ

0

ðt

f ð2Þ ðk; tÞ ¼ pa ðk; tÞ þ dtpa ðk; t tÞV k pa ðk; tÞ 0

ðt

ðt

þ dt dt0 pa ðk; t tÞV k pa ðk; t t0 ÞV k pa ðk; t0 Þ 0

ð76Þ

0

and so on. From the convolution, ðt

ðt

A B ¼ dtAðt tÞBðtÞ ¼ dtAðtÞBðt tÞ 0

ð77Þ

0

using A B C ¼ ðA BÞ C ¼ A ðB CÞ

ð78Þ

we arrive at the formal solution Pðk; tÞ ¼

1 X

pa ðV k pa Þn

ð79Þ

n¼0

This procedure is analogous to perturbation theory, with V k P playing the role of the interaction term (see, for instance, Ref. 66, Chapter 16). Applying a Laplace Transformation, namely, 1 ð

Pðk; uÞ ¼

dt expðutÞPðk; tÞ 0

ð80Þ

fundamentals of le´vy ﬂight processes

461

to Eq. (72), we obtain Pðk; uÞ ¼ pa ðk; uÞ þ pa ðk; uÞV k Pðk; uÞ

ð81Þ

where pa ðk; uÞ ¼

1 u þ ka

ð82Þ

is the Fourier–Laplace transform of the homogeneous Le´vy stable PDF. Thus, we obtain the equivalent of the solution (79) in ðk; uÞ-space: Pðk; uÞ ¼

1 X

½pa ðk; uÞV k n pa ðk; uÞ

ð83Þ

n¼0

This iterative construction scheme for the solution of the fractional Fokker– Planck equation will be useful below. E.

Existence of a Bifurcation Time

For the unimodal initial condition Pðx; 0Þ ¼ dðxÞ we now prove the existence of a ﬁnite bifurcation time t12 for the turnover from a unimodal to a bimodal PDF. At this time, the curvature at the origin will vanish; that is, it is a point of inﬂection: q2 PðxÞ qx2

¼0

ð84Þ

x¼0;t¼t12

Introducing 1 ð

JðtÞ ¼

dkk2 Pðk; tÞ

ð85Þ

0

Eq. (84) is equivalent to (note that the characteristic function is an even function) Jðt12 Þ ¼ 0

ð86Þ

The bifurcation can now be obtained from the iterative solution (83); we consider the speciﬁc case c ¼ 4. From the ﬁrst-order approximation 1 1 P1 ðk; uÞ ¼ 1 þ Vk ð87Þ u þ ka u þ ka where Vk ¼ k

q3 qk3

ð88Þ

462

aleksei v. chechkin et al.

Combining these two expressions, we have P1 ðk; uÞ ¼

1 ka2 þ aða 1Þð2 aÞ a uþk ðu þ ka Þ3 þ 6a2 ða 1Þ

k2a2 ðu þ ka Þ

6a3 4

k3a2 ðu þ ka Þ5

or, on inverse Laplace transformation, a3 ka t P1 ðk; tÞ ¼ e 1 t4 k3a2 þ a2 ða 1Þt3 k2a2 4 t2 a2 þ aða 1Þð2 aÞ k 2

ð89Þ

ð90Þ

The ﬁrst approximation to the bifurcation time t12 is then determined via Eq. (85); that is, we calculate 1 ð

ð1Þ

dk k2 P1 ðk; t12 Þ ¼ 0

ð91Þ

0

to obtain ð1Þ t12

¼

a=ð2þaÞ 4ð3=aÞ 3ð3 aÞð1=aÞ

ð92Þ ð1Þ

In Fig. 9, we show the dependence of this ﬁrst approximation t12 as a function of the Le´vy index a (dashed line), in comparison to the values determined from the numerical solution of the fractional Fokker–Planck equation (38) shown as the dotted line. The second-order iteration for the PDF, P2 ðk; tÞ, can be obtained with maple6, whence the second approximation for the bifurcation time is found by analogy with the above procedure. The result is displayed as the full line in Fig. 9. The two approximate results are in fact in surprisingly good agreement with the numerical result for the exact PDF. Note that the second approximation appears somewhat worse than the ﬁrst; however, it contains the minimum in the a-dependence of the t12 behavior. 1.

Trimodal Transient State at c > 4.

we have already proved the existence of a bimodal stationary state for the quartic ðc ¼ 4Þ Le´vy oscillator. This bimodality emerges as a bifurcation at a critical time t12 , at which the curvature at the origin vanishes. This scenario is changed

fundamentals of le´vy ﬂight processes

463

t12

1.0

0.5

0 1.0

1.5 a

2.0

Figure 9. Bifurcation time t12 versus Le´vy exponent a for external potential exponent c ¼ 4:0. Black dots: bifurcation time deduced from the numerical solution of the fractional Fokker–Planck equation [Eq. (38)] using the Gru¨nwald–Letnikov representation of the fractional Riesz derivative (see ð1Þ ð2Þ appendix). Dashed line: ﬁrst approximation t12 ; solid line: second approximation t12 .

for c > 4, as displayed in Fig. 10: There exists a transient trimodal form of the PDF. Thus, there are obviously two time scales that are relevant: the critical time for the emergence of the two off-center maxima, which are characteristic of the stationary state; and a second one, which corresponds to the relaxing initial central hump—that is, the decaying initial distribution Pðx; 0Þ ¼ dðxÞ. The formation of the two off-center humps while the central one is still present, as detailed in Fig. 11. The existence of a transient trimodal state was found to be typical for all c > 4. 2.

Phase Diagrams for n-Modal States

The above ﬁndings can be set in the context of the purely bimodal case discussed earlier. A convenient way of displaying the n-modal character of the PDF in the presence of a superharmonic external potential of the type (39) is the phase diagram shown in Fig. 12. There, we summarize the ﬁndings that for 2 < c 4 the bifurcation occurs between the initial monomodal and the stationary bimodal PDF at a ﬁnite critical time, whereas for c > 4, a transient trimodal state exists. Moreover, we also include the shaded region, in which c is too small to ensure a ﬁnite variance. In Fig. 13, in complementary fashion the temporal domains of the n-modal states are graphed, and the solid lines separating these domains correspond to the critical time scales tcr ð¼ t12 ; t13 ; t32 Þ. Again, the transient nature of the trimodal state is distinctly apparent.

464

aleksei v. chechkin et al.

Figure 10. Time evolution of the PDF governed by the fractional Fokker–Planck equation (38) in a superharmonic potential (26) with exponent c ¼ 5:5, and for Le´vy index a ¼ 1:2, obtained from numerical solution using the Gru¨nwald–Letnikov method explained in the appendix. Initial condition is Pðx; 0Þ ¼ dðxÞ. The dashed lines indicate the corresponding Boltzmann distribution. The transitions between 1 ! 3 ! 2 humps are clearly seen. This picture of time evolution is typical for c > 4. On a ﬁner scale, we depict the transient trimodal state in Fig. 11.

F.

Consequences

By combining analytical and numerical results, we have discussed Le´vy ﬂights in a superharmonic external potential of power c. Depending on the magnitude of this exponent c, different regimes could be demonstrated. Thus, for c ¼ 2, the character of the Le´vy noise imprinted on the process, is not altered by the external potential: The resulting PDF has Le´vy index a, the same as the noise itself, and will thus give rise to a diverging variance at all times. Conversely, for c > 2, the variance becomes ﬁnite if only c > ccr ¼ 4 a. Because the PDF no

465

fundamentals of le´vy ﬂight processes

0.74

0.75

0.76

0.78

0.79

0.81

0.83

0.84

0.87

0.89

0.90

0.93

Figure 11. The transition 1 ! 3 ! 2 from Fig. 10 on a ﬁner scale (c ¼ 5:5, a ¼ 1:2).

2.0

a

1

1

2

3

2

1.5

1.0 2

4

C

6

8

Figure 12. ðc; aÞ map showing different regimes of the PDF. The region with inﬁnite variance is shaded. The region c < 4 covers transitions from 1 to 2 humps during the time evolution. For c > 4, a transition from 1 to 3, and then from 3 to 2 humps occurs. In both cases, the stationary PDF exhibits 2 maxima. Compare Fig. 13.

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aleksei v. chechkin et al.

1.0 2

tcr

3

0.5 1

0 2

4

C

6

8

ðc; tÞ map showing states of the PDF with different number of humps and the transitions between these. Region 1: The PDF has 1 hump. Region 2: The PDF exhibits 2 humps. Region 3: Three humps occur. At c < 4, there is only one transition 1 ! 2, whereas for c > 4, there occur two transitions, 1 ! 3 and 3 ! 2.

Figure 13.

longer belongs to the set of Le´vy stable PDFs and acquires an inverse power-law asymptotic behavior with power m ¼ a þ c 1. Obviously, moments of higher order will still diverge. Apart from the ﬁnite variance, the PDF is distinguished by the observation that it bifurcates from the initial monomodal to a stationary bimodal state. If c > 4, there exists a transient trimodal state. This interesting behavior of the PDF both during relaxation and under stationary conditions, depending on a competition between Le´vy noise and steepness of the potential is in contrast to the universal approach to the Boltzmann equilibrium, solely deﬁned by the external potential, encountered in classical diffusion. One may demand the exact kinetic reason for the occurrence of the multiple humps. Now the nontransient humps seem to coincide with the positions of maximum curvature of the external potential, which at these points changes almost abruptly for larger c from a rather ﬂat to a very steep slope. Thus one may conclude that the random walker, which is driven towards these ﬂanks by the anomalously strong Le´vy diffusivity, is thwarted, thus the PDF accumulates close to these points. Apart from this rudimentary explanation, we do not yet have a more intuitive argument for the existence of the humps and their bifurcations, we also remark that other systems exist where multimodality occurs, for instance, in the transverse ﬂuctuations of a grafted semiﬂexible polymer [67]. We will later return to the issue of ﬁnite variance in the discussion of the velocity distribution of a Le´vy ﬂight. The different regimes for c > 2 can be classiﬁed in terms of critical quantities, in particular, the bifurcation time(s) tcr ð¼ t12 ; t13 ; t32 Þ and the critical

fundamentals of le´vy ﬂight processes

467

external potential exponent ccr . Le´vy ﬂights in superharmonic potentials can then be conveniently represented by phase diagrams on the ðc; aÞ and ðc; tcr Þ plains. The numerical solution of both the fractional Fokker–Planck equation in terms of the Gru¨nwald–Letnikov scheme used to ﬁnd a discretized approximation to the fractional Riesz operator exhibits reliable convergence, as corroborated by direct solution of the corresponding Langevin equation. Our ﬁndings have underlined the statement that the properties of Le´vy ﬂights, in particular under nontrivial boundary conditions or in an external potential are not fully understood. The general difﬁculty, which hampers a straightforward investigation as in the regular Gaussian or the subdiffusive cases, is connected with the strong spatial correlations associated with such problems, manifested in the integrodifferential nature of the Riesz fractional operator. Thus it is not easy to determine the stationary solution of the process. We expect, since diverging ﬂuctuations appear to be relevant in physical systems, that many hitherto unknown properties of Le´vy ﬂights remain to be discovered. Some of these features are discussed in the following sections. IV.

FIRST PASSAGE AND ARRIVAL TIME PROBLEMS FOR LE´VY FLIGHTS

The ﬁrst passage time density (FPTD) is of particular interest in random processes [14,68–70]. For Le´vy ﬂights, the ﬁrst passage time density was determined by the method of images in a ﬁnite domain in reference [71], and by similar methods in reference [72]. These methods lead to results for the ﬁrst passage time density in the semi-inﬁnite domain, whose long-time behavior explicitly depends on the Le´vy index a. In contrast, a theorem due to Sparre Andersen proves that for any discrete-time random walk process starting at x0 6¼ 0 with each step chosen from a continuous, symmetric but otherwise arbitrary distribution, the ﬁrst passage time density asymptotically decays as n3=2 with the number n of steps [70,73,74], being fully independent of the index of the Le´vy ﬂight—that is, universal. In the case of a Markov process, the continuous time analogue of the Sparre Andersen result reads [69,70] pðtÞ t3=2

ð93Þ

The analogous universality was proved by Frisch and Frisch for the special case in which an absorbing boundary is placed at the source of the Le´vy ﬂight at t > 0 [75], and numerically corroborated by Zumofen and Klafter [76]. In the following, we demonstrate that the method of images is generally inconsistent with the universality of the ﬁrst passage time density, and therefore cannot be applied to solve ﬁrst passage time density-problems for Le´vy ﬂights. We also

468

aleksei v. chechkin et al.

show that for Le´vy ﬂights the ﬁrst passage time density differs from the PDF for ﬁrst arrival. The discussion will be restricted to the case 1 < a < 2 [77]. A.

First Arrival Time

By incorporating in the fractional diffusion equation (21) a d-sink of strength pfa ðtÞ, we obtain the diffusion-reaction equation for the non-normalized density function f ðx; tÞ, q qa f ðx; tÞ ¼ D f ðx; tÞ pfa ðtÞdðxÞ qt qjxja

ð94Þ

from which by integration over all space, we may deﬁne the quantity d pfa ðtÞ ¼ dt

1 ð

f ðx; tÞ dx

ð95Þ

1

that is, pfa ðtÞ is the negative time derivative of the survival probability. By deﬁnition of the sink term, pfa ðtÞ is the PDF of ﬁrst arrival: once a random walker arrives at the sink, it is annihilated. By solving equation (94) by standard methods (determining the homogeneous and inhomogeneous solutions), it is straightforward to calculate the solution f in terms of the propagator P of the fractional diffusion

equation (21) with initial condition Pðx; 0Þ ¼ dðx x0 Þ yielding f ðx; tÞ ¼ eikx0 þ pðuÞ =ðu þ Djkja Þ, whence pfa ðtÞ satisﬁes the chain rule (pfa implicitly depending on x0 ) ðt Pðx0 ; tÞ ¼ pfa ðtÞPð0; t tÞ dt

ð96Þ

0

which corresponds to the m domain relation pfa ðuÞ ¼ Pðx0 ; uÞ=Pð0; uÞ. Equation (96) is well known and for any sufﬁciently well-behaved continuum diffusion process is commonly used as a deﬁnition of the ﬁrst passage time density [14,70]. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ For Gaussian processes with propagator Pðx; tÞ ¼ 1= 4pDt expðx2 =½4DtÞ, one obtains by direct integration of the diffusion equation with appropriate boundary condition the ﬁrst passage time density [70] x0 x2 pðtÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ exp 0 4D 4pDt3

ð97Þ

including the asymptotic behaviour pðtÞ t3=2 for t x20 =ð4DÞ. In this Gaussian case, the quantity pfa ðtÞ is equivalent to the ﬁrst passage time density. From a random walk perspective, this occurs because individual steps all have the same increment, and the jump length statistics therefore ensure that the

469

fundamentals of le´vy ﬂight processes

walker cannot hop across the sink in a long jump without actually hitting the sink and being absorbed. The behaviour is very different for Le´vy jump length statistics: There, the particle can easily cross the sink in a long jump. Thus, before eventually being absorbed, it can pass by the sink location many times, and therefore the statistics of the ﬁrst arrivalÐ will be different from those of the 1 1 ﬁrst passage. In fact, with Pðx; uÞ ¼ ð2pÞ1 1 eikx ðu þ Djkja Þ dk , we ﬁnd 1 ð ð1 cos kx0 Þ=ðu þ Dka Þdk pfa ðuÞ ¼ 1

0

ð98Þ

1 ð

1=ðu þ Dka Þdk Since 1 ð

Ð1 0

0 a 1

ðu þ Dk Þ dk ¼ pu

1=a1

=ðaD1=a sinðp=aÞÞ and

1 cos kx0 ðð2 aÞ sinðpð2 aÞ=2Þxa1 0 ; u þ Dka ða 1ÞD

for u ! 0; a > 1

0

we obtain the limiting form 11=a 1þ1=a ~ pfa ðuÞ 1 xa1 ðaÞ D 0 u

ð99Þ

~ ðaÞ ¼ að2 aÞ sinðpð2 aÞ=2Þ sinðp=aÞ=ða 1Þ. We note that the where same result may be obtained using the exact expressions for Pðx0 ; uÞ and Pð0; uÞ in terms of Fox H-functions and their series expansions [78]. The inverse Laplace transform of the small u-behavior (99) can be obtained by completing (99) to an exponential, and then computing the Laplace inversion using the 1;0 identity ez ¼ H0;1 ½zjð0; 1Þ in terms of the Fox H-function [78], for which the exact Laplace inversion can be performed [79]. Finally, series expansion of this result leads to the long-t form xa1 0 ð100Þ pfa ðtÞ CðaÞ 11=a D t21=a 1ÞÞ. with CðaÞ ¼ að2 aÞð2 1=aÞ sinðp½2 a=2Þ sin2 ðp=aÞ=ðp2 ðap ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Clearly, in the Gaussian limit, the required asymptotic form pðtÞ x0 = 4pDt3 for the ﬁrst passage time density is consistently recovered, whereas in the general case the result (100) is slower than in the universal ﬁrst passage time density behavior embodied in Eq. (93), as it should be since the d-trap used in equation (94) to deﬁne the ﬁrst arrival for Le´vy ﬂights is weaker than the absorbing wall used to properly deﬁne the ﬁrst passage time density. For Le´vy ﬂights, the PDF for ﬁrst arrival thus scales like (100) (i.e., it explicitly depends on the index a of the underlying Le´vy process), and, as shown below, it differs from the corresponding ﬁrst passage time density.

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log t*p(t)

α=1.2, x0=0.0, w=0.3 α=1.2, x0=0.3, w=0.3 α=1.2, x0=1.0, w=0.3 α=1.2, x0=10.0, w=1.0 α=1.8, x0=10.0, w=0.25 t-2+1/α , α=1.2 t-2+1/α, α=1.8 t-3/2

0.01 10

100

1000

10000

log t Figure 14. First arrival PDF for a ¼ 1:2 demonstrating the t2þ1=a scaling, for optimal trap width w ¼ 0:3. For comparison, we show the same scaling for a ¼ 1:8, and the power-law t3=2 corresponding to the ﬁrst passage time density. The behavior for large w ¼ 1:0 shows a shift of the decay toward the 3=2 slope. Note that the ordinate is lg tpðtÞ. Note also that for the initial condition x0 ¼ 0:0, the trap is activated after the ﬁrst step, consistent with Ref. [76].

Before calculating this ﬁrst passage time density, we ﬁrst demonstrate the validity of Eq. (100) by means of a simulation the results of which are shown in Fig. 14. Random jumps with Le´vy ﬂight jump length statistics are performed, and a particle is removed when it enters a certain interval of width w around the sink; in our simulations we found an optimum value w 0:3. As seen in Fig. 14 (note that we plot lg tpðtÞ!) and for analogous results not shown here, relation (100) is satisﬁed for 1 < a < 2 , whereas for larger w, the slope increases. B.

Sparre Anderson Universality

To corroborate the validity of the Sparre Anderson universality, we simulate a Le´vy ﬂight in the presence of an absorbing wall—that is, random jumps with Le´vy ﬂight jump length statistics exist along the right semi-axis—and a particle is removed when it jumps across the origin to the left semi-axis. The results of such a detailed random walk study are displayed in Figs. 15 and 16. The expected universal t3=2 scaling is conﬁrmed for various initial positions x0 and

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fundamentals of le´vy ﬂight processes

log t*p(t)

0.1

x_0=0.10 x_0=1.00 x_0=10.0 x_0=100.0 t**(-1.5) t**(-1.5) t**(-1.5) t**(-1.5) First arrival Images method

0.01

0.001

1

10

100

1000

10000

log t Figure 15. Numerical results for the ﬁrst process time density process on the semi-inﬁnite domain, for an Le´vy ﬂight with Le´vy index a ¼ 1:2. Note abscissa, is tpðtÞ. For all initial conditions x0 ¼ 0:10 1.00, 10.0, and 100.0 the universal slope 3=2 in the log10 –log10 plot is clearly reproduced, and it is signiﬁcantly different from the two slopes predicted by the method of images and the direct deﬁnition of the ﬁrst process time density.

Le´vy stable indices a. Clearly, the scaling for the ﬁrst arrival as well as the image method–ﬁrst passage time density derived below are signiﬁcantly different. The following qualitative argument may be made in favor of the observed universality of the Le´vy ﬂight–ﬁrst passage time density: The long-time decay is expected to be governed by short-distance jump events, corresponding to the central region of very small jump lengths for the Le´vy stable jump length distribution. However, in this region the distribution function is, apart from a prefactor, indistinguishable from the Gaussian distribution, and therefore the long-time behavior should in fact be the same for any continuous jump length distribution lðxÞ. In fact, the universal law (93) can only be modiﬁed in the presence of non-Markov effects such as broad waiting time processes or spatiotemporally coupled walks [45,46,70,80,81]. In terms of the special case covered by the theorem of Frisch and Frisch [75], in which the absorbing boundary coincides with the initial position, we can understand the general situation for ﬁnite x0 > 0, as in the long-time limit, the distance x0 becomes negligible in comparison to the diffusion length hjxðtÞji t1=a :

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log t*p(t)

alpha=2.0 alpha=1.5 alpha=1.0 alpha=0.6 t**(-1.5) ditto ditto ditto

0.01

0.001 100

1000 log t

10000

Figure 16. Same as in Fig. 15, for a ¼ 2:0, 1.5, 1.0, and 0.6, and for the initial condition x0 ¼ 10:0. Again, the universal t3=2 behavior is obtained.

therefore the asymptotic behavior is necessarily governed by the same universality. C.

Inconsistency of Method of Images

We now demonstrate that the method of images produces a result, which is neither consistent with the universal behavior of the ﬁrst passage time density (93) nor with the behavior of the PDF of ﬁrst arrival (100), Given the initial condition dðx x0 Þ, the solution fim ðx; tÞ for the absorbing boundary value problem with the analogous Dirichlet condition fim ð0; tÞ ¼ 0 according to the method of images is given in terms of the free propagator P by the difference [69,70] fim ðx; tÞ ¼ Pðx x0 ; tÞ Pðx þ x0 ; tÞ

ð101Þ

that is, a negative image solution originating at x0 balances the probability ﬂux across the absorbing boundary. The corresponding pseudo-ﬁrst passage time density is then calculated just as Eq. (95). For the image solution in the ðk; uÞ domain, we obtain fim ðk; uÞ ¼

2i sinðkx0 Þ u þ Djkja

ð102Þ

fundamentals of le´vy ﬂight processes

473

for a process which starts at x0 > 0 and occurs in the right half-space. In u space, the image method–ﬁrst passage time density becomes 1 ð

pim ðuÞ ¼ 1 u

1 ð

dx 1

0

dk ikx 2i sin kx0 e 2p u þ Djkja

ð103Þ

After some transformations, we have 2 pim ðuÞ ¼ 1 p

1 ð

sin xs1=a x0 =D1=a dx xð1 þ xa Þ

ð104Þ

0

In the limit of small Ðs, this expression reduces to pim ðuÞ 1 ðaÞx0 D1=a u1=a , 1 with ðaÞ ¼ ð2=pÞ 0 ð1 þ xa Þ1 dx ¼ 2=ða sinðp=aÞÞ. In like manner, we ﬁnd the long-t form pim ðtÞ 2ð1=aÞ

x0 1=a paD t1þ1=a

ð105Þ

for the image method–ﬁrst passage timepdensity. ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ In the Gaussian limit a ¼ 2, expression (105) produces pim ðtÞ x0 = 4pDt3 , in accordance with Eq. (97). Conversely, for general 1 < a < 2, pðtÞ according to Eq. (105) would decay faster than t3=2 . The failure of the method of images is closely related to the strongly nonlocal character of Le´vy ﬂights. Under such conditions, the random variable x x0 is no longer independent of x þ x0 , so that the method of images is not appropriate. The proper dynamical formulation of a Le´vy ﬂight on the semi-inﬁnite interval with an absorbing boundary condition at x ¼ 0, and thus the determination of the ﬁrst passage time density, has to ensure that in terms of the random walk picture jumps across the sink are forbidden. This objective can be consistently achieved by setting f ðx; tÞ 0 on the left semi-axis, i.e., actually removing the particle when it crosses the point x ¼ 0. This procedure formally corresponds to the modiﬁed dynamical equation qf ðx; tÞ D q2 ¼ qt k qx2

1 ð

0

f ðx0 ; tÞ jx x0 ja1

dx0

q2 F ðx; tÞ qx2

ð106Þ

in which the fractional integral is conﬁned to the semi-inﬁnite interval. Here, we have written pa k ¼ 2ð2 aÞ cos ð107Þ 2

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After Laplace transformation and integrating over x twice, one obtains 1 ð

Kðx x0 ; uÞ f ðx0 ; uÞ dx0 ¼ ðx x0 Þðx x0 Þ xpðuÞ F ð0; uÞ

ð108Þ

0

where pðtÞ is the FPTD and the kernel Kðx; uÞ ¼ uxðxÞ ðkjxja1 Þ. This equation is formally a Wiener–Hopf equation of the ﬁrst kind [82]. After some manipulations similar to those applied in Ref. 76, we arrive at the asymptotic expression pðuÞ ’ 1 Cu1=2 ;

where C ¼ const

ð109Þ

in accordance with the expected universal behavior (93) and with the ﬁndings of reference [76]. Thus, the dynamic equation (106) governs the ﬁrst passage time density problem for Le´vy ﬂights. We note that due to the truncation of the fractional integral it was not possible to modify the well-established Gru¨nwald– Letnikov scheme [61] to numerically solve Eq. (106) with enough computational efﬁciency to obtain the direct solution for f ðx; tÞ. V.

BARRIER CROSSING OF A LE´VY FLIGHT

The escape of a particle from a potential well is a generic problem investigated by Kramers [84] that is often used to model chemical reactions, nucleation processes, or the escape from a potential well 84. Keeping in mind that many stochastic processes do not obey the central limit theorem, the corresponding Kramers escape behavior will differ. For subdiffusion, the temporal evolution of the survival behavior is bound to change, as discussed in Ref. 85. Here, we address the question how the stable nature of Le´vy ﬂight processes generalizes the barrier crossing behavior of the classical Kramers problem [86]. An interesting example is given by the a-stable noise-induced barrier crossing in long paleoclimatic time series [87]; another new application is the escape from traps in optical or plasma systems (see, for instance, Ref. 88). A.

Starting Equations

Here, we investigate barrier crossing processes in a reaction coordinate xðtÞ governed by a Langevin equation [Eq. (25)] with white Le´vy noise a ðtÞ. Now, however, the external potential VðxÞ is chosen as the (typical) double-well shape a b VðxÞ ¼ x2 þ x4 2 4

ð110Þ

compare, for instance, Ref. 89. For convenience, we introduce dimensionless variables t ! t=t0 and x ! x=x0 with t0 ¼ mZ=a and x20 ¼ 1=ðbt0 Þ and

fundamentals of le´vy ﬂight processes 1=a

1=a1

dimensionless noise strength D ! Dt0 =x0 (by a ðt0 tÞ ! t0 that we have the stochastic equation dxðtÞ

¼ x x3 þ D1=a a ðtÞ dt

475 a ðtÞ) [43], so

ð111Þ

Here, we restrict our discussion to 1 a < 2. B.

Brownian Motion

In normal Brownian motion corresponding to the limit a ¼ 2, the survival probability S of a particle whose motion at time t ¼ 0 which is initiated in one of the potential minima xmin ¼ 1, follows an exponential decay SðtÞ ¼ exp ðt=Tc Þ with mean escape time Tc , such that the probability density function pðtÞ ¼ dS=dt of the barrier crossing time t becomes pðtÞ ¼ Tc1 expðt=Tc Þ

ð112Þ

The mean crossing time (MCT) follows the exponential law Tc ¼ C expðh=DÞ

ð113Þ

where h is the barrier height (equal to 1/4 for the potential (110)) in rescaled variables, and the prefactor C includes details of the potential [84]. We want to determine how the presence of Le´vy stable noise modiﬁes the laws (112) and (113). C.

Numerical Solution

The Langevin equation [Eq. (111)] was integrated numerically following the procedure developed in Ref. 90. Whence, we obtained the trajectories of the particle shown in Fig. 17. In the Brownian limit, we reproduce qualitatively the behavior found in Ref. 89. Accordingly, the ﬂuctuations around the positions of the minima are localized in the sense that their width is clearly smaller than the distance between the minima and barrier. In contrast, for progressively smaller stable index a, characteristic spikes become visible, and the individual sojourn times in one of the potential wells decrease. In particular, we note that single spikes can be of the order of or larger than the distance between the two potential minima. From such single trajectories we determine the individual barrier crossing times as the time interval between a jump into one well across the zero line x ¼ 0 and the escape across x ¼ 0 back to the other well. In Fig. 18, we demonstrate that on average, the crossing times are distributed exponentially, and thus follow the same law (112) already known from the Brownian case. Such a result has been reported in a previous study of Kramers’ escape driven by Le´vy noise [91]. In fact, the exponential decay of the survival probability

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Figure 17.

Typical trajectories for different stable indexes a obtained from numerical integration of the Langevin equation [Eq. (111)]. The dashed lines represent the potential minima at 1. In the Brownian case a ¼ 2, previously reported behavior is recovered [89]. In the Le´vy stable case, occasional long jumps of the order of or larger than the separation of the minima can be observed. Note the different scales.

-7 -8

ln p(t)

-9 -10 -11 -12 -13 0

1000

2000

3000

4000

5000

t

Figure 18. Probability density function pðtÞ of barrier crossing times for a ¼ 1:0 and D ¼ 102:5 0:00316. The dashed line is a ﬁt to Eq. (112) with mean crossing time Tc ¼ 1057:8 17:7.

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fundamentals of le´vy ﬂight processes

S observed in a Le´vy ﬂight is not surprising, given the Markovian nature of the process. Due to the Le´vy stable properties of the noise a , the Langevin equation [Eq. (111)] produces occasional long jumps, by which the particle can cross the barrier. Large enough values of the noise a thus occur considerably more frequently than in the Brownian case with Gaussian noise (a ¼ 2), causing a lower mean crossing time. The numerical integration of the Langevin equation (111) was repeated for various stable indices a, and for a range of noise strengths D. From these simulations we obtain the detailed dependence of the mean crossing time Tc ða; DÞ on both of the parameters, a and D. As expected, for decreasing noise strength, the mean crossing time increases. For sufﬁciently large values of 1=D and ﬁxed a, a power-law trend in the double-logarithmic plot is clearly visible. These power-law regions, for the investigated range of a are in very good agreement with the analytical form Tc ða; DÞ ¼

CðaÞ DmðaÞ

ð114Þ

over a large range of D. Equation (114) is the central result of this study. It is clear from Fig. 19, that this relation is appropriate for the entire a-range studied 5.5 5 4.5

log Tc

4 3.5 3

α = 2.00 α = 1.95 α = 1.90 α = 1.80 α = 1.60 α = 1.40 α = 1.20 α = 1.20

2.5 2 1.5 1

1.5

2

2.5 log 1/D

3

3.5

Figure 19. Escape time Tc as a function of noise strength D for various a. Above roughly lg 1=D ¼ 1:5, a power-law behavior is observed that corresponds to Eq. (114). The curve [Eq. (113)] for a ¼ 2:0 appears to represent a common envelope.

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aleksei v. chechkin et al. 1.6

2

µ(α) fitted by 1+0.401 (α-1)+0.105 (α-1) log C(α)

1.14 1.12

1.4

1.1

µ(α)

1.06

1

log C(α)

1.2 1.08

1.04 0.8 1.02 0.6

1

0.98 0.4 1

1.2

1.4

α

1.6

1.8

2

Scaling exponent m as function of stable index a. The constant behavior mðaÞ 1 over the range 1 a / 1:6 is followed by an increase above 1.6, and it eventually shows an apparent divergence close to a ¼ 2, where Eq. (113) holds. Corresponding to the right ordinate, we also plot the decadic logarithm of the amplitude CðaÞ.

Figure 20.

in our simulations. For larger noise strength, we observe a breakdown of the power-law trend, and the curves seem to approach the mean crossing time behavior of the Brownian process (a ¼ 2) as a common envelope. A more thorough numerical analysis of this effect will be necessary in order to ascertain its exact nature. The main topic we want to focus on here is the behavior embodied in Eq. (114). We note from Fig. 19 that for a ranging roughly between the Cauchy case a ¼ 1 and the Holtsmark case a ¼ 3=2, the exponent m is almost constant; that is, the corresponding lines in the log–log plot are almost parallel. The behavior of both the scaling exponent m and the prefactor C as a function of the stable index a becomes clear in Fig. 20. There, we recognize a slow variation of m for values of a between 3/2 and slightly below 2, before a steeper rise in close vicinity of 2. This apparent divergence must be faster than any power, so that in the Gaussian noise limit a ¼ 2, the activation follows the exponential law (113) instead of the scaling form (114). The mðaÞ results are ﬁtted with the parabola indicated in the plot where, for the analytical results derived below, we forced the ﬁt function to pass through the point mð1Þ ¼ 1. D.

Analytical Approximation for the Cauchy Case

In the Cauchy limit a ¼ 1, we can ﬁnd an approximate result for the mean crossing time as a function of noise strength D. To this end, we start with the

479

fundamentals of le´vy ﬂight processes

rescaled fractional Fokker–Planck equation [20,46,54,57,71], corresponding to equation (111), qPðx; tÞ q

qa ¼ x þ x3 Pðx; tÞ þ D Pðx; tÞ qt qx qjxja

ð115Þ

Rewriting Eq. (115) in continuity equation form qPðx; tÞ=qt þ qjðx; tÞqx ¼ 0, that is equivalent to qPðk; tÞqt ¼ ikjðk; tÞ in k space, we obtain for the ﬂux the expression q3 q a1 jðkÞ ¼ 3 i þ i D signðkÞjkj Pðk; tÞ ð116Þ qk qk To obtain an approximate expression for the mean crossing time, we follow the standard steps [92] and for large values of 1=D make the constant ﬂux approximation assuming that the ﬂux across the barrier is a constant, j0 , corresponding to the existence of a stationary solution Pst ðxÞ. By integration of the continuity equation, it then follows that equation (112) is satisﬁed, and Tc ¼ 1=j0 . Due to the low Ð 0 noise strength, we also assume that for all relevant times the normalization 1 Pst ðxÞ ¼ 1 obtains. In this constant ﬂux approximation, we obtain from equation (116) the relation d 3 Pst ðkÞ dPst ðkÞ D signðkÞPst ðkÞ ¼ 2pij0 dðkÞ þ dk3 dk in the Cauchy case a ¼ 1. With the ansatz Pst ðkÞ ¼ C1 ez

k

ð117Þ

þ C2 eðz Þ

k

for k > < 0,

3

we ﬁnd the characteristic equation ðz Þ þ z D ¼ 0 solved by the Cardan pﬃﬃﬃ

pﬃﬃﬃﬃﬃﬃﬃ expressions z ¼ 12 ðu þ v Þ þ 12 i 3ðu v Þ, with u3þ ¼ D 1 þ 1þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4=½27D2 Þ=2 ¼ v3 and v3þ ¼ D 1 1 þ 4=½27D2 =2 ¼ u3 . Matching the left and right solutions at k ¼ 0, requiring that Pst ðkÞ 2 R, and assuming that Pst ðkÞ in the constant ﬂux approximation is far from the fully relaxed (t ! 1) solution, we obtain the shifted Cauchy form Pst ðkÞ ¼

j0 þ ; 2þ ð x þ Þ2 þþ2

pﬃﬃﬃ 3 1 ð118Þ ; þ ¼ ðuþ þ vþ Þ; ¼ ðuþ vþ Þ 2 2 Ð0 With the normalization 1 Pst ðxÞ dx ¼ 1, we arrive at the mean crossing time p 2 Tc ¼ 1 þ arctan ð119Þ 4þ p þ

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For D 1, þ D=2 and 1, so that Tc p=D. In comparison with the numerical result corresponding to Fig. 18 with Tc ¼ 1057:8 for D ¼ 0:00316, we calculate from our approximation Tc 994:2, which is within 6% of the numerical result. This good agreement also corroborates the fact that the constant ﬂux approximation appears to pertain to Le´vy ﬂights. E.

Discussion

We observe from numerical simulations an exponential decrease of the survival probability SðtÞ in the potential well, at the bottom of which we initialize the process. Moreover, we ﬁnd that the mean crossing time assumes the scaled form (114) with scaling exponent m being approximately constant in the range 1 a / 1:6, followed by an increase before the apparent divergence at a ¼ 2, that leads back to the exponential form of the Brownian case, Eq. (113). An analytic calculation in the Cauchy limit a ¼ 1 reproduces, consistently with the constant ﬂux approximation commonly applied in the Brownian case, the scaling Tc 1=D, and, within a few percent error, the numerical value of the mean crossing time Tc . Employing scaling arguments, we can restore the dimensionality into expression (114) for the mean crossing time. From our model potential (110), where we absorb the friction factor pﬃﬃﬃﬃﬃﬃﬃﬃmZ via a ! a=ðmZÞ and b ! b=ðmZÞ, we ﬁnd that the minima are xmin ¼ a=b and the barrier height V ¼ a2 =ð4bÞ . In terms of the rescaled prefactors a and b with dimensions ½a ¼ sec1 and ½b ¼ sec1 cm2 , we can now reintroduce the dimensions via t0 ¼ 1=a and x20 ¼ b=a. In the domain where Tc 1=D (i.e., mðaÞ 1), we then have the scaling Tc

xa0 ða=bÞa=2 jxmin ja ¼ ¼ D D D

ð120Þ

by analogy with the result reported in Ref. 91. However, we emphasiz two caveats based on our results: (i) The linear behavior in 1=D is not valid over the entire a-range. For larger values, a ’ 1:6, the scaling exponent mðaÞ assumes nontrivial values; then, the simple scaling used to establish Eq. (120) has to be modiﬁed. It is not immediately obvious how this should be done systematically. (ii) From relation (120) it cannot be concluded that the mean crossing time is independent of the barrier height V, despite the fact that Tc depends on the distance jxmin j from the barrier only. The latter statement is obvious from the expressions for xmin and V derived for our model potential: The location of the minima relative to the barrier is in fact coupled to the barrier height. Therefore, a random walker subject to Le´vy noise senses the potential barrier and does not simply move across it with the characteristic time given by the free mean-square displacement. Apparently, the activation for the mean crossing time as a function of noise strength D varies only as a power law instead of the standard exponential behaviour.

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481

The time dependence of the probability density dSðtÞ=dt for ﬁrst barrier crossing time of a Le´vy ﬂight process is exponential, just as the standard Brownian case. This can be understood qualitatively because the process is Markovian. From the governing dynamical equation (115), it is clear that the relaxation of modes is exponential, compare Ref. 46. For low noise strength D, the barrier crossing will be dominated by the slowest time-eigenmode ’ el1 t with eigenvalue l1 . This is indeed similar to the ﬁrst passage time problem of Le´vy ﬂights discussed in the previous section. VI.

DISSIPATIVE NONLINEARITY

The alleged ‘‘pathology’’ of Le´vy ﬂights is related to their divergent variance, unless conﬁned by a steeper than harmonic external potential. There indeed exist examples of processes where the diverging variance does not pose a problem: for example, diffusion in energy space [93], or the Le´vy ﬂight in the chemical coordinate of diffusion along a polymer chain in solution, where Le´vy jump length statistics are invoked by intersegmental jumps, which are geometrically short in the embedding space [94]. Obviously however, for a particle with a ﬁnite mass moving in Euclidian space, the divergence of the variance is problematic.7 There are certain ways of overcoming this difﬁculty: (i) by a time cost through coupling between x and t, producing Le´vy walks [45,98], or (ii) by a cutoff in the Le´vy noise to prevent divergence [99,100]. While (i) appears a natural choice, it gives rise to a nonMarkov process. Conversely, (ii) corresponds to an ad hoc measure. A.

Nonlinear Friction Term

Here, we pursue an alternative, physical way of dealing with the divergence; namely, inclusion of nonlinear dissipative terms. They provide a mechanism, that naturally regularizes the Le´vy stable PDF PðV; TÞ of the velocity distribution. Dissipative nonlinear structures occur naturally for particles in a frictional environment at higher velocities [101]. A classical example is the Riccati equation MdvðtÞ=dt ¼ Mg KvðtÞ2 for the motion of a particle of mass M in a gravitational ﬁeld with acceleration g [102], autonomous oscillatory systems with a friction that is nonlinear in the velocity [101,103], or nonlinear corrections to the Stokes drag as well as drag in turbulent ﬂows [104]. The occurrence of a non-constant friction coefﬁcient gðVÞ leading to a nonlinear dissipative force 7 Note that in fact the regular diffusion equation includes a similar ﬂaw, although less signiﬁcant: Due to its parabolic nature, it features an inﬁnite propagation speed; that is, even at very short times, there exists a ﬁnite value of Pðx; tÞ for large jxj. In that case, this can be removed by invoking the telegrapher’s (Cattaneo) equation [95–97]. (Editor’s note: For a critical discussion of this procedure, see Risken [12, p. 257 et seq.)

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gðVÞV was highlighted in Klimontovich’s theory of nonlinear Brownian motion [105]. In what follows, we show that dissipative nonlinear structures regularize a stochastic process subject to Le´vy noise, leading to ﬁnite variance of velocity ﬂuctuations and thus a well-deﬁned kinetic energy. The velocity PDF PðV; tÞ associated with this process preserves the properties of the Le´vy process for smaller velocities; however, it decays faster than a Le´vy stable density and thus possesses a physical cutoff. In what follows, we start with the asymptotic behavior for large V and then address the remaining, central part of PðV; TÞ, that preserves the Le´vy stable density property. B.

Dynamical Equation with Le´vy Noise and Dissipative Nonlinearity

The Langevin equation for a random process in the velocity coordinate V is usually written as [59] dVðtÞ þ gðVÞVðtÞ ¼ a ðtÞ dt

ð121Þ

with the constant friction g0 ¼ gð0Þ. a ðtÞ is the a-stable Le´vy noise deﬁned in terms of a characteristic function (see Section I). The characteristic function of the velocity PDF PðV; tÞ, Pðk; tÞ FfPðV; tÞg is then governed by the dynamical equation [59] qPðk; tÞ qPðk; tÞ ¼ g0 k Djkja Pðk; tÞ qt qk

ð122Þ

This is exactly the V-space equivalent of the Le´vy ﬂight in an external harmonic potential discussed in the introduction. Under stationary conditions the characteristic function assumes the form Djkja Pst ðk; tÞ ¼ exp ð123Þ g0 a So that the PDF PðV; tÞ converges toward a Le´vy stable density of index a. This stationary solution possesses, however, a diverging variance. To overcome the divergence of the variance hV 2 ðtÞi, we introduce into Eq. (121) the velocity-dependent dissipative nonlinear form gðVÞ for the friction coefﬁcient [101,105]. We require gðVÞ to be symmetric in V [105], assuming the virial expansion up to order 2N gðVÞ ¼ g0 þ g2 V 2 þ þ g2N V 2N

; g2N > 0

ð124Þ

The coefﬁcients g2n are assumed to decrease rapidly with growing n (n 2 N). To determine the asymptotic behavior, it is sufﬁcient to retain the highest power 2N.

fundamentals of le´vy ﬂight processes

483

More generally, we will consider a power gn jVjn with n 2 Rþ and gn > 0. We will show that, despite the input driving Le´vy noise, the inclusion of the dissipative nonlinearity (124) ensures that the resulting process possesses a ﬁnite variance. To this end, we pass to the kinetic equation for PðV; tÞ, the fractional Fokker– Planck equation [20,46,54,60,64] qPðV; tÞ q qa PðV; tÞ ¼ ðVgðVÞPðV; tÞÞ þ D qt qV qjVja

ð125Þ

The nonlinear friction coefﬁcient gðVÞ thereby takes on the role of a conﬁning potential: while for g0 ¼ gð0Þ the drift term Vg0 , as mentioned before, is just the restoring force exerted by the harmonic Ornstein–Uhlenbeck potential, the next higher-order contribution g2 V 3 corresponds to a quartic potential, and so forth. The fractional operator qa =qjVja in Eq. (125) for the velocity coordinate for 1 < a < 2 is explicitly given by [20,64] d a PðVÞ d2 a ¼ k dV 2 djVj

1 ð

PðV 0 Þ

1

jV V 0 ja1

dV 0

ð126Þ

by analogy with the x-domain operator (31), with k being deﬁned in Eq. (107). C.

Asymptotic Behavior

To derive the asymptotic behavior of PðV; tÞ in the presence of a particular form of gðVÞ, it is sufﬁcient to consider the highest power, say, gðVÞ gn jVjn . In particular, to infer the behavior of the stationary PDF ÐPst ðVÞ for V ! 1, it is 1 reasonable to assume that we can truncate the integral 1 dV 0 in the fractional 0 operator (126) at the pole V ¼ V, since the domain of integration for the remaining left-side operator is much larger than the cutoff right-side domain. Moreover, the remaining integral over ð1; V also contains the major portion of the PDF. For V ! þ1, we ﬁnd in the stationary state after integration over V, gn V

nþ1

d Pst ðVÞ ’ Dk dV

ðV 1

Pst ðV 0 Þ ðV V 0 Þa1

dV 0

ð127Þ

We then use the ansatz Pst ðVÞ C=jVjm , m > 0. With the approximation ÐV Ð1 ÐV 0 0 a1 dV 0 V 1a 1 Pst ðV 0 ÞdV 0 V 1a 1 Pst ðV 0 ÞdV 0 ¼ 1 Pst ðV Þ=ðV V Þ V 1a we obtain the asymptotic form Pst ðVÞ ’

Ca D gn jVjm

;m ¼ a þ n þ 1

ð128Þ

484

aleksei v. chechkin et al. -2

α = 1.5; γ2 = 0.0; γ4 = 0.00; Ntra = 15,000 , N = 25,000 α = 1.5; γ2 = 0.1; γ4 = 0.00; Ntra = 30,000 , N = 25,000 α = 1.2; γ2 = 0.1; γ4 = 0.01; Ntra = 100,000, N = 50,000 Slope -1.5 Slope -3.5 Slope -5.2

ln V Pst(V)

-4

-6

-8

-10

-12 0.5

1

1.5

2

2.5

3 ln V

3.5

4

4.5

5

5.5

Figure 21.

Power-law asymptotics of the stationary PDF, ln–ln scale. We observe the expected scaling with exponent m from Eq. (128). In the graph, we also indicate the number Ntra of trajectories of individual length N simulated to produce the average PDF.

valid for V ! 1 due to symmetry. We conclude that for all n > ncr ¼ 2 a the variance hV 2 i is ﬁnite, and thus a dissipative nonlinearity whose highest power n exceeds the critical value ncr counterbalances the energy supplied by the Le´vy noise a ðtÞ. D.

Numerical Solution of Quadratic and Quartic Nonlinearity

Let us consider dissipative nonlinearity up to the quartic order contribution, gðVÞ ¼ g0 þ g2 V 2 þ g4 V 4 . According to the previous result (128), the stationary PDF for the quadratic case with g2 > 0 and g4 ¼ 0 falls off like Pst ðVÞ jVja3 , and thus 8a 2 ð0; 2Þ the variance hV 2 i is ﬁnite. Higher-order moments such as the fourth-order moment hV 4 i are, however, still inﬁnite. In contrast, if g4 > 0, the fourth-order moment is ﬁnite. We investigate this behavior numerically by solving the Langevin equation (121); compare Ref. 64 for details. In Fig. 21 we show the asymptotic behavior of the stationary PDF Pst ðVÞ for three different sets of parameters. Clearly, in all three cases the predicted powerlaw decay is obtained, with exponents that, within the estimated error bars agree well with the predicted relation for m according to Eq. (128).8 8

From the scattering of the numerical data after repeated runs, see Fig. 7.

485

fundamentals of le´vy ﬂight processes -2

γ2 = 0.0001, γ4 = 0 γ2 = 0, γ4 = 0.000001 Slope -1 Slope -3 Slope -5

-4

ln V Pst(V)

-6 -8 -10 -12 -14

1

2

3

4 ln V

5

6

7

Figure 22. Stationary PDF Pst ðVÞ for g0 ¼ 1:0 and (i) g2 ¼ 0:0001 and g4 ¼ 0, and (ii) g2 ¼ 0 and g4 ¼ 0:000001, with a ¼ 1:0. The lines indicate the slopes 1, 3, and 5.

E.

Central Part of PðV; tÞ

The nonlinear damping (124) mainly affects larger velocities, while smaller velocities (V 1) are mainly subject to the lowest-order friction gð0Þ. We therefore expect that in the central region close to V ¼ 0, the PDF PðV; TÞ preserves it Levy stable density character. This is demonstrated in Fig. 22, where the initial power-law decay of the Levy stable density eventually gives way to the steeper decay caused by the nonlinear friction term. In general, the PDF shows transitions between multiple power laws in the case when several higher-order friction terms are retained. The turnover point from the unaffected Levy stable density to steeper decay caused by nonlinear friction depends on the ratio g0 : g2n , where 2n is the next higher-order nonvanishing friction coefﬁcient. In Fig. 23, we show the time evolution of the variance hV 2 ðtÞi for various combinations of Le´vy index a and magnitude g2 of the quadratic nonlinearity (g0 ¼ 1:0 and g4 ¼ 0:0). For all cases with ﬁnite g2 (g2 ¼ 0:1), we ﬁnd convergence of the variance to a stationary value. For the two smaller a values (1.2 and 1.5), we observe some ﬂuctuations; however, these are comparatively small with respect to the stationary value they oscillate around. For a ¼ 1:8, the ﬂuctuations are hardly visible, and in fact the stationary value is practically the same as in the Gaussian case a ¼ 2:0. In contrast, the case with vanishing

486

aleksei v. chechkin et al. α = 2.0; γ2 = 0.0 α = 1.2; γ2 = 0.1 α = 1.5; γ2 = 0.1 α = 1.8; γ2 = 0.1 α = 1.2; γ2 = 0.0

1400

2

1000 1.5 800 600

1

2

1200

2

(α = 1.2, γ2 = 0)

2.5

1600

400 0.5 200 0

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t Figure 23. Variance hV 2 ðtÞi as function of time t, with the quartic term set to zero, g4 ¼ 0 and g0 ¼ 1:0 for all cases. The variance is ﬁnite for the cases a ¼ 2:0; g2 ¼ 0:0; a ¼ 1:2; g2 ¼ 0:1; a ¼ 1:5; g2 ¼ 0:1; and a ¼ 1:8; g2 ¼ 0:1. These correspond to the left ordinate. For the case a ¼ 1:2; g2 ¼ 0:0, the variance diverges and strong ﬂuctuations are visible; note the large values of this curve corresponding to the right ordinate.

g2 (and a ¼ 1:2) clearly shows large ﬂuctuations requiring a right ordinate whose span is roughly two orders of magnitude larger than that of the left ordinate. Similarly, in Fig. 24, we show the fourth order moment hV 4 ðtÞi as a function of time. It is obvious that only for ﬁnite g4 (g4 ¼ 0:01 and a ¼ 1:8) the moment converges to a ﬁnite value that is quite close to the value for the Gaussian case (a ¼ 2:0) for which all moments converge. In contrast to this behavior, both examples with vanishing g4 exhibit large ﬂuctuations. These are naturally much more pronounced for smaller Le´vy index (a ¼ 1:2, corresponding to the right ordinate). F.

Discussion

Strictly speaking, all naturally occurring power-laws in fractal or dynamic patterns are ﬁnite. Scale-free models nevertheless provide an efﬁcient description of a wide variety of processes in complex systems [16,20,46,106]. This phenomenological fact is corroborated by the observation that the power-law properties of Le´vy processes persist strongly even in the presence of cutoffs [99]

487

fundamentals of le´vy ﬂight processes α = 2.0; α = 1.8; α = 1.2; α = 1.8;

7

900

γ2 = 0.0; γ4 = 0.00 γ2 = 0.1; γ4 = 0.01 γ2 = 0.1; γ4 = 0.00 γ2 = 0.1; γ4 = 0.00

800 700

6

600 500 4 400 300

2

4

3

4

5

(α = 1.2, γ4 = 0.0)

8

200

1

100 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

Figure 24. Fourth-order moment hV 4 ðtÞi as function of t, with g0 ¼ 1. hV 4 ðtÞi converges to a

ﬁnite value for the two cases a ¼ 2 (Gaussian) and a ¼ 1:8 with g4 ¼ 0:01. The other two examples with vanishing quartic contribution (g4 ¼ 0) show large ﬂuctuations—that is, diverging hV 4 ðtÞi. Note that the case a ¼ 1:2 and g4 ¼ 0 corresponds to the right ordinate.

and, mathematically, by the existence of the generalized central limit theorem due to which Le´vy stable laws become fundamental [69]. A categorical question is whether in the presence of Le´vy noise, there exists a physical cause to remove the consequential divergences. A possible, physically reasonable answer is given by introducing a nonconstant friction coefﬁcient gðVÞ, as occurs in various classical systems. Here, we present a concise derivation of the regularization of a stochastic process in velocity space driven by Le´vy stable noise, in the presence of dissipative nonlinearities. These dissipative nonlinearities remove the divergence of the kinetic energy from the measurable subsystem of the random walker. In idealized mathematical language, the surrounding heat bath provides an inﬁnite amount of energy through the Le´vy noise, and the coupling via the nonlinear friction dissipates an inﬁnite amount of energy into the bath, and thereby introduces a natural cutoff in the kinetic energy distribution of the random walker subsystem. Physically, such divergences are not expected, but correspond to the limiting behavior associated with large numbers in probability theory. In this section, we showed that both statements can be reconciled, and that Le´vy processes are indeed physical.

488

aleksei v. chechkin et al. VII.

SUMMARY

A hundred years after Einstein’s seminal work [4], the theory of stochastic processes has been put on solid physical and mathematical foundations, at the same time playing a prominent role in many branches of science [36,107–109]. Le´vy ﬂights represent a widely used tool in the description of anomalous stochastic processes. By their mathematical deﬁnition, Le´vy ﬂights are Markovian and their statistical limit distribution emerges from independent identically distributed random variables, by virtue of the central limit theorem. Despite this quite straightforward deﬁnition, Le´vy ﬂights are less well understood than one might at ﬁrst assume. This is due to their strongly nonlocal character in space, these long-range correlations spanning essentially the entire available geometry; as exempliﬁed by the inﬁnite range of the integration boundaries in the associated fractional operator. In this review, we have addressed some of the fundamental properties of random processes, these being the behaviour in external force ﬁelds, the ﬁrst passage and arrival behaviour, as well as the Kramers-like escape over a potential barrier. We have examined the seemingly pathological nature of Le´vy ﬂights and showed that dissipative non-linear mechanisms cause a natural cutoff in the PDF, so that with a ﬁnite experimental range the untruncated Le´vy ﬂight still provides a good description. These investigations have been almost entirely based on fractional diffusion and Fokker–Planck equations with a fractional Riesz derivative and have turned out to be a convenient basis for mathematical manipulations, while at the same time being easy to interpret in the context of a dynamical approach. Acknowledgments We would like to thank Iddo Eliazar and Igor M. Sokolov for helpful discussions.

VIII.

APPENDIX. NUMERICAL SOLUTION METHODS

In this appendix, we brieﬂy review the numerical techniques, which have been used in this work to determine the PDF from the fractional Fokker–Planck equation [Eq. (38)] and the Langevin equation [Eq. (37)]. A.

Numerical solution of the fractional Fokker–Planck equation [Eq. (38)] via the Gru¨nwald–Letnikov Method

From a mathematical point of view, the fractional Fokker–Planck equation [Eq. (38)] is an ﬁrst-order partial differential equation in time, and of nonlocal, integrodifferential kind in the position coordinate x. It can be solved numerically via an efﬁcient discretization scheme following Gru¨nwald and Letnikov [110–112].

fundamentals of le´vy ﬂight processes

489

Let us designate the force component on the right-hand side of Eq. (38) as q dV P ð129Þ F ðx; tÞ qx dx and the diffusion part as a ðx; tÞ q Pa D qjxj

ð130Þ

With these deﬁnitions, we can rewrite Eq. (38) in terms of a discretisation scheme as Pj;nþ1 Pj;n j;n ¼ Fj;n þ D t

ð131Þ

where we encounter the term Pjþ1;n Pj1;n c2 Fj;n ¼ xj ðc 1ÞPj;n þ xj 2x

ð132Þ

which is the force component of the potential VðxÞ ¼ jxjc =c. Here, t and x are the ﬁnite increments in time and position, such that tn ¼ ndt and xj ¼ jx, for n ¼ 0; 1; . . . ; N and j ¼ 0; 1; . . . ; J, and Pj;n Pðxj ; tn Þ. Due to the inversion symmetry of the kinetic equation (38), it is sufﬁcient to solve it on the right semiaxis. In the evaluation of the numerical scheme, we deﬁne xJ such that the PDF in the stationary state is sufﬁciently small, say, 103 , as determined from the asymptotic form (64). In order to ﬁnd a discrete time and position expression for the fractional Riesz derivative in Eq. (130), we employ the Gru¨nwald–Letnikov scheme [110–112], whence we obtain j;n ¼ D

J X

1 xq Pjþ1q;n þ Pj1þq;n a 2ðxÞ cosðpa=2Þ q¼0

where xq ¼ ð1Þq with

a q

a aða 1Þ . . . ða q þ 1Þ=q!; ¼ 1; q

ð133Þ

ð134Þ

q>0 q<0

ð135Þ

and 1 < a 2. Note that in the limiting case a ¼ 2 only three coefﬁcients differ from zero, namely, x0 ¼ 1, x1 ¼ 2, and x2 ¼ 1, corresponding to the

490

aleksei v. chechkin et al.

Figure 25. Coefﬁcients xq in Gru¨nwald-Letnikov approximation for different values of the Le´vy index a ¼ 1:9, 1.5, and 1.1.

standard three-point difference scheme for the second order derivative, d 2 gðxj Þ=dx2 ðgjþ1 2gj þ gj1 Þ=ðxÞ2 . In Fig. 25, we demonstrate that with decreasing a, an increasing number of coefﬁcients contribute signiﬁcantly to the sum in Eq. (133). This becomes particularly clear in the logarithmic representation in the bottom plot of Fig. 25. We note that the condition m t=ðxÞa < 0:5

ð136Þ

is needed to ensure the numerical stability of the discretisation scheme. In our numerical evaluation, we use x ¼ 103 , and therefore the associated time increment t 105 . . . 106 , depending on the actual value of a. The initial condition for Eq. (131) is P0;0 ¼ 1=x. In Fig. 26, the time evolution of the PDF is shown together with the evolution of the force and diffusion components deﬁned by Eqs. (129) and (130),

fundamentals of le´vy ﬂight processes

Figure 26.

491

Further details of the Gru¨nwald–Letnikov scheme. Left: Time evolution of the PDF as obtained by numerical solution of Eq. (131) at c ¼ 4 and a ¼ 1:2. Right: Time evolution of the diffusion component (130) (thick lines), and of the force term (129) (thin lines).

492

aleksei v. chechkin et al.

respectively. Accordingly, at the initial stage of the relaxation process, the diffusion component prevails. The force term grows with time, until in ! D . This is particularly visible in the bottom right the stationary state F part of Fig. 26, which corresponds to the stationary bimodal state shown to the left. B.

Numerical Solution of the Langevin Equation [Eq. (25)]

An alternative way to obtain the PDF is to sample the trajectories determined by the Langevin equation [Eq. (25)]. To this end, Eq. (37) is discretized in time according to xnþ1 ¼ xn þ Fðxn Þt þ ðtÞ1=a a ðntÞ

ð137Þ

with tn ¼ nt for n ¼ 0; 1; 2; . . . , and where Fðxn Þ is the dimensionless force ﬁeld at position xn . The sequence fa ðntÞg is a discrete-time approximation of a white Le´vy noise of index a with a unit scale parameter. That is, the sequence of independent random variables possessing the characteristic function ^ p ¼ expðjkja Þ. To generate this sequence fa ðntÞg, we have used the method outlined in Ref. 113. References 1. T. L. Carus, De rerum natura (50 BC), On the Nature of Things, Harvard University Press, Cambridge, MA, 1975. 2. J. Ingenhousz, Nouvelles expe´riences et observations sur divers objets de physique, T. Barrois le jeune, Paris, 1785. 3. R. Brown, Philos. Mag. 4, 161; (1828); Ann. Phys. Chem. 14, 294; (1828). 4. A. Einstein, Ann. Phys. 17, 549 (1905); ibid. 19, 371 (1906); R. Fu¨rth, Ed., Albert Einstein— Investigations on the Theory of the Brownian Movement, Dover, New York, 1956. (Einstein actually introduced the name ‘‘Brownian motion,’’ although he did not have access to Brown’s original work.) 5. J. Perrin, Comptes Rendus (Paris) 146, 967 (1908); Ann. Chim. Phys. VIII 18, 5 (1909). 6. A. Fick, Ann. Phys. (Leipzig) 94, 59 (1855). 7. A. Westgren, Z. Phys. Chem. 83, 151 (1913); ibid. 89, 63 (1914); Arch. Mat. Astr. Fysik 11, 00 (1916); Z. Anorg. Chem. 93, 231 (1915); ibid. 95, 39 (1916). 8. E. Kappler, Ann. Phys. (Leipzig) 11, 233 (1931). 9. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). 10. P. Le´vy, Processus stochastiques et mouvement Brownien, Gauthier–Villars, Paris, 1965. 11. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North–Holland, Amsterdam, 1981, 2nd ed., 1992, Reprinted 1997. 12. H. Risken, The Fokker–Planck Equation 2nd ed., Springer-Verlag, Berlin, 1989.

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77. A. V. Chechkin, R. Metzler, J. Klafter, V. Y. Gonchar, and L. V. Tanatarov, J. Phys. A 36, L537 (2003). 78. A. M. Mathai and R. K. Saxena, The H–Function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd., New Delhi 1978. 79. W. G. Glo¨ckle and T. F. Nonnenmacher, J. Stat. Phys. 71, 755 (1993). 80. H. Scher, G. Margolin, R. Metzler, J. Klafter, and B. Berkowitz, Geophys. Res. Lett. 29, 10.1029/2001GL014123 (2002). 81. R. Metzler and J. Klafter, Biophys. J. 85, 2776 (2003). 82. F. D. Gakhov, Boundary Value Problems, Pergamon Press, Oxford, 1966. 83. H. A. Kramers, Physica 7, 284 (1940). 84. P. Ha¨nggi, P. Talkner, and M. Bokrovec, Rev. Mod. Phys. 62, 251 (1990). 85. R. Metzler and J. Klafter, Chem. Phys. Lett. 321, 238 (2000). 86. A. V. Chechkin, V. Y. Gonchar, J. Klafter, and R. Metzler, Europhys. Lett., 00, 000 (2005). 87. P. D. Ditlevsen, Geophys. Res. Lett. 26, 1441 (1999). 88. J. Fajans and A. Schmidt, Nucl. Instrum. & Methods in Phys. Res. A 521, 318 (2004). 89. P. Ha¨nggi, T. J. Mroczkowski, F. Moss, and P. V. E. McClintock, Phys. Rev. A 32, 695 (1985). 90. A. V. Chechkin and V. Y. Gonchar, Physica A 27, 312 (2000) 91. P. D. Ditlevsen, Phys. Rev. E 60, 172 (1999) 92. Y. L. Klimontovich, Statistical Theory of Open Systems, Vol. 1, Kluwer Academic Publishers, Dordrecht, 1995. 93. G. Zumofen and J. Klafter, Chem. Phys. Lett. 219, 303 (1994). 94. I. M. Sokolov, J. Mai, and A. Blumen, Phys. Rev. Lett. 79, 857 (1997). 95. G. Cattaneo, Atti. Sem. Mat. Fis. Univ. Modena 3, 83 (1948). 96. P. C. de Jagher, Physica A 101, 629 (1980). 97. G.Zumofen and J. Klafter, Phys. Rev. E 47, 851 (1993). 98. M. F. Shlesinger, J. Klafter, and Y. M. Wong, J. Stat. Phys. 27, 499 (1982). 99. R. N. Mantegna and H. E. Stanley, Phys. Rev. Lett. 73, 2946 (1994). 100. I. M. Sokolov, A. V. Chechkin, and J. Klafter, Physica A 336, 245 (2004). 101. N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear oscillations, Hindustan Publishing Corp., Delhi, distributed by Gordon & Breach, New York, 1961. 102. H. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962. 103. A. A. Andronow, C. E. Chaikin, and S. Lefschetz S., Theory of Oscillations, Princeton University Press, Princeton, NJ, 1949. 104. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics, Pergamon Press, London, 1966. 105. Y. L. Klimontovich, Turbulent Motion and the Structure of Chaos: A New Approach to the Statistical Theory of Open Systems, Kluwer, Dordrecht, NL, 1992. 106. J. M. Halley, S. Hartley, A. S. Kallimanis, W. E. Kunin, J. J. Lennon, and S. P. Sgardelis, Ecology Lett. 7, 254 (2004). 107. E. Frey and K. Kroy, Ann. Phys. 14, 20 (2005).

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CHAPTER 10 DISPERSION OF THE STRUCTURAL RELAXATION AND THE VITRIFICATION OF LIQUIDS KIA L. NGAI Naval Research Laboratory, Washington, DC 20375, USA RICCARDO CASALINI Naval Research Laboratory, Washington, DC 20375, USA; and Chemistry Department, George Mason University, Fairfax, Virginia 20030, USA SIMONE CAPACCIOLI Dipartimento di Fisica and INFM, Universita` di Pisa, I-56127, Pisa, Italy; and CNR-INFM Center ‘‘SOFT: Complex Dynamics in Structured Systems,’’ Universita` di Roma ‘‘La Sapienza,’’ I-00185 Roma, Italy MARIAN PALUCH Institute of Physics, Silesian University, 40-007 Katowice, Poland C. M. ROLAND Naval Research Laboratory, Washington, DC 20375, USA

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

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I. Introduction II. Invariance of the a-Dispersion to Different Combinations of T and P at Constant ta A. Molecular Glass-Formers B. Amorphous Polymers C. Implication of T–P Superpositioning of the a-Dispersion at Constant ta D. Invariance of the a-Dispersion to Different T–P at Constant ta Investigated by Techniques Other than Dielectric Spectroscopy III. Structural Relaxation Properties Are Governed by or Correlated with the a-Dispersion IV. The Primitive Relaxation and the Johari–Goldstein Secondary Relaxation V. The JG Relaxation and Its Connection to Structural Relaxation A. Pressure Dependence of tJG B. Invariance of tJG to Variations of T and P at Constant ta C. Non-Arrhenius Temperature Dependence of tb Above Tg D. Increase of tb on Physical Aging E. JG Relaxation Strength and Its Mimicry of Enthalpy, Entropy, and Volume F. The Origin of the Dependences of Molecular Mobility on Temperature, Pressure, Volume, and Entropy is in tJG or t0 VI. The Coupling Model A. Background B. The Correspondence Between t0 and tJG C. Relation Between the Activation Enthalpies of tJG and ta in the Glassy State D. Explaining the Properties of tJG E. Consistency with the Invariance of the a-Dispersion at Constant ta to Different Combinations of T and P F. Relaxation on a Nanometer Scale G. Component Dynamics in Binary Mixtures H. Interrelation Between Primary and Secondary Relaxations in Polymerizing Systems I. A Shortcut to the Consequences of Many-Molecule Dynamics and a Pragmatic Resolution of the Glass Transition Problem VII. Conclusion Acknowledgments References

I.

INTRODUCTION

The glass transition, which refers to the dramatic slowing down of kinetic processes, such as viscous ﬂow and molecular reorientations, is a general phenomenon found in organic, inorganic, metallic, polymeric, colloidal, and biomolecular materials. On decreasing temperature T or increasing pressure P, the structural relaxation time ta of supercooled liquids, or analogously the local segmental relaxation time of polymers, becomes increasingly large. Eventually the molecules cannot attain their dynamic equilibrium conﬁgurations and vitriﬁcation commences. The science and technology of glass formation has a long history, with the ﬁrst recorded recipe for glass appearing a few millennia ago in Babylon

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(in present-day Iraq), while modern glasses are formed from many different starting materials, including not only inorganic glasses but also organic glasses, metallic glasses, and many plastics. The use of glass is widespread, and the glassmaking industry contributes signiﬁcantly to the world economy. In spite of the long history and technological signiﬁcance of glass, a universally accepted, fundamental understanding of the dynamics of materials undergoing vitriﬁcation is still lacking. There is not even a consensus concerning the factors governing the dramatic slowing down of the structural relaxation and related kinetic processes, such as viscous ﬂow and molecular reorientations. This situation is highly unusual—most problems in the physical and materials sciences have been solved within a few decades—and is a testament to the complexity of the structural relaxation process in the precursor supercooled liquid. Development of a microscopic and quantitatively accurate theory of the glass transition, applicable to real materals, has become even more challenging with the improvement of experimental techniques and the introduction of new ones; these have led to the discovery of an increasing number of general properties of the dynamics of glass-forming materials spanning the range from picoseconds to years [1,2]. Some of these general properties are discussed in sections to follow in this chapter. Acceptance of a theory or model of the glass transition requires more than offering a description of selected properties. A theory is viable only if it can explain, or at least be consistent with, all the known general properties. The most obvious dynamic property, being the immediate cause of vitriﬁcation, is the divergence of ta with decreasing temperature at constant pressure. This temperature dependence can be represented in the vicinity of the glass transition by the Vogel–Fulcher–Tammann–Hesse (VFTH) equation [3–5] or the equivalent Williams–Landel–Ferry (WLF) equation [6]. The factor governing the divergence of ta was identiﬁed as unoccupied volume in the free volume model [6,7], and conﬁgurational entropy in the Adam and Gibbs [8] entropy model. An increase of pressure, similar to a temperature decrease, reduces both the free volume and conﬁgurational entropy and slows down structural relaxation. The free volume model has been extended to include the effect of hydrostatic pressure [6,7,9], and an extension of the Adam–Gibbs model for elevated pressures has been proposed [10–16]. Offshoots of these two classic approaches to describing vitriﬁcation have been proposed and are generally more sophisticated, without deviating from the basic idea that free volume or entropy is the factor controlling ta (T,P). However, such models do not address other general properties of the dynamics of glassformers—for example, the dispersion of the structural relaxation times ta . The dispersion is ignored or at best considered as an afterthought. If derived, the dispersion is obtained separately from ta , entailing other factors or assumptions. These are the common traits of not only the free volume and conﬁgurational entropy models, but practically all known theories and models of the glass transition, notwithstanding any differences in the underlying physics.

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Accordingly, since the dispersion and ta are obtained independently as separate and unrelated predictions, in such models the dispersion (or the time/frequency dependence) of the structural relaxation bears no relation to the structural relaxation time. This means it cannot govern the dynamic properties. As have been shown before [2], and will be further discussed in this chapter, several general properties of the dynamics are well known to be governed by or correlated with the dispersion. Therefore, neglect of the dispersion means a model of the glass transition cannot be consistent with the important and general properties of the phenomenon. The present situation makes clear the need to develop a theory that connects in a fundamental way the dispersion of relaxation times to ta and the various experimental properties. This chapter is organized as follows: 1. At the outset, we emphasize the fundamental importance of the dispersion of the structural relaxation by presenting a recently discovered experimental fact for many glass-formers—that is, that the dispersion of the structural relaxation remains unchanged for widely different combinations of temperature and pressure, provided that the most probable structural relaxation time ta is constant. Certainly ta can be constant for different combinations of temperature and pressure because of their compensating effects on the molecular mobility, even though the speciﬁc or free volume, entropy or conﬁgurational entropy, and static structure factor may change. However, if the dispersion of the structural relaxation is derived independently of ta , it is not expected to be constant for these same combinations of T and P because the two quantities do not necessarily have the same dependence on volume, entropy, and so on. This apparently general property implies that the dispersion of the structural relaxation is deﬁned by ta, or at least ta and the dispersion have to be coupled predictions of any viable theoretical interpretation. If the dispersion of the structural relaxation is derived independently of ta , as in conventional theories, it is unlikely that ta would uniquely deﬁne the dispersion. This observation and the conclusions drawn from it have important consequences for the study of the glass transition. 2. The dispersion is shown to govern or correlate with various dynamic properties, further indicating that the dispersion plays a fundamental role in the glass transition. In the case of amorphous polymers, the dispersion of the structural relaxation (i.e., of the local segmental relaxation) even inﬂuences the relation of the relaxation to mechanisms at longer times and larger length-scales and the viscoelastic spectrum as a whole. 3. Although vitriﬁcation is related directly to structural relaxation, processes that occur at earlier times may underly structural relaxation and thus ultimately the glass transition itself. The following scenario is presented for the evolution of the dynamics as a function of time. At short times, molecules are mutually caged and cannot relax by reorientation or translation. The cage starts to decay

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(loss of near-neighbor order) at intermediate times by independent (primitive) relaxation, associated with a relaxation time t0 identiﬁable with the relaxation time tJG of a special class of secondary relaxation which we call the Johari– Goldstein (JG) b-relaxation [17–18,19,20]. The primitive relaxation is the initiator of the ensuing many-molecule relaxation dynamics, which evolve in time to become increasingly ‘‘cooperative,’’ merging eventually into structural relaxation. There is a strong correlation between the ratio ta =tJG and the dispersion of the structural relaxation at any ﬁxed ta . This correlation suggests that the dispersion originates from the primitive or JG relaxation, through the evolution of the many-molecule relaxation dynamics. This is another indication that the dispersion of the structural relaxation is a fundamental quantity, which along with ta is a consequence of the many-molecule relaxation dynamics. 4. Experimental data are presented to show that the JG relaxation mimics the structural relaxation in its volume–pressure and entropy–temperature dependences, as well as changes in physical aging. These features indicate that the dependences of molecular mobility on volume–pressure and entropy–temperature have entered into the faster JG relaxation long before structural relaxation, suggesting that the JG relaxation must be considered in any complete theory of the glass transition. 5. A model having predictions that are consistent with the aforementioned experimental facts is the Coupling Model (CM) [21–26]. Complex many-body relaxation is necessitated by intermolecular interactions and constraints. The effects of the latter on structural relaxation are the main thrust of the model. The dispersion of structural relaxation times is a consequence of this cooperative dynamics, a conclusion that follows from the presence of fast and slow molecules (or chain segments) interchanging their roles at times on the order of the structural relaxation time ta [27–29]. The dispersion of the structural relaxation can usually be described by the Kohlrausch–William–Watts (KWW) [30,31] stretched exponential function, fðtÞ ¼ exp½ðt=ta ÞbKWW

ð1Þ

The fractional exponent bKWW can be rewritten as (1 n), where n is the coupling parameter of the CM. The breadth of the dispersion is reﬂected in the magnitude of n and increases with the strength of the intermolecular constraints. The dispersion and the structural relaxation time are simultaneous consequences of the many-molecule dynamics, and hence they are related to each other. The intermolecularly cooperative dynamics are built upon the local independent (primitive) relaxation, and thus a relation between the primitive relaxation time t0 and ta is expected to exist. The CM does not solve the many-body relaxation problem but uses a physical principle to derive a relation between ta and t0 that involves the dispersion parameter, n. This deﬁning relation of the CM

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has many applications. Although t0 is not predicted, nor are its T and P dependences described by the model, it can be calculated from the parameters, ta and n, of the measured structural relaxation. The primitive relaxations have properties similar to the Johari–Goldstein (JG) secondary relaxation (but they are not identical if the JG relaxation is interpreted in the conventional sense and not in the CM way) and good correspondence between their relaxation times, t0 and tJG , is found for many glass-formers [32–44]. From t0 tJG and the aforementioned properties of tJG , we can infer that t0 also has properties that mimic ta . Moreover, from t0 tJG and the CM relation (to be given later), both the dispersion of the structural relaxation and tJG are shown to be invariant to changes of temperature and pressure while ta is constant, in accord with experimental ﬁndings. Furthermore, we show that the difﬁcult problem of tackling the structural relaxation time ta and its properties, which involves complex many-body interactions, is made easy through the CM by starting from the tractable independent (primitive) relaxation time t0 . The intermolecular interactions/constraints of a glass-former can be changed by various means including (a) conﬁnement in nanometer space, location at a free surface, and at interfaces with another material, (b) mixture with another glass-former, and (c) increase in the number of covalent bonds by polymerization or chemical cross-links. Some of these modiﬁcations of a glass-former introduce factors that inﬂuence the dispersion, other than intermolecular interactions/constraints. An example is concentration ﬂuctuation in a mixture. Therefore, in some cases, it is more appropriate to consider the change in the coupling parameter of the CM instead of the dispersion. We discuss the changes of dynamics with various modiﬁcations of the material and explain the changes by the CM. II.

INVARIANCE OF THE a-DISPERSION TO DIFFERENT COMBINATIONS OF T AND P AT CONSTANT sa

Studies of molecular dynamics have focused on the effect of temperature due largely to experimental convenience. Isobaric measurements of relaxation times and viscosities are carried out routinely as a function of temperature. From these experiments it is well established that the shape of the adispersion (i.e., the KWW stretch exponent bKWW ), when compared at Tg or some other reference value of ta , varies among different glass-formers [45,46]. Many experimental studies have shown also that for a given material, very often the distribution of relaxation times systematically broadens with decreasing temperature [47–50]. Less common than temperature studies at ambient pressure are experiments employing hydrostatic pressure, although dielectric measurements at high pressure were carried out nearly half a century ago [31,51–60]. Recently,

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pressure has been employed as an experimental variable in broadband dielectric spectroscopy [12,40,44,61–96]. The relaxation time at ambient pressure can be maintained constant at elevated pressure P by raising the temperature T. Various combinations of P and T can be chosen for which the a-loss peak frequency na (and ta ) are the same. One important fact emerging from these pressure studies is that at a constant value of the structural relaxation time ta or frequency na, the dispersion of the structural relaxation is constant [97–99]. In cases where the height of the a-loss peak, e00max , changes somewhat, the dispersions at a ﬁxed na have to be compared after the measured dielectric loss e00 ðnÞ is normalized by e00max. Very generally it is found that for a given material at a ﬁxed value of ta , the relaxation function is constant, independent of thermodynamic conditions (temperature and pressure). Alternatively stated, temperature-pressure superpositioning works for the dispersion of the structural a-relaxation at constant ta . Lack of superposition may occur at frequencies sufﬁciently high compared with na . Such deviation can be attributed to the contribution to dielectric loss from a resolved or unresolved secondary relaxation at higher frequencies, whose dielectric relaxation strength does not have the same P and T dependences as the a-relaxation. In order to demonstrate convincingly that this is a general experimental fact of glass-formers, experimental data for many different materials and (for a particular material) experimental data for several dielectric relaxation times are presented herein. The glass-formers include both molecular liquids and amorphous polymers of diverse chemical structures. All show the property of temperature–pressure superpositioning of the dispersion of the structural a-relaxation at constant ta . A.

Molecular Glass-Formers

Numerous molecular glass-forming liquids that have narrow dispersions of the a-relaxation and have an excess wing on the high-frequency ﬂank, but otherwise no resolved secondary relaxation in their dielectric spectra. There are experimental results [38,39,100–103] indicating that the excess wing is an unresolved Johari–Goldstein secondary relaxation [17,18,19,20,104]. The materials include cresolphthalein-dimethylether (KDE) [40], phenylphthaleindimethylether (PDE) [71], propylene carbonate (PC) [72], chlorinated biphenyl (PCB62) [12], phenyl salicylate (salol) [73], 3,30 ,4,40 -benzophenonetetracarboxylic dianhydride (BPTCDaH) [44], 1,10 -di(4-methoxy-5-methylphenyl)cyclohexane (BMMPC) [74]. For these materials, due to the unresolved secondary relaxation, a strong dependence of the shape of the dispersion on T and P (with ta varying) is especially evident; see, for example, refs. 12 and 40. The fact that at a ﬁxed value of ta the dispersion of the a-relaxation is constant, independent of T and P, is demonstrated in Fig. 1a for KDE, Fig. 1b for PC, and Fig. 1c for PCB62

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Figure 1. Dielectric loss data at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time for (a) cresolphthalein-dimethylether (KDE) ta , (b) propylene carbonate (PC) (loss normalized to the value of the maximum of the a-loss peak), (c) chlorinated biphenyl (PCB62), (d) diisobutyl phthalate (DiBP), and (e) Dielectric loss of dipropyleneglycol dibenzoate (DPGDB). Loss normalized to the value of the maximum of the a-loss peak. The dc conductivity contribution has been subtracted. Triangles are isothermal measurements at T ¼ 253 K and P ¼ 48, 72, 94, 115, 142, 163 MPa (from right to left). Black symbols are isobaric measurements done at P ¼ 0:1 MPa and T ¼ 244, 240, 236.7, 233, 229, 226 K (from right to left). The spectrum at T ¼ 226 K has been shifted along the x axis by multiplying frequency by a factor 1.3. (f) Dielectric loss of benzoyn isobutylether (BIBE) at different T and P. The dc conductivity contribution has been subtracted. Spectra obtained at higher P are normalized to the value of the maximum of the loss peak obtained at the same frequency at atmospheric pressure. From right to left: Black lines are atmospheric pressure data at T ¼ 271 K (a), 263 K (b), 253 K (c), 240 K (d), 236 K (e), 230 K (f), 228 K (g), 226 K (h), 223 K (i), 220.5 K (j), 218 K (k). Symbols represent high-pressure data: T ¼ 278:5 K and P ¼ 32 MPa (a), 65 MPa (b), 118 MPa (c), 204 MPa (d), 225 MPa (e), 320 MPa (h), 370 MPa (j), 396 MPa (k); T ¼ 288:2 K and P ¼ 350 MPa (f), 370 MPa (g), 423 MPa (i), 450 MPa (j); T ¼ 298 K and P ¼ 330 MPa (d), 467 MPa (h).

(experimental details of these measurements can be found respectively in Refs.12, 40, and 72). In each ﬁgure, data are used to show that this property holds for more than one value of ta . The same results are found for PDE, BPTCDaH, BMMPC, and salol and are shown in Figs. 2, 3, 4, and 5, respectively.

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Figure 2. Dielectric loss data of phenylphthalein-dimethylether (PDE) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta .

Some molecular glass-formers have a resolved secondary relaxation whose peak frequency is practically pressure independent; these are not Johari– Goldstein (JG) processes (according to the deﬁnition given in Ref. 38). The slower JG relaxation is not resolved from the a-relaxation in the equilibrium liquid state, but in some cases it can be observed in the glassy state. Such liquids include 1,10 -bis(p-methoxyphenyl)cyclohexane (BMPC) [75], diethyl phthalate, (DEP) [76], di-n-butyl phthalate (DBP) [77], diisobutyl phthalate (DiBP) [77], di-isooctal phthalate (DiOP) [78], decahydroisoquinoline (DHIQ) [79], dipropyleneglycol dibenzoate (DPGDB) [80], benzoin-isobutylether (BIBE) [81], the epoxy compounds including diglycidyl ether of bisphenol-A (EPON828) [82], 4,4’-methylenebis(N,N-diglycidylaniline) (MBDGA) [83,84], bisphenol-A-propoxylate(1 PO/phenol)diglycidylether) (1PODGE) [85], N,N-diglycidyl-4-glycidyloxyaniline (DGGOA) [86], and N,N-diglycidylaniline (DGA) [86]. For all members of this class of glass-formers, a constant dispersion is associated with a ﬁxed value of ta , independent of thermodynamic conditions (T and P). We show this (for more than one value of ta ) with data in Fig. 1d for DiBP, Fig. 1e for DPGDB, and Fig. 1f for BIBE. The experimental details for these can be found in respectively Refs. 77, 80, and 81. The same

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Figure 3.

property holds also for DiBP (Fig. 1d), BMPC (Fig. 6), DEP (Fig. 7), DiOP (Fig. 8), DHIQ (Fig. 9), EPON828 (Fig. 10), 1PODGE (Fig. 11), and MBDGA, DGGOA, and DGA (Fig. 12). Earlier dielectric studies under elevated pressure [20,105–110] had found temperature–pressure superpositioning at constant ta in a few molecular glassformers including ortho-terphenyl (OTP), di(2-ethylhexyl) phthalate, tricresyl phosphate, polyphenyl ether, and reﬁned naphthenic mineral oil, although the temperature and pressure ranges are not as wide as achieved in more recent measurements.

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Figure 4. Dielectric loss data of 1,10 -di(4-methoxy-5-methylphenyl)cyclohexane (BMMPC) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant arelaxation time ta . The dashed line is the imaginary part of the one-sided Fourier transform of the KWW function with bKWW ð1 nÞ ¼ 0:55. The logarithmic ordinate scale makes evident the presence of an excess wing at higher frequencies.

B.

Amorphous Polymers

Dielectric relaxation measurements under pressure have been carried out on several amorphous polymers, and for all cases studied the dispersion of the local segmental relaxation (i.e., the structural a-relaxation) conforms to temperature–pressure superpositioning at constant ta . These polymers include: polyvinylmethylether (PVME) [87]; poly(vinylacetate) (PVAc) [88]; poly(ethylene-co-vinyl acetate) (EVA, having 70 wt% vinyl acetate) [89]; polymethylphenylsiloxane (PMPS) [90]; poly(methyltolylsiloxane) (PMTS) [91]; 1,2polybutadiene (1,2-PBD, also referred to as polyvinylethylene, PVE) [92]; poly(phenyl glycidyl ether)-co-formaldehyde (PPGE) [93]; 1,4-polyisoprene (PI) [94]; poly(propylene glycol) (molecular weight: 4000 Da), PPG-4000 [95]; poly(oxybutylene), POB [96]; and poly(isobutyl vinylether), PiBVE [111]. Constant dispersions at a ﬁxed value of ta independent of thermodynamic conditions (T and P) are shown for more than one ta in Fig. 13a for PVAc, Fig. 13b for PMTS, Fig. 13c for PPGE, and Fig. 13d for POB. Experimental details for these measurements can be found in Refs. 88, 91, 93, and 96, respectively.

508

kia l. ngai et al.

10 0

salol

ε ''

fβ(510) 10–1

fβ(0.1) T = 223 K, P = 0.1 MPa T = 297 K, P = 380 MPa T = 309 K, P = 510 MPa T = 323 K, P = 640 MPa

10–2 10–3

10–1

101 103 frequency [Hz]

105

Figure 5. Dielectric loss data of phenyl salicylate (salol) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta . The dashed line represents extrapolation of the power law for frequencies just past the maximum.

Note that for POB there is a dielectrically active normal mode at lower frequencies, which has different P and T dependences than the local segmental mode. The T–P superposition at ﬁxed ta holds also for PVME (Fig. 14), EVA (Fig. 15), PMPS (Fig. 16), PVE (Fig. 17), PPG-4000 (Fig. 18), and PiBVE (Fig. 19). For PMPS this superposition is maintained even when ta varies. As illustrated in some of these ﬁgures, all the a-loss peaks are well-ﬁtted by the one-sided Fourier transform of the KWW over the main part of the dispersion. Thus, the experimental fact of constant dispersion at constant ta can be restated as the invariance of the fractional exponent bKWW (or the coupling parameter n) at constant ta . In other words, ta and bKWW (or n) are co-invariants of changing thermodynamic conditions (T and P). If w is the full width at half-maximum of the dielectric loss peak normalized to that of an ideal Debye loss peak, there is an approximate relation between w and n given by n ¼ 1:047ð1 w1 Þ [112]. Hydrogen-bonded networks or clusters, if present, are modiﬁed at elevated pressure and temperature, changing the structure of the glass-former in the process. This occurs, for example, in glycerol [113], propylene glycol dimer and trimer (2PG and 3PG) [101,102], and m-ﬂuoraniline [44]. These

dispersion of the structural relaxation Isotherm T = 253.5 K e2p = 300 bar e2p = 398 bar e2p = 506 bar e2p = 702 bar e2p = 904 bar e2p = 1104 bar eps2p = 1280 bar Ambient pressure: Eps2T-25.00

0.1

ε' '

509

0.01

BMPC 1E-3 10–3 10–2 10–1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 f [Hz] Figure 6. Master curve of the dielectric loss data of 1,10 -bis(p-methoxyphenyl)cyclohexane (BMPC). The spectra measured under pressure were shifted on the frequency scale to superpose with the a-loss peak at T ¼ 248 K and ambient pressure. The secondary relaxation of BMPC is not a JG relaxation (its loss peak frequency is pressure-insensitive), and it is not temperature–pressuresuperposable along with the a-loss peak.

hydrogen-bonded glass-formers do not obey temperature–pressure superpositioning at constant ta , since not just the relaxation time, but the material itself, is changing with changes in T and P. Such behavior is shown for glycerol and threitol in Fig. 20, 2PG in Fig. 21, and m-ﬂoraniline in Fig. 22. In higher members of the polyols, such as xylitol and sorbitol, the departure from T–P superpositioning at constant ta is small compared with the lower member glycerol. This is shown for xylitol in Fig. 23.

C.

Implication of T–P Superpositioning of the a-Dispersion at Constant sa

We now discuss the impact of this general property on theories and models of the glass transition. The primary concern of most theories is to explain the temperature and pressure dependences of the structural relaxation time ta . The dispersion (n or bKWW ) of the structural relaxation is either not addressed or

510

kia l. ngai et al.

ε ' ' / ε ' 'max

1

α−process

T = 190.2 K; P = 0.1 MPa T = 293.6 K; P = 1.05 GPa

β−process 0.1

1.2 decade

diethyl phthalate 0.01 10–2 10–1 10 0

10 1

10 2 10 3 f [Hz]

10 4

10 5

10 6

10 7

Figure 7. Dielectric loss data of diethyl phthalate (DEP) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta .

else considered separately with additional input not involved in arriving at ta . For example, the original free-volume models and the Adam–Gibbs model treat the variation of relaxation times with T and P but do not predict the distribution of molecular relaxation rates. Additional input such as a speciﬁc ﬂuctuation and distribution of some parameter must be introduced to generate a distribution of relaxation times consistent with the empirical KWW time correlation function, as done in some modern versions of these theories. It is not difﬁcult for any of model to ﬁnd combinations of T and P such that the predicted ta (T,P) is constant. However, it is unlikely that the same combinations will also keep the predicted dispersion or bKWW (T,P) constant. For one glass-former it may be possible to introduce additional assumptions to force both ta (T,P) and bKWW (T,P) to be simultaneously constant. However, this would not be a worthwhile undertaking since ta (T,P) and bKWW (T,P) are simultaneously constant for many glassformers, with different physical and chemical structures and broadly different sensitivities to temperature and density [62,114,115]. Thus, the experimental observations (i.e., simultaneous constancy of ta (T,P) and bKWW (T,P)) has direct impact on the theoretical efforts to understand the glass transition, past and present. We are able to conclude that conventional theories and models, in which the structural relaxation time does not deﬁne or govern the dispersion of the

dispersion of the structural relaxation

ε''

1

511

T = 30 C, p = 1 GPa T = 76C P = 1.6 GPa T = -74 C, p = 1 bar

0.1

DiOP di-isooctyl phthalate

10–310–210–1100 101 102 103 104 105 106 107 108 f [Hz]

Figure 8. Dielectric loss data of di-isooctal phthalate (DiOP) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant loss peak frequency na or equivalently at constant a-relaxation time ta .

structural relaxation, cannot explain the simultaneous constancy of ta (T,P) and the dispersion of the a-relaxation or bKWW (T,P). Revision is required to bring them to consistency with this general experiment fact. D. Invariance of the a-Dispersion to Different T–P at Constant sa Investigated by Techniques Other than Dielectric Spectroscopy The spectra for molecular and polymeric glass-formers shown above to demonstrate the invariance of the a-dispersion to T and P at constant ta , were

512

kia l. ngai et al. decahydroisoquinoline (DHIQ) 1

β KWW =0.4

ε ''/ ε ''max

T = 232 K P = 0.5 GPa

β-process

T = 183 K P = 0.1 MPa

10

–3

10

–1

10

1

10

3

10

5

10

7

f [Hz]

Figure 9. Dielectric loss data of decahydroisoquinoline (DHIQ) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta .

acquired by dielectric spectroscopy. This reﬂects the utility of dielectric spectroscopy in investigating broadband dynamics under pressure, particularly with recent instrumental developments. [62]. Therefore, an extensive database of broadband dielectric spectra of different materials in different T–P conditions is available, enabling an assessment of the T–P superpositioning. Although we believe the phenomena to be quite general, there is a paucity of data from other experimental techniques. However, some results are available, as described below. For polymeric systems, only a few Photon Correlation Spectroscopy (PCS) studies have been carried out, and these were done more than 20 years ago. Within the experimental resolution, the shape of the a-relaxation was found to be essentially invariant to temperature and pressure at ﬁxed ta , and time–

513

dispersion of the structural relaxation EPON828 1

ε"/ ε''max

T = 6 C, P = 0.1 MPa T = 20 C, P = 110 MPa

0.1 102

103

104 105 Frequency [Hz]

106

107

Figure 10. Dielectric loss data of diglycidyl ether of bisphenol-A (EPON828) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant arelaxation time ta .

temperature–pressure (tTP) superposition was invoked. The investigated systems were poly(ethylacrylate) [116,117], poly(methylacrylate) [118], and polystyrene [119]. A recent PCS experiment done on polypropyleneoxide also found that tTP superposition held. [120]. The available studies of molecular glass-formers are wider, for both materials and techniques. For ortho-terphenyl (OTP), the Kohlrausch parameters for the a-relaxation by PCS at different temperatures and pressures [121,122] have been reported. The stretching parameters bKWW versus a-relaxation time ta for different T and P fall, within the experimental uncertainty, on a single curve, with bKWW slightly decreasing with increasing ta . On the other hand, a shape invariance of the a-relaxation has been observed for OTP by speciﬁc heat spectroscopy under elevated pressure [123]. Additionally, experiments done on

514

kia l. ngai et al.

10

2

bisphenol-A-propoxylate(1 PO/phenol)diglycidylether

259 K, 0.1 MPa 266 K, 34.5 MPa 273 K, 75.0 MPa 280 K, 134 MPa

1

ε''

10

10

0

–1

10

10–2

10–1

10 0

10 1

10 2

10 3

10 4

10 5

frequency [Hz] Figure 11. Dielectric loss data of bisphenol-A-propoxylate(1 PO/phenol)diglycidylether) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant arelaxation time ta .

OTP by neutron scattering at different T and P revealed a negligible dependence of the shape of the structural relaxation, while the static structure factor yielded a master curve only for isochronal conditions—that is, for constant relaxation time ta [124]. Salol is another system for which the invariance under pressure of the a-relaxation shape was reported. Recent PCS experiments [125,126] revealed that the correlation functions acquired at different pressures up to 180 MPa and at room temperature superposed. The stretching parameter bKWW was 0.68, in agreement with the PCS measurements done at ambient pressure [127,128]. Very accurate PCS measurements for different T and P conditions were carried out on different molecular glass-forming systems by Patkowski and co-workers, including epoxy oligomers [129–131], and the van der waals liquids PDE [132,133], BMPC [134], and BMMPC [135]. In most of the investigated systems, master curves are obtained for bKWW (T,P) of the a-relaxation plotted versus ta (T,P), with the value decreasing (broader dispersion) as the dynamics slow (longer ta (T,P).). This is another evidence that the a-dispersion is directly related to the relaxation time.

dispersion of the structural relaxation

p = 538 MPa T = 297 K

101

ε''

515

p = 0.1 MPa T = 276 K

100 MBDGA 10–1 p = 79 MPa T = 273 K

ε''

101

p = 0.1 MPa T = 252 K

100 DGGOA 10–1 p = 98 MPa T = 263 K

101

ε''

p = 0.1 MPa T = 299K 100 DGA 10–1 10–3 10–2 10–1 100 101 102

103 104 105 106 107

frequency [Hz] Figure 12. Dielectric loss data of MBDGA (top), DGGOA (middle), and DGA (bottom) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant arelaxation time ta . Open symbols represent the relaxation curves measured at ambient pressure, and solid symbols represent those at elevated pressures.

We can conclude that the result from dielectric spectroscopy—that the adispersion is invariant to T and P at constant ta —appears to be quite general, with respect to both the material and the experimental technique. The limitation is only that sufﬁciently broad spectra must be obtained under different conditions of temperature and pressure.

516

kia l. ngai et al. (a) PVAc

0

0

ε"

10

(c) PPGE

10

363.1 K, 0.1 MPa 363.1 K, 200 MPa 363.1 K, 100 MPa 393.1 K, 350 MPa 393.1 K, 400 MPa 393.1 K, 250 MPa

-1

10

–3

–2

–1

0

1

2

3

4

5

6

7

10 10 10 10 10 10 10 10 10 10 10 10

-1

–3

8

(b) PMTS

–2

10 10 10 10

–1

0

1

2

3

4

5

6

7

10 10 10 10 10 10 10 10 10

8

0

(d) POB

ε"

10

280 K, 60 MPa 270 K, 0.1 MPa 292 K, 60 MPa 281 K, 0.1 MPa

-1

10

10

-2 -1

10

-3

297.4 K, 952 MPa 204.6 K, 0.1 MPa 297.4 K, 763 MPa 213.9 K, 0.1 MPa 297.4 K, 395 MPa 237.9 K, 0.1 MPa

260 K, 0.1 MPa 303 K, 137 MPa 265 K, 0.1 MPa 303 K, 118 MPa 283 K, 0.1 MPa 313 K, 81 MPa

10 –3 –2 –1 0 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 10 10 10 frequency [Hz]

–3

–2

–1

10 10 10 10 0 101 10 2 10 3 10 4 10 5 10 6 10 7 10 8 frequency [Hz]

Figure 13. Dielectric loss data at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta , for (a) poly(vinylacetate) (PVAc), (b) poly(methyltolylsiloxane) (PMTS), and (c) poly(phenyl glycidyl ether)-co-formaldehyde (PPGE); d)poly(oxybutylene) (POB). In all cases, spectra obtained at higher P are normalized to the value of the maximum of the loss peak obtained at the same frequency at atmospheric pressure.

III. STRUCTURAL RELAXATION PROPERTIES ARE GOVERNED BY, OR CORRELATED WITH, THE a-DISPERSION Not only does ta uniquely deﬁne the dispersion, as shown herein, but also many properties of ta are governed by, or correlated with, the dispersion of the structural relaxation or the fractional exponent bKWW ð¼ 1 nÞ of the KWW function that describes it. Some examples of these properties have been described in reviews [2,22]. Here we mention brieﬂy a few examples. 1. The steepness or ‘‘fragility’’ index, deﬁned by m d log10 ta = dðTg =TÞjTg =T¼1 , at ambient pressure increases with increasing n [45]. Glass-formers in general conform broadly to this correlation, although many exceptions are known, and in particular the correlation breaks down at elevated pressure [98]. Strict applicability may require restricting considerations

517

dispersion of the structural relaxation 1.0

o

PVME

P = 0.1MPa T = 250.6 K o P = 211 MPa T = 282.7 K o P = 334 MPa T = 296.1 K o P = 629 MPa T = 322.4 K

ε"/ε"max

0.8

0.6

0.4

0.2

0.0 –3 10

–2

10

–1

10

0

1

10 10 Frequency [Hz]

2

10

3

10

Figure 14. Dielectric loss data of polyvinylmethylether (PVME) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta .

to materials belonging to the same family, such as the polyols (glycerol, threitol, xylitol, and sorbitol) [136] or carbon-backbone amorphous polymers [46,137–139]. This is because, as discussed later, the temperature dependence of ta is governed not only by the dispersion, but also by the speciﬁc volume, V and entropy S. Chemically different glass-formers can have widely different dependence of ta on V [114,115]. Thus, the correlation between m and n can break down among chemically dissimilar glass-formers. An example is propylene carbonate, when considered together with glycerol, threitol, xylitol, and sorbitol. Among these, propylene carbonate has the narrowest dielectric relaxation dispersion (i.e., smallest n), but its m is larger than that of glycerol and threitol [36,140]. The correlation also breaks down in the same glass-former under different pressures [98]. In general, m decreases with P [141]. From the previous section, since n or the dispersion is invariant at ta ¼ 102 s (a time customarily used to deﬁne Tg at any given pressure), the fact that m decreases with P means that the correlation between m and n necessarily breaks down.

518

kia l. ngai et al. poly(ethylene-co-vinyl acetate) 1.0

EVA

max

0.8

0.6

0.4 63.1°C, 386 MPa 42.3°C, 232 MPa 23.0°C, 99 MPa 7.0°C, 0.1 MPa

0.2

0.0 –1 10

0

10

1

10

2

3

10 10 Frequency [Hz]

4

10

5

10

6

10

Figure 15. Dielectric loss data of poly(ethylene-co-vinyl acetate) (EVA, with 70 wt% vinyl acetate) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta .

2. Quasielastic neutron scattering experiments and molecular dynamics simulations on polymeric and nonpolymeric glass-formers have found that the dependence of ta on the scattering vector Q is given by Q2=ð1nÞ [24,142–151]. Hence the Q dependence of ta is governed by the breadth of the dispersion or n. Such Q dependence of the relaxation time is also shared by other interacting systems including suspensions of colloidal particles [152], semidilute polymer solutions [153–155], associating polymer solutions [156,157], and polymer cluster solutions [158]. 3. The temperature dependence of ta observed over more than 12 decades from sub-nanoseconds to 100 s cannot be ﬁt by a single Vogel–Fulcher– Tammann–Hesse (VFTH) equation [159,160]. At short times and temperatures higher than TA , ta has an Arrhenius dependence. Below TA, ta has a VFTH dependence, (VFTH)1, which is no longer adequate when temperature falls below TB. A second VFTH equation, (VFTH)2, has to be used to describe ta for T < TB . At TA, n(TA) is small. There is a signiﬁcant increase of the rate of change of n(T) with decreasing temperature when crossing TB. The difference

519

dispersion of the structural relaxation 10 –1 Poly(methylphenylsiloxane)

scaled ε

''

10 –2

1.5 MPa 21.4 MPa 42.4 MPa 72.2 MPa 59.8 MPa

10 –3

–2

10

–1

10

0

10

1

10

2

3

4

5

10 10 10 10 shifted frequency (Hz)

6

10

7

10

Figure 16. Dielectric loss curves of polymethylphenylsiloxane (PMPS) measured at a constant temperature (T ¼ 273 K) and different pressures. The data have been shifted to superimpose onto the data for P ¼ 42:4 MPa.

between (VFTH)1 and (VFTH)2 correlates with the width of the a-relaxation dispersion or n(Tg), if Tg is deﬁned uniformly as the temperature at which ta reaches an arbitrarily chosen long time, say 102 s [161–164]. The crossover from (VFTH)1 to (VFTH)2 was observed isobarically also at elevated pressures. The crossover temperature TB generally increases with applied pressure P, but the value of ta or the viscosity at the crossover is the same for a given glassformer [141,165,166].Two examples, BMMPC and PCB62, are shown in Fig. 24 and 25, respectively. All glass-formers studied have the same dispersion

520

kia l. ngai et al. 8 1,2- polybutadiene 7

1000 ε'' / ε''max

6 5 4 3 2 T = 273 K, p = 21.1 MPa T = 278 K, p = 41.3 MPa T = 283 K, p = 61.4 MPa

1 0 1

10

10

2

3

10

10 f / f max

4

10

5

10

6

Figure 17. Dielectric loss data of 1,2-polybutadiene (1,2-PBD) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta .

at constant ta , independent of T and P, a general property discussed in Section II. Hence, the dispersion is invariant at the crossover from (VFTH)1 to (VFTH)2 when the latter is observed for different combinations of T and P. 4. The rotational diffusion coefﬁcient, Dr, of a probe molecule in a glassformer follows the temperature dependence of the Debye–Stokes–Einstein (DSE) equation [167–171], Dr

1 kT ¼ 6htc i 8pZrs3

ð2Þ

Here Z is the shear viscosity, htc i is the mean rotational correlation time, and rs the spherical radius of the probe molecule. On the other hand, the translational diffusion coefﬁcient, Dt, of the probe molecule is given by the Stokes–Einstein (SE) relation [167–171], Dt ¼

kT 6pZrs

ð3Þ

521

dispersion of the structural relaxation 1.0

PPG4000

0.8

ε"/ε"α

0.6

0.4 T = 293 K 318.8 MPa 612.5 MPa T = 268 K 79.5 MPa 317.6 MPa

0.2

0.0 –2 10

–1

10

0

10 ω τα

1

10

2

10

Figure 18. Dielectric loss data of poly(propylene glycol) (PPG-4000, molecular weight: 4000 Da) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta .

Thus, the combined SE and the DSE equations predict that the product Dt tc ðDt tc ÞSE;DSE should equal 2rs2 =9. Measurements of probe translational diffusion and rotational diffusion made in glass-formers have found that the product Dt t can be much larger than this value, revealing a breakdown of the Stokes–Einstein (SE) relation and the Debye–Stokes–Einstein (DSE) relation. There is an enhancement of probe translational diffusion in comparison with rotational diffusion. The time dependence of the probe rotational time correlation functions r(t) is well-described by the KWW function, rðtÞ ¼ rðtÞ ¼ rð0Þ exp½ðt=tc ÞbKWW

ð4Þ

The ratio Dt tc =ðDt tc ÞSE;DSE evaluated at T ¼ Tg is a measure of the degree of breakdown of the SE and DSE relations for various combinations of probes

522

kia l. ngai et al.

PiBVE

s1 s2 s3 s4 s5 s6

0.12 -log ~4 max

0.12 -logmax~3 0.09

ε"

0.09

0.06 s1: T = 293 K, P = 20 MPa s2: T = 298 K, P = 30 MPa s3: T = 303 K, P = 60 MPa s4: T = 313 K, P = 60 MPa s5: T = 323 K, P = 90 MPa s6: T = 323 K, P = 150 MPa

0.06

3

10

10

4

0.03 5

103

104

105

s1 s2 s3 s4 s5

0.1

ε"

s1: T = 293 K, P = 60 MPa s2: T = 298 K, P = 90 MPa s3: T = 303 K, P = 90 MPa s4: T = 308 K, P = 120 MPa s5: T = 313 K, P = 150 MPa s6: T = 323 K, P = 210 MPa

102

10

0.15 -log ~2 max

s1 s2 s3 s4 s5 s6

0.05 s1: T = 293 K, P = 100 MPa s2: T = 298 K, P = 120 MPa s3: T = 303 K, P = 150 MPa s4: T = 308 K, P = 180 MPa s5: T = 313 K, P = 210 MPa

10

0

10

1

2

10 10 f (Hz)

3

10

4

10

5

Figure 19. Dielectric loss data of poly(isobutyl vinylether) PiBVE at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the arelaxation at constant a-loss peak frequency na or equivalently at constant a-relaxation time ta , for three different ta or the corresponding loss peak frequency na. Data supplied by G. Floudas [K. Mpoukouvalas, G. Floudas, B. Verdonck, and F. E. Du Prez, Phys. Rev. E 72, 011802 (2005).].

and host glass-formers [167–171]. A strong correlation was observed at T ¼ Tg between the quantity Dt tc =ðDt tc ÞSE;DSE and the dispersion of the probe rotational correlation functions r(t) (see Fig. 26). A more enhanced probe translation compared with probe rotation is found for hosts having correlation functions that are more dispersive—that is, larger n or smaller bKWW values. Hence, the dispersion is related to the degree of breakdown of SE and DSE relations, a general property of glass-forming liquids. The variation of the dispersion was traced to the difference between the probe rotation time and the host structural relaxation time [172]. When the probe is identical to the host, probe diffusion becomes selfdiffusion in a neat glass-former. By extrapolating the results of probe/host systems, the breakdown of SE and DSE relations in neat glass-formers is expected, and the correlation between Dt tc =ðDt tc ÞSE;DSE and n ð1 bKWW Þ

dispersion of the structural relaxation (a)

glycerol

10

H

523

CH2OH OH CH2OH

ε ''

1

0.1

0.1 MPa, 201 K 1.8 GPa, 255 K

(b)

threitol

10

H HO

CH 2OH OH H CH 2OH

ε '' 1

0.1 10 –3

0.1 MPa, 239 K 0.54 GPa, 257 K 0.1 MPa, 224 K 0.54 GPa, 241 K

10 –1

10 1 10 3 f [Hz]

10 5

10 7

Figure 20. Dielectric loss data of glycerol and threitol at various combinations of temperature and pressure as indicated to demonstrate the departure of invariance of the dispersion of the a-relaxation at constant loss peak frequency na or equivalently at constant a-relaxation time ta .

should still hold. Actually, the breakdown of SE and DSE relations in neat glassformers was discovered more than three decades earlier in 1,3-bis-(1-naphthyl)5-(2-naphthyl)benzene (TNB) and 1,2-diphenylbenzene (OTP) [160,173–177] and was recently reconﬁrmed using modern techniques [178]. The enhancement of translational/diffusional motions has been ascribed to spatially heterogeneous dynamics [178,179]. In this view, regions of differing dynamics give rise to nonexponential relaxation (dispersion) in ensemble average measurements. The decoupling between self-diffusion and rotation occurs because Dt is related to an average over 1/t of the distribution, emphasizing the fast dynamics, while tc is related to an average over t of the distribution, which would be determined primarily by the slowest molecules. Experimentally, the product Dt tc of TNB is equal to ðDt tc ÞSE;DSE for T 1:28Tg , and it increases montotonically with decreasing temperature to reach a value about 400 times ðDt tc ÞSE;DSE . Within this view, an increasing product Dt tc is associated with the enlarging relaxation time dispersion (i.e.,

524

kia l. ngai et al. 0

10

ε ''/ ε " max

PG dimer

–1

10

199 K 0.1 MPa 216.7 K 222.8 MPa KWW ( β = 0.67)

–2

10

KWW

–4

–3

–2

–1

0

1

2

3

4

5

10 10 10 10 10 10 10 10 10 10

frequency [Hz] Figure 21. Dielectric loss data of PG dimer at various combinations of temperature and pressure as indicated to demonstrate the departure of invariance of the dispersion of the a-relaxation at constant loss peak frequency na or equivalently at constant a-relaxation time ta .

Figure 22.

Dielectric loss spectrum of m-FA at 279 K and 1.69 GPa (&), 1.60 GPa (^), 1.52 GPa data (), and 1.4 GPa data (~). Dielectric loss spectrum of mFA at ambient pressure and 174 K (&), 177 K data (), and 180 K (). The dashed lines are ﬁts to the data at 279 K and under GPa pressures by the one-sided Fourier transform of the KWW function. The solid lines are similar ﬁts to the ambient pressure data. The vertical arrows indicate the calculated primitive relaxation frequencies, n0 , for all the data sets.

525

dispersion of the structural relaxation 11 10

10

9 8 1

7

ε ''

6 10

5

0

10

2

10

4

10

6

4 3 Xylitol

2 1

T = 258 K, P = 1 atm. T = 293 K, P = 1 GPa

0 –2

10

10

–1

10

0

1

10

10

2

10

3

4

10

10

5

10

6

7

10

f [Hz] Figure 23. Dielectric loss data of xylitol at various combinations of temperature and pressure as indicated to demonstrate the much smaller departure of invariance of the dispersion of the a-relaxation at constant loss peak frequency na or equivalently at constant a-relaxation time ta .

decreasing bKWW ) as T is lowered towards Tg. Therefore, this explanation requires a concomitant temperature dependence of the dispersion. However, dielectric measurements of the dispersion of the a-relaxation of TNB found it to be independent of temperature in the range Tg < T < 1:23Tg [180]. Thus, the decoupling of rotational translational dynamics cannot be explained in the manner described by spatially heterogeous dynamics, which, like other properties including the a-dispersion per se, is a consequence of the manymolecule a-relaxation dynamics. Although the dispersion is consistent with spatial dynamic heterogenity, the former is not a derived consequence of the latter. Both are parallel consequences of many-molecule relaxation. An alternative explanation is based on the dispersion and its dependence on the dynamic variables probed (i.e., rotation versus translation) [172]. The dispersion of rotational diffusion and of the shear viscoelastic response is broader than that of translational diffusion or the mean-square displacement of the molecule. It is a consequence of the cooperative dynamics that the dispersion of different dynamic variables for the same substance can be different and the

526

kia l. ngai et al. 4

t

ns

BMMPC

–4

MP a

=

co

0.1

20

0M Pa 60

Log(τ[s])

0 –2

V

0M Pa

2

–6 –8 –10 0.7 0.6

φT

0.5 0.4 0.3 0.2 1.6

2.0

2.4

2.8

3.2

3.6

4.0

1000/T[K] Figure 24. Upper panel: Dielectric relaxation time for BMMPC experimental data for 0.1 MPa, other isobars (200 and 600 MPa), and the isochore at V ¼ 0:9032 ml/g were calculated. Dotted line indicates the average of log10 ðtB Þ ¼ 6:1 for the different curves. Lower panel: Stickel function, with low- and high-T linear ﬁts, done over the range 4:68 < log10 ðt½sÞ < 3:85 and 8:55 < log10 ðt½sÞ < 6:4, respectively. Vertical dotted lines indicate the dynamic crossover.

dynamic variable having a broader dispersion usually has a stronger temperature dependence [181–186]. Thus, the breakdown of SE and DSE relations in glassforming liquids is a special case of a general phenomenon [172,187]. 5. The a-relaxation involves cooperative and heterogeneous dynamics of many molecules (or chain segments), which at any temperature deﬁne a lengthscale Ldh . The dispersion of the a-relaxation is also a consequence of the many-body dynamics. Naturally we expect a larger Ldh to be associated with a broader dispersion, because both quantities directly reﬂect the intermolecularly cooperative dynamics. This correlation is borne out by comparing bKWW with Ldh for glycerol, ortho-terphenyl, and poly(vinylacetate) as obtained by multidimensional 13C solid-state exchange NMR experiments [188]. 6. Amorphous polymers have relaxation processes transpiring at longer times and longer length-scales than the local segmental relaxation, which is the analog of the structural a-relaxation of molecular glass formers. For

dispersion of the structural relaxation

527

4 PCB62 t

a

=

MP

V

co

0 .1

–4

ns

MP a

a MP

–2

400

Log(τ[s])

0

200

2

–6 –8 0.8

φT

0.6

0.4

0.2 1.5

2.0

2.5

3.0

3.5

4.0

1000/T[K] Figure 25. Upper panel: dielectric relaxation times for PCB62 experimental data for 0.1 MPa; other isobars and isochoric curve at V ¼ 0:6131 ml/g were calculated. Dotted line indicates the average of log10 ðtB Þ ¼ 5:9 for the different curves. Lower panel: Stickel function, with low- and high-T linear ﬁts, done over the range 5:42 < log10 ðt½sÞ < 2:169 and 8:87< log10 ðt½sÞ < 5:98, respectively. The vertical dotted lines in both panels represent the dynamic crossover.

unentangled linear polymers, these processes are referred to as the Rouse modes. For entangled polymers, the long-time processes include the Rouse modes of chain units between entanglements and the terminal, entangled chain modes. It has been well documented that the Rouse and the terminal relaxation times have a weaker temperature dependence than ta of the local segmental relaxation, leading to breakdown of time–temperature superpositing in the viscoelastic response of amorphous polymers [189–209]. The width of the dispersion of the local segmental relaxation, or the bKWW appearing in the exponent of the KWW ﬁtting function, determines the difference in the temperature dependences. For example, from light scattering measurements, polystyrene has bKWW ¼ 0:36 [210] and polyisobutylene has bKWW ¼ 0:55 [211]. The breakdown of time–temperature superposition of viscoelastic data is much more pronounced in polystyrene than in polyisobutylene [190,191, 204,208].

528

kia l. ngai et al.

log(DTτc/(DTτc)DSE/SE) @ T=Tg

4

3

2

1

0 1.0

0.8

0.6 β (T=Tg)

0.4

0.2

Figure 26. Correlation between enhanced translation log½Dt tc Þ=ðDt tc ÞSE;DSE at Tg and the KWW exponent of the probe rotational correlation function at Tg in four matrices: OTP, TNB, polystyrene (PS), and polysulfone (PSF). The probes are tetracene, rubrene, anthracene, and BPEA. The symbols represent PS/tetracene (closed circle), PS/rubrene (open circle), PSF/tetracene (closed triangle), PSF/ rubrene (open triangle), OTP/tetracene (cloded square), OTP/rubrene (open square), OTP/antharcene (open diamond), OTP/BPEA (closed diamond), TNB/tetracene (closed hourglass), and TNB/rubrene (open hourglass). Figure adapted from data in the following references: M. T. Cicerone, F. R. Blackburn, and M. D. Ediger, J. Chem. Phys. 102, 471 (1995); M. T. Cicerone and M. D. Ediger, J. Chem. Phys. 104, 7210 (1996); F. R. Blackburn, C.-Y. Wang, and M. D. Ediger, J. Phys. Chem. 100, 18249 (1996); M. D. Ediger, J. Non-Cryst. Solids 235–237, 10 (1998)].

IV. THE PRIMITIVE RELAXATION AND THE JOHARI–GOLDSTEIN SECONDARY RELAXATION In addition to the the structural a-relaxation, there are faster relaxation processes originating at earlier times. At short times, molecules are caged by neighboring molecules and cannot relax by reorientation or translation. The cage (or local liquid structure) is not ﬁxed but ﬂuctuates with time, giving rise to low loss which has no characteristic time scale [35,36,212]. This situation continues until some local independent (primitive) relaxation of the entire molecule (or a local segment in the case of a polymer) takes place, whereupon cages decay commences. The primitive relaxation should be observed as a secondary relaxation process, which is the precursor of the many-molecule relaxation dynamics. The latter evolves with time, becomes increasingly ‘‘cooperative’’—

dispersion of the structural relaxation

529

that is, involving more and more molecules (larger length-scale)—and eventually achieves the structural a-relaxation described by a KWW correlation function [Eq. 1]. This interpretation of the evolution of the dynamics is supported by experimental data for colloidal particles obtained by confocal microscopy [213]. Flexible glass-formers having internal degrees of freedom also exhibit intramolecular motions, which involve only some atoms in the the molecule. These intramolecular processes are also secondary relaxation, but they are not universal and are invariably faster than the primitive relaxation. Thus, if more than one secondary relaxation is observed for a glass-former, the slowest is the important JG relaxation. Since the primitive relaxation involves the motion of the entire molecule and initiates the structural a-relaxation, the secondary JG relaxation to which it corresponds can have properties analogous to those of the structural relaxation. An example is the sensitivity of the secondary relaxation time to applied pressure, which is found in all JG relaxations but not in the non-JG secondary relaxations [38,101,102]. Rigid small-molecule glass-formers offer the best cases to test for the existence of the JG relaxation. There is no intramolecular degree of freedom in a rigid molecule, and hence any secondary relaxation must involve all atoms and thus comprise the JG (or CM primitive) relaxation. Johari and Goldstein found secondary relaxations in toluene and chlorobenzene, both rigid molecules [17–19]. Similarly, secondary relaxations have been found in polymers with no substantial side group (such as 1,4-polybutadiene, polyvinylchloride [214], and polyisoprene [35]), plastic crystals [215,216], phosphate-silicate glasses [217], the molten salt 0.4Ca(NO3)2–0.6KNO3 (CKN) [218], and metallic glasses [219–223]. These are all likely JG relaxations because they cannot be intramolecular motions. For glass-formers in general, rigid molecules or not, criteria were given [38] for identifying secondary JG relaxations that are the CM primitive relaxation. All criteria bear some relation to the properties of the structural relaxation. Naturally, the CM primitive relaxation time, t0 , should correspond roughly to the experimentally observed JG relaxation time, tJG . However, in the context of the CM, the JG relaxation should not be interpreted as a Cole–Cole distribution of relaxation times in addition to the a-relaxation represented by the one-sided Fourier transform of the KWW function. The measured JG spectrum is the result of an evolution of dynamics that includes all processes starting at short times with the primitive relaxation, along with the continuous buildup of many-molecule dynamics with time, and ending up with the structural relaxation described by the KWW function. Multidimensional NMR, dielectric hole burning, light scattering, and solvation experiments [29,224,225] give evidence of such evolution of dynamics. For a review of other works see Ref. 29. These experiments have shown that the structural relaxation is dynamically heterogeneous. There are both rapid and slowly moving molecular units which exchange roles at a time t ta .

530

kia l. ngai et al.

The evolution of the many-molecule dynamics, with more and more units participating in the motion with increasing time, is mirrored directly in colloidal suspensions of particles using confocal microscopy [213]. The correlation function of the dynamically heterogeneous a-relaxation is stretched over more decades of time than the linear exponential Debye relaxation function as a consequence of the intermolecularly cooperative dynamics. Other multidimensional NMR experiments [226] have shown that molecular reorientation in the heterogeneous a-relaxation occurs by relatively small jump angles, conceptually simlar to the primitive relaxation or as found experimentally for the JG relaxation [227]. Correspondences between t0 and the observed JG relaxation times, tJG , are further discussed in a later section, wherein experimental data are reviewed which show that tJG approximately equals t0 of the CM for many glass-formers. One prominent experimental observation is that the separation between the JG and a-relaxation times in logarithmic scale, [logðta Þ logðtJG Þ], correlates with the width of the a-relaxation dispersion or n [32,36,38]. This empirical correlation can be derived theoretically from the CM based on the result that tJG t0 . V.

THE JG RELAXATION AND ITS CONNECTION TO STRUCTURAL RELAXATION

From the view that the a-relaxation is the product of the cooperative dynamics originating from the JG or primitive relaxation, it is natural to expect that the properties of the JG relaxation will mimic those of the structural relaxation. We consider the relaxation time as well as the relaxation strength of the JG relaxation. A.

Pressure Dependence of sJG

The a-relaxation time ta increases with pressure at constant temperature although this sensitivity to pressure, or the density dependence of ta , varies among glass-formers, depending on their chemical structure. The JG relaxation time tJG also increases with applied pressure, although less so than ta . On the other hand, secondary relaxations that are not of the JG kind (involving intramolecular degrees of freedom) usually have little or no pressure dependence. Dipropyleneglycol dibenzoate (DPGDB) [80] and benzoinisobutylether (BIBE)[80] are good examples. Of the two secondary relaxations in BIBE (Fig. 1f), the slower one is the JG relaxation. The increase of tJG with applied pressure and the lack of it for the faster secondary relaxation time are evident in the ﬁgure. Figure 1e shows only the slower JG secondary relaxation in DPGDB, which is sensitive to pressure, unlike the faster secondary relaxation (shown in Fig. 27).

531

dispersion of the structural relaxation DPGDB @ T= 253 K

0

P = 1 bar P = 1418 bar P = 1630 bar P = 2834 bar P = 4424 bar

ε ''

10

10

–1

10

–1

10

0

10

1

10

2

10 ν [Hz]

3

10

4

10

5

10

6

Figure 27. Dielectric loss of DPGDB versus frequency at 253 K under high pressure. The dotted line is the ﬁt to the a-loss peak of data taken at 1418 bars by the KWW function and the value of coupling parameter n used in the ﬁt is 0.37. The arrows indicate the locations of the calculated frequency n0 of the primitive process.

Glass-formers that have small differences between ta and tJG at Tg are useful for determining the pressure and temperature dependence of tJG above Tg because of the wide temperature range over which the JG relaxation can be observed in the liquid state. However, proximity of the two relaxations often obscures the weaker JG relaxation under the high-frequency ﬂank of the a-relaxation; only an ‘‘excess wing’’ is observed in the dielectric loss spectrum. This situation occurs in many glass-formers, such as KDE, PDE, PC, PCB62, salol, BPTCDaH, and BMMPC, as previously discussed in Section II.A. The fact that the a-loss peak and the excess wing are superposable for different combinations of T and P at a ﬁxed value of ta (see Fig. 1a–1c and 2–5) implies that the excess wing (i.e., the submerged JG relaxation) shifts with changes in pressure at constant temperature. This property is shown explicitly in Figs. 28 and 29 for the dielectric loss spectra of BMMPC and salol, respectively. There are also glass-formers that have a resolved secondary relaxation that is not the JG relaxation according to the established criteria [38], but lack an apparent JG peak in their loss spectra at ambient pressure. These glassformers include BMPC [75], dibutyl phthalate (DBP) [77], diethyl phthalate (DEP) [76], 2PG, 3PG [101,102], m-ﬂuroaniline (m-FA) [44], and bis-5hydroxypentylphthalate (BHPP) [228,229]. One criterion is the lack of a pressure dependence of their relaxation times, as shown for BMPC in Fig. 30. NMR measurements of molecular motion in BMPC had shown [230] that the

532

kia l. ngai et al.

dielectric loss

1

0.1

0.01 10

–3

10

–2

10

–1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

Frequency (Hz) Figure 28. Dielectric loss for BMMPC at 288.8 K at pressures equal to 0.1 (rightmost curve), 12.0, 27.9, 51.8, 80.9 ( fmax ¼ 0:03 Hz), 101.7, 129.9, 185.5, and 218.7 MPa (lowest curve).

secondary relaxation is intramolecular in nature (arising from the rotation of the methoxyphenol moiety) and hence not the JG relaxation. In these glass-formers, the JG relaxation is unresolved from the structural relaxation peak at ambient pressure. However, in oligomers of propylene glycol, the JG relaxation has been resolved under the appropriate combination of pressure and temperature [101,102]. Figure 31 shows the dielectric spectrum for 3PG at T ¼ 220:5 K (open symbols); with increasing pressure an excess wing develops between the a-relaxation and the pressure-invariant secondary relaxation. This excess wing eventually is transformed into a distinct peak. The pressure dependence of the resolved JG relaxation time and the lack of any P dependence for the faster secondary relaxation time are evident in Fig. 32. The fast secondary relaxation of m-FA seen at ambient pressure originates from the hydrogen-bonded clusters and not the entire m-FA molecule. Hence it is not the JG relaxation [44]. These hydrogen-bonded clusters are reﬂected by the presence of a prepeak in the static structure factor from neutron scattering at ambient pressure [231,232]. The hydrogen-bonded clusters are suppressed at high temperatures and elevated pressures, causing the disappearance of the fast secondary relaxation and the emergence of the JG relaxation peak [44]. A similar situation pertains for BHPP at elevated pressures and temperatures [229].

533

dispersion of the structural relaxation

dielectric loss

1

0.1

0.01 10 –3 10 –2 10 –1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

Frequency (Hz)

Figure 29. Representative dielectric loss curves for salol measured at 36 C and pressures equal to (from right to left) 0.334, 0.352, 0.383, 0.414, 0.431, 0.460, 0.495, 0.528, 0.566, and 0.590 GPa.

T=253.68 K

ε ''

0.1

11 bar 98 bar 198 bar 300 bar 398 bar 506 bar 702 bar 904 bar 1104 bar 1280 bar

0.01

1E-3 10 –2

10 –1

10 0

10 1

10 2

10 3

10 4

10 5

10 6

f [Hz]

Figure 30. Representative dielectric loss spectra of BMPC obtained under isothermal conditions and applied pressure as indicated.

534

kia l. ngai et al. 10

T = 277.5 K and P = 1.31 GPa

0

ε"

10

1

10

–1

T = 277.5 K and P = 1.61 GPa

T = 277.5 K and P = 1.76 GPa

10

–2

10–4 10–3 10–2 10–1 100 101 102 103 104 105 106 107

Frequency [Hz] Figure 31. Dielectric loss of 3PG at T ¼ 220:5K (open symbols), measured at pressures equal to (from right to left): 33.4, 61.9, 93.0, 120.7, 150.0, 180.2, 209.3, 237.5, 268.6, 297.2, 331.3, 373.4, 415.3, 447.2, 463.7, 510.2, and 591.3 MPa. The closed symbols are measured at T ¼ 277:5 K and three pressures as indicated. 105

α

Tri-PPG

103 101

JG

τ [s]

JG 10–1 10–3 10–5

β

10–7 0

200

400

600

800

1000

P [MPa] Figure 32. Relaxation times obtained from ﬁtting the 3PG spectra at T ¼ 218:4 (squares) and 245.2 K (triangles).

dispersion of the structural relaxation

ε ''

BIBE

535

(a)

10 0

10 –1

10 –2 I

(b)

ε ''

10 0 II 10 –1

10 –2 10 –2 10 –1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 ν [Hz] Figure 33. Dielectric loss of BIBE versus frequency at different pressures and temperatures. (a) Open circles: T ¼ 288:2 K and P ¼ 3502, 3699, 4234, 4507, 5165 bars (from right to left); solid squares P ¼ 1 bar and T ¼ 226.1, 223.0, 220.5 K (from right to left); solid lines are KWW ﬁtting curves for a-process (coupling parameter n ¼ 0.33–0.38). Arrows indicate the frequency location of JG process according to the CM predictions. (b) Comparison of different spectra with the structural peak in position I or II. The I peak includes: T ¼ 278.4 K and P ¼ 3202 bars (open triangles), T ¼ 298 K and P ¼ 4666 bars (open diamonds), and T ¼ 226.1 K and P ¼ 1 bar (solid squares). The II peak includes: T ¼ 278.4 K and P ¼ 3699 bars (open triangles), T ¼ 288.2 K and P ¼ 4507 bars (open circles), and T ¼ 220.5 K and P ¼ 1 bar (solid squares).

B.

Invariance of sJG to Variations of T and P at Constant sa

In Section II, we have shown that for a given material the dispersion of the structural a-relaxation is the same for various combinations of T and P as long as ta is constant. From the same experimental data, tJG is also found to be invariant to changes in temperature and pressure at constant ta . BIBE and DPGDB in Figs. 1f and 1e, respectively, are examples of this invariance of tJG . A clearer demonstration for BIBE is given in Fig. 33. Although the ratio tJG =ta is constant, the entire dispersion encompassing the a-relaxation and the JG relaxation may not be exactly the same for different combinations of T and P. This is because the

536

kia l. ngai et al.

relaxation strengths of the JG relaxation and the a-relaxation do not necessarily change in exactly the same manner with changes in T and P. Many glass-formers have an unresolved JG relaxation, appearing as an excess wing in the loss spectrum. For these materials, the invariance of tJG to changes in temperature and pressure at ﬁxed ta is manifested by the superpositioning of the a-loss peak together with the excess wing. Examples of such glass-formers include KDE, BMMPC, BMPC, salol, diglycidyl ether of bisphenol-A (Epon828), DEP,DBP, PCB62, and PDE (see examples in Figs.1a–1d and 2–5). C.

Non-Arrhenius Temperature Dependence of sb Above Tg

It is generally found that tb of all secondary relaxations, both JG and non-JG, has an Arrhenius temperature dependence in the glassy state; that is, tJG ¼ t1 expðEa =RTÞ with constant t1 and Ea. The JG relaxation tends to merge with the a-relaxation above the glass transition temperature, Tg, as inferred by assuming that the Arrhenius temperature dependence persists into the equilibrium liquid state. The actual temperature dependence of tJG at temperatures above Tg is of central importance to any theoretical explanation of the origin of the JG relaxation. Unfortunately, above Tg the situation is less clear because of the difﬁculty in resolving the JG relaxation from the proximate a-process. By ﬁtting ambient pressure dielectric spectra of the overlapping aand b-peaks of sorbitol in this region to the sum of two functions, several groups have concluded that the Arrhenius dependence observed below Tg changes into a stronger temperature dependence above Tg [233–235]. Representative results are shown in Fig. 34 (open squares for a-relaxation and open diamonds for JG relaxation) [236]. The deduced b-relaxation has a temperature dependence above Tg which departs from the Arrhenius temperature dependence below Tg. Another example is ambient pressure dielectric data of poly(vinylacetate) [49,237], with deconvolution of the two relaxations achieved either by the superposition method or by the convolution method [238]. Such results, however, are somewhat inconclusive because the temperature dependence of the unresolved JG relaxation is deduced from a somewhat arbitrary ﬁtting procedure. Moreover, the assumption that the JG relaxation has some distribution (e.g., Cole–Cole function) to be added on to the a-process (represented by an empirical Cole– Davidson or Havriliak–Negami distribution) assumes independence of the two processes; however, this is incompatible with the interpretation of structural relaxation as the evolution of the primitive (JG) dynamics. A clearer picture is offered by sorbitol, for which the JG peak is well-resolved for pressures above 0.5 GPa over a range of temperatures above Tg [239]; representative results are shown in Fig. 34. The JG relaxation times can be determined directly (no deconvolution required) at high pressures, showing clearly that the temperature dependence changes from one Arrhenius relation below Tg to a more sensitive one above Tg. Similar results are found in dielectric measurements at high

537

dispersion of the structural relaxation 2 sorbitol open P = 0.1 MPa half P = 0.59 GPa solid P = 1.8 GPa

0

–2

10

log τ [s]

τα

3

–4

ε ''

2 1

–6

3.0

log10f [Hz]

0

Tg

2

3.5

4.0

4.5

3

4

5.0

5

6

5.5

–1

1000 / T [K ]

Figure 34. Isobaric a-relaxation times at P ¼ 0:1 MPa (&), 0.59 GPa (half-ﬁlled squares), and 1.8 GPa pressure (&), along with the corresponding JG b-relaxation times at ambient ( ) and elevated pressure 0.59 GPa (half-ﬁlled circles) and () for sorbitol. The slope of tJG is independent of pressure, although it differs markedly for low versus high temperatures. Inset shows the JG peak in the dielectric loss at P ¼ 1:8 GPa for temperatures from 273 K to 343 K, in 5-degree increments (bottom to top). The a-peak is too low in frequency to appear within the measured frequency range.

pressure on another polyol, xylitol [239], and in 17.2% chlorobenzene in decalin [67]. The implication is that a change in T dependence of tJG above Tg is a general feature of glass-forming liquids. The temperature dependence of tJG above Tg is much stronger than the dependence below Tg. Experiments at ambient pressure, in which the JG relaxation of dipropylglycol dibenzoate [240] and picoline mixed with tristyrene [241,242] were resolved both above and below Tg, also showed that tJG has a stronger temperature dependence in the equilibrium liquid state than in the glass. For glass-formers that have an unresolved JG relaxation, which appears as an excess wing, the temperature dependence of tJG above Tg can be inferred from the shift of the excess wing with temperature. Since ta is non-Arrhenius above Tg and the excess wing is on the high-frequency side of the a-peak, the shift of the excess wing with temperature will also be non-Arrhenius, albeit less than that of ta . This non-Arrhenius behavior of the excess wing for T > Tg has been shown for glycerol, propylene carbonate, propylene glycol [39], and KDE [40]. Thus, for all JG relaxations, hidden or resolved, tJG has different temperature dependences above and below Tg.

538

kia l. ngai et al. 3.5

T = 173.1 K

DPGDB

q = 30 K/min

3.0

93 s

ε''x10

2

745 s 1353 s

2.5

3752 s 7272 s eq

2.0 1.5 1.0 10

–2

10

–1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

ν [Hz]

Figure 35. Time evolution of the secondary dielectric relaxation loss spectrum of DPGDB on isothermal annealing at 173.15 K after rapid cooling from 300 K. From top to bottom, the data were obtained after the sample has been annealed isothermally at 173.15 K for times, ta , equal to 93 s, 745 s, 1353 s, 3752 s, and 7272 s elapsed after the thermal stabilization. Solid circles represent the spectrum obtained by slowly cooling the sample at 0.05 K/min. Vertical arrows show the frequencies of the maximum loss for the JG b- and the g-processes.

D. Increase of sb on Physical Aging Glasses usually densify on physical aging with a concomitant increase in the structural relaxation time. Early on it was noted by Johari [243] that the JG relaxation in the glass was affected by thermal history, such as the cooling rate used to vitrify the liquid or the aging time. A recent study [240] of the JG relaxation in dipropyleglycol dibenzoate (DPGDB) found the thermal history of the glass to exert a strong inﬂuence on the JG relaxation time. The increase in tJG with aging mimics the behavior of ta , as illustrated in Figs. 35 and 36. Although the change in tJG is less than that of ta , the effect on the JG brelaxation is much greater than on the faster, non-JG g-relaxation. Similar results have been found for xylitol [244] and sorbitol [245]. Since aging increases the separation of the a- and JG-relaxations, the excess wing seen in many glass-formers can be resolved into a distinct JG peak by long-time physical aging. This was shown for glycerol, propylene carbonate (PC), and propylene glycol (PG) [39,100,246], which all have an excess wing and no other secondary relaxation. The samples were annealed at constant temperature below Tg for up to ﬁve weeks, during which the excess wing was transformed into a shoulder; that is, a nascent JG relaxation peak. These changes of the dielectric loss of PC and PG with aging time are shown in Figs. 37 and 38 respectively. An even longer aging time of 3 months gives rise to a distinct peak, instead of the shoulder, in glycerol [247].

539

dispersion of the structural relaxation 6

-log10(τ [Hz])

DPGDB

τγ

4 2 0 –2

aging

τβ

τα

–4 –6 n=0.43

–8

4.0

4.5

5.0

5.5

6.0

6.5

7.0

1000/T[K]

Figure 36. Relaxation map of DPGDB. Logarithm of the experimentally determined relaxation time versus 1000/T, of the a-process (ﬁlled and open circles, from ﬁtting spectra and shift factors, respectively), JG b-process (ﬁlled triangles) and g-process (open squares), compared to simulation data (lines) for a- and b-relaxation obtained by numerical solution of the Hodge model and the CM model, respectively. Solid lines refer to slow cooling (0.05 K/min), whereas dashed lines refer to fast cooling (30 K/min). Open triangles and diamonds are for the b-relaxation during the aging process, after a cooling at rate of 30 K/min and 9 K/min, respectively. Dotted lines represent the VFTH equation of the equilibrium liquid. Vertical lines (at Tg ¼ 220 K and Tx ¼ 198:15 K) delineate three regions from left to right for the slow-cooled system: equilibrium liquid region (T ¼ Tf ), ‘‘delayed’’ region (Tg > Tf ðTÞ > Tx ) where the system has fallen out of equilibrium but Tf changes with T, and the isostructural glassy state (T < Tx ) where Tf is equal to 213.8 K and constant on the laboratory time scale. 0.3

O 0.1

log10 [t (s)] 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 6.5

O

n = 0.27

∆ log να

O

CH 3

ε ¢¢

0.03

PC, 152 K 10

–2

10

–1

∆log νβ 10

0

10

1

2

10

3

10

4

10

5

10

ν (Hz) Figure 37. Shifts of the a-relaxation and the excess wing of propylene carbonate at 152 K after aging for the periods of time as indicated.

540

kia l. ngai et al.

–1

propylene glycol

∆ log να

10

aging 157 K

log10 t(s)

n = 0.30 time time

ε¢¢

∆log νβ

–2

10

2 2.5 3 3.5 4 4.5 5 5.5 5.75

1 -1

10

0

10

10

1

2

10

n (Hz)

3

10

4

10

ν (Hz)

Figure 38. Shifts of the a-relaxation and the excess wing of propylene glycol at 157 K after aging for the periods of time as indicated.

2PG, DBP, dioctalphthalate (DOP), and several other glass-formers each have a resolved secondary g-relaxation; however, these are not JG relaxations, as evidenced by the invariance of the relaxation times to pressure. In such materials, the JG relaxation must lie between the a- and g-relaxations (ta < tJG < tg ), making its resolution more difﬁcult than for PC and PG which have no grelaxation. Consequently, even the existence of an excess wing may be questioned, at least at ambient pressure. However, again, physical aging further separated the a- and JG-processes, enabling the excess wing to become clearly visible; results are shown for 2PG in Fig. 39 and DBP in Fig. 40. Note that the position of the g-relaxation is unchanged with aging. This is strong evidence that the g-relaxation has no relation to the glass transition. Properties of the grelaxation, such as a small decrease of its peak frequency with temperature, have been reported and an asymmetric double potential model proposed to explain them [248,249]. However interesting in their own right, it is important to distinguish among the various types of secondary relaxations. Notwithstanding the prominence of the g-relaxation in some liquids (see, for example, the spectrum of DOP in Fig. 41), it has no inﬂuence on vitriﬁcation. In glass-formers such as DOP, physical aging (Fig. 42) or hydrostatic pressure are necessary to reveal the existence of the JG process as an excess wing. E.

JG Relaxation Strength and Its Mimicry of Enthalpy, Entropy, and Volume

The b-relaxations in 16.6 mol% chlorobenzene-decalin mixture [250], 5-methyl2-hexanol [251], and D-sorbitol [233,252], are all of the JG kind. The relaxation

541

dispersion of the structural relaxation 10

1

Di-PPG T = 216.7 K

0

ε"

10

10

–1

after aging 10–4 10–3 10–2 10–1 100

101

102

103

104

105

106

Frequency [Hz]

Figure 39. Dielectric loss for PPG dimer at pressures (from right to left) of 67.6, 248.7, 335.7, 520, and 510 MPa. The last one was measured after 12 hours of aging. There is a pressureindependent secondary peak at 104 Hz, which exhibits a negligible response to aging.

strength, eb , of the JG relaxation in these glass-formers is found to change on heating through the glass transition temperature in a manner similar to that of the changes observed in the enthalpy H, entropy S, and volume V. The derivative of eb with respect to temperature, deb =dT, increases from relatively low values below Tg to higher values above Tg . This is the same behavior observed for the speciﬁc heat Cp and the thermal expansion coefﬁcient, which are proportional to the derivatives dH=dT and dV=dT, respectively. The rotation angle for the motions underlying the JG relaxation, and hence eb , likely depends on the speciﬁc volume and the entropy. Thus, it is expected that the rate of change of eb with temperature should be similar to that of these thermodynamic quantities. For the same reason, the angle of rotation or eb is expected to depend on the thermal history of the glass, a denser glass having a smaller eb. It should be noted, however, that similar changes are observed also for some secondary relaxations that are not JG [253]. F. The Origin of the Dependences of Molecular Mobility on Temperature, Pressure, Volume, and Entropy Is in sJG or s0 The properties of the JG relaxation discussed in Sections V.A–V.E call to mind the properties of the structural a-relaxation associated with vitriﬁcation. The properties discussed in Sections V.A, V.C, V.D, and V.E indicate that pressure P,

542

kia l. ngai et al. DBP aging at T = 168 K 1s 4h 8h 24 h

dielectric loss

10–1

10–2

10–4 10–3 10–2 10–1 100

101

102

103

104

105

106

7

10

f [Hz]

Figure 40. Dielectric loss of dibutylphthalate (DBP) at ambient pressures measured after aging for different periods of time as indicated. The excess wing becomes evident after aging for a sufﬁciently long time.

temperature T, and their conjugate variables, volume V and entropy S, govern the mobility of the JG relaxation in the equilibrium liquid state as well as in the glassy state. Since the JG relaxation transpires before the a-relaxation, we are led to conclude that pressure P (volume V ) and temperature T (entropy S) ﬁrst enter into the determination of molecular mobility at the level of the JG relaxation, well before the emergence of the a-relaxation. The dependences of the arelaxation on temperature, pressure, volume, and entropy are derived from those of the JG relaxation after many-molecule dynamics have transformed the latter progressively with time into the former. Many molecules are involved in the arelaxation, particularly at lower temperatures or higher pressures for which ta is longer. Evidence of the involvement of many molecules comes from the the heterogeneous dynamics engendering the dispersion. The width of the adispersion and the length-scale of the heterogenous dynamics are convenient measures of the intensity of the many-molecule dynamics. In contrast, the JG relaxation time, or more accurately the primitive relaxation time, corresponds to the individual and independent relaxation of molecules. The involvement of many molecules in the a-relaxation ampliﬁes the original but weaker dependences of tJG or t0 on P (V ) and T (S), and it naturally gives rise to the much stronger corresponding dependences of ta . The dependences of ta on P (V ) and T (S) are the results of two contributing factors: (1) the many-molecule

543

dispersion of the structural relaxation 0

10

o

T = -82 C o T = -84 C o T = -86 C o T = -88 C o T = -90 C o T = -92 C o T = -94 C o T = -96 C

DOP 10

0 -1

ε"

10

-2

10

-3

10 –1

10

-2

10

-1

0

10

10

1

10

2

10

3

4

10

10

5

10

6

10

7

10

ε ''

frequency [Hz]

γ - process

10 –2

"excess wing"

10 –4 10 –3 10 –2 10 –1 10 0

10 1

10 2

10 3

10 4

10 5

10 6

10 7

frequency [Hz] Figure 41. The inset shows isothermal dielectric loss spectra of DOP at ambient pressure. The grelaxation is the only resolved secondary relaxation. The main ﬁgure is obtained by time– temperature superposition.

dynamics manifested by the dispersion and (2) the originating dependences of tJG on the thermodynamic variables P (V) and T (S). These results suggest that a viable theory of glass transition has to start from the dependence of tJG or a0 on pressure (volume) and temperature (entropy) and then implement the manymolecule dynamics to arrive ﬁnally at the a-relaxation. Likewise, quantities derived from dependences of ta on P (V) and T (S), such as the ‘‘steepness’’ or the ‘‘fragility’’ index m ¼ dðlog10 ta Þ=dðTg =TÞ T¼Tg , are determined by both thermodynamics and many-body dynamics. These two factors may not affect m in the same proportion. For example, in some glassformers the many-molecule dynamics may not be severe (i.e., smaller n), but the dependences of tJG on the thermodynamic variables are strong, or vice versa. This scenario is observed for PC, KDE, and PDE, which have narrow dielectric dispersions (smaller n), but their m values are larger compared to other glassformers [140]. Another example is the fact that m usually decreases with applied pressure. For any chosen ta , at elevated pressure the temperature can be raised to maintain ta constant. In particular, the P and T combinations can be chosen to

544

kia l. ngai et al.

DOP 10

o

–1

aging at T = –96 C

ε ''

2.7 hour 8.3 hour 13.8 hour 42.4 hour

"excess wing" 10

–2

–4

10

–3

10

–2

10

–1

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

frequency [Hz] Figure 42. Dielectric loss for DOP at ambient pressures measured after aging for different periods of time as indicated. The excess wing becomes evident after aging.

have the same speciﬁc volume (isochoric condition). Since ta is the same, it follows from Section II that the dispersion is also the same, and hence the inﬂuence of many-molecule dynamics is the same for the three cases: at ambient pressure, at a ﬁxed elevated pressure (isobaric condition), or at isochoric condition. The temperatures necessary to attain the same ta in all three cases are solely controlled by the thermodynamic factor. In Fig. 43, log10(ta ) is plotted against reciprocal temperature scaled by the respective Tg of the three cases. One can see that at any constant ta , the scaled reciprocal temperature Tg =T and the ‘‘fragility’’ index m of a glass-former change with thermodynamic condition. The fragility decreases with elevating pressure, and this decrease is spectacular when the glass-former is constrained to constant volume. These changes of m in the same glass-former illustrate the role of thermodynamic variables in determining ta and m, beyond the effects due to many-molecule dynamics (as reﬂected in the breadth of the dispersion). The classiﬁcation of glass-formers according to their fragility is only useful to the extent that m is related in a straightforward fashion to a single fundamental factor. Unfortunately, our results indicate that this is generally not the case; at least two distinctly different factors, the thermodynamics and the dispersion, determine m and the nature of a Tg-scaled Arrhenius plot. Thus, notwithstanding its popularity, the use of m to classify glass-formers can lead to inconsistent conclusions.

545

dispersion of the structural relaxation 2 PC

salol

BMMPC

0

-4 co ns t

-6

V=

V=

Log(τ[s])

-2

0.1 MPa 300 MPa 600 MPa

-8 2 PDE

co V=

t ns

t ns co

0.1 MPa 200 MPa 600 MPa

0.1 MPa 300 MPa 600 MPa

KDE

PCB62

0

Log(τ[s])

-2 -4 -6

c V=

c V=

st on

st on

V=

0.8

Tα /T

1.0

0.1 MPa 200 MPa 600 MPa

0.1 MPa 400 MPa

0.1 MPa 400 MPa

-8

t ns co

0.8

Tα /T

1.0

0.8

1.0

Tα /T

Figure 43. Isobaric dielectric relaxation times for salol, PC, BMMPC, PDE, KDE, and PCB62 versus Ta =T, where t(Ta ) ¼ 10 s. Isochors were calculated at the volume at which t ¼ 10 s at atmospheric pressure; V ¼ 0:7907 (salol), 0.7558 (PC), 0.9067 (BMMPC), 0.7297 (PDE), 0.7748 (KDE), and 0.6131 (PCB62) ml/g. [Adapted from R. Casalini and C. M. Roland, Phys.Rev. B 71, 014210 (2005).]

We have seen in Section V.B that the relation between the JG relaxation and the a-relaxation, expressed in terms of the ratio ta =tJG , is independent of the combination of P and T as long as ta is constant. This property in conjuction with the results of Section II lead to the following conclusion: For any given glass-former, tJG , in conjuction with the dispersion of the a-relaxation (or bKWW ), is invariant to changes in the thermodynamic variables P (V ) and T (S ), provided that ta is constant. In other words, tJG and bKWW together uniquely deﬁne ta and vice versa, independent of the thermodynamic variables. This characteristic of tJG is another indication of the fundamental importance of the JG relaxation to the vitriﬁcation process. A theory of the glass transition is neither fundamental nor complete if it neglects the JG relaxation. Recalling the discussion in Section II, this statement also applies to models in which the dispersion of the a-relaxation is either ignored or is not uniquely deﬁned by ta.

546

kia l. ngai et al. VI.

THE COUPLING MODEL A.

Background

One of the manifestations of many-molecule dynamics is the dispersion of the structural a-relaxation or the Kohlrausch function [Eq. 1] used to represent the time correlation function with the fractional exponent, bKWW ¼ ð1 nÞ. We have discussed in previous sections that this dispersion is fundamental because it deﬁnes, governs, or correlates with the properties of the structural a-relaxation and even the JG relaxation. However, it is not easy to construct a theory or model that captures the many-molecule relaxation dynamics in real glass-formers and has predictions consistent with the experimental facts or that are falsiﬁable by experiment. Such a theory or model does not exist at the present time. This state of affairs is not surprising because the intermolecular potential in glass-forming liquids is anharmonic and motions of an Avogadro’s number of molecules in phase space are chaotic and thus difﬁcult to describe [254]. The problem is compounded by the fact that relaxation is an irreversible process and a method that can fully describe the many-molecule relaxation process and its evolution with time is lacking. Most theories of supercooled liquids and the glass transition avoid the issue and focus their attention on the connection between dynamics and thermodynamics [255]. In neglecting to address the many-molecule relaxation dynamics, these models suffer the consequence that the obtained dispersion of the a-relaxation does not deﬁne, govern, or correlate with other properties of the a-relaxation. The authors of the present chapter, as well, have no solution to the complete many-molecule relaxation problem. Nevertheless, one of us did recognize the importance of many-molecule relaxation in interacting systems 26 years ago in initiating what is now known as the Coupling Model (CM). At that time and continuing until a few years ago, the CM has been concerned only with the terminal a-relaxation. No consideration was given to processes transpiring at early times. This conceptual model [256–262] indicated that the slowing down of the averaged a-relaxation rate is caused by interactions and constraints between molecules. Rigorous solutions of simple, coupled systems have given support to the premise of the CM and reinforced its generality. For instance, the energy relaxation in interacting arrays or lattices of nonlinear oscillators with anharmonic or dissipative coupling has been found to be slowed down by interactions and, under particular coupling conditions, to follow a stretchedexponential law [260–268]. Most importantly, in many cases a crossover from the faster single exponential relaxation to a stretched function was observed at a some crossover time [260–268]. Such a dynamic crossover should also take place in the case of glass-formers because many-molecule dynamics caused by intermolecular interaction cannot transpire instantly. Thus, the slowing down

dispersion of the structural relaxation

547

starts at some time, tc, the magnitude of which is determined by the strength of the molecular interaction potential. A system with weaker interaction has a longer tc [24]. At inﬁnitely weak interaction, there is no slowing down and tc ! 1. In the case of glass-formers, the many-molecule a-dynamics are heterogeneous and complicated, so that only averages over these heterogeneities are considered in the CM. A schematic description of the dynamic evolution can be described as follows: The CM recognizes that all attempts of relaxation have the same primitive (i.e., independent) relaxation rate W0 ¼ t1 0 , but the many-molecule dynamics, starting at tc, forestall all attempts of molecules to be simultaneously successful, resulting in faster and slower relaxing molecules or heterogeneous dynamics. However, when averaged, the effect is equivalent to the slowing down of t1 0 by another, time-dependent, multiplicative factor. The time-dependent rate W(t) has the product form, f(t)t1 0 , where f(t) is a decreasing function with values less than unity. In particular, the slowing-down factor f(t)1 was found to be dependent on time according to a sublinear power law, and hence W(t)/ tn t1 0 , where n is the coupling parameter of the CM and 0 n < 1. The stronger the intermolecular interaction, the greater the slowing effect of the many-molecule dynamics and the larger the coupling parameter n. Therefore, the corresponding correlation function of the model is the Kohlrausch stretched exponential function [Eq. (1)], which holds only for t > tc . At times shorter than tc, there is no slowing down and the correlation function is the linear exponential, fðtÞ ¼ expðt=t0 Þ

ð5Þ

where t0 is the primitive relaxation time of a molecule, unimpeded by other molecules, and has normal properties such as the Q2 dependence on the scattering vector Q. For polymers, because the repeat units are bonded along the chain, equation (5) has to be replaced by the Hall–Helfand function [269,270]. It was assumed that the crossover from fðtÞ ¼ expðt=t0 Þ to the Kohlrausch function takes place continuously in a narrow neighborhood of tc. All the above are supported by solutions of much simpliﬁed interacting or coupled systems [260–262], and the crossover leads to a relation between ta and a0 given by ta ¼ ðtcn t0 Þ1=ð1nÞ

ð6Þ

Clear evidence of a crossover from the primitive relaxation to Kohlrausch relaxation was reported in several other systems, comprised of larger units than molecular glass-formers and having weaker interactions. The crossover times tc of these systems are usually much longer than a picosecond (up to tens of microseconds) [24,152–158]. Vibrations and librations do not contribute to the measured quantity in this longer time regime, and the crossover of the correlation function can be clearly observed in these systems. Unfortunately, this is not the case for structural relaxation of inorganic, organic, and polymeric glass-formers,

548

kia l. ngai et al.

which have shorter tc on the order of picoseconds. Nonetheless, at sufﬁciently high temperatures, the structural relaxation dominates the intermediate scattering function obtained by neutron scattering experiments and molecular dynamics simulations of polymeric and small-molecule liquids, and the crossover can be seen at 1–2 ps [143,150,271–275]. The transport coefﬁcients, including viscosity and conductivity, assume the Arrhenius temperature dependence of the primitive relaxation when the relaxation time becomes less than 2 ps [276]. These properties indicate that tc is equal to about 2 ps for molecular and polymeric glass-formers. The Lennard-Jones potential, VðrÞ ¼ 4e½ðs=rÞ12 ðs=rÞ6 , is often used to model the intermolecular potential of molecular and polymeric glass-formers in molecular dynamic simulations. The unit of time from the Lennard-Jones potential is given by ðms2 =48eÞ1=2, which gives 1 ps for typical parameters. The experimentally determined value of tc is comparable to the Lennard-Jones unit of time [277]. The crossover from the primitive relaxation to Kohlrausch relaxation was seen also at about 1 ps in the many-ion dynamics of molten, crystalline and glassy ionic conductors [24]. The earlier works focused on the structural a-relaxation with the Kohlrausch correlation function, and they ignored the processes that precede it. In order of appearance, these processes include the vibrations inside cages (Boson peak), ﬂuctuations of cages giving rise to the nearly constant loss, and cage decay due to the emergence of the local primitive relaxation, which is related to the Johari–Goldstein secondary relaxation (see below). The primitive relaxation is the building block of the many-molecule dynamics, which increase in lengthscale with time to become the terminal a-relaxation with the maximal possible length-scale and the Kohlrausch function as its correlation function. Only recently, the Coupling Model (CM) has been extended beyond the Kohlrausch structural a-relaxation to incorporate some of the earlier processes [26,32– 38,41,240,276,278,279]. Most important is the local primitive relaxation, which is an observable process occurring at times much shorter than ta . However, since the primitive relaxation is the initiator and building block of the ensuing many-molecule dynamics, it will not be resolved in the measured loss spectrum as a Debye process suggested by its exponential correlation function [Eq. (5)]. Nevertheless, the primitive relaxation frequency, n0 1=2pt0 , should correspond to the characteristic frequency of some observed features in the spectrum, which we shall see is the universal Johari–Goldstein secondary relaxation frequency. Thus, the primitive relaxation not only manifests itself as a genuine relaxation process at shorter times, but also plays the other distinctly different role when considering the Kohlrausch relaxation that leads to Eq. (6). It is important for the reader to recognize that there is no contradiction in the fact that the primitive relaxation plays two separate and distinctly different roles. A deﬁnitive step in extending the CM is to include the relaxation processes occurring before the a-relaxation and to establish the existence of the primitive

dispersion of the structural relaxation

549

relaxation at times shorter than ta . To do this, a connection is made between the primitive relaxation time t0 and the characteristic relaxation time of the Johari– Goldstein (JG) secondary relaxation tJG . In the literature, it is common to interpret the JG secondary peak in the spectrum as due to a local process having a Cole–Cole distribution of relaxation times. The CM interprets the spectrum differently, comprised of the primitive relaxation and thereafter the emerging many-molecule relaxation processes with length-scales that increase with time. However, we continue to use the term JG relaxation; henceforth, it should be interpreted in the sense of the CM. We are not identifying the primitive relaxation with a broad distribution of local processes, but only using the JG relaxation time tJG as an indicator or estimate of t0. Such a connection is expected from the similar characteristics of the two relaxation processes, including their local nature, involvement of essentially the entire molecule, common properties, and the fact that both serve as the precursor to the arelaxation [38]. Multidimensional NMR experiments [226,280] have shown that the dynamically heterogeneous molecular reorientations of a-relaxation occur by relatively small jump angles with exponential time dependence, which is exactly the role played by the primitive relaxation of the CM. Furthermore, from one- and two-dimensional 2H NMR studies [227,281], the JG relaxation in toluene-d5 and polybutadiene-d6 also involves small angle jumps of similar magnitude at temperatures above Tg. This similarity in size of the jump angles of the primitive relaxation and the JG relaxation further supports the connection between these two relaxation processes. Hence, tJG ðT; PÞ t0 ðT; PÞ

ð7Þ

On combining Eqs. (6) and (7), we obtain a relation between ta and tJG given by ta ðT; PÞ ¼ ½tcn tJG ðT; PÞ1=ð1nÞ

ð8Þ

which should be valid for any temperature T and pressure P. A similar relation between the secondary relaxation time and the a-relaxation time but different in quantitative details was given by Cavaille et al. [282] in their model of relaxation in glass-formers. The CM describes the evolution of the molecular dynamics chronologically as follows. At very short times all molecules are caged. Cages decay by local and independent (primitive) relaxation. At short times, only a few primitive relaxations are occurring and they appear separately in space as localized motions just like the JG relaxation. At times beyond t0 or tJG, more units tend to independently relax and when they can no longer be considered as isolated events, intermolecular interactions and mutual constraints impose a degree of cooperativity (or dynamic heterogeneity) on the motions. The degree of dynamic heterogeneity and the corresponding length-scale continues to increase with time as more and more units participate in the motion, as suggested by the

550

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evolution of the dynamics of colloidal particles with time seen by confocal microscopy [213]. These time-evolving processes contribute to the observed response at time longer than t0 or tJG and are responsible for the broad dispersion customarily identiﬁed as the JG relaxation. After sufﬁciently long times, t t0 or tJG, the fully cooperative a-relaxation with the averaged correlation function having the Kohlrausch form [Eq. (1)] is attained. In this terminal regime, the a-relaxation has the maximum dynamic heterogeneity length-scale, Ldh . Naturally we expect a larger Ldh is associated with a broader dispersion or a larger n, because both quantities are proportional to the intensity of the many-molecule dynamics. This correlation is borne out by comparing n with Ldh for glycerol, ortho-terphenyl, and poly(vinylacetate) obtained by multidimensional 13C solid-state exchange NMR experiments [188]. These experiments found that at about T ¼ Tg þ 10 K, the number of molecules per slow domain is 390 monomer units for poly(vinylacetate), 76 molecules for ortho-terphenyl, and only 10 molecules for glycerol. The coupling parameter n deduced from the Kohlrausch ﬁt to the dielectric spectra near T ¼ Tg is 0.53, 0.50, and 0.29 for poly(vinylacetate) [49], ortho-terphenyl [233], and glycerol [36,100], respectively. NMR experiments also conﬁrm that the distribution of relaxation rates is narrower for glycerol than it is for either poly(vinylacetate) or ortho-terphenyl [188]. In the extended CM, the JG relaxation is just part of the continuous evolution of the dynamics. The JG relaxation should not be represented by a Cole–Cole or Havriliak–Negami distribution, as customarily assumed in the literature, and considered as an additive contribution to the distribution obtained from the Kohlrausch a-relaxation. Nevertheless, the JG relaxation may be broadly deﬁned to include all the relaxation processes that have transpired with time up until the onset of the Kohlrausch a-relaxation. Within this deﬁnition of the JG relaxation, experiments performed to probe it will ﬁnd that ‘‘essentially’’ all molecules contribute to the JG relaxation and the motions are dynamically and spatially heterogeneous as found by dielectric hole burning [180,283] and deuteron NMR [284] experiments. This coupling model description of the JG relaxation may help to resolve the different points of view of its nature between Johari [285] and others [180,226,227,280,281,283,284]. The broad width of the JG relaxation can be accounted for by the broad transition from the cage dynamics (revealed by the nearly constant losses) to the fully cooperative Kohlrausch relaxation [36]. In fact, it is important to recall that the time scale where JG relaxation takes place usually exceeds tc, especially in molecular glass-formers, whereas the onset of many-molecule dynamics starts at tc 2 ps. So, when tc t0 ta , the molecules are essentially all caged at short time and then a gradual development of cooperativity occurs when increasing numbers of molecules are ready to reorient independently. In the region tc t ta all the molecules are attempting to make independent

dispersion of the structural relaxation

551

relaxation, but not all of them are successful because of the interaction with or the constraints by the surrounding molecules. The most probable relaxation time tJG , related to the few molecules that have the chance to independently reorient, should be close to t0 , the primitive relaxation time of the independent reorientation if molecular interactions would be negligible. Evidently, the primitive and the JG relaxation processes are not identical, but they are closely related. Therefore the check of a correspondence between the JG relaxation time and the primitive relaxation time t0 is of paramount importance to test the predictions of the CM in glass-forming systems. B.

The Correspondence Between s0 and sJG

In the previous subsection, we have provided conceptually the rationale and experimentally some data to justify the expectation that the primitive relaxation time t0 of the CM should correspond to the characteristic relaxation time of the Johari–Goldstein (JG) secondary relaxation tJG . Furthermore, it is clear from the CM relation, ta ¼ ðtcn t0 Þ1=ð1nÞ , given before by Eq. 6 that t0 mimics ta in behavior or vice versa. Thus, the same is expected to hold between tJG and ta . This expectation is conﬁrmed in Section V from the properties of tJG . The JG relaxation exists in many glass-formers and hence there are plenty of experimental data to test the prediction, tJG ðT; PÞ t0 ðT; PÞ. Broadband dielectric relaxation data collected over many decades of frequencies are best for carrying out the test. The ﬁt of the a-loss peak by the one-sided Fourier transform of a Kohlrausch function [Eq. (1)] determines n and ta , and together with tc 2 ps, t0 is calculated from Eq. 6 t0 ¼ ðtc Þn ðta Þ1n

ð9Þ

The result is then compared with tJG of the resolved JG relaxation appearing at higher frequencies in the isothermal or isobaric spectrum. Remarkably, for all small molecule and polymer glass-formers tested, including those mentioned in this chapter, the relation tJG ðT; PÞ t0 ðT; PÞ ¼ tcn ½ta ðT; PÞ1n

ð10Þ

with tc ¼ 2 ps holds [32,33,34,36,39,40,41,80,240]. Here we demonstrate this with three examples of the correspondence between the calculated t0 and the experimental tJG in the dielectric loss spectra of the small-molecule glass-formers DPGDB (Fig. 27), and BIBE (Fig. 33), and the polymer polyisoprene (Fig. 44). The separation between n0 1=2pt0 and na 1=2pta , (in units of hertz) on a logarithmic scale is given by log10 n0 log10 na ¼ nð10:9 log10 na Þ

ð11Þ

552

kia l. ngai et al.

ε"

10–2

10–3

10–4 10–3 10–2 10–1 100 101 102 103 104 105 frequency (Hz) Figure 44. Dielectric loss spectra of PI. The data at 216.0 K (^), 211.15 K (&), 208.15 K ( ), and 204.15 K (r) were obtained using the IMass Time-Domain Dielectric Analyzer. All the other data, which start at 10 Hz and continue up to 100 kHz, were taken with the CGA-83 Capacitance Bridge. There is good agreement of the CGA-83 data at 216.7 K (^), 212.7 K (&), 208.7 (), and 204.7 K (!) with the IMass data at 216.0 K (^), 211.15 K (&), 208.15 K ( ), and 204.15 K (r), respectively, after the latter have been shifted horizontally by an amount determined from the VFTH temperature dependence of the a-relaxation frequency, in order to account for the slight differences in temperature. The other eight spectra were obtained only using the CGA-83. The spectra that show a-loss maxima correspond (from right to left) to T ¼ 236:7 (*), 232.7 (~), 228.7 (þ) and 224.7 K (~). The lower three CGA-83 curves, which show b-loss peaks, were taken (starting from the bottom) at 169.7 (~), 181.7 (þ), and 200.7 K (*). The vertical arrows mark the locations of the calculated primitive relaxation frequencies, n0 , at (from right to left) 212.7 K (&), 208.7 K (), and 204.7 K (!). The locations of these n0 should be compared with the secondary relaxation peaks at these temperatures.

If n0 nJG holds, then we have log10 nJG log10 na nð10:9 log10 na Þ

ð12Þ

which says that the separation between nJG and na is smaller for smaller n at constant na . Hence, glass-formers with small n will have nJG too close to na and their JG relaxations will be hidden by the high-frequency ﬂanks of the more intense a-loss peaks. The JG relaxation is unresolved and appears as an excess wing instead. In fact, for many such glass-formers, the calculated primitive relaxation frequency, n0 , falls within the excess wing. Some small n glass-formers show also a secondary relaxation with peak frequency ng much higher than n0 . If these secondary relaxations were the JG relaxations, then the data would be counter examples to n0 nJG . However, in all such cases, the position of these secondary relaxations do not change much

dispersion of the structural relaxation

553

or at all on applying pressure, and hence they are not the JG relaxation. The JG relaxation is still in the excess wing with its relaxation time nJG located in between na and ng . In some cases, the excess wing can be transformed to a resolved JG relaxation by applied pressure or by physical aging like that found in dipropylene glycol and tripropylene glycol [101,102] (see Figs. 31 and 32). The resolved JG relaxation time nJG , unlike ng , is pressure-dependent and is approximately the same as the calculated n0 . m-Fluoroaniline (m-FA) is another example of this class of glass-formers, but it stands out because we have a clue on the origin of the additional secondary relaxation. Elastic neutron scattering measurement and computer simulation of m-FA [231,232] have found the presence of hydrogen-bond-induced clusters of limited size in m-FA at ambient pressure and temperature of dielectric measurement. The resolved secondary relaxation in m-FA originates from the hydrogen-bond-induced clusters and hence is not the JG relaxation involving the entire m-FA molecule. Neutron scattering experiments have also found that the hydrogen-bond-induced clusters are suppressed under high pressure and elevated temperature. If the additional secondary relaxation were indeed coming from the hydrogen-bond-induced clusters, the dielectric relaxation spectrum taken at high pressure and elevated temperature would be different from that at ambient pressure and lower temperatures. By performing high-pressure/high-temperature dielectric relaxation measurements on m-FA, we ﬁnd changes in the entire spectrum as a consequence of the suppression of the hydrogen-bond-induced clusters and alteration of the physical structure [44]. The a-loss peak is broadened, the excess wing is curtailed to the extent that its existence is in doubt, and a new secondary relaxation emerges to replace the one seen at ambient pressure (see Fig. 22). The spectrum on the whole resembles that of the closely related molecular glass-former, toluene, the secondary relaxation of which is deﬁnitely the JG relaxation because (a) toluene is a rigid molecule and (b) its characteristic frequency is nearly the same as the calculated n0 (indicated by the upward pointing vertical arrow in Fig. 45). The characteristic frequency of the new secondary loss peak of m-FA also is in good agreement with the calculated n0 (indicated by the downward pointing vertical arrow in Fig. 22) and is pressure-dependent. Thus, the secondary relaxation that emerges after the hydrogen bonded clusters have been suppressed at high temperature and pressure is a genuine JG relaxation. The results of m-FA indicate that, for other glass-formers having similar dielectric spectrum (i.e., narrow a-loss peak, presence of an excess wing, and a resolved secondary relaxation at higher frequencies), the well-resolved secondary relaxation should not be identiﬁed as the JG relaxation. The secondary relaxations in some of these glass-formers, such as di-n-butyl phthalate (DBP) [77] and di-ethyl phthalate [76], do not arise from hydrogen-bonding, and unlike m-FA they are not suppressed by high pressure and temperature. However, the lack of pressure dependence in their

554

kia l. ngai et al. 10 1

10 0 m-FA (T = 279 K, P = 1.5 GPa)

NH 2

dielectric loss

10 0

CH 3

10 –1

10 –2

10–1

F

10–2

Toluene (T = 119 K, P = 0.1 MPa)

10

–2

10

–1

10

0

10

1

2

10 10 f [Hz]

10–3 3

10

4

10

5

10

6

10

7

Figure 45. The dielectric loss spectrum of m-FA at high pressure and temperature compared with toluene at ambient pressure. The a-loss peak frequency is nearly the same in the two cases. The vertical arrows indicate the calculated primitive relaxation frequencies, n0 , for the two cases.

relaxation times is another good indication that they are not the JG relaxation. The genuine JG relaxation of DBP is identiﬁed with a new excess wing, which emerges after physical aging for a period of time (see Fig. 40). These examples are sufﬁcient for us to warn against the practices of either (a) referring to all observed secondary relaxations as JG relaxations without applying any criterion or (b) the other extreme of not distinguishing the secondary relaxation that bears relation to the structural a-relaxation from others that do not. Both extremes are unreasonable and detrimental to the search of secondary relaxations that may play a fundamental role in glass transition. Polymers that have bulky repeat units can have multiple secondary relaxations. If more than one secondary relaxation is found, then the slowest one has to be the JG relaxation, assuming that the latter is resolved. Excellent illustrations of this scenario are found by dielectric relaxation studies of aromatic backbone polymers such as poly(ethylene terephthalate) (PET) and poly(ethylene 2,6-naphthalene dicarboxylate) (PEN) [43]. The calculated t0 from the parameters, n and ta , of the a-relaxation are in good agreement with the experimental value of tJG obtained either directly from the dielectric loss spectra or from the Arrhenius temperature dependence of tJG in the glassy state extrapolated to Tg. The example of PET is shown in Fig. 46.

555

dispersion of the structural relaxation 103 102 101 100 10 –1 10 –2 –3 τ(sec) 10 10 –4 10 –5 10 –6 10 –7 10 –8 10 –9 10–10 2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

–1

1000/T (K ) Figure 46. Relaxation map of PET showing the primary relaxation and three secondary relaxations. The calculated primitive relaxation time are represented by stars. [A. Sanz, A. Nogales, and T. Ezquerra, paper presented at the 5th International Discussion Meeting on Relaxation in Complex Systems, July 7–13, 2005 and to be published in J. Non-Cryst. Solids 2006.]

C.

Relation Between the Activation Enthalpies of sJG and sa in the Glassy State

In the previous section, at temperatures above Tg, the Johari–Goldstein relaxation time has been shown to correspond well to the primitive relaxation time, and both are related to the structural a-relaxation time by Eq. (10). This equation should continue to hold at temperatures below Tg. However, testing this relation in the glassy state is difﬁcult because of either the scarcity or the unspeciﬁed thermal history of the data on the a-relaxation time ta . In fact, a reliable characterization of the structural relaxation can be acquired only at equilibrium, and such condition is rarely satisﬁed below Tg. Glassy systems are nonergodic, and their properties can depend on aging time and thermal history. Anyway, for glasses in isostructural state with a constant ﬁctive temperature Tf, both ta and t0 as well as tJG should have Arrhenius T-dependences with activation enthalpies Ea , E0 , and EJG respectively. Eq. (10) leads us to the relation ð1 nÞEa ¼ E0 ¼ EJG

ð13Þ

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between the activation enthalpies. This relation was predicted by another model of relaxation in glass-formers [282,286–288], although it differs from the CM in quantitative details. When checking this prediction, it is important to bear in mind that the JG relaxation times tJG , obtained at all temperatures to determine EJG , are derived from experiment under isostructural condition with Tf being held constant. A wrong conclusion could result from data of tb obtained in glassy states having different Tf coming from different cooling rates, annealing temperatures, and waiting times. In fact, it has been demonstrated in physical aging experiments [289] that the structural states induced in three different types of thermal history (isothermal, isochronal, and isostructural) are very different. Thus, to verify Eq. (13), it is important to know the thermal history and the fulﬁllment of isostructural condition. This was done for dipropylene glycol dibenzoate (DPGDB) [240]. Knowing Ea of the isostructural state and n ¼ 0:43, the product (1 n)Ea has the value of 48.4 kJ/mol. On the other hand, by ﬁtting the experimental JG b-relaxation times (Fig. 36) in the lower- temperature region to an Arrhenius equation, its activation energy EJG has the value of 48.0 0.6 kJ/mol. There is good agreement between (1 n)Ea and EJG . Unfortunately, most of the articles concerning JG relaxations do not report any information about thermal history, and one cannot be sure if the relaxation times were derived isostructurally. The simple relation [Eq. (13)], has been tested for other systems where genuine JG relaxations have been reported in the literature, although the thermal history followed to reach the glassy state is not given. The results are shown in Fig. 47 [80], where the product (1 n)Ea is plotted against the experimentally measured activation energy EJG of the JG brelaxation. A linear regression of data yields an angular coefﬁcient of 0.99 0.01, meaning a remarkable agreement between experiments and model predictions, in spite of possible errors due to unknown thermal history of the glassy state. The data reported in Fig. 47 include the glass-formers reported in Ref. 34, some low-molecular-mass van der Waals glass-formers (like OTP, isopropyl benzene, and toluene), some hydrogen-bonding systems (like glucose and n-propanol), the mixture benzyl chloride in toluene, and the polymer PMMA. Additionally, Fig. 47 displays also the JG b-relaxation of some epoxy compounds having a multiple secondary relaxation scenario (only the slowest secondary relaxation is the genuine JG) [290,291], the data of the JG b-relaxation for the polymer PET [292], and the recently discovered JG b-relaxation of polyisoprene [35]. D. Explaining the Properties of sJG (i). Pressure Dependence and Non-Arrhenius Temperature Dependence Above Tg. The CM equation ta ðT; P; V; SÞ ¼ ½tcn t0 ðT; P; V; SÞ1=ð1nÞ ½tcn tJG ðT; P; V; SÞ1=ð1nÞ

ð14Þ

557

dispersion of the structural relaxation 80

Eβ th=(1-n)Eα(Tf ) [kJ/mol]

PMMA PIS

60

BIBE OTP DPGDB

40 Toluene

Polymers VW molecules OH-systems ClB/tol epoxies

20

0 0

20

40

60

80

Eβ expt.[kJ/mol] Figure 47. Linear correlation between the experimental activation energy in the glassy state for the JG b-relaxation process (abscissa) and the activation energy predicted by the CM for the primitive relaxation (ordinate). Symbols for simple van der Waals molecules, H-bonded systems, polymers, chlorobenzene/toluene mixture, and epoxy oligomers are shown in the ﬁgure. The solid line is a linear regression of data (linear coefﬁcient 0.99 0.01).

CH 3

8

HO

Si

O

H

CH 3

log (f p [Hz])

6

E A =48 kJ/mol

ν0

4 Bulk 20.0 nm 7.5 nm 5.0 nm

2 0

Bulk να

–2 4.5

Figure 48.

5.0

5.5

6.0 6.5 1000 K / T

, n = 0.48 7.0

7.5

Loss peak frequency, fp , of PDMS in bulk or in nanopores.

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relates tJG to ta . Here we write out explicitly all the possible dependences of the relaxation times on the thermodynamic variables P, V, and T and entropy S, which are related by ðqS=qPÞT ¼ ðqV=qTÞP [293]. From Eq. (14), the properties of the JG relaxation discussed in Section V can be immediately explained. For example, the properties of tJG , including the pressure dependence and the non-Arrhenius temperature dependence (in the equilibrium liquid state), all readily follow from the corresponding properties of ta and vice versa from their relation. Any dependence of ta is stronger than the corresponding dependence of tJG because, in Eq. (14), the former is obtained from the latter by raising it to the power 1/(1 n), which is larger than one. (ii) Physical Aging. The same is true for the increase of relaxation times on physical aging of the glassy state. Let us denote by tJG (0) and ta (0) the two relaxation times at t ¼ 0 when the aging experiment starts, and by tJG (tage) and ta (tage) the two relaxation times after tage. From Eq. (14), we see that the shift factors of the two relaxation times are then related by log½ta ðtage Þ=ta ð0Þ ¼

log½tJG ðtage Þ=tJG ð0Þ 1n

ð15Þ

The shift factor of ta is larger than that of tJG by the factor 1/(1 n). This quantitative prediction is consistent with the physical aging data of propylene carbonate (Fig. 37), propylene glycol (Fig. 38), and glycerol (not shown). As can be seen from the dielectric aging data and the value of n of propylene carbonate and propylene glycol in Figs. 37 and 38, the relation between two shift factors (indicated by the lengths of the horizontal arrows) are consistent with the Eq. (15). In principle, the Kohlrausch parameter, and therefore the coupling parameter n, could be dependent on tage in Eq. (15) [294]. Anyway, it is noteworthy that in a recent aging experiment done on similar samples [244], negligible changes of the Kohlrausch parameter were reported during aging, thereby strengthening the substantial validity of Eq. (15). A glass is in a nonequilibrium state. At temperatures not far below Tg and when the structural a-relaxation time is not too long, given time the structure changes, driving the glass toward the equilibrium liquid state. It is the structural a-relaxation that can change the structure of the glass during aging. The JG or the primitive relaxation is local, and by itself it cannot effect structural changes. In a recent article [244], the dielectric loss data of several glasses obtained after they have been aged for different lengths of time, tage, were analyzed using two assumed equations to describe the changes of the dielectric loss e00 at any frequency n and the loss peak frequency na with tage. These equations were successful in ﬁtting the dependence of e00 on tage for many frequencies n higher than na , although two adjustable parameters est and eeq have to be used in the ﬁts for each n. The accomplishment led

dispersion of the structural relaxation

559

the authors of Ref. 244, to state: ‘‘It is the structural rearrangement during aging, which in a direct way (by shifting the a-peak to lower n) inﬂuences e00 (tage) in the a-peak region and in a more indirect way (by varying the structural ‘‘environment’’ felt by the relaxing entities) in the other regions.’’ By ‘‘other regions,’’ these authors mean the excess wing or the JG relaxation. We agree with this statement because in Section V the relaxation time of the JG relaxation (either resolved or hidden under the excess wing) has been shown to be pressure dependent in the equilibrium liquid state and nJG is therefore density-dependent. After the glass has been densiﬁed by aging for the time period tage, nJG is shifted along with na to lower frequencies. However, we do not agree with the conclusion of the authors that the excess wing and the JG relaxation shift to lower frequencies by exactly the same amount as na for all tage. This conclusion was drawn from their good ﬁts to the data, but the ﬁts involve two adjustable parameters est and eeq for each n. The aging data of propylene carbonate (PC) and propylene glycol used in their ﬁts are the same as that shown in Figs. 37 and 38, where we can see directly without any ﬁt that the excess wing does not shift by the same amount as the high-frequency ﬂank of the a-loss peak. The difference between the two shifts is not large because PC has n ¼ 0:27 [see Eq. (15)], which may explain the success of the ﬁts performed in Ref. 244, particularly with the luxury of two adjustable parameters for each n. Xylitol has a larger n ¼ 0:46, and the shift of the a-loss peak toward lower frequencies is more signiﬁcant than the resolved JG relaxation. Aging xylitol at 243 K, the e00 (tage) data for n ¼ 1 kHz belongs to the JG relaxation. On the other hand, the e00 (tage) data for n lower than 10 Hz are in the domain of the a-loss peak. The authors of Ref. 244 pointed out an anomaly in their ﬁts to the xylitol data: The observed decreases of e00 (tage) with increasing tage for n ¼ 1 kHz and 100 kHz are much milder than that of e00 (tage) for n less than 10 Hz. This anomalous behavior is likely due to the fact that actually the shift of nJG with aging time is smaller than that of na , and the good agreement of ﬁt with data is probably due to redundant free parameters. Although the structural change in aging is effected by the a-relaxation, in the glassy state the JG or primitive relaxation is still the elementary process from which the structural a-relaxation is formed out of many-molecule dynamics. The following describes how aging proceeds by feedback between the changes of the a-relaxation and the JG or the primitive relaxation with time. Let aging start at t ¼ 0. After a small increment of time, t1 , there is a small change in the structure or ﬁctive temperature Tf effected by the structural a-relaxation with relaxation time ta (t ¼ 0) given by ta ðt ¼ 0Þ ¼ ½tcn tJG ðt ¼ 0Þ1=ð1nÞ. Since the JG or the primitive relaxation is sensitive to the structure through volume and entropy, their relaxation times tJG (t1 ) or t0 (t1 ) at t ¼ t1 increase slightly from tJG (t ¼ 0) or t0 (t ¼ 0), in response to the change of the ﬁctive temperature

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from Tf to Tf Tf 1 . In the next time period ½t1 ; t1 þ t2 , the structural is further changed by the structural a-relaxation with relaxation time now given by ta (t1 ), which is given by ta ðt1 Þ ¼ ½tcn tJG ðt1 Þ1=ð1nÞ. The new ﬁctive temperature is Tf Tf 1 Tf 2 . The JG or primitive relaxation responds to this step change of structure and are further increased to tJG (t1 þ t2 ) or t0 (t1 þ t2 ) at t ¼ t1 þ t2 . Repeating this argument many times to reach the aging time tage, the results for ta (tage) and tJG (tage) are determined. This description illustrates the feedback between the changes of ta and tJG during aging, but still the primitive or the JG relaxation is the precursor of the structural relaxation at all times. (iii) Invariance of tJG to Variations of T and P at Constant ta . We have seen in Section II that the dispersion of the a-relaxation (or n) is invariant to different combinations of temperature and pressure so long they keep ta constant. Accepting this as an empirical fact, Eq. (14) immediately explains the experimental ﬁnding discussed in Section V.B that tJG is also invariant to changes in temperature and pressure while ta is maintained constant. The T and P dependences of ta of many glass-formers can be recasted as the dependence of log(ta ) on the product variable, T 1 V g [114,115]. Here V is the speciﬁc volume and g is a material-speciﬁc constant, which is found to vary over a broad range 0:14 g 8:5 for the glass-formers investigated to date. Combining this with the invariance of tJG to changes in T and P while maintaining ta constant, it follows that log(tJG ) must also be a function of T 1 V g However, when rewritten as functions of T 1 V g , the established relation between ta and tJG , ta ðT 1 V g Þ ¼ ½tcn t0 ðT 1 V g Þ1=ð1nÞ

or

½tcn tJG ðT 1 V g Þ1=ð1nÞ

ð16Þ

still holds. This is a CM prediction that both ta and tJG (or t0 ) depend on the same variable T 1 V g , but their dependences are different and related by the equation above. This prediction has not yet been tested due to lack of data of resolved JG relaxation over large range of variations of T and P at the present time. Nevertheless, dielectric relaxation data of a few polymers, showing not only the a-relaxation but also the longer time normal chain modes [94,96], offer quantitative tests of the prediction that ta and t0 are different functions of the same product variable T 1 V g and they are related by Eq. (16). This is because, in the application of the CM to polymer dynamics and viscoelasticity [25,191,202,205,206,208,209,295], the normal modes and the local segmental mode have the same primitive monomeric friction coefﬁcient 0 ðT 1 V g Þ. Based on this, the breakdown of thermorheological simplicity of the viscoelastic spectrum of amorphous polymers discussed in Section III, (paragraph 4), was

dispersion of the structural relaxation

561

explained based on the difference between the coupling parameter na and nn of the local segmental mode and the normal modes, respectively. While ta is governed by the friction coefﬁcient, a ðT 1 V 1 Þ ¼ ½0 ðT 1 V 1 Þ1=ð1na Þ

ð17Þ

n ðT 1 V g Þ ¼ ½0 ðT 1 V g Þ1=ð1nn Þ

ð18Þ

tn is governed by

Thus, ta and tn are both functions of T 1 V g , but their dependences on T 1 V g are different. Usually, na is larger than nn , and certainly in the case of nn ¼ 0 for the Rouse normal modes of unentangled polymers. Thus, a stronger dependence of ta on T 1 V g than tn is predicted by the two equations given above. The prediction can be tested quantitatively after the values of na and nn have been determined by analysis of the dielectric spectra. In fact, dielectric relaxation measurements of the a-relaxation and the normal mode were obtained for various T and P combinations on polypropylene glycol (PPG), 1,4-polyisoprene (PI), and poly(oxybutylene) (POB)94,96. Both ta and tn were shown to yield master curves when plotted against the parameter T 1 V g with the same value of g. However, the dependences of ta and tn on T 1 V g are not the same. Neither is their dependences on V at constant T, or on T at constant P. The experimental ﬁndings were explained by the CM equations, Eqs. (17) and (18) [207]. The values of na and nn were determined from the dielectric spectra, and the stronger dependences of ta on T 1 V g than tn were accounted for quantitatively. E. Consistency with the Invariance of the a-Dispersion at Constant sa to Different Combinations of T and P According to the CM interpretation of the evolution of dynamics with time, the primitive relaxation is the fundamental building block of the many-molecule dynamics that ultimately ends up in the a-relaxation with its correlation function having the Kohlrausch form [Eq. (1)]. Experimental data have shown that the origin of the dependences of molecular mobility on T, P, V, and S is in tJG or t0. The corresponding dependences of the a-relaxation time ta are not original but derived from those of tJG or t0 and given by the CM equation, Eq. (14). Different combinations of T and P can certainly be found to maintain tJG or t0 constant. As a consequence of Eq. (14), n must be the same in order that ta can be kept constant for all these combinations of T and P. Thus, the invariance of the a-dispersion to different combinations of T and P at constant ta , found experimentally in many glass-formers (see Section II), can be derived from the CM.

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kia l. ngai et al. F.

Relaxation on a Nanometer Scale

Supercooled liquids conﬁned in nanometer-size glass pores may not be chemically bonded or physically interacting with the walls if the surfaces of the walls has been rendered passive by treatment such as with silane. There are several studies of the relaxation of the nanoconﬁned liquids in silanized glass pores. Examples include the liquids 1,2-diphenylbenzene (also known as ortho-terphenyl (OTP)) [296–297,298], salol [299], and the polymers poly(dimethyl siloxane) and poly(methylphenyl siloxane) [300,301]. Under such conditions, the molecules nearer the smooth wall have larger reduction of intermolecular coupling because their neighboring molecules on the side of the wall have been removed. Another factor that contributes to the reduction of intermolecular coupling is when the pore size becomes comparable or smaller than the many-molecule dynamics length-scale, and hence reduces the extent of the many-molecule dynamics inside the pores. Possible lower density of the liquid in the pores than in the bulk directly affects the primitive relaxation time through its dependence not only on density but also on the coupling parameter, because the molecules are further apart. Reduction of intermolecular coupling in the CM means that the coupling parameter n(T) becomes smaller than the bulk value nb(T). Had this effect been the only factor, the dispersion of the liquid in the pore will be narrower than in the bulk. However, there is an additional complication. The cooperative dynamics of molecules situated at various distances from the wall are necessarily affected differently and thus n is position-dependent. The overall observed relaxation dispersions of conﬁned liquids are additionally broadened by the spatial distribution of the molecules in the pores. This is a dominant effect on the observed dispersion because all molecules inside the pore are sampled by experiment, and it could eclipse the effect of the reduction of n(T) of the molecules at distances nearer the smooth wall. Hence, we cannot obtain the reduced values of n(T) of molecules nearer the wall from the experimentally observed width of the dispersion of the liquid conﬁned in pores. Nevertheless, we expect that n(T) to decrease with the decreasing size of the pores. The CM equation can be rewritten as ta ¼ t0 ðt0 =tc Þn=ð1nÞ or ta ¼ tJG ðtJG =tc Þn=ð1nÞ. Since t0 or tJG is usually much larger than tc ¼ 2 ps and n=(1 n) is a monotonic decreasing function with decreasing n, ta (T) of nano-conﬁned liquids decreases on decreasing the size of the pores. Consequently, the difference between ta and t0 or tJG becomes smaller [298,302,303]. This trend is shown in Fig. 48 by the dielectric relaxation data of PDMS conﬁned in silanized glass pores of various sizes. If in sufﬁciently small pores n ! 0, then ta ! t0 or tJG, and the characteristics of the a-relaxation will not be very different from that of the JG relaxation. The location of the primitive frequency n0 corresponding to t0 calculated from the bulk ta and n ¼ 0:48

563

dispersion of the structural relaxation

1.0

TMDSC; Dielectric;

X = ∆Cp X = ∆ε

Bulk

0.6

X/X

Bulk

0.8

5 nm

0.4

0.2

0.0 0.1

1

10

100

1000

10000

Pore Size [nm] Figure 49. Dielectric strength e as well as the thermal capacity Cp of the a-relaxation of PDMS normalized to the bulk values are plotted against pore size. The data show mark decrease of e and Cp from the bulk value on decreasing the pore size.

values at one temperature is indicated in Fig. 49, and note that they differ by about ﬁve orders of magnitude. On the other hand, ta of PDMS in 5-nm pores at the same temperature is only a factor of 4 longer than t0 , and its temperature dependence is weaker than ta of bulk PDMS, having an apparent activation enthalpy of 48 kJ/mol commonly found for JG relaxation of polymeric glassformers. Moreover, the dielectric strength e as well as the thermal capacity Cp of the a-relaxation undergo a marked decrease from the bulk value on decreasing the pore size as shown in Fig. 49. At 5 nm, the dielectric strength and Cp of the a-relaxation are signiﬁcantly smaller than the same quantities of bulk PDMS. They become similar to that observed for the JG relaxation of bulk glass-formers, in comparison with the a-relaxation [304], indicating once more that in nanometer-size pores the a-relaxation is not much different from the JG relaxation. The data of OTP conﬁned in silanized nano-pores and comparison with t0 is shown in Fig. 50. Again, there is good indication that at small pore size, ta tends to be close to t0 . The results in Figs. 48–50 demonstrate the veracity of the primitive relaxation and its relaxation time t0 . They are actually observed as the a-relaxation and ta when the size of the glass-former is scaled down to a few nanometers. This transformation of the a-relaxation of the bulk glass-former to the primitive or

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Figure 50. Temperature dependencies of the various relaxation times of OTP. Filled circles are a-relaxation times of bulk OTP obtained by photon correlation spectroscopy; open diamonds are JG relaxation times obtained by dielectric spectroscopy; open circles are the primitive relaxation times to of bulk OTP calculated by Eq. (10). The photon correlation spectroscopy relaxation times of OTP conﬁned in 7.5-nm pores (~); 5.0-nm pores (!); 2.5-nm pores (&).

the JG relaxation occurs because the many-molecule dynamics in the bulk are suppressed when the size is reduced to the nanometer scale. Similar effects were found in thin poly(methylphenylsiloxane) (PMPS) ﬁlms with thickness of the order of 1.5 to 2.0 nm intercalated into galleries of silicates (Fig. 51) [302,303,305]. The observed a-relaxation time ta in the PMPS ﬁlm is much shorter than in the bulk at the same temperature. The root-mean-squared end-to-end distance of the PMPS chains is estimated to be of the order of 3 nm, which is about twice the thickness of the ﬁlms. It can be expected that there are signiﬁcant induced orientations in the chains due to severe chain conﬁnement. This, together with the extremely small thickness of the ﬁlm, suggests that we may now have large reduction of intermolecular coupling of the local segmental relaxation. Hence, in the thin ﬁlm, the coupling parameters n may be reduced to approach zero value, and the observed a-relaxation time ta (T) becomes nearly equal to the primitive relaxation time t0 (T) calculated from the parameters, ta (T) and n, of bulk PMPS. This expected change of ta (T) is supported by the experimental data shown in Fig. 51 [302], like in the case of PDMS conﬁned in nanometer-size glass pores discussed in the previous paragraph. Another similar case is thin supported polystyrene ﬁlms [306]. The changes of ta (T) on decreasing ﬁlm thickness from 90 nm to 6 nm are similar to that found in PDMS conﬁned in glass pores when the pore size is decreased from the bulk to 5 nm (Fig. 49) or in PMPS thin ﬁlms (Fig. 51).

dispersion of the structural relaxation

565

Figure 51. Temperature dependencies of the various relaxation times of PMPS obtained by dielectric spectroscopy. (!) a-Relaxation rates of bulk PMPS. () a-Relaxation rates of 1.5-nm thin ﬁlms of PMPS. (&) Primitive relaxation rate of bulk PMPS calculated by Eq. (10).

Some studies of the dynamics of thin polymer ﬁlms go beyond the local segmental relaxation to include the modes of longer length scales, such as the Rouse modes, and the terminal modes if the polymer is entangled. In bulk amorphous polymers, it is well established experimentally that the temperature and pressure dependences of the more global viscoelastic modes are different from that of the local segmental relaxation [25,94,96,189,193,202,206, 207,295,307–309]. In considering the viscoelastic data of thin polymer ﬁlms, one should be mindful of these facts in bulk polymers. When analysing or interpreting the data of thin ﬁlms, it is logical to require the theory or model used to be able to account for, or at least consistent with, these known bulk polymer properties. Otherwise, the theory constructed for thin ﬁlms will not be worthwhile. The Coupling Model (CM) satisﬁes this requirement (see Section III, paragraph 6). Extended to consider polymer thin ﬁlms, the CM predicts an increase of the mobility of the local segmental motions as discussed in the prededing section, but predicts the lack of such a change for the mobility of the

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P8-ONS/PDE

1.0 R/OTP

0.9

R/PIB

0.8

BPEA/OTP

βA = (1-nA)

R/TNB

0.7 R/PS

0.6

T/PIB T/OTP

0.5

T/TNB R/PSF

0.4

T/PS A/OTP

0.3 0.2

T/PSF

–2

–1

0 1 2 Log 10 (τA / ταB)

3

4

Figure 52. Plot of the stretch exponent bA of the probe correlation function against the ratio, tA =taB , of the probe correlation time tA to the host a-relaxation time taB . T/PS denotes the probe tetracene (T) in the host polystyrene (PS). Other probes are anthracene (A), BPEA, rubrene (R), and the PS spheres (PS-ONS). The other hosts are polysulfone (PSF), tri(naphthal benzene) (TNB), orhto-terphenyl (OTP), polyisobutylene (PIB), and phenylphthalein-dimethylether (PDE).

Rouse modes and the diffusion of entire polymer chains [303]. These predictions of the CM are in accord with experiments and computer simulations cited in Refs. 303 and 310. Since the discovery more than a decade ago of the deviation from the bulk glass-transition temperature due to conﬁnement of glass-formers in nanometer spaces [311–313], many experiments have shown that there can be signiﬁcant changes of Tg as well as other dynamic properties in the nanoconﬁned materials. However, the changes vary greatly depending on the nature of the interfaces, the chemical structure of the nanoconﬁned glass-former, the experimental methods used, and, in the case of polymers, the length scale of the dynamics probed [314–317]. Just for Tg alone, it can decrease, increase, or remain the same depending upon the experimental or simulation conditions. As concluded in the

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most recent and comprehensive review [317a], ‘‘. . . the existing theories of Tg are unable to explain the range of behaviours seen at the nanometre size scale, in part because the glass transition phenomenon itself is not fully understood.’’ We fully agree with the above quoted remark by Alcoutlabi and McKenna, and furthermore we can identify the reason why the existing theories of Tg fail to explain the range of behaviors of nanoconﬁnement. Most theories of dynamics of nanoconﬁnement are based on conventional theories of glass transition for bulk materials. But, as has been shown in previous sections, conventional theories and models of glass transition are not able to account for some general dynamic properties of bulk glass-formers. What causes the conventional theories to be deﬁcient already for bulk glass-formers is the lack of an adequate treatment of the many-molecule dynamics or at least taking into account the dispersion of the structural a-relaxation as a fundamental element. Thus, it is unsurprising to ﬁnd that, when applied to nanoconﬁnement, the conventional theories have limited success in explaining the range of behaviors of nanoconﬁnement. On the other hand, the coupling model is able to explain the range of behaviors seen at the nanometer size scale [317b]. G.

Component Dynamics in Binary Mixtures

In general, when two miscible glass-formers A and B are mixed, the dynamics of the component molecules A and B are changed from that in their respective pure states. The problem of predicting the dynamics of each component has attracted much attention particularly in miscible mixtures of two polymers. Perhaps the ﬁrst published model addressing the component dynamics of binary polymer blends was by Roland and Ngai (RN) [318–324]. The coupling model’s description of homopolymer dynamics was extended to blends by incorporating dynamic heterogeneity, due both to the intrinsic mobility differences of the components and to the local compositional heterogeneity from concentration ﬂuctuations. The dynamics of any relaxing species in a blend is determined by its chemical structure, as well as by the local environments, since the latter govern the intermolecular cooperativity and the associated coupling parameters of the relaxation. Thus, the relaxation of a given species reﬂects its intrinsic mobility and the degree of intermolecular coupling (measured by the coupling parameter n) imposed by neighboring segments (molecules, if the mixture is composed of nonpolymeric glassformers), the latter obviously ﬂuctuates and is composition dependent. The important ansatz of the blend model is that each environment i of A is associated with a coupling parameter, nAi, the magnitude of which depends on the intermolecular constraints on A imposed by the molecules in the environment. The correlation function is given by Ai ðtÞ ¼ exp½ðt=tAi Þ1nAi

ð19Þ

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with the relaxation time tAi given by tAi ¼ ½tcnAi t0 1=ð1nAi Þ

ð20Þ

This ansatz can be rationalized by some theoretical considerations [325,326].It is also supported by the experimental data at very low concentrations of the component A where the study is reduced to the dynamics of the probe A in host B. Each probe molecule experiences the same environment, which eliminates the complications from concentration ﬂuctuations. We have mentioned in Section III, paragraph 4, that the probe rotational correlation function indeed has the Kohlrausch form. The differential between the probe rotational time tA and the host a-relaxation time taB is gauged by their ratio, tA =taB . As expected, the slower the host B compared with the probe A, the larger the coupling parameter, nA ð1 bA Þ, obtained from the stretch exponent bA of the measured probe correlation function. The experimental data are shown in Fig. 52. For more details, see Ref. 172. Another support of the CM ansatz for blends come from molecular dynamics simulations. Molecular dynamics simulations of a binary LJ liquid thin ﬁlm conﬁned by frozen conﬁgurations of the same system has been performed [327]. The ﬁlm thickness was 15.0 in units of the length parameter of the LennardJones potential, with the ﬁlm center at a distance z ¼ 7:5 from the conﬁning walls. An important feature of the simulation is the interaction of the mobile particles in the ﬁlm with the immobile particles comprising the conﬁning walls. Interaction between the mobile and the immobile particles occurs via the Lennard-Jones potential, the same as between mobile particles. The particles in the immediate vicinity of the wall are highly constrained (i.e., strongly coupled) because of the neighboring immobile particles of the wall (the analog of the component B that has much higher Tg in mixture with A). Thus, the ansatz of the blend model leads to the expectation that particles in layers closer to the wall will have a large n. The advantage of simulation is that the self part of the intermediate scattering function, Fs(q,z,t), can be calculated for any layers at a distance z from the wall. All particles at the same distance from the wall are equivalent. Remarkably, for all z, Fs(q,z1,t) has the Kohlrausch form, Fs ðq; z; tÞ ¼ AðzÞexp ½t=tðzÞ1nðzÞ

ð21Þ

with n(z) decreasing with increasing distance z from the wall, and at the center of the ﬁlm it become the same as the coupling parameter of bulk binary LJ liquid. Furthermore, the relaxation time t(z) and the coupling parameter n(z) obtained from the simulation were shown [302] to follow the relation nðzÞ tðzÞ ¼ ½tc t0 1=½1nðzÞ approximately for all z, with the same primitive relaxation time t0 for all z. Thus, the dynamics of the layers are consistent

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with the CM ansatz for component dynamics of blends. When all particles in the conﬁned thin ﬁlm are included in the calculation, the intermediate scattering function, Fs(q,t), does not have the Kohlrausch stretched exponential time dependence. Instead it exhibits a long-time tail, which when Fourier transformed causes asymmetric broadening toward low frequencies. This effect is similar to the dielectric frequency dispersion of a polymer blended with a less mobile component, such as PVME mixed with polystyrene [328], and described by the blend model by a superposition of Kohlrausch functions with different ni and ti . Various observed phenomena have been explained by the blend model, such as component dynamics differing qualitatively from that of the pure components, thermorheological complexity, unusual concentration dependences, asymmetric broadening of the relaxation function, and the emergence of a secondary relaxation peak. For a review see Ref. 329, where supporting evidence for the change of coupling parameter of a component in the blend and its distribution due to concentration ﬂuctuations are given. In the intervening years, other models of component dynamics in polymer blends have been proposed. Fischer and co-workers (FZ) [330,331] developed a model that speciﬁcally addressed in terms of subvolumes the effect of local composition on the glass transition temperature. Kumar and co-workers [332,333] extended the FZ concentration ﬂuctuation model. While concentration ﬂuctuations still give rise to subvolumes, each governed by a local glass temperature, Kumar et al. invoke the idea that experimental probes of the dynamics only sense composition ﬂuctuations that occur over a certain cooperative volume. Rather than a constant length scale, the cooperative volume is governed by the local composition. Lodge and McLeish [334,335] observed that, when making comparisons to experimental results, the cooperative length scale required to ﬁt the data is too large, on the order of 10 nm or more in both the models of Fischer et al. and Kumar et al. Lodge and McLeish proposed an alternative model based on the Kuhn length, lK , of the polymer chain. The segmental relaxation rate of a segment of component A, in a binary blend of polymers A and B, is determined by the composition of its local volume with a length scale of lK of A. Because of chain connectivity, this local volume is, on average, richer in A units than the average bulk concentration fA (‘‘self-concentration’’ effect). This enhanced local concentration of A is given by aeff ¼ fA þ fsA ð1 fA Þ, where fsA is the ‘‘self-concentration’’ of A, determined by the volume fraction occupied by A repeat units in a volume of size ðlK Þ3 . Concentration ﬂuctuations and the dispersion of component dynamics have not been taken into consideration because the objective of the Lodge and McLeish model is to predict the component relaxation times. For polymer blends, the proposal of taking into consideration of ‘‘self-concentration’’ effect is reasonable, although not necessarily in the manner proposed by Lodge and McLeish. However, this effect cannot be the ultimate factor that governs component dynamics in

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polymer blends. This is because the ‘‘self-concentration’’ effect is absent in mixtures of nonpolymeric molecular glass-former, and yet their component dynamics are similar in many respects to polymer blends. Understandably, the component dynamics in binary mixtures are more complicated to unravel than the dynamics of a neat glass-former. The solution to the problem will require additional inputs, making it more difﬁcult to test any proposed theory against the observed component dynamics stringently. Actually, a cogent as well as stringent test of any theory of mixtures already exists without even having to compare the predictions of the theory with the observed component dynamics. We just have to ask the question, What does the theory predict when the concentration of one component vanishes and the mixture is reduced to a neat glass-former? Many general properties of neat glass-formers have been given in the previous sections, and the test is whether the predictions of the theory in the neat glass-former limit are consistent with these properties. One outstanding example is the invariance of the dispersion of the structural arelaxation to different combinations of temperature and pressure while ta is kept constant. In the blend theory of Fischer et al., it is assumed that within each subvolume the shape of the complex dielectric sucseptibility for one component is the same as that of the component in the pure state and can be described by the empirical Havriliak–Negami function. Thus, the blend theory of Fischer et al. is built upon a theory of glass transition of neat glass-formers in which the dispersion and the relaxation time ta of the a-relaxation are independent inputs, a priori bearing no relations with each other. Hence, there is no guarantee that they are co-invariants for different combinations of T and P. The emphasis of the blend theories of Kumar et al. and Lodge and McLeish is on predicting the component relaxation times and not on the dispersions of the components. Obviously these two theories have not much to say about (a) the dispersion of a component in its pure state and (b) the fact that many general properties of ta of pure glass-formers are either governed by or correlated with the dispersion, as shown throughout this chapter. When the concentration of one component in a binary blend is small, the problem is transformed to the study of the dynamics of a probe molecule in a host glass-former. This is a simpler problem than component dynamics in a blend because of the absence of concentration ﬂuctuation. Nevertheless, the extent of deviation of the probe dynamics from the Stokes–Einstein and the Debye– Stokes–Einstein relations (see Fig. 26) and the frequency separation between the a- and b-relaxations of the probe molecule (see Fig. 53) is correlated with the dispersion of the probe molecule. None of these properties in simpler situations can be addressed by the three blend theories discussed in this paragraph. The logical basis of these theories can be questioned because they attack a more complicated problem (i.e., the component dynamics in a blend) before resolving or at least addressing the outstanding phenomenologies found in a simpler problem (i.e., the dynamics of neat glass-formers or a probe molecule). On the

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Figure 53. The dielectric loss of 2-picoline in mixtures with tri-styrene at different concentrations obtained at different temperatures but similar a-relaxation times of the 2-picoline component. For clarity, each spectrum is shifted by a concentration-dependent factor kc. Data from T. Blochowicz and E. A. Ro¨ssler, Phys. Rev. Lett. 92, 225701 (2004).

other hand, this criticism does not apply to the Coupling Model (CM) constructed for component dynamics in blends. It is an extension of the CM for the dynamics of neat glass-former and probe dynamics in a host glass-former, polymeric or nonpolymeric, the general properties of which have either been explained or shown to be consistent with the predictions of the CM. Anomalous component dynamics in blends and mixtures have been observed which cannot be explained by the models of Fischer et al., Kumar et al., and Lodge and McLeish [329]. A recent example is the anomalous dynamics of d4PEO found in the d4PEO/PMMA polymer blends [336]. While this anomaly cannot be understood by the other models, it has an immediate explanation from the CM [337]. The CM has a quantitative relation between the a-relaxation frequency na and the Johari–Goldstein (JG) relaxation frequency nJG or the primitive relaxation frequency n0 given by Eqs. (10)–(12), which accounts well for the experimental data of many neat glass-formers. For any ﬁxed value of na , the separation between the a-relaxation and the JG relaxation, log(nJG ) log(na ), is proportional to n. The change of dynamics of a lower Tg component A when mixed with another component B occurs partly due to an increase of its most probable coupling parameters, nA , according to the CM. The increase of nA is larger at higher concentration of the component B because of the increasing numbers of the less mobile molecules B in the neighborhoods of the A molecules. The immediate consequence of this precept of the CM theory for blends, in ðAÞ ðAÞ ðAÞ conjunction with the CM relation log10 nJG log10 na nA ð10:9 log10 na Þ,

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ðAÞ

is that the separation distance, log10 nJG log10 na , between the a-relaxation and ðAÞ the JG relaxation of component A at constant log10 na , should increase with increasing concentration of component B. On the other hand, the most probable coupling parameters, nB, of the higher Tg component B will decrease with increasing concentration of the component A. From the same CM relation, the ðBÞ ðBÞ separation distance, log10 nJG log10 na , between the a-relaxation and the JG ðBÞ relaxation of component B at constant log10 na should decrease with increasing concentration of the lower Tg component A. These are predictions that can be either veriﬁed or falsiﬁed by experimental measurements. In the following we shall cite some examples in which these predictions have been veriﬁed. (i) The JG and the a-Relaxations of Poly(n-butyl methacrylate) in Poly(nbutyl-methacrylate-stat-styrene) Random Copolymers. The copolymers of poly(n-butyl methacrylate) (PnBMA) and polystyrene (PS) is one example [338,339]. Here PnBMA is the lower Tg component A. Neat PnBMA has n ¼ 0:47 and a JG relaxation. On increasing the styrene content from 0 to 66 mol% in the copolymer, a monotonic increase of nA of the PnBMA component leads to a concomitant increase in the separation of the JG relaxation from the segmental relaxation, both of the PnBMA component. This changes were observed in the dielectric relaxation experiment. (ii) The JG and the a-Relaxations of Poly(vinyl methyl ether) in Blends with Polystyrene. Neat poly(vinyl methyl ether) (PVME) does not have a resolved JG relaxation. In the mixture with polystyrene (PS) that has a higher Tg, an increase of nA of the PVME component is expected. For blends with large enough PS ðAÞ ðAÞ concentrations, the distance, log10 nJG log10 na , will be sufﬁcient large for the JG relaxation of PVME to be resolved as a distinct peak. This prediction is borne out by dielectric relaxation data for blends with 70% or higher PS concentrations [340]. For these compositions, a new relaxation loss peak appears, at a frequency intermediate between that of the a-peak and the highfrequency non-JG secondary relaxation peak of the PVME component. The new peak has an Arrhenius temperature dependence at temperatures below TgA of the PVME component, and its dielectric strength increases with temperature (see Fig. 5 of Ref. 340). These characteristics indicate that the relaxation is indeed a JG process. The isochronal dielectric data at 1 kHz for 70% PVME mixed with 30% poly(2-chlorostyrene) [341] also show a resolved JG secondary relaxation, and a similar explanation applies. (iii) The JG and the a-Relaxations of Picoline in Mixtures with Tri-styrene or Ortho-terphenyl. Neat picoline is a small molecule glass-former A having a smaller n ¼ 0:36 [241,242,342]. As a result and in accordance with Eq. (12), its JG relaxation is not well separated from the a-relaxation, showing up as an excess wing. When mixed with a higher-Tg glass-former B such as tri-styrene or OTP [241,242], the JG b-relaxation of picoline was resolved at sufﬁciently high

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dispersion of the structural relaxation 100% wt.TBP 60% wt. TBP 40% wt. TBP 25% wt. TBP 16% wt. TBP

169 K 1

10

K*ε ''

0 188 K

10

203 K –1 215 K

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10

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–2

10

–1

0

10

1

10

2

10 f[Hz]

10

3

10

4

10

5

6

10

7

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Figure 54. Dielectric loss spectra with the same maximum peak frequency for different concentrations of tert-butylpyridine (wt%) in tristyrene. For clarity, each spectrum is shifted vertically by a concentration dependent factor K: K ¼ 7, 2, 1.3, 1, and 0.98 for 100%, 60%, 40%, 25%, and 16% TBP, respectively. The x axis is the real measurement frequency, except for the spectra of 100% and 16% TBP, where horizontal shifts of frequency by factors of 1.75 and 0.80, respectively, have been applied.

concentrations of B molecules. The separation between the JG b-relaxation and aðAÞ ðAÞ ðAÞ relaxation of the picoline component, log10 nJG log10 na at constant log10 na , was observed to increase monotonically with concentration of B in the mixtures (Fig. 53). The experimental data are in accord with the predictions of the CM [342]. Exactly the same behavior was recently found in the JG b-relaxation and arelaxation of tert-butylpyridine, a polar rigid molecule with low Tg ¼ 164 K, when mixed with the apolar tri-styrene (Tg ¼ 232 K) by Kessairi et al. [343] Sample data are shown in Fig. 54. (iv) The JG and the a-Relaxations of Sorbitol in Mixtures with Glycerol. A dielectric study of mixtures of sorbitol with glycerol [344] reported on the changes of the dynamics of both the a-relaxation and the JG relaxation of sorbitol, the component B that has a higher Tg than glycerol, the component A. Neat sorbitol has a larger n ¼ 0:52, along with a resolved JG relaxation well separated from the a-relaxation [235]. The CM predicts increasing reduction of the coupling parameter nB of sorbitol on increasing the concentration of the ðBÞ ðBÞ more mobile glycerol molecules, and decreasing separation, log10 nJG log10 na , between the a-relaxation and the JG relaxation of the sorbitol component B at ðBÞ constant log10 na . The experimental data was shown to be in accord with the CM prediction [136]. Similar results are found in mixtures of xylitol with water

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[345]. The separation between the a-relaxation and the JG relaxation of xylitol ðBÞ at constant log10 na of xylitol is reduced with addition of water. H.

Interrelation Between Primary and Secondary Relaxations in Polymerizing Systems

Linear polymers or polymer networks are built upon small molecular repeat units by bonded interactions. These bonded interactions reinforce the intermolecular constraints on the primitive motion and increase the coupling parameter n or equivalently the width of the dispersion of the structural a-relaxation. The comparison of the dynamics of a monomer with the oligomers and the highermolecular-weight linear polymers or networks formed by physical or chemical cross-links is a worthwhile undertaking because essentially the chemical structure of the repeat unit remains the same. The change in the intermolecular constraints or the coupling parameter of the CM is made evident experimentally by the change of the dispersion of the structural a-relaxation. The polyols— glycerol, threitol, xylitol and sorbitol—have the number of carbon atoms on the ‘‘backbone’’ increase from 3 to 6, and their n values are 0.29, 0.36, 0.46, and 0.52, respectively. Concomitant with the increase of n, the separation between the a-relaxation and the JG relaxation, log10 nJG log10 na , is observed to increase as expected by the CM [36]. A similar trend is found in a series of propylene glycols: its dimer, trimer, heptamer, and poly(propylene oxide) [278]. The same is found in a series of oligo(propyleneglycol) dimethyl ethers [346], where again both n and log10 nJG log10 na increases with molecular weight. However, there is hardly any change in the ‘‘fragility’’ index m. This ﬁnding is another indication that m is not a fundamental quantity and its correlation with n or the dispersion can break down as cautioned before in Section III, paragraph 1. Naturally there is no correlation between log10 nJG log10 na and m. This was mentioned by Mattsson et al. [346], and for this aspect they cited Ref. 32, implying that that this reference proposed a correlation between log10 nJG log10 na and m. This implication is inappropriate because throughout Ref. 32 as well as in this chapter, the emphasis is on the correlation between log10 nJG log10 na and n. Tri-styrene has been mentioned before in this section. It has a much narrower a-relaxation dispersion [242] compared with that of higher-molecular-weight polystyrene. Tristyrene has no resolved JG relaxation because n is small and hence the separation distance, log10 nJG log10 na , is small. On the other hand, polystyrene has a resolved JG relaxation [194,347] because it has a broad a-relaxation dispersion and n ¼ 0:64 from photon correlation spectroscopic measurement [210]. The changes of molecular dynamics from the starting material of lowmolecular-weight epoxy resins to the fully polymerized state during polymerization reaction are well brought out by the large number of experimental studies of the evolution of the primary and secondary relaxations over the past

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two decades by many researchers [207,348–375]. The most widely studied starting material is diglycidyl ether of bisphenol-A (DGEBA) under the trade name EPON828. The neat diepoxide EPON828 is an example of molecular glass-formers that have two secondary relaxations and polymerization offers an excellent opportunity of showing the difference in their properties and which one is the true JG relaxation as well as elucidating the changing relation between the JG relaxation and the a-relaxation with the degree of polymerization. There is a tremendous amount of published work on the evolution of the primary and the two secondary relaxations with reaction time, or the number of covalent bonds formed. Dielectric spectra of epoxy systems have been measured at various stages of the polymerization process and the properties of the secondary and primary relaxations as well as their changes from stage to stage are known. In time order, the stages are given as follows: (1) the neat EPON828 [290] or another neat glass-forming tri-epoxy triphenylolmethane triglycidyl ether (TPMTGE) with trade name Tactix742 [291]; (2) the unreacted mixtures of EPON828 with aniline, cyclohexylamine (CHA), or p-aminodicyclohexylmethane (PACM), or unreacted mixtures of Tactix742 with the monoamines 3-chloroaniline or 4-chloroaniline [355,357,358,361,364–367]; (3) the partially polymerized products at different instants of reaction; and (4) the completely polymerized product. The observed molecular dynamics at each stage as well as the changes from one stage to another present multiple challenges for interpretation and explanation. The results are general, independent of the structure of the starting molecular liquids and the polymerizing or cross-linking agents used. Here we present one example to discuss the evolution of the dynamics in several stages. Additional examples can be found in a recent work [207]. The relaxation map of Fig. 55 shows the temperature dependence of the most probable relaxation times ta , tb , and tg of neat EPON828 obtained. The dielectric a-loss peak of neat EPON828 was well-ﬁtted by the one-sided Fourier transform of the KWW function with n ¼ 0:47. It is temperature-independent near Tg and together with ta (T), the corresponding t0 (T) is calculated by Eq. (10). The calculated values of t0 (T) at T ¼ 256 and 259 K near Tg are shown in Fig. 55 by the two solid inverted triangles. They are in good correspondence with the value of tb (T) obtained by extrapolating the data below Tg to these two temperatures, and thus the good correspondence between t0 (T) and tb (T) is veriﬁed in neat EPON828, as well as in Tactix 742 (not shown), like found in other molecular and polymeric glass-formers. This indicates that slow b-relaxation in EPON828 and Tactix742 is truly of the JG kind. On the other hand, the g-relaxation appearing at higher frequencies seems to be related to intramolecular motions involving mainly the epoxide end groups. On mixing 1 mol of EPON828 with 1 mol of aniline (which has a lower Tg) and before reaction starts, ta of the EPON828 component becomes shorter as

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10

γ 8 neat DGEBA

DGEBA/aniline

α

polymerized

JG

4

α

n=0.70

log10(1/τmax[s])

6

β β α

DGEBA/aniline unreacted

n=0.47

2

0

increasing Tg

-2 2.5

3.0

3.5

4.0

4.5

5.0

–1

1000 / T [K ]

Figure 55. The relaxation map for neat EPON828, for an unreacted mixture of one mole of EPON828 with 1 mole of aniline, and for partially polymerized and completely polymerized EPON828/aniline mixtures. There are two secondary relaxations in addition to the primary arelaxation. Shown are the reciprocals of the a-relaxation time ta , the JG b-relaxation time tb , and the non-JG relaxation time tg . The two solid inverted triangles indicate 1/t0 of neat EPON828 calculated at two temperatures just above its Tg. For the legends of other symbols and lines, as well as for the references from which the data are taken, see text.

expected in the presence of the more mobile aniline molecules. More interesting is that the JG relaxation time tb also becomes shorter in the unreacted mixture, while the g-relaxation time tg is basically insensitive to mixing before reaction starts. These changes of ta and tb are shown in Fig. 55. The fact that tb of EPON828 changes in the presence of aniline is another example of the JG relaxation mimicking the behavior of ta discussed in Section V.

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After reaction has started and polymers with increasing molecular weight are formed, tb increases monotonically with time, mimicking the changes of ta , although the changes of ta are much larger. This trend is demonstrated in Fig. 55 by tb and ta of the fully polymerized product represented by the dashed line and the solid circles, respectively. There is an increase of the conﬁgurational restriction of molecular motion and a decrease of the speciﬁc volume with the number of covalent bonds formed during polymerization, causing a decrease of conﬁgurational entropy or free volume. Hence the changes of tb in parallel to ta during polymerization can be considered as another evidence of the sensitivity of the JG relaxation time to (conﬁgurational) entropy or free volume long before the a-relaxation transpires. Accompanying the increase of ta with formation of polymer networks or linear-chain polymers from small epoxy resin molecules is the increase of the width of the dispersion of the a-relaxation time. When interpreted in terms of the CM, this is equivalent to the increase of the coupling parameter n of the cooperative many-molecule a-relaxation. The CM predicts that the separation between the JG relaxation and the a-relaxation given by (logta logtb ) increases with coupling parameter n or N, the number of covalent bonds formed by reaction. This prediction is in qualitative agreement with the changes in the relation between the two relaxation times at all stages of the experiment. This is illustrated in Fig. 56 for the completely polymerized product, where the experimentally determined separation distances (logta logtb ) corresponds to the large value of n ¼ 0:70 (compared with n ¼ 0:47 of EPON828) and the broad width of the a-relaxation. I.

A Shortcut to the Consequences of Many-Molecule Dynamics and a Pragmatic Resolution of the Glass Transition Problem

From the numerous experimental evidences accumulated over the years including those discussed in this chapter, there is no doubt that (free) volume and (conﬁgurational) entropy play important roles in determining the structural relaxation time. Recent experiments with applied pressure have already shown that the relative importance of temperature and volume can vary greatly from one glass-former to another [114]. Glass-formers of different chemical composition and physical structures exist in great numbers. A theory that can account quantitatively the temperature, pressure, volume, and entropy contributions to the structural relaxation rate for any glass-former has to be extremely difﬁcult to come by. Besides, as we have shown throughout this chapter, many dynamic and kinetic properties of the glass-forming substances cannot be explained by a theory based on these thermodynamic variables alone. Dispersion of the structural a-relaxation has to be considered as another factor because, as we have seen from multiple experimental facts, it either governs or correlates with the dynamic and kinetic properties of the a-relaxation, and even the relation

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of the a-relaxation to its precursor, the Johari–Goldstein (JG) b-relaxation. Experiments have shown that a-relaxation is dynamically heterogeneous, involving slowly and fast relaxing molecules exchanging roles in time scales of the order of the a-relaxation time [27,29,213]. These are clear evidences that many-body dynamics are involved in the a-relaxation. Since the dispersion of the a-relaxation is some average of the heterogeneous dynamics, it is clear that the dispersion of the a-relaxation originates from the many-molecule dynamics. Therefore, a viable theory of glass transition must incorporate the many-molecule dynamics. Unfortunately, many-molecule relaxation is an unsolved problem at the present time. A workable theory of glass transition has to take into account the many-molecule dynamics as well as the inﬂuence of the thermodynamic variables on a fundamental basis. Considering the difﬁculties to implement these two aspects of the problem, a perfectly rigorous theory that can explain all the salient properties will not be available for some time to come. In the mean time, it is not a bad idea to have an imperfect or even a schematic theory that can assimilate the fundamental factors and has predictions that are consistent with all the experimental facts. A schematic theory based on the Coupling Model (CM) is proposed as follows. From the evidences given before, the JG relaxation time tJG , or equivalently the primitive relaxation time t0 of the CM, is temperature, pressure, volume, and entropy dependent and has the VFTH temperature dependence at temperatures above Tg. Thus, T, P, V, and S have entered into the determination of molecular mobility of the JG or the primitive relaxation. Time being a natural variable, along with the fact that tJG or t0 precedes ta , implies that the source of the dependence of molecular mobility on T, P, V, and S originates from tJG or t0. The T, P, V, and S dependences inherent in the JG or the primitive relaxation is passed onto the a-relaxation through the cooperative many-molecule relaxation process occurring at later times. The T, P, V, and S dependences of ta is stronger than that of tJG because the a-relaxation stems from cooperative dynamics involving more than one molecule with length scale that increases with decreasing temperature, while the JG or the primitive relaxation is a local process arising from the independent relaxation of individual molecules. Construction of the schematic theory starts by ﬁrst simply writing down the dependence of tJG or equivalently t0 on T,P,V,S, explicitly as tJG (T,P,V,S) or t0 (T,P,V,S), without providing the actual dependence. For any particular glassformer, the dependence is best obtained from experiment data on the JG relaxation, if available, or deduced from that observed on the a-relaxation via Eq. (10). Until a theory that can provide accurate account of tJG (T,P,V,S) becomes available for that glass-former, realistically this is the best that one can do. Next we implement the many-molecule dynamics, which eventually gives rise to the a-relaxation. Due to the absence of a theory of the many-molecule dynamics from ﬁrst principles, we use the shortcut to ta by the relation between

dispersion of the structural relaxation

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ta and t0 or tJG by Eq. (14) of the CM. To carry on further, we need to know the coupling parameter n appearing in the fractional exponent, 1 n, of the Kohlrausch function. Numerous experimental data and molecular dynamics simulations have shown at long times that the observed correlation functions are invariably well approximated by the Kohlrausch function. Thus, there is no need to have a theory that just assures us that the correlation function is approximately a Kohlrausch function but nothing else. Moreover, no existing theory can gives us the exact coupling parameter n for any real glass-former anyway. We just shrewdly take what the experiment gives us for the value of n by ﬁtting the dispersion of the a-relaxation to the Kohlrausch function, at any temperature or pressure [376]. Since tc has already been determined to be approximately 2 ps for molecular liquids and polymers, the relation between the measured ta and t0 or tJG is quantitative and completely speciﬁed. We write it out here again, including dependence on other variables Q, such as the scattering vector of neutron scattering, ta ðT; P; V; S; QÞ ¼ ½tcn t0 ðT; P; V; S; QÞ1=ð1nÞ or ½tcn tJG ðT; P; V; S; QÞ1=ð1nÞ ð22Þ Eq. (22) gives quantitatively ta, as well as its dependences on the thermodynamic variables and other parameters Q, from t0 or tJG. Since the exponent, 1/(1 n), is larger than one, it is clear that ta has a stronger dependences on T, P, V, and S than the corresponding dependences of t0 or tJG. The normal dependence of t0 or tJG on Q is modiﬁed by the superlinear power, 1/(1 n). The result is that ta having an unfamiliar or anomalous Q dependence. If Q is the scattering vector of neutron scattering, then the normal Q2 dependence of t0 will become the anomalous Q2=ð1nÞ dependence of ta observed in experiments [144–151]. The isobaric dependence of t0 or tJG on T, P, V, and S is the origin of the deviation of molecular mobility from Arrhenius temperature dependence in the equilibrium liquid state, which is magniﬁed in ta . Possible dependence of n on temperature can make the temperature dependence of ta even more complicated, such as the need of more than one VFTH equation to ﬁt it [159,160], while only one is sufﬁcient for t0 [48,162]. As demonstrated before, when used judiciously, Eq. (22) can explain other observed properties of ta and the transport coefﬁcients such as viscosity, diffusion coefﬁcient, and dc conductivity, especially the anomalous ones. The origin of several observed correlations between n and the properties of ta , as well as the ratio (tJG =ta ), are easily traced back to the dependence of ta on n in Eq. (22). Admittedly, the schematic theory described above is far short of a rigorous solution to the problem. It does not provide any method to calculate n and tJG (T,P,V,S) or t0 (T,P,V,S). Nevertheless, simple as it is, the schematic

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theory has the beneﬁt that its many predictions are consistent with the experimental data. But, since a rigorous solution of many-molecule dynamics does not exist at the present time, there is no need to apologize for the shortcomings of the CM. Although far short of solving the many-molecule dynamics problem, the CM is an expedient way or a shortcut to predict or explain the observed properties of ta . We have shown in Section III these properties are either governed by or correlated with the dispersion. Explanation of these properties and more [1,2] can be found in several publications [22,25,26,48,162,377]. The achievements of the CM using basically the same deﬁning equation in explaining the relaxation behavior of several other classes of interacting systems are noteworthy [22,155,157,276]. The problems in some other systems actually are simpler than the dynamics of glass-forming liquids because there volume and entropy do not change with temperature and the relaxation times have Arrhenius temperature dependence. An example is the dynamics of ions in crystalline and glassy ionic conductors [276]. In some recent works, the activation energy of t0 deduced from the CM equations is the actual energy barrier that the ion has to surmount in migration, and the preexponential factor of t0 correspond to the ion vibrational frequency [378–380]. The simple relation between ta (or its equivalent for other interacting systems) and the primitive relaxation time t0 is the basis for all the CM explanations of the experimental data, particularly the anomalous properties which are more difﬁcult to resolve. This feat of the CM is due to it having captured the many-molecule dynamics through the dispersion or the coupling parameter n, along with the fact that the properties of t0 are normal and usually known. On the other hand, conventional theories of glass transition usually have no application to the other coupled or interacting systems where volume and entropy do not change. VII.

CONCLUSION

One purpose of this chapter is to show by a variety of general experimental ﬁndings that the dispersion of the structural a-relaxation plays a fundamental role in governing the dynamic and kinetic properties of the a-relaxation. The most direct evidence comes from a recently discovered general experimental property of glass-formers: For a given material at a ﬁxed value of ta , the dispersion of the structural relaxation is constant, independent of thermodynamic conditions (T and P); that is, the shape of the a-relaxation function depends only on the relaxation time. If the dispersion of the structural relaxation is derived independently of ta , it is an unlikely that it would be uniquely deﬁned by ta. Most conventional theories and models of the glass transition either do not address the dispersion or derive it independently of ta ; hence, they are in need of revision. We point out that the dispersion of the a-relaxation originates from the

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dynamically heterogeneous many-molecule dynamics. Thus, a proﬁtable way to revise conventional theories and models of glass transition is the incorporation of many-molecule dynamics, or at least the effects they have on the a-relaxation. The current trend of research in glass transition is in systems more complicated than bulk materials. Examples are nanoconﬁned glass-formers with the presence of surfaces or interfaces and miscible mixtures of two glass-formers. The tendency in these research communities is to take over the conventional theories of glass transition and apply them to these more complicated cases. Success in explaining the new phenomena is doubtful because of the conventional theories applied are already inadequate even for the simpler case of neat bulk materials. Although glass transition is conventionally deﬁned by the thermodynamics and kinetic properties of the structural a-relaxation, a fundamental role is played by its precursor, the Johari–Goldstein (JG) secondary relaxation. The JG relaxation time, tJG , like the dispersion of the a-relaxation, is invariant to changes in the temperature and pressure combinations while keeping ta constant in the equilibrium liquid state of a glass-former. For any ﬁxed ta , the ratio, tJG =ta , is exclusively determined by the dispersion of the a-relaxation or by the fractional exponent, 1 n, of the Kohlrausch function that ﬁts the dispersion. There is remarkable similarity in properties between the JG relaxation time and the a-relaxation time. Conventional theories and models of glass transition do not account for these nontrivial connections between the JG relaxation and the a-relaxation. For completeness, these theories and models have to be extended to address the JG relaxation and its remarkable properties. The Coupling Model (CM) is an exception. It recognizes the importance of many-molecule dynamics that give rise to as well as govern the a-relaxation and its dispersion. In addition, it recognizes the need to consider the faster primitive or independent relaxation from which the many-molecule dynamics are built. Although by no means a solution of the problem of many-molecule relaxation, it ﬁnds a physical principle that makes a quantitative relation between ta and the primitive relaxation time t0 given by ta ¼ ½tcn t0 1=ð1nÞ. The relation involves the dispersion of the a-relaxation or its measure by the Kohlrausch exponent, 1 n. An immediate consequences of this deﬁning relation of the CM is that the dispersion of the a-relaxation does govern ta . As a tool, this relation has been used to explain many anomalous properties of ta from the normal properties of t0 . This relation also implies that t0 has properties similar to that of ta , although the latter is more prominent because of the nonlinear exponent, 1/(1 n), in the relation. A further development of the CM is the recognition that the primitive relaxation time should correspond well to the JG relaxation time, i.e., t0 tJG . This expected correspondence has been veriﬁed for many different kinds of glass-formers, and the favorable results can be considered as a positive

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reality check of the primitive relaxation. The observed similarity in properties of the JG relaxation to the a-relaxation is immediately explained by that of the primitive relaxation and the CM relation. We have emphasized in this chapter the importance of the recent general experimental ﬁndings that the dispersion of the a-relaxation is invariant to different combinations of T and P at constant ta . This property can be derived from the CM. In spite of the many accomplishments of the CM, it has not solved the many-molecule relaxation problem and should not be considered to be a solution of the glass transition problem. These problems are too complex to be solved rigorously for some time to come. Based on the CM, a pragmatic and schematic theory can be constructed to ﬁll this void, providing a shortcut to reach the anomalous properties discussed in this chapter, which would otherwise remain unexplained. This schematic theory may be used as a stepping stone to reach a ﬁnal solution of the problem. Acknowledgments The authors thanks all their collaborators in the works cited in this chapter. The work at the Naval Research Laboratory was supported by the Ofﬁce of Naval Research, the work at the Universita` di Pisa was supported by I.N.F.M. and by MIUR (Coﬁn2002), and the work at Silesian University was supported by the Polish Committee for Scientiﬁc Research (KBN) (Project 2005–2007; No 1PO3B 075 28).

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336. T. R. Lutz, Y. He, M. D. Ediger, H. Cao, G. Lin, and A. A. Jones, Macromolecules 36, 1724 (2003). 337. K. L. Ngai and C. M. Roland, Macromolecules 37, 2817 (2004). 338. S. Kahle, J. Korus, E. Hempel, R. Unger, S. Ho¨ring, K. Schro¨ter, and E. Donth, Macromolecules 30, 7214 (1997). 339. K. L. Ngai, Macromolecules 32, 7140 (1999). 340. C. Lorthioir, A. Alegria, and J. Colmenero, Phys. Rev. E 68, 031805 (2003). 341. O. Urakawa, Y. Fuse, H. Hori, Q. Tran-Cong, and K. Adachi, Polymer 42, 765 (2001). 342. S. Capaccioli and K. L. Ngai, J. Phys. Chem. B 109, 9727 (2005). 343. K. Kessairi, S. Capaccioli, D. Prevosto, M. Lucchesi, and P. Rolla, to be published. 344. K. Duvvuri and R. Richert, J. Phys. Chem. B 108, 10451 (2004). 345. T. Psurek, S. Maslanka, M. Paluch, R. Nozaki, and K. L. Ngai, Phys. Rev. E 70, 011503 (2004). 346. J. Mattsson, R. Bergmann, P. Jacobsen, and L. Bo¨rjesson, Phys. Rev. Lett. 94, 165701 (2005). 347. S. E. B. Petrie, J. Macromol. Sci. Phys. B 12, 225 (1976). 348. M. B. M. Mangion, G. P. Johari, J. Polym. Sci., Part B: Polym. Phys. 28, 71(1990); 28, 1621 (1990); 29, 437 (1991). 349. G. P. Johari, Dynamics of irreversibly forming macromolecules, in Disorder Effects in Relaxational Processes, R. Richert and A. Blumen, eds., Springer-Verlag, Berlin, 1994, p. 627. 350. A. Lee and G. B. McKenna, Polymer 31, 423 (1990). 351. A. Lee and G. B. McKenna, Polymer 29, 1812 (1988). 352. D. J. Plazek and I.-C. Chay, Polym. Sci., Part B: Polym. Phys. Ed. 29, 17 (1991). 353. I. Alig, G. P. Johari, J. Polym. Sci., Part B: Polym. Phys. 31, 299 (1993). 354. I. Alig, D. Lellinger, G. P. Johari, J. Polym. Sci., Part B: Polym. Phys. Ed. 30, 791 (1992). 355. M. G. Parthun and G. P. Johari, Macromolecules 29, 3254 (1992). 356. M. G. Parthun and G.P. Johari, J. Chem. Phys. 102, 6301 (1995). 357. M. G. Parthun and G. P. Johari, J. Chem. Phys. 103, 7611 (1995). 358. M. G. Parthun and G. P. Johari, J. Chem. Phys. 103, 440 (1995). 359. S. Monserrat, J. L. Gomez-Ribelles, and J. M. Meseguer, Polymer 39, 3801 (1998). 360. J. M. Hutchinson, D. McCarthy, S. Monserrat, and P. Cortes, J. Polym. Sci., Part B: Polym. Phys. Ed. 34, 229 (1996). 361. D. A. Wasylyshyn and G. P. Johari, J. Chem. Phys. 104, 5683 (1996). 362. D. A. Wasylyshyn, M. G. Parthun, and G. P. Johari, J. Mol. Liq. 69, 283 (1996). 363. D. A. Wasylyshyn and G. P. Johari, J. Polym. Sci., Part B: Polym. Phys. 35, 437 (1997). 364. E. Tombari, C. Ferrari, G. Salvetti, and G. P. Johari, J. Phys.: Condens. Matter. 9, 7017 (1997). 365. C. Ferrari, E. Tombari, G. Salvetti, and G. P. Johari, J. Chem. Phys. 110, 10599 (1999). 366. E. Tombari, C. Ferrari, G. Salvetti, and G. P. Johari, J. Phys. Chem. Phys. 1, 1965 (1999). 367. E. Tombari, G. Salvetti, and G. P. Johari, J. Chem. Phys, 113, 6957 (2000). 368. D. Lairez, J. R. Emery, D. Durand, and R. A. Pethrick, Macromolecules 25, 7208 (1992). 369. R. Casalini, A. Livi, P.A. Rolla, G. Levita, and D. Fioretto, Phys. Rev. B 53, 564 (1996). 370. G. Levita, A. Livi, P. A. Rolla, and C. Culicchi, J. Polym. Sci. Part B: Polym. Phys. 34, 2731 (1996). 371. G. Gallone, S. Capaccioli, G. Levita, P. A. Rolla, and S. Corezzi, Polym. Int., 50, 545 (2001).

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CHAPTER 11 MOLECULAR DYNAMICS IN THIN POLYMER FILMS ANATOLI SERGHEI AND FRIEDRICH KREMER Fakultat fu¨r Physik und Geowissenschaften, Universita¨t Leipzig, 04103 Leipzig, Germany

CONTENTS I. Introduction II. Preparation of Thin Polymer Films III. Confinement-Induced Mode in Thin Films of cis-1,4-Polyisoprene A. Experiment B. Molecular Assignment C. Simulations D. Summary: Experiment and Simulations IV. Confinement Effects on the Molecular Dynamics in Thin Films of Hyperbranched Polyester V. Confinement Effects in Thin Polystyrene Films VI. Conclusions Acknowledgments References

I.

INTRODUCTION

The glass transition is ubiquitous. It is observed in a multitude of materials as diverse as metals, inorganic and organic liquids, liquid crystals, polymers or colloidal suspensions. Glass-forming systems have been studied for decades using a variety of experimental tools for measuring microscopic or macroscopic physical quantities. The conjecture that the glass transition has an inherent length scale, along with the rise of nanotechnology, has led to numerous studies on conﬁned glassy dynamics. Of special interest in this context are thin polymer

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

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ﬁlms. The present chapter contributes to this discussion and places emphasis on the following topics: 1. What are the experimental requirements ensuring reproducible preparation and reproducible measurements of thin polymer ﬁlms? 2. The glass transition temperature can be determined from microscopic (e.g., relaxation time of ﬂuctuating dipoles) or macroscopic (e.g., thermal expansion coefﬁcient) quantities. Do these different approaches necessarily yield coinciding results? 3. Does a shift of the glass transition temperature reﬂect a change in the relaxation time distribution or in its mean value only? This makes a fundamental difference underlining the power of microscopic techniques like Broadband Dielectric Spectroscopy. 4. Does (one-dimensional) conﬁnement in thin polymer ﬁlms give rise to novel dynamic modes? 5. What is the inﬂuence of molecular architecture of polymers on their dynamics in thin layers? II.

PREPARATION OF THIN POLYMER FILMS

One of the experimental techniques especially suitable for the investigation of thin polymer ﬁlms is Broadband Dielectric Spectroscopy (BDS) [1]. It does not exhibit sensitivity loss with decreasing sample amount (because the capacitance is inversely proportional to the ﬁlm thickness), and it provides a direct experimental access to molecular relaxations of polymers over a wide frequency and temperature range. On the other hand, this technique requires special sample preparation for thin ﬁlms, because of the need to have metal electrodes and good electrical contacts at both interfaces. Spin-coating, one of the most commonly employed methods for the preparation of thin polymer ﬁlms, has been successfully adapted for dielectric studies [2–5]. A detailed presentation of the preparation procedure together with other important aspects such as the characterization of the sample geometry, reproducibility of the sample preparation, thermal stability of thin polymer ﬁlms, water adsorption, thickness determination, annealing, and oxidation can be found in Ref. 6. In the following we brieﬂy review the most relevant aspects of the preparation of thin polymer ﬁlms. The preparation procedure is schematically illustrated in Fig. 1. Glass plates of 1 cm 1 cm 1 mm, previously cleaned in an ultrasound alkaline bath, rinsed in pure acetone, and dried under nitrogen ﬂow, are used as support substrates. After cleaning, aluminium electrodes (width 0.5 mm, height 60 nm) are deposited on the glass substrate by thermal evaporation in high vacuum

597

molecular dynamics in thin polymer ﬁlms thin polymer film

electrode evaporated on the polymer film 20 nm

Glass substrate (a)

support electrode

(b)

Figure 1. (a) Scheme showing the preparation of thin polymer ﬁlms. (b) Optical image (0.7 0.6 mm2 ) of a thin polystyrene ﬁlm (89 nm) between two aluminum electrodes.

(106 mbar). Subsequently, thin polymer ﬁlms are spin-coated from solution at a moderate spinning-rate (2000 rotations/min), to avoid possible chain-breaking effects. The ﬁlm thickness is adjusted by changing the concentration of the polymer in solution. After spin-coating, the samples are annealed at temperatures above the bulk glass transition (i.e., Tg þ 50 K) in an (oil-free) high vacuum (106 mbar) for at least 10 hours. The ﬁnal step in the preparation procedure is the evaporation of a second aluminum counterelectrode on top of the polymer ﬁlm. Measurements by Atomic Force Microscopy (in tapping-mode) reveal a typical root-mean-square roughness of 3 nm for the aluminum electrode and 0.3 nm for the surface of the polymer ﬁlm. The ratio between the actual area of the rough interfaces and their corresponding 2D projection is very close to unity (<1.04). This ensures a uniform electrical ﬁeld between the electrodes, because the deviations from parallel interfaces (corresponding to an ideally ﬂat geometry) are smaller than 0.5 . The half-width of the thickness distribution, calculated using the geometrical proﬁle measured by AFM, is typically 4 nm. ‘‘Edge’’ effects, arising in dielectric experiments from the nonuniformity of the applied electrical ﬁeld at the geometrical boundaries, are negligible, since the form factor (the ratio between the width of the electrode and the ﬁlm thickness) is typically 104 for thin ﬁlms, and hence much larger than in bulk measurements. The preparation procedure is perfectly reproducible (Fig. 2): Measuring different samples prepared under identical experimental conditions reveals, within the margins of 0.5% of experimental accuracy, coincident dielectric spectra. In terms of their dielectric response, thin polymer ﬁlms turn out to be thermally stable if kept in an inert atmosphere (i.e., ﬂow of pure nitrogen), even for long times at temperatures well above the glass transition. An example is given in Fig. 3 for a thin polystyrene ﬁlm of 89 nm: After 24 hours at 180 C in a pure nitrogen atmosphere, the sample was measured again and no changes in the dielectric response were detected.

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27 nm 28 nm 2400 Hz 970 Hz 380 Hz 150 Hz

2.0 1.5 1.0 0.5 0.0 380

400

420

440

460

480

500

T [K] Figure 2. Dielectric loss versus temperature at different frequencies, as indicated, for two thin ﬁlms of hyperbranched polyesters of 27 nm and 28 nm.

After evaporating the aluminum counter electrode and prior to dielectric measurements, a second annealing step is performed for several hours at temperatures above Tg under nitrogen ﬂow in the cryostat of the dielectric spectrometer. Thin polymer ﬁlms exhibit a pronounced increase of the surface-to-volume ratio with decreasing ﬁlm thickness. As a consequence, their exposure area to the ambient environment is orders of magnitude larger than that in the bulk. This makes them extremely sensitive to the presence of water vapor and oxygen, whose adsorption can drastically alter, through plasticizer and oxidation effects, 0.020 0.015 0.010 before after 31 kHz 12 kHz 4.9 kHz 1.9 kHz

0.005 0.000 360

380

400 T [K]

420

440

Figure 3. Dielectric loss versus temperature at different frequencies, as indicated, for a thin polystyrene ﬁlm of 89 nm, before and after 24 hours of annealing at 180 C in a pure nitrogen atmosphere.

molecular dynamics in thin polymer ﬁlms 3.2

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in ambient air under N2 flow after anneal.

2.8

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–1

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0

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6

Figure 4. e0 and e00 versus frequency at 25 C for a thin PS ﬁlm of 223 nm, measured in ambient air, under pure nitrogen ﬂow and after annealing 2 hours at 135 C.

the dynamics of the polymers under study. Even if some polymers are hydrophobic in the bulk, concerns about possible water adsorption effects in thin ﬁlms are justiﬁed, because of the preponderant role of the interface in conﬁnement. For instance, a negligible water adsorption is reported in the handbook of polymers [7] for polystyrene in the bulk: 0.05% at 23 C and 50% relative humidity. In spite of this, strong water adsorption–desorption effects are observed in thin polystyrene ﬁlms. An example is given in Fig. 4 for a thin polystyrene ﬁlm of 223 nm: The dielectric loss decreases by more than one decade when the sample is measured in a dry nitrogen atmosphere and after 2 hours of annealing at 135 C. A pronounced decrease in e0 (20% at 1 Hz) is also detected. Conversely, on a time scale of minutes, both the real part of the complex sample capacitance (and correspondingly e0 ) and the dielectric loss increase when a polystyrene thin ﬁlm (20 nm) is replaced from a dry nitrogen atmosphere and exposed to ambient water vapor at room temperature (Fig. 5). Even more pronounced effects are observed when such an adsorption experiment is performed at temperatures above the glass transition (Fig. 6). A thin PS ﬁlm of 63 nm is kept at 180 C in vacuum (4 mbar). The real part of the complex sample capacitance, a parameter sensitive to any changes in the sample geometry and e0 , remains constant, proving sample stability under these conditions. After 2–3 hours, Millipore water is injected in the vacuum chamber

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C [pF]

788

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time [min] Time dependence of the dielectric loss and sample capacitance at 24 C and 0.1 MHz for a thin PS ﬁlm of 20 nm in a pure nitrogen atmosphere and in ambient air.

Figure 5.

through a vacuum-tight membrane. Before injection, the water was previously degassed for at least 1 hour in vacuum. In the presence of water vapor, at the same annealing temperature, a sudden increase of the real part of the complex sample capacitance is observed ﬁrst, due to water adsorption. Then, the capacitance starts to increase continuously in time, accompanied by a gradual change of the sample geometry. This phenomenon, related to the formation of a periodical pattern in thin polymer ﬁlms, is initiated by an enhanced mobility in

C [pF]

300

290 vacuum 280

1

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3

vacuum + water vapors

4 5 time [h]

6

7

Figure 6. Time dependence of the sample capacitance at 180 C and 0.1 MHz for a thin PS ﬁlm of 63 nm in vacuum (4 mbar) and in the presence of water vapor.

molecular dynamics in thin polymer ﬁlms

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the presence of water vapor and oxygen and will be discussed in detail in Section V. It will be shown that the mobility is enhanced as a consequence of oxygen-induced chain scissions that cause a decrease of the average molecular weight. III.

CONFINEMENT-INDUCED MODE IN THIN FILMS OF CIS-1,4-Polyisoprene

One attempt to explain the reductions of the glass transition temperature Tg observed in thin free-standing polystyrene ﬁlms [8–10] represents the sliding model proposed by de Gennes [10–12]. For chain loops in contact with free interfaces (Fig. 7) the existence of a novel relaxation process is assumed, the socalled sliding mode, responsible for the reported faster dynamics. This model implies that the relaxation of the chain loops, taking place on a length scale comparable to that of the polymer chains, has an impact on the dynamic glass transition—that is, on the dynamics of the monomer segments. Thus, in essence, a coupling between the segmental dynamics and the dynamics of the chain loops is supposed, giving rise to a faster dynamic glass transition. A type-A polymer (in accord with the nomenclature given in Ref. 13), in our study cis-1,4-polyisoprene (PIP), provides a natural choice to probe the validity of this concept. The monomer dipole moment of PIP is tilted with respect to the direction of the polymer chain (Fig. 8a). Its parallel component adds up along the chain contour giving rise to a total dipole moment related to the end-to-end distance of the polymer coil (Fig. 8b). Hence, two relaxation processes are dielectrically active for PIP: the segmental mode, corresponding to the dynamic glass transition and representing ﬂuctuations of 2–3 monomer units [14], and the normal mode, originating from ﬂuctuations of the end-to-end vector of the polymer chains. For the latter, a strong molecular weight dependence is reported in the literature [15,16], with a transition from a Rouse regime (t Mw2) to a reptation regime (t Mw3.4) at the critical molecular weight Mw*, which for PIP has a value of 104 g/mol. free interface

polymer chain

Figure 7. Polymer chain in contact with a free interface; the chain loops formed between the contact points are supposed to exhibit a ‘‘sliding-mode’’ motion.

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HH p

CH

HH p

C

C

C

CH2

p II C

HH

CH3

C chain

R EE

H

Figure 8. Scheme showing (a) the monomer of cis-1,4-polyisoprene and (b) a type-A polymer, having a dipole moment related to the end-to-end distance of the polymer chain.

A.

Experiment

Thin ﬁlms of PIP are prepared by spin-coating using the procedure described in detail in Section II. Four different molecular weights with a narrow polydispersity (1.02–1.04) are investigated (Table I, containing additionally the radii of gyration and the corresponding end-to-end distances). Broadband dielectric spectroscopy (High-Resolution Alpha Analyzer from Novocontrol GmbH, 0.1 Hz to 10 MHz, 200 K to 350 K) is employed to study the molecular dynamics of thin PIP ﬁlms down to thicknesses smaller than the end-to-end distance of the polymer chains. As expected, two relaxation processes are observed for PIP in the bulk: the segmental mode, related to the dynamic glass transition representing the dynamics of the polymer segments, and the normal mode, sensing the chain dynamics (Fig. 9). In thin PIP ﬁlms, for thicknesses comparable to the size of the polymer chains, a novel relaxation process—a conﬁnement-induced mode [5]—shows up (Fig. 10). TABLE I Molecular Weights and the Corresponding Polydispersities, End-to-End Distances, and Radii of Gyration of the cis-1,4-Polyisoprenes Investigated in the Experiment Mw ðkg=molÞ 34 52 60 108

Mw =Mn 1.02 1.04 1.02 1.02

REE ðnmÞ 18 22 24 32

Rg ðnmÞ 7 9 10 13

molecular dynamics in thin polymer ﬁlms –1

10

segmental

215 K

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290 K

mode

300 K 310 K

–2

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–1

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5

Figure 9. Dielectric loss e00 versus frequency for a thin PIP ﬁlm of 150 nm (Mw ¼ 52; 000 g=mol) at different temperatures, as indicated, showing the segmental and the normal mode.

0.030

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0.025

0.020 0.015 0.010 0.005

0.020

segmental mode

200 250 300 T [K]

normal mode

350

0.015 confinement-induced mode

0.010 200

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280

320

normal mode 360

T [K] 00

Figure 10. Dielectric loss e versus temperature at 970 Hz for a thin PIP ﬁlm of 61 nm (Mw ¼ 35; 000 g=mol), showing the segmental, the normal, and the conﬁnement-induced mode. Inset: Dielectric loss e00 versus temperature at 970 Hz for a PIP ﬁlm of 677 nm (Mw ¼ 35; 000 g=mol), showing the segmental and the normal mode.

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0.02

0.01 confinement induced mode normal segmental mode mode

0.00

200

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300 T [K]

conductivity

350

400

Dielectric loss e00 versus temperature at 30 Hz showing the segmental, the normal and the conﬁnement-induced mode for thin PIP ﬁlms (Mw ¼ 52; 000 g=mol) of different thicknesses, as indicated.

Figure 11.

The overall dynamics of PIP in the context of dependence on the conﬁnement size is illustrated in Fig. 11, showing the temperature dependence of the dielectric loss of PIP (Mw ¼ 52 kg/mol) down to a ﬁlm thickness smaller than the end-toend distance of the polymer chains. The conﬁnement-induced mode becomes faster with decreasing ﬁlm thickness and approaches the segmental mode, while its relaxation strength increases at the expense of the normal mode, which decreases (Figs. 11 and 12).

0.06 –1.8

105 nm 54 nm 32 nm

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–2.4 –1

0

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thickness d [nm] Figure 12. Dielectric strength of the normal mode versus ﬁlm thickness at 332.5 K for an Mw of 52, 000 g/mol. Inset: Dielectric loss e00 versus frequency at 330 K showing the normal mode of thin PIP ﬁlms (Mw ¼ 52; 000 g=mol) of different thicknesses, as indicated. The dotted line indicates the position of the peak maxima.

molecular dynamics in thin polymer ﬁlms 7 6 5 4

466 150 56 32 19 12

nm nm nm nm confinement induced nm nm mode

605

segmental mode

3 2 1

normal mode

0 2.8

3.2

3.6

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4.4

1000/T [1/K] Figure 13. Relaxation rate of the segmental, normal, and conﬁnement-induced mode versus inverse temperature for thin PIP ﬁlms (Mw ¼ 52; 000 g=mol) of different thicknesses, as indicated.

The usual ﬁtting procedure is employed to analyze the dielectric spectra. Alternatively also applied when necessary, ﬁtting in the temperature representation (e00 versus temperature at a constant frequency) is using a modiﬁed Havriliak–Negami function. For samples where both procedures are applicable, no differences in the ﬁtting results are observed. The effect of conﬁnement on the overall dynamics of PIP is illustrated in Fig. 13, in a plot of relaxation rate as a function of inverse temperature. The relaxation rate of the normal mode remains unchanged down to a thickness comparable to the end-toend distance of the polymer chain. No shifts of the dynamic glass transition are observed, even for a strong chain conﬁnement (i.e., thicknesses smaller that the end-to-end distance). The conﬁnement-induced mode becomes orders of magnitude faster with increasing conﬁnement. The molecular weight dependence of these three relaxation processes at a constant ﬁlm thickness is also investigated (Fig. 14). The segmental mode, taking place on a length scale of 2–3 monomer units [14], shows, as expected, no molecular weight dependence. The normal mode, originating from ﬂuctuations of the end-to-end distance of the polymer chains, exhibits a molecular weight dependence similar to that of bulk PIP. The conﬁnementinduced mode also shows no molecular weight dependence. The raw dielectric spectra (e00 versus temperature at a ﬁxed frequency) for a thin PIP ﬁlm of 45 nm and different molecular weights are presented in the inset of Fig. 14. The scaling for the relaxation rate of the conﬁnement-induced mode as a function of ﬁlm thickness is given in Fig. 15, in a double logarithmic plot. The

anatoli serghei and friedrich kremer 8

MW = 60 Kg/mol MW = 52 Kg/mol MW = 34 Kg/mol

7 6

0,02 ''

606

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5

SM

4

CIM 200 240 280 320 T [K]

3 2 normal mode 1 0 2.5

3.0

segmental mode confinement-induced mode 3.5

4.0

4.5

5.0

1000/T [1/K] Figure 14. Relaxation rate of segmental, normal, and conﬁnement-induced mode versus inverse temperature for a thin PIP ﬁlm of 45 nm and different molecular weights, as indicated. Inset: The corresponding raw data, i.e. dielectric loss e00 vs. temperature at 96 Hz for the same thickness and same molecular weights.

inherent uncertainty that is observed is due to the width of the CIM, making the evaluation of the relaxation rate difﬁcult. Fitting with a linear dependence provides a slope of 6:0 1:0, corresponding to the scaling law:tCIM d 6 , where tCIM represents the relaxation time of the conﬁnement-induced mode and d the ﬁlm thickness.

7 6 max

5

slope= –6.0±1.0

4 3 2 1 0 –1 10

100 film thickness d [nm]

Figure 15.

Relaxation rate of the conﬁnement-induced mode as a function of ﬁlm thickness. The dotted line represents a linear ﬁt with a slope of 6.0.

607

molecular dynamics in thin polymer ﬁlms B.

Molecular Assignment

A close look at the experimental results affords a deeper insight into the nature of the novel conﬁnement-induced mode. Its location between the normal and the segmental mode proves that it takes place on a length-scale intermediate between a few monomer units and the end-to-end distance of the polymer chains. Additionally, the fact that its relaxation strength increases with decreasing ﬁlm thickness indicates that the conﬁnement-induced mode originates from interactions with the interfaces. To characterize it we assume that the polymer segments coming in contact with a conﬁning interface are immobilized. This leads to an interruption of the end-to-end ﬂuctuation of the immobilized PIP chains; thus only their terminal subchains that remain are still able to ﬂuctuate in this case. The conﬁnement-induced mode is attributed to the end-to-end ﬂuctuations of the terminal subchains, tethered at one end to the immobilizing interface (Fig. 16). The immobilized chains do not contribute any longer to the normal mode but to the conﬁnement-induced mode, thus explaining the steep decrease in the dielectric strength of the normal mode and the increase in dielectric strength of the conﬁnement-induced mode with decreasing ﬁlm thickness. A similar model of immobilization (dynamic stickers) was proposed by Petychakis et al. in a study of polyisoprene conﬁned in nanopores [17] in order to explain a faster and broadened normal mode observed in conﬁnement; nevertheless, no conﬁnement-induced mode was detected in their experiment. It has been shown in the literature that the relaxation rate of a tethered chain is a factor of four smaller than that of a free chain having the same molecular weight [18].

Immobilized chain segments

TSC R EE

d TSC R EE

Figure 16.

Scheme illustrating the molecular conﬁguration of the conﬁnement-induced mode. RTSC EE represents the end-to-end distance of the terminal subchains formed by the immobilization of the polymer segments at the interface, while d is the ﬁlm thickness.

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Simulations

In order to lend additional support to the molecular model presented above, simulations are carried out [19], treating the polymer chains as ideal random walks in three dimensions taking place between two impenetrable interfaces (Fig. 17). The length of one monomer segment is denoted by a. Values close to the molecular weights used in the experiment (i.e., 500 segments) are chosen for the total length of the walks (representing the molecular weight of the PIP chain). The simulations are accomplished via an off-lattice, one step (walk) being deﬁned by two independent random angles (the azimuthal and the longitudinal angles of the spherical polar coordinates). The walks penetrating the boundaries are rejected. In order to have good statistics, 104 independent chains per unit of thickness are simulated. We follow in our simulations the same idea as in the molecular model: We assume that the chain segments coming in contact with a conﬁning interface are immobilized. Therefore, in our case, the effect of conﬁnement consists basically in pinning the chains at the immobilizing interface in addition to the rejection of the interface-penetrating random walks. The chain loops formed between the pinning points (and also the terminal subchains) are thus approximated by ideal random walks. Single-segment as well as multisegment contacts with the interfaces are allowed. Keeping the molecular weight constant and varying the

18 16 14 RTSC EE

12 Z [a]

10 8

chain

REE

6 4 2 0 –2 0

2

4

6

8 10 X [a]

12

14

16

Figure 17. Simulations of conﬁned polymer chains as ideal random walks between two hard impenetrable interfaces. Two populations exist: free (nonimmobilized) chains, which contribute to the normal mode, and immobilized chains, which contribute to the conﬁnement-induced mode via ﬂuctuations of their terminal subchains.

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thickness d, the effects of conﬁnement are investigated. Additionally, different walk lengths are considered in order to examine the chain conﬁguration independence on the molecular weight. In the experiment, the normal mode is related to the ﬂuctuations of the endto-end vector of free (nonimmobilized) chains, while the novel relaxation process was assigned to the ﬂuctuations of the end-to-end distance of terminal subchains (hence, the conﬁnement-induced mode is by its nature also a ‘‘normal mode,’’ not of the whole chain but of the terminal subchains only). The dynamics of these two relaxation processes is related to the corresponding endto-end distances, consequently, their distribution, in dependence on thickness and molecular weight, is simulated in our study. Taking into account the existing indications for asymmetric interactions at the interfaces (e.g., three-layer model in Ref. 4), two distinct cases must be analyzed: (i) Both interfaces immobilize the chains or (ii) the chains are immobilized only at one interface while at the other one they are reﬂected (hardwall behavior). In Fig. 18 the distribution of the end-to-end distance of terminal subchains is shown when both interfaces immobilize the chain segments. No shifts of the maximum position of the distribution are observed with decreasing ﬁlm thickness. This result would imply that the conﬁnement-induced mode does not exhibit thickness dependence, in contrast to the experiment. In the other case, when one interface immobilizes chain segments while at the other interface the chains are reﬂected, a conﬁnement-induced and thickness-dependent distribution of end-to-end distances for the terminal

1.2 d = infinite d = 75 a d = 36 a d = 30 a d = 24 a d = 18 a d = 12 a

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Figure 18. Distribution of the end-to-end distance of terminal subchains formed by the immobilization of the polymer chains at both interfaces, for different ﬁlm thicknesses, as indicated.

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Figure 19. Distribution of the end-to-end distance of terminal subchains for polymer chains immobilized at one interface and reﬂected at the other one.

subchains is observed (Fig. 19). It arises from terminal subchains which are in contact with both interfaces (immobilized at one interface and reﬂected at the other) and its maximum position is shifted to lower values with decreasing ﬁlm thickness. Thus, the terminal subchains become on average shorter, in full agreement with experiment: The conﬁnement-induced mode becomes faster with increasing conﬁnement. Summarizing, two conditions must be fulﬁlled in order to obtain from the simulations a conﬁnement-induced and thickness-dependent distribution of the end-to-end distance for terminal subchains. First, a chain should be in contact with both interfaces. This happens only when the ﬁlm thickness becomes comparable to the size of the chains and, obviously, explains why the conﬁnement-induced mode does not exist in the bulk. Second, the interactions at the interfaces should be asymmetric: One interface should immobilize the polymer chains, while the second one should only reﬂect them. This asymmetry could be induced by the nonequivalent preparation of the electrodes in the experiment: While one interface is prepared by spin-coating, the other one is prepared by evaporation of aluminium on top of the polymer ﬁlm (see Section II for details). A similar picture of asymmetry was found in studies on thin PS ﬁlms, with a preparation procedure identical with ours. For thin PS ﬁlms capped between two aluminum electrodes a three-layer model was proposed, in which, in addition to a middle-layer having bulk properties, a dead (immobilized) layer and a liquid-like layer were assumed to be present at the interfaces. The molecular-weight dependence has also been investigated in our simulations. For a constant thickness of d ¼ 18a the end-to-end distribution of the terminal subchains, dependent on the molecular weight of the chain, was

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normalized probability

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Mn = 500 Mn = 1000

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40

50

60

[a]

Figure 20. Distribution of the end-to-end distance of terminal subchains for a ﬁxed ﬁlm thickness of d ¼ 18a and for different molecular weights, as indicated. The chains are in contact with both interfaces, immobilized at one interface and reﬂected at the other one.

simulated. No molecular-weight dependence for the maximum position of the distribution is observed (Fig. 20), in full agreement with the experiment. Since increasing the molecular weight, increases also the probability of the chain to meet the surface, and so, the number of contact points N of the chain with the immobilizing interface also increases, leaving the average length of the terminal subchains MwTSC Mw =N constant. In Fig. 21 the distribution of the end-to-end distance for free (nonimmobilized) chains is given, which corresponds to the normal-mode relaxation. The maximum position is practically not affected by the conﬁnement down to a thickness comparable with the size of the coil (d ¼ 36a), while for smaller thicknesses it is shifted to lower values. Thus the relaxation rate of the normal mode remains constant down to thicknesses comparable to the size of the PIP chains, which was indeed observed in the experiment. It indicates further that even in ﬁlms as thin as the chain size a certain fraction of free (nonimmobilized) chains is still present, exhibiting a bulk-like normal-mode dynamics. This is easily comprehendible since the simulations reveal that only 30% of the chains are immobilized when d ¼ 2REE (immobilization is supposed at one interface). A steep decrease of the relative number of free chains is observed with decreasing ﬁlm thickness, which indicates a decrease of the relaxation strength of the normal mode, in agreement with experiment. The simulations yield MTSC d1:7 for the scaling of the molecular weight MTSC of the terminal subchains with thickness d. Assuming for the terminal subchains Rouse-like dynamics (the conﬁnement-induced mode is faster

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Figure 21. Distribution of the end-to-end distance of free (nonimmobilized) chains for a molecular weight of Mn ¼ 500 segments and for different ﬁlm thicknesses, as indicated. Inset: Relative number of free (nonimmobilized) chains as a function of ﬁlm thickness. The dotted line marks the value 1 for Nfree =Ntotal.

than the normal mode corresponding to the critical molecular weight M* of 2 polyisoprene), one ﬁnds with tTSC MTSC the relation tTSC d3:4 , in contrast 6:01 . To resolve this discrepancy more reﬁned to experiment, where tTSC d molecular dynamics simulations are required, for instance modelling the structure and mobility of polymers close to interfaces [20]. D. Summary: Experiment and Simulations The main ﬁnding of the present study is the discovery of a novel relaxation process in thin ﬁlms of cis-1,4-polyisoprene, called a conﬁnement-induced mode, attributed to ﬂuctuations of terminal subchains formed by the immobilization of the polymer segments at one of the two conﬁning interfaces. In the context of our molecular model and with the help of the simulations, one can understand most of the features observed in the experiment: 1. The conﬁnement-induced mode does not exist in the bulk because it arises from chains in contact with both interfaces. Thus it appears only when the ﬁlms thickness become comparable to the size of the PIP chains. 2. The conﬁnement-induced mode becomes faster with decreasing ﬁlm thickness because the terminal subchains become on average shorter. 3. The conﬁnement-induced mode exhibits no molecular weight dependence because increasing chain length increases also its probability to touch the

molecular dynamics in thin polymer ﬁlms

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interface; this increases the number of immobilised segments, which in turn maintains the length of the terminal subchains constant. The relaxation strength of the conﬁnement-induced mode increases because the relative number of immobilized chains increases with decreasing ﬁlm thickness. Conversely, the relaxation strength of the normal mode decreases because the relative number of free (nonimmobilized) chains decreases with decreasing ﬁlm thickness. The relaxation rate of the normal mode is not shifted with decreasing ﬁlm thickness, because even for thicknesses comparable to the end-to-end distance of the PIP coils a certain fraction of free chains still exists which are not in contact with the conﬁning interfaces and they exhibit bulk-like normal mode dynamics. The segmental mode is not shifted in its relaxation rate with decreasing ﬁlm thickness because it takes place on a length scale much smaller than the conﬁnement size. From the experiment, the relaxation time of the conﬁnement-induced mode depends on thickness as tCIM d 6 . Assuming Rouse dynamics, one obtains tCIM d3:4 from the simulations. In order to explain this discrepancy, more subtle simulations are required.

IV. CONFINEMENT EFFECTS ON THE MOLECULAR DYNAMICS IN THIN FILMS OF HYPERBRANCHED POLYMERS Hyperbranched polymers [21,22] belong to the general class of dendritic macromolecules [23]. Due to their highly branched structure and large number of end groups (Fig. 22a), they possess properties that are notably different from those of their linear counterparts. Developed from the idea of simplifying the complicated and time-consuming stepwise synthesis of dendrimers, but without losing their dendritic character, hyperbranched polymers have potential applications in ﬁelds like blends, additives, and coatings [24]. Special interest has developed in using these materials as thin layers [25–30]—for instance, for the development of chemical sensors. Nevertheless, until recently [31], their molecular dynamics in thin ﬁlms has not been investigated. In the present section, hyperbranched polyester (POHOAc) of type AB1B2 (having –OH and –OCOCH3 as terminal groups, see Fig. 22b), with an average molecular weight of 13,500 g/mol and a degree of branching of 50%, are investigated in thin ﬁlms by Broadband Dielectric Spectroscopy, calorimetry, and capacitive scanning dilatometry. Conﬁnement effects on the molecular dynamics are discussed in detail, with special focus on the dynamics of the glass transition.

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Figure 22. Schemes showing (a) the architecture of hyperbranched polymers and (b) the hyperbranched aromatic polyesters with two terminal groups ( OH and OCOCH3).

The synthesis of POH (hyperbranched polyesters with hydroxil end groups), the precursor of POHOAc, is described elsewhere [32,33]. Measurements by differential scanning calorimetry (DSC) on heating with 20 K/min reveal a glass transition temperature Tg of 431 K for the bulk (calculated as the mid-point in the step of the heat capacity) and no indications of crystallinity. Thin ﬁlms were prepared by spin-coating from a THF solution, using the preparation procedure described in detail in Section II. The dielectric measurements were performed between 0.1 Hz and 10 MHz, in a temperature interval ranging from 150 K up to 510 K, using a High-Resolution Alpha Analyzer (Novocontrol GmbH). In accordance to the data reported in the literature for bulk hyperbranched polyesters [34,35], three relaxation processes are also observed in thin POHOAc ﬁlms, (Fig. 23): the alpha relaxation process, representing the dynamic glass transition, the beta process, attributed to the relaxation of the ester groups, and the gamma relaxation process, originating from ﬂuctuations of the OH end groups. The latter two, which are broad and not well-separated from one another, are only distinguishable in the temperature representation of the dielectric spectra (inset, Fig. 23). The assignment of the alpha relaxation was additionally conﬁrmed by ACcalorimetric measurements on thin ﬁlms of POHOAc [31]. The effect of conﬁnement on the overall dynamics of POHOAc is given in Fig. 24, showing the dielectric spectra for different ﬁlm thicknesses, ranging from 310 nm down to 17 nm. While the beta and the gamma relaxations, as local processes, are not affected with decreasing ﬁlm thickness (except a decrease of the dielectric

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Figure 23. Dielectric loss e00 versus frequency at different temperatures, as indicated, showing the alpha relaxation process for a thin ﬁlm of hyperbranched polyesters of 17 nm. Inset: Dielectric loss e00 versus temperature at 49 kHz, showing the beta and the gamma relaxation processes for the same sample. Dotted lines serve as a guide for the reader.

strength for the smallest thickness of 17 nm; see inset, Fig. 24), the alpha relaxation shows a pronounced effect: The low frequency wing decreases in its dielectric strength, the maximum position being consequently shifted to higher frequencies. 0.08 0.07 0.06 0.05

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Figure 24. Dielectric loss e00 versus frequency at 478 K showing the alpha relaxation process of POHOAc for different ﬁlm thicknesses, as indicated. Inset: Dielectric loss e00 versus temperature at 49 kHz, showing the beta and the gamma relaxation processes for the 310-, 67-, and 17-nm-thick samples.

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Figure 25. Relaxation rate of the alpha and beta relaxation process of POHOAc versus inverse temperature for different ﬁlm thicknesses, as indicated.

Using the usual ﬁtting procedure [1], the dependence of the relaxation rate on the inverse temperature for the alpha and beta relaxation process is extracted (Fig. 25). The dynamic glass transition becomes more than one order of magnitude faster with increasing conﬁnement, corresponding to a shift of 36 K to lower temperatures (Fig. 26). The thickness dependence of both the alpha relaxation time (at a constant temperature of 427 K) and the maximum

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Figure 26. Relaxation time of the alpha relaxation process (at 427 K) and the maximum temperature position of the alpha peak (at 0.3 Hz) as a function of ﬁlm thickness. Inset: The relaxation time distribution of POHOAc at 510 K for different thicknesses, as indicated.

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molecular dynamics in thin polymer ﬁlms

temperature position of the alpha relaxation peak (at a constant frequency of 0.3 Hz) indicates that the conﬁnement effects on the dynamics of POHOAc (Fig. 26) start at much larger thicknesses (i.e. 200 nm) than for linear polymers. Since the size of these dendritic macromolecules is smaller than 10 nm, this ﬁnding must be attributed to the peculiar architecture of POHOAc. Applying the Tikhonov regularization procedure, the relaxation time distribution of the dynamic glass transition of POHOAc is determined (inset, Fig. 26), after the contribution of the dc-conductivity to the dielectric loss is eliminated by ﬁtting with Havriliak–Negami functions. With decreasing ﬁlm thickness a gradual suppression of the slower relaxation modes is observed, while no changes on the high-frequency side of the time distribution are detected. The freezing-out of the slower relaxation modes is attributed to immobilization in conﬁnement of the polymer segments located at the periphery of the hyperbranched macromolecular structures [31]. This interpretation is experimentally supported by the decrease of the dielectric strength of the alpha relaxation and by the increase of the refractive index n (measured at l ¼ 630 nm by spectroscopic ellipsometry), indicating a sample densiﬁcation with decreasing ﬁlm thickness (Fig. 27). This aspect facilitates a deeper insight into the mechanism of the Tg reductions observed in thin polymer ﬁlms [36–42]. Faster dynamics, by deﬁnition an increase of the average relaxation rate, can be achieved in two ways: Either the polymer segments (at the molecular level), are relaxing faster

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Figure 27. Dielectric strength of the alpha relaxation process of POHOAc at 500 K as a function of ﬁlm thickness. Inset: Refractive index of POHOAc at l ¼ 630 nm in dependence on the ﬁlm thickness.

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in conﬁnement than in the bulk, or the slower relaxation modes are gradually suppressed (frozen-out) in thin ﬁlms. These are two fundamentally different conﬁnement effects, although they both produce on average a ‘‘faster’’ dynamics. To distinguish between them, a quantitative determination of the relaxation time distribution independence on the conﬁnement size is required. Since in the majority of the studies this could not be done in the literature because of methodological limitations, the molecular mechanism underlying the reported shifts of the glass transition temperature in thin polymer ﬁlms still remains an open problem. To give a particular example: In the last decade many investigations reported a faster dynamics in thin polystyrene ﬁlms [36–42]. Nevertheless, an important aspect has not yet been addressed—that is, the question of whether this effect is due to a freezing-out of the slower relaxation modes in thin ﬁlms or to the fact that the polystyrene segments are indeed relaxing faster in conﬁnement than in the bulk. Simultaneously with dielectric measurements, the glass transition temperature Tg of thin POHOAc ﬁlms was determined by capacitive scanning dilatometry. Its principle, as usually applied in the literature [43,44], is illustrated in Fig. 28a,b for a ﬁlm thickness of 310 nm. The real part of the sample capacitance is measured as a function of temperature in a spectral region not affected by dielectric dispersions—that is, between 380 K and 460 K and at 105 Hz. At this frequency, the contribution of the alpha relaxation to the real part of the complex permitivity is negligible in this temperature interval. The beta and gamma processes, observed at much lower temperatures (Fig. 23), are weak and do not depend on thickness, and their overall contribution is negligible. The slope in the temperature dependence of the sample capacitance is proportional to the thermal expansion coefﬁcient along the normal to the ﬁlm interfaces [65,66]. A break in slope—or, alternatively, a step in the thermal expansion coefﬁcient—is observed at 427 K. It corresponds to the transition of POHOAc from a glassy to a liquid state and thus marks the position of the glass transition temperature Tg. Equivalent results are achieved by considering the ﬁrst and the second numerical derivatives of the sample capacitance as a function of temperature (Fig. 28b). The Tg value obtained in this way (427 K) is in good agreement with the Tg determination by differential scanning calorimetry (DSC) on bulk POHOAc (431 K, calculated as the mid-point in the step of the heat capacity; see Fig. 28c). The difference of 4 K between the dilatometric and the calorimetric measurements is due to the different heating rates of these two experiments (20 K/min for DSC, while 10 K/h for dilatometry). A slight increase (10 K) of Tg as determined by dilatometry is observed with decreasing ﬁlm thickness. This is illustrated in the inset in Fig. 28a, giving the second numerical derivative of the sample capacitance as a function of temperature for two thicknesses, 310 nm and 17 nm. An increase of dilatometric Tg is often assigned in the literature to slower molecular dynamics in conﬁnement.

molecular dynamics in thin polymer ﬁlms

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Figure 28. (a) Normalized sample capacitance versus temperature at 1.2 MHz for a ﬁlm thickness of 310 nm; the dotted lines serve as a guide for the reader. (b) The corresponding ﬁrst and second numerical derivatives with respect to the temperature T. The Tg is deﬁned as the maximum of the second derivative. (c) the normalized heat ﬂow versus temperature, as measured by Differential Scanning Calorimetry. Inset, part a: The second derivative of the Cnorm: versus temperature dependence for a ﬁlm thickness of 310 nm and 17 nm.

A thickness dependence of both dielectric and dilatometric determinations (normalized in respect to the values corresponding to 310 nm) is presented in a common plot in Fig. 29: While the temperature position of the alpha relaxation peak (at 0.3 Hz) in the dielectric measurements is shifted with 35 K to lower temperatures, indicating faster dynamics in thin ﬁlms, the glass transition temperature Tg determined by dilatometry slightly increases. This is the ﬁrst study reporting diverging thickness dependencies for conﬁnement effects in thin polymer ﬁlms, simultaneously investigated by different experimental techniques. This discrepancy reﬂects the fact that different microscopic and macroscopic techniques for characterizing the dynamic glass transition rely on completely different physical quantities (i.e., relaxation of polar entities in

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Figure 29. Comparison dielectric spectroscopy versus dilatometry: Tg Tgref (determined by dilatometry) versus ﬁlm thickness, where Tgref is the value of Tg for the 310-nm-thick sample and Ta Taref at 0.6 Hz versus ﬁlm thickness, where Ta represents the maximum temperature position of the alpha relaxation peak and Taref corresponds to the thickness of 310 nm.

the case of broadband dielectric spectroscopy or thermal expansion coefﬁcient in the case of dilatometry). It is by no means self-evident that these different techniques deliver coincident results; this is especially true for thin polymer ﬁlms. V.

CONFINEMENT EFFECTS IN THIN POLYSTYRENE FILMS

Among the polymers whose glass transition was investigated in the conﬁnement of thin ﬁlms in the last decade, polystyrene has been the most preferred. Starting with the study of Keddie and Jones [36], numerous investigations have revealed reductions of the glass transition temperature Tg in thin polystyrene ﬁlms [36–42]. Nevertheless, the overview emerging from the data reported in the literature is still controversial, since many recent studies exist indicating no shifts of Tg in thin polystyrene ﬁlms [45–51]. A factor to which not enough attention has been paid in many experiments is related to the impact of the experimental ambient conditions on the preparation and measurements of thin polymer ﬁlms. A careful inspection of the studies reported in the literature reveals a variety of experimental conditions under which annealing and measurements of thin polystyrene ﬁlms were performed: ambient air (at various humidities), vacuum, high vacuum, ultra-high vacuum, nitrogen ﬂow. Due to an increased surface-to-volume ratio in thin polymer ﬁlms, their exposure area to ambient gas atmosphere is orders of magnitude larger than in

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the bulk. As a consequence, thin polymer ﬁlms are extremely sensitive to the presence of water vapor and oxygen, whose adsorption can drastically alter, through plasticizer and oxidation effects, the dynamics of the polymers under study. Such ambient effects will now be illustrated for thin polystyrene ﬁlms annealed above the glass transition Tg alternatively in vacuum, in a pure nitrogen atmosphere, in the presence of water vapor, or under atmospheric conditions (ambient air). It will be experimentally shown that these preparative conditions have a pronounced impact on the stability of the samples, on their surface topography, and on the molecular mobility. This ﬁlms of polystyrene (Mw ¼ 700; 000 g=mol, polydispersity 1.05) were prepared by spin-coating from a toluene solution using the procedure described in detail in Section II. In order to characterize the dynamic glass transition in different experimental environments, several techniques have been used: broadband dielectric spectroscopy, capacitive scanning dilatometry, ac calorimetry. The dielectric measurements were performed using a High-Resolution Alpha Analyzer (Novocontrol GmbH) in a frequency range covering eight decades (102–107 Hz) and in a wide temperature interval (320–440 K). The temperature stability was set to be better than 0.1 C. When kept at 180 C in air, thin PS ﬁlms undergo a gradual change of their initially ﬂat geometry, which ends up with the formation of a speciﬁc pattern (Fig. 30). The pattern development is caused by the compressive stress originating from the difference in the thermal expansion coefﬁcient between the polymer and the upper metal layer [52,53]. In contrast to the measurements performed in air, in a pure nitrogen atmosphere, after an identical thermal treatment, no changes of the sample geometry are detected (Fig. 30). annealing at 180˚C in a pure nitrogen atmosphere

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Figure 30. Optical image (top view) of a thin PS ﬁlm of 123 nm after different annealing times at 180 C in a pure nitrogen atmosphere (upper diagram) and in ambient air (lower diagram).

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Figure 31. Optical image (a) and the corresponding surface topography by AFM (b) of a PS ﬁlm of 239 nm as prepared (left column), after 4 hours at 180 C in a pure nitrogen atmosphere (middle column) and after 4 hours at 180 C under air ﬂow (right column). Rq represents the root-meansquare roughness. (c) Linecuts in the AFM images presented above.

Complementary measurements by atomic force microscopy were involved in order to inspect the sample geometry after annealing in different experimental environments. After 4 hours at 180 C in air, the initial sample geometry of a thin PS ﬁlm of 239 nm is changed and an undulating pattern is developed, with a well-deﬁned lateral wavelength (Fig. 31). Figure 31b shows the corresponding surface topographies, measured by atomic force microscopy. By waving a typical initial value of 2–3 nm, the root-mean-square roughness (parameter Sq in Fig. 31) increases steeply and reaches 129 nm after the formation of the

molecular dynamics in thin polymer ﬁlms

623

pattern. In a pure nitrogen atmosphere, under the same experimental conditions, no pattern is developed: Both optical microscopy and atomic force microscopy reveal in these cases similar results as in Fig. 31a, thus no deformations of the sample geometry. This is clearly illustrated in Fig. 31c, which shows the vertical surface proﬁles measured by AFM for the sample of 239 nm as prepared and after 4 hours at 180 C in a nitrogen atmosphere and in air, respectively. The deformation of the sample geometry during the pattern formation, while no changes are detected in a pure nitrogen atmosphere, suggests that thin PS ﬁlms exhibit an enhanced mobility in ambient air. To prove this, several experimental techniques were employed. The molecular dynamics of thin PS ﬁlms is investigated under different experimental environments by broadband dielectric spectroscopy [1]. In a pure nitrogen atmosphere, no changes of the dynamic glass transition are detected for a thin PS ﬁlm of 84 nm during annealing at 140 C (40 C above the bulk glass transition), proving sample stability under these conditions (Fig. 32). On the

Dielectric loss versus frequency at 140 C showing the dynamic glass transition for a thin PS ﬁlm of 84 nm in a pure nitrogen atmosphere (a) and in air (b).

Figure 32.

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Figure 33. Normalized dielectric loss versus frequency at 122 C, showing the alpha relaxation process of a thin PS ﬁlm of 75 nm as prepared, and after different annealing times at 180 C in air, as indicated. Inset: The corresponding relaxation rate in dependence on the annealing time at 180 C in air.

contrary, annealing at the same temperature in ambient air results in a faster alpha relaxation process or, correspondingly, in a reduction of the glass transition temperature Tg (Fig. 32). This effect, which is manifested in ambient air and not in a nitrogen atmosphere, appears to be irreversible; that is, after annealing in air, the initial dynamics cannot be recovered when the samples are measured again under nitrogen ﬂow, or even after keeping them a long time in high vacuum. Much more pronounced effects are observed at higher temperatures: The alpha relaxation peak of a thin PS ﬁlm of 75 nm (measured at 122 C) is shifted to higher frequencies after keeping the sample at 180 C in ambient air (Fig. 33). Consequently, the average relaxation rate of the alpha relaxation process increases in time by more than one order of magnitude during this thermal treatment (inset, Fig. 33). For thin PS ﬁlms of 63 nm, using the usual ﬁtting procedure [1], the relaxation rate as a function of inverse temperature is extracted (Fig. 34), after different annealing steps in air and in pure nitrogen. While unchanged after 24 hours at 180 C in a nitrogen atmosphere, the dynamic glass transition becomes one decade faster when the sample is annealed in air. This corresponds to a shift to lower temperatures of the maximum position of the alpha relaxation peak (inset, Fig. 34). The glass transition temperature of thin PS ﬁlms after different annealing times at 180 C in ambient air was also measured by capacitive scanning dilatometry [43,44] (Fig. 35). It provides a Tg determination on an experimental time scale much longer than in conventional dielectric measurements, since the

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Figure 34. Relaxation rate versus inverse temperature for a 63-nm-thick PS ﬁlm as prepared, after 24 hours annealing at 180 C in a pure nitrogen atmosphere, and after 1, 2, 5, and 10 hours annealing at the same temperature in air. Inset: The corresponding alpha relaxation at 1.2 kHz in spectra of e00 versus temperature.

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Figure 35. Glass transition temperature Tg (determined by dilatometry) and relaxation time t at 131 C as a function of annealing time in air at 180 C for a ﬁlm thickness of 63 nm. The dotted lines serve as a guide for the reader. Inset: Dilatometric determination of the glass transition temperature. Upper: Normalized capacitance Cnorm versus temperature at 106 Hz (the solid lines represent linear dependencies, the dotted line marks the position of the glass transition temperature). Lower: The corresponding ﬁrst and second numerical derivatives of Cnorm (in arbitrary units) as a function of temperature.

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Figure 36. Glass transition temperature Tg (determined by ac calorimetry at 20 Hz) for a ﬁlm thickness of 93 nm with respect to dependence on the annealing time at 200 C in ambient air. Inset: The real and the imaginary part of the complex heat capacity (in arbitrary units) at 20 Hz as a function of temperature.

effective rate of cooling/heating during an dilatometric experiment is of the order of 0.2 C/min. Tg reductions of 10 C are detected, corresponding to a shift of one decade in the average relaxation rate of the dynamic glass transition. The aluminum electrodes appear to play no role in the observed Tg reductions. Measurements by ac calorimetry at 20 Hz of thin PS ﬁlms prepared by spin-coating on a SiN membrane and not covered by an upper aluminum layer reveal results similar to those for the samples prepared in a sandwich geometry: a decrease in time of the glass transition temperature during annealing at high temperatures in ambient air (Fig. 36). The details of the ac calorimetry technique developed for investigating the glass transition of ultrathin polymer ﬁlms can be found elsewhere [54]. The origin of the enhanced mobility is indicated by IR-spectroscopic measurements. Following annealing for 24 hours at 200 C of bulk PS sample (4 mm) in ambient air, a new spectroscopic band is observed in the spectra (Fig. 37). It is identiﬁed as originating from carbonyl C O groups. This indicates an oxidation effect, since oxygen is not present initially in the structure of the polystyrene macromolecule. As a consequence, a pronounced reduction of the average molecular weight has to be considered, caused by oxygen-induced chain scissions. According to the formula of Flory and Hugens [55], a reduction of the glass transition temperature must be expected in this case, and, correspondingly, an enhanced mobility. This hypothesis has been checked by keeping PS thin ﬁlms at 180 C in a pure oxygen atmosphere: A similar pattern as in Fig. 30 is developed after few hours. No changes in the IR spectra are observed when the same annealing procedure is performed in a pure nitrogen atmosphere (Fig. 37).

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aromatic ring stretching

absorbance

vibrations

0.2

3 C-O band 2

1 0.0 1200

1400

1600

1800

2000

wavelength [cm–1]

Figure 37. Infrared spectra (spectral resolution of 8 cm1) for PS in the bulk (sample thickness 4 mm): (1) as prepared, (2) after 24 hours at 200 C in a pure nitrogen atmosphere, and (3) after 24 hours at 200 C in air. The curves are vertically shifted for clarity.

Such degradation effects were reported for bulk PS a long time ago [56–58], even in the absence of oxygen (in high vacuum). At temperatures above 260 C, a decrease of the sample viscosity was observed in time, caused by a decrease of the average molecular weight induced by chain scissions. This ﬁnding was associated with the existence of the so-called ‘‘weak-links’’ in the polystyrene chains. When such experiments are performed not in high vacuum but in ambient air, much more pronounced effects are expected and at much lower temperatures, because of the presence of oxygen, which diminishes the probability of chain recombination after scission. In the case of thin and ultrathin polystyrene ﬁlms, the drastic increase of the exposure area originating from the increase of the surface-to-volume ratio represents another important factor that favors the manifestation of oxidation effects. They cannot be suppressed (presumably only slowed down) even when the free interface of a thin PS ﬁlm is covered by a thin aluminum layer (50 nm thick). As observed in our experiments, the diffusion of oxygen takes place laterally, initiated at the edges of the upper aluminum electrode and progressively advancing inside the sandwiched PS ﬁlm (Fig. 30). This can happen in regions far away from the edges, as well, when the presence of dust particles or impurities (before evaporating the upper aluminum layer) may produce, on a micrometric scale, small irregularities (ruptures) of the electrode, favoring subsequent oxygen diffusion. Thus, in the absence any capping layers, for thin PS ﬁlms having a free interface, much more pronounced effects are expected. Two experimental techniques are employed to investigate the dynamic glass transition for thin polystyrene annealed at least 12 hours at 150 C in an oil-free

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Tg (by dilatometry) [K]

380

370

360

350 20

40

60 80 thickness [nm]

100

120

Figure 38. Glass transition temperature Tg determined by capacitive scanning dilatometry for thin PS ﬁlms as a function of ﬁlm thickness.

high vacuum (106 mbar) and measured in a pure nitrogen atmosphere. Down to a thickness of 20 nm, no Tg reductions are detected by capacitive scanning dilatometry with an effective cooling rate of 0.2 C/min (Fig. 38). Measurements by broadband dielectric spectroscopy also reveal no shifts of the dynamic glass transition (inset, Fig. 39). In consequence, the average relaxation rate of the dynamic glass transition remains unchanged for all thicknesses investigated in the present study (Fig. 39).

1.0

6 0.5

5

113 nm

380

400 420 T[K]

59 nm 33 nm 30 nm

4

20 nm 2.24

2.32

2.40 2.48 1000/T [1/K]

2.56

2.64

Figure 39. Relaxation rate of the alpha process versus inverse temperature for different thin PS ﬁlms, as indicated. Inset: Dielectric loss (normalized) versus temperature at 31 kHz skHz showing the alpha relaxation of the same samples.

molecular dynamics in thin polymer ﬁlms VI.

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CONCLUSIONS

In relation to the questions addressed in the introduction, the following answers are given, based on experimental results obtained for three polymer systems investigated in thin layers using broadband dielectric spectroscopy, capacitive scanning dilatometry, and calorimetry: 1. (a) Due to a strongly enhanced surface-to-volume ratio which increases the area of exposure to ambient environment, thin polymer ﬁlms are extremely sensitive to preparative and measurement conditions. Although important for a variety of phenomena investigated in thin ﬁlms (dewetting, Tg shifts, surface dynamics, pattern formation), this concern has not been seriously taken into account in many studies reported so far. In our investigation of thin polystyrene ﬁlms, we show that the dynamic glass transition becomes faster in the presence of oxygen and water vapor, due to chain scissions and plasticizer effects. (b) Thin polystyrene ﬁlms annealed in high vacuum and measured in a pure nitrogen atmosphere do not show any shifts of their dynamic glass transition down to a thickness of 20 nm. This experimental ﬁnding is in sharp contrast with numerous other investigations on thin polystyrene ﬁlms performed in ambient air, revealing reductions of the glass transition temperature Tg. 2. As shown for conﬁned hyperbranched polyesters, different experimental techniques do not provide necessarily similar results when applied to thin ﬁlms because they measure different microscopic or macroscopic physical quantities. 3. A molecular model of the mechanisms underlying the shifts of the dynamic glass transition in conﬁnement is possible only by a quantitative determination of the relaxation time distribution in dependence on the conﬁnement size. This important aspect is exempliﬁed in our study of hyperbranched polyesters and isotactic poly(methylmethacrylate). Two possible mechanisms giving rise to a faster dynamic glass transition in thin polymer ﬁlms are put forward: Either (a) at a molecular level, the polymer segments are relaxing faster in conﬁnement than in the bulk, or (b) the slower relaxation modes are gradually suppressed (frozen-out) in conﬁnement, resulting in an increase of the average relaxation rate and, consequently, in a reduction of the glass transition temperature Tg. Due to methodological limitations, the vast majority of the studies on conﬁned polymers reported in the literature cannot make a clear distinction between the two above-mentioned scenarios. This recommends broadband dielectric spectroscopy, with its ability to determine quantitatively the relaxation time distribution of polar entities, as an ideal tool for the investigation of conﬁnement effects. 4. (a) As exempliﬁed with the conﬁnement-induced mode discovered in thin ﬁlms of cis-1,4-polyisoprene, conﬁnement can give rise to novel relaxation

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processes which do not exist in the bulk. For PIP, this is due to the presence of the conﬁning interfaces altering the dynamics of the adjoining polymer segments and leads to an interruption of the end-to-end ﬂuctuation of the PIP chains. (b) The concept of ‘‘chain conﬁnement,’’ proposed to explain the shifts of the glass transition temperature in thin polymer ﬁlms, appears not to be valid for cis-1,4-polyisoprene. Thus, despite a strong chain conﬁnement (as proven by the existence of the conﬁnement-induced mode), no shifts of the dynamic glass transition in thin PIP ﬁlms are observed, even for ﬁlm thicknesses smaller than the end-to-end distance of the PIP chains. (c) Flexible polymer chains conﬁned in thin ﬁlms, such as those of PIP, can be well-described by ideal random walks in three dimensions taking place between two impenetrable interfaces. Most of the experimental ﬁndings characterizing the dynamics of PIP in thin ﬁlms were recovered in simulations of polymer chains as ideal random walks. 5. Conﬁnement effects in thin ﬁlms of hyperbranched polyester show up at thickness much larger than those reported for linear polymers. This must be attributed to their special macromolecular architecture. Acknowledgments We acknowledge the contribution of L. Hartmann to the measurements on iso-PMMA. We would like also to thank Y. Mikhailova, K.-J. Eichhorn, and B. Voit for providing the hyperbranched polyesters, H. Huth and C. Schick for performing the AC-calorimetric measurements, and L. Ha¨ussler for performing the DSC measurements.

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AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that the author’s work is referred to although his name is not mentioned in the text. Numbers in italic show the pages on which the complete references are listed. Letter in boldface indicates the volume. Abbes, K., B:556(286), 590 Abel, J., B:174–175(94), 280 Abetz, V., A:11(42), 118, B:569(331), 591 Abou, B., A:259–260(12), 303(12,56), 307(12,56), 315(12), 317(12), 319(12), 321,323 Abragam, A., A:150(81), 154(81), 246 Abraham, R. J., B:54(69), 90 Abramowitz, M., B:316(51), 319(51), 323(51), 332(51), 334(51), 341(51), 367–369(51), 381–382(51), 385(51), 389–390(51), 403(51), 421(51), 414–426(51), 436 Adachi, K., B:572(341), 592 Adachi, K. B., B:601(15), 631 Adam, G., B:499(8), 582 Adam, M., B:518(158), 547(158), 586 Adams, G., A:13(63), 94(63), 103(63), 118, 156(100), 246 Adib, A. B., A:410(85), 472 Adichtchev, S., A:52(203), 62(203), 122, 164(136), 170(181,183), 175(136,183), 177(136), 179(136,183,230), 181–182(183), 183(136), 188–189(136,230), 191(230), 194(230), 202–203(230), 205(230), 209(230), 216(183), 222(183), 223(136,230), 230(183), 231(410), 232(183), 235(410), 247–248, 250, 255 Adler, J., B:135(48), 278 Adler, S. L., A:436(109), 473 Adolf, D., A:11(39), 118 Afanasyev, V., B:445(39), 493 Affouard, F., A:173(210), 249, B:548(277), 590 Agmon, N., A:52(206), 112(206), 123 Agranat, A. J., A:24(126), 40(177–180), 45(179), 46(257), 47–48(179), 93–94(179),

96(179), 101(179,257), 102(257), 120, 122, 124 Aharony, A., A:38(152), 62(152), 65–66(152), 72(152), 80(152), 121, B:96–97(1), 132–133(1), 135(1,48), 138(1), 147(1), 156(1), 160(1), 277–278 Akagi, Y., A:234(414), 255 Alba-Simionesco, C., A:170(184), 179(229), 182(229), 210(343), 223(229), 224(378–379), 249–250, 253–254, B:518(149), 532(231–232), 553(231–232), 579(149), 586,589 Alcoutlabi, M., B:566–567(317), 591 Alder, B. J., A:142(57), 245 Alegrı´a, A., A:11(41), 106(263), 118, 124, 180(234), 202(318), 250, 252, B:502(50), 503(69), 518(142), 558(294), 567(322), 572(340), 584, 586, 590–592 Alencar, A. M., B:4(7), 13(24), 87–88 Alexander, S., A:10(24), 117 Alexandrov, Y., A:6(9), 12(47), 33(149), 35(47), 56(47), 68(225), 72(47), 117–118, 121, 123, B:235(204), 283 Alexandrowicz, Z., B:137(56), 279 Alig, I., B:575(353–354), 592 Alivisatos, A. P., A:358(1), 469 Allain, M., B:132–133(31), 278 Allegrini, P., A:374(35), 392(59), 396(59), 401(71), 403(59,71), 404(71), 415(93), 419(93), 424(96), 425–426(100), 429(96,102), 430(96), 431(96,102), 433(96), 439(59), 440(59,71,117), 441(59), 444(123), 451(146), 452(150), 454(96), 462(117), 464(166), 466(100), 467(96), 468(59,93,100), 470–473, 476, B:35(49), 89

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

633

634

author index

Almgren, R. F., B:209(142), 281 Altemeier, W. A., B:17(30), 88 Alvarez, F., A:11(41), 118, B:518(146,151), 548(273), 579(146,151), 586, 590 Alvarez-Gomariz, H., A:135(38), 241 Amaral, L. A. N., B:69(89), 91 Ambegaokar, V., A:261(26), 322 Amma¨la¨, P., B:80(102), 91 Anastasiadis, S. H., B:564(305), 569(332), 591 Andersen, H. C., A:156(111–113), 165(112–113), 221(111–113), 247 Anderson, J. E., A:153–154(92), 246 Anderson, J. M., A:24(129), 120 Anderson, M. W., A:163(135), 225(135), 247 Anderson, P. W., A:368(20–21), 442(20), 470 Anderson, R. M., B:445(38), 493 Andersson, O., B:503(64), 584 Andersson, S. P., B:503(64), 584 Andjelic, S., B:575(372), 593 Andrianov, A., A:16(86), 18–25(86), 119 Andronow, A. A., B:481(103), 495 Angell, C. A., A:13(55–58,65), 48(185), 118, 122, 130(4,7), 157(124–125),165(7), 172(7,125,205–207), 173(207,214), 231(7), 243, 247, 249, B:499(1), 502(45), 516(45), 580(1), 582–583 Annunziato, M., A:390(51), 411(51), 413(51,87), 414(89), 415(87), 471–472, B:76(98), 91 Anoardo, E., A:148(75), 246 Antropova, T., A:38(156), 39–42(156), 58(156), 60(156), 63–64(156), 103(156), 121 Aouadi, A., A:179(229), 182(229), 223(182), 250 Aquino, G., A:337(30), 355, 424(96–97), 425–426(97), 429(96,102), 430(96), 431(96,102), 432(105), 433(96), 434(105), 454(96–97), 466(105), 467(96,105), 472 Arbabi, S. A., B:147(76), 208(135,138), 210(161), 212(168–169), 214(161), 279, 281–282 Arbe, A., A:106(263), 124, 180(234), 190(277), 201(277,302–304), 202(277,303,318), 204(277), 236(277,303), 250–252, B:518(142–148,151), 536(237), 548(143,271,273), 579(144–148,151), 586, 589–590 Arita, I., A:106(265), 110(265), 124

Arkhipov, V., A:33(147), 38(155), 40(155), 52(206), 58(155), 63(155), 96–100(155), 112(206), 121, 123 Arndt, M., A:9(14), 26(14), 117, B:562(299), 590 Arnedo, A., 46(59), 89 Arrhenius, S., A:12(48), 118 Arsenin, V. Y., A:9(13), 117 Asahina, M., B:502(56), 584 Asami, K., A:16(95), 18(95), 119 Aseltine, J. A., B:325(56), 436 Ashwin, S. S., A:223(371), 254 Aslangul, C., A:261(25), 266(25), 286–287(25), 297–299(51), 321–322 Assaad, F. F., A:68(223), 123 Astrand, P. O., A:52(205), 112(274), 122, 124 Atkinson, R. P. D., B:445(38), 493 Attene´, P., B:527(196), 588 Aubry, S., B:546(266), 589 Auty, R. P., A:88(245), 124 Avakian, P., A:139(49), 245 Axelrod, E., A:26–27(132), 30(132), 38(153,157–161), 40(153), 41(153,160–161), 42(153,157–161), 44(160), 58(153), 63(153,157–161), 64(157–160), 98–100(153), 103(153), 120–121 Axelrod, N., A:26–27(132), 30(132), 120 Ayala-Orozco, B:445–446(41), 493 Babloyantz, A., B:25–26(43), 89 Bachhuber, K., A:112(272), 124 Bacry, E., B:46(59), 89 Bahar, B:602(14), 605(14), 631 Bailberg, I., A:42(167), 121 Baker, G. A., A:68(222), 123 Balagurov, B. Y., B:179(110), 280–281 Baldassari, A., A:346(29), 355 Baldus, O., A:214(348), 253 Bale, R. C., B:97(10), 277 Balescu, R., B:460(66), 494 Balin, I., A:52–54(208), 82–85(208), 86(244), 89(244), 90(208,244), 91(244), 123–124 Balucani, U., A:132(22), 134(22), 244 Banin, U., A:358(1), 469 Bao, J. Z., A:23–24(125), 120 Bao, L-R., B:519(164), 587 Bao, L. R., A:216(364), 254 Baranger, M., A:447(137), 473 Barbi, F., A:456(157), 474

author index Barenblatt, G. I., B:5(15), 88 Barkai, E., A:11(34), 117, 334(26–27), 338(31), 340(31), 344(34), 346(34,37), 349(26,40), 350(26), 353(26), 354(34), 355–356, 406(76–78), 424(95), 426(95), 433(95,107), 434(107), 454(95), 472, B:41(54), 89, 237(209–210), 253(236), 257(236), 283–284, 297(30), 300(37), 302–303(37), 326(57), 372–374(30), 379–380(30), 394(30,37), 398(30), 400(30), 413(30), 416(30), 421(30), 435–436, 446(52), 448(58), 494 Barrachough, G. G., B:183(114), 280 Barrat, J.-L., A:156(110), 247, A:259(14), 321 Barrochi, F., A:182(247), 223(247), 250 Barshilia, H. C., A:145(61),179(231), 183(231), 193(231), 230(231), 245, 250 Barthel, J., A:13(53–54), 21(111), 112(54,272), 118, 120, 124 Barthelmy, M., B:188(128), 281 Bartolini, P., A:147(69), 167(150), 177(219), 182(69,247), 223(247), 224(69,150,388–389,392–393), 245, 248, 250, 255 Bartos, J., A:216(363), 254 Bartsch, E., A:156(118), 171–172(201), 180(233,235), 215(201), 216(233), 247, 249–250 Barvik, I., A:436(111), 473 Basche´, T., A:334(19), 355 Baschnagel, J., B:292–293(19), 297–299(19), 303(19), 305(19), 435, 612(20), 631 Bassingthwaighte, J. B., B:8(21), 23(21), 42(21), 67(21), 88 Ba¨ssler, H., A:14(77), 119 Bauer, C., B:618(44), 631 Baughman, R. H., B:209(155), 282 Bawendi, M. G., A:328(2–3,7), 331(3,7), 335(28), 344(3), 353(7), 354–355, 358(3–4), 432(4), 470 Beck, C., A:452–453(154), 474 Becker, H. F., B:17(32), 88 Beckham, H. W., A:201(308–309), 252 Bedeaux, D., A:378(38), 460(38), 470 Beevers, M. S., A:147(66), 170(66), 245 Behrends, R., A:52(204,208), 53–54(208), 82–85(208), 90(208), 122–123 Beiner, M., A:190(274), 201(274), 204(324), 251, 253, B:502(41), 548(41), 551(41), 556(290), 575(290), 583, 590

635

Bel, G., A:344(34), 346(34), 354(34), 356 Belge, G., B:613(30), 631 Bellazzini, J., A:401(71), 403–404(71), 440(71), 471 Bellini, T., A:11(38), 67(38), 118 Bellisent-Funel, M. C., B:532(231), 553(231), 589 Bello, A., A:11(40), 118 Bellon, L., A:259(8), 303(8), 307(8), 321 Bellour, M., A:303(55), 307(55), 323 Belov, D. V., B:167(79), 279 Belov, V. P., B:125(24), 136(24), 138(24), 149(24), 278 Ben-Avraham, D., B:443(15), 493 Ben-Chorin, M., A:10–11(25), 38(163), 41(25), 42(163), 63(163), 117, 121 Ben-Ishai, P., A:26–27(132), 30(132), 40(179), 45(179), 46(257), 47–48(179), 93–94(179), 96(179), 101(179,257), 102(257), 120, 122, 124 Benassi, P., A:143(58), 245 Benci, V., A:451(146), 474 Bendler, J. T., A:14(71–73), 118 Benetatos, P., B:466(67), 494 Bengtzelius, U., A:14(75), 119, 156(104), 246 Benini, G., A:212(346), 253 Benkhof, S., A:130(6), 163–164(6), 165(6,142), 167(6), 168(6,142,153), 170(153,183), 175(183), 177(220), 179(142,183), 181–182(179), 184(142,259–260), 186–187(142), 188(6,142), 189(142), 190–191(6,142), 192(259–260), 194(142), 200(6), 201(6,153,285), 202(6,142,153,285,315), 203(6,315), 204(6), 205(6,142,153,285), 206(6,142), 209(6,142), 210(220), 216(183), 222(183), 230(183), 231(6), 232(6,183), 234(6,220), 236(6,153,285,315), 243, 247–248, 250–252 Benson, D. A., B:445(42), 493 Beran, J., B:31(45), 89 Berberian, J. G., A:16(79–81), 18(79–81), 19(80), 20–21(79–81), 23(79–81,116), 119–120 Bercu, V., A:219–220(369), 231(405), 254–255 Berg-Sørensen, K., B:443(19), 493 Berge´, P., A:381(39), 471 Berger, T., A:179(229), 182(229), 208(338), 223(182), 250, 253, B:514(128), 586 Bergman, D. J., B:132–133(28), 144(28), 174–175(92, 95,107), 181(92), 187(123,125–126), 188(121–123), 208(136,139), 212(167), 278–282

636

author index

Bergman, R., A:147(65), 169(65,156), 170(65), 184(65,256), 185(65), 194(65,256), 204(65), 205(156), 207(65), 208(65,256), 209(156), 216–218(65), 231(65), 245, 248, 251, B:503(103), 585 Bergmann, R., B:574(346), 592 Berkowitz, B., B:443(18), 471(80), 493, 495 Bernasconi, J., A:10(24), 117, B:144(62), 167(62), 189(62), 279 Berne, B. J., A:132–133(25), 140(25), 171(25), 184(25), 244 Bero, C. A., A:170(182), 248, B:527(191,199), 560(191), 587–588 Bershtein, V. M., A:201(294), 252 Bersie, A., B:225(175), 228(175), 282 Bertault, M., A:223(376), 254 Bertolini, D., A:16(83), 18(83), 20–21(83), 23–24(83), 119 Best, A., B:514(126,133), 586 Bettin, R., A:444(125), 473 Beuaﬁls, S., A:223(376), 254 Beyerlein, D., B:613(29), 631 Bhargava, V., B:80(103,106), 91 Bhatnagar, P. L., B:418(100), 437 Bianco, S., A:425(99), 466(99), 472 Bianucci, M., A:409–411(84), 472 Bikaki, A., B:546(266), 589 Binder, K., A:156(99), 246, B:129(26), 224(26), 233(26), 278, 568(327), 591, B:612(20), 631 Birch, J. R., A:16(93), 119 Bird, N. F., B:132(45), 278 Bishop, D. J., A:14(76), 119 Bitton, G., A:24(126), 120 Bizzarri, A. R., A:334(20), 355 Blacher, S., B:174–175(108), 280 Blackburn, F. R., B:522(168,170), 587 Blackburn, M. J., B:80(105), 91 Blackham, D. V., A:24(130), 120 Blaszczyk, J. W., B:22(40), 89 Blochowicz, T., A:52(203), 62(203), 122, 130(6), 163(6), 164(6,136), 165(6,142), 167(6), 168(6,142,153), 169(157), 170(153,183), 175(136,183), 177(136,220), 179(136,142,183,230), 181–182(183), 183(136), 184(142,259), 186–187(142), 188(6,136,142,230,271,273,275,331), 189(136,142,230,275), 190(6,142,275), 191(6,142,230,275), 192(259), 193(275), 194(142,230), 195–199(275), 200(6,275),

201(6,153,285), 202(6,142,153,157,230,285,315), 203(6,230,315), 204(6), 205(6,142,153,157,230,285,331), 206(6,142), 209(6,142,157,230,331), 210(220), 213(331), 216(183), 222(183), 223(136,230), 230(183,401), 231(6), 232(6,183), 234(6,220), 236(6,153,285,315), 243, 247–248, 250–253,255, B:537(241–242), 572(241–242), 574(242), 589 Bloembergen, N., A:150(82), 246 Blumen, A., A:10(32), 54(210), 117,:123, 432(104), 467(104), 472, B:235–236(187–189), 237(189,212), 283, 446(45), 471(45), 481(45,94), 494–495 Boal, D. H., B:209(148), 281 Boeffel, C., A:201(308–309), 248 Boese, D., A:16(90), 119, 184(255), 194(255), 251, B:531(230), 588 Bogoliubov, N. N., B:481–482(101), 495 Bo¨hmer, R., A:13(57), 16(92), 48(188), 51(188), 118–119, 122, 130(11–12), 148(11–12), 149(11), 150(11,349), 152–153(11), 154(11–12), 171(189,199), 172(206–207), 173(207,214–215), 188(331), 200(11), 205(331), 209(12,331), 210(343), 213(12,189,331,347), 214(11–12,349), 235(11), 236(424), 244, 249, 253, 256, B:501(29), 502(45), 516(45), 529(29), 530(226), 549(236,280), 550(236,280,284), 578(29), 583, 588, 590 Bohnenberger, J., A:328(1), 354 Bohren, C. F., B:183(112), 280 Bokov, A. A., A:102(258), 124 Bokrovec, M., B:474–475(84), 495 Bol, A. A., A:349(46), 356 Bolcano, F., B:97(17), 278 Bologna, M., A:389–392(50), 394(60), 396–397(60), 398(50), 407(80), 409(82), 413(80), 424–426(97), 449(141), 453(155), 454(97), 467(155), 468(60,155), 471–474, B:39(52), 40(53), 54(52), 55(53), 62(53), 64–65(53), 77(53), 87(53), 89, 292–293(20), 313(20), 324(20), 435 Bonci, L., A:373(31), 374(35), 436(110), 439(112), 444(31,122–123), 445(31,112), 446(112), 447(31,122), 470, 473 Bonello, B., A:223(376), 254 Bonn, D., A:303(56), 307(56), 323

author index Bonvallet, P. A., A:171(197), 249, B:523(178), 587 Boon, J., B:627(58), 632 Bordat, P., A:173(210), 249, B:548(277), 590 Bordewijk, P., A:7–9(11), 11(11), 27–28(11), 117, 134–136(34), 147(34), 179(34), 202(34), 244, B:286(6), 290–291(6), 293(6), 434 Bordi, F., A:32(145), 120 Bo¨rjesson, L., A:169(156), 184(256), 194(256), 205(156), 208(256,338), 209(156), 215(355), 248, 251, 253–254, B:574(346), 592 Borkovec, M., B:327(59), 380(59), 427(59), 436 Bo¨se, D., B:601(16), 608(17), 631 Bose, T. K., A:16(84), 18(84), 20(84), 21(84,109), 23–24(84), 119–120 Bo¨ttcher, C. J. F., A:4(7), 6(7), 7(7,11), 8–9(11), 11(11), 13(7), 27–28(11), 105(7), 117, 134–136(34), 147(34), 179(34), 202(34), 244, B:286(6), 290–291(6), 293(6), 434 Bottreau, A. M., A:21(107), 120 Bouchaud, J.-P., A:259–260(2), 276(2), 320, 328(5), 331(5), 336(5), 337(48), 344(5), 346(29), 351(5), 353(5), 354–356, 405(75), 425(98), 432–433(98), 456(98), 472, 253(222), 284, B:443(16), 445(16), 486(16), 493 Bourret, R. C., A:415(92),472 Bouwstra, S., B:209(153), 282 Bowden, N., B:620–621(52), 632 Bowtell, R., A:115(280), 125 Boyd, R., B:556(292), 590, 614(34), 631 Brabley, R. M., B:137(51), 279 Brace, D. D., A:147(70), 182(70), 224(70,390–391), 226(70), 245, 255 Brand, R., A:13(59), 52(201), 118, 122, 140(52), 169(158), 184(258,262), 191–193(258), 201(258), 202(316), 208(158,258), 236(316), 245, 248, 251–252, B:502(39), 503(39,100), 529(216), 537(39), 538(39,100), 550(100), 551(39), 583,585,588 Brandrup, J., B:599(7), 630 Branka, A. C., B:209(143,147), 213(143), 281 Bravard, S. P., B:556(292), 590 Bregechka, V. F., B:97(17), 178 Breuer, H. P., A:446(129), 473 Brinkmann, D., A:148(77), 246 Broadbent, S. R., B:131(27), 278

637

Broda, E., B:3(2), 53(2), 87 Brodin, A., A:146(64), 147(65), 163–164(64), 166(64), 169–170(65), 171(64), 173–174(216), 177–178(64), 179(64,227,231), 180(64,227), 182(64), 183(64,231), 184–185(65), 193(231), 194(64–65), 196(64), 204(65), 207–208(65), 216(64–65), 217–218(65), 230(231), 231(65), 245, 249–250 Brokmann, X., A: 328(5), 331(5), 336(5), 344(5), 351(5), 353(5), 354, 425(98), 432–433(98), 456(98), 472 Brouers, F., B:174–175(106), 280 Brow, R. K., A:177(222), 233(222), 250 Brown, R., B:440(3), 492 Brown, W. F. J., B:289(13), 347(76), 434, 436 Bruchez, M., A:358(1), 469 Bruening, M. L., B:613(25), 631 Bruggman, D A. G., B:160(84), 163(84), 279 Brumer, P., A:373(30), 470 Brus, L. E., A:332(15), 355 Brychkov, Y. A., B:265(217), 283, 313(48), 423–424(48), 435 Buch, V., A:85(242), 123 Buchanan, C. I., B:19(37), 89 Buchenau, U., A:145(60), 245, B:518(143,145), 548(143,145), 586 Buchman, T. G., B:85–86(108), 92 Buchner, R., A:13(53–54), 21(111), 112(54,272), 118, 120, 124 Buckingham, A. D., A:147(67), 245 Budil, D. E., A:218(367), 254 Bug, A., B:174–175(96–97), 280 Bug, A. L. R., A:33(148,150), 70(150), 121 Buiatti, M., A:391(52), 409(52), 462(164), 471, 474 Buisson, L., A:259(9), 321 Buldyrev, S. V., B:4(7), 13(24), 87–88, 445(39–40), 493 Bulgadaev, S. A., B:187(125), 281 Bunde, A, A:66(220), 109(220), 123, B:17(32), 88, 443(30), 493 Burns, A., A:16(79), 18(79), 20–21(79), 23(79), 119 Burstein, A. I., B:364(84), 436 Butaud, P., A:276(41), 278(41), 322 Ca´ceres, M. O., A:415(90), 472 Caddock, B. D., B:209(145–146), 281 Cahill, D. G., A:235(418), 255

638

author index

Caillaux, A., B:518(149), 579(149), 586 Cakir, R., A:461(161), 463(161), 474 Calcott, P. D. J., A:41(165), 121 Caldeira, A. O., A:261(23), 321 Callen, H. B., A:134–135(31), 244, 269(32–33), 322 Cameron, G. G., B:627(56), 632 Cametti, C., A:32(144–145), 33(151), 121 Campana, A., A:89(256), 102(256), 124 Campisi, M., A:407(80), 413(80), 472 Cang, H., A:224(391), 225(395–397), 255 Canham, L. T., A:41(164–165), 42(164), 121 Cannistraro, S., A:334(20), 355 Cantelli, R., A:89(256), 102(256), 124 Cao, H., B:571(336), 592 Capaccioli, S., B:499(10–11,14,16), 502(34), 503(79–81,99), 505(79–81), 517(136), 530(80), 537–538(240), 548(240), 551(34,80,240), 556(34,80,240,290–291), 572(342), 573(136,342–343), 575(290–291,371), 582–583, 585–586, 589–590, 592 Capek, V., A:436(111), 473 Cappaccioli, S., A:177(219), 201(287), 202(314,317), 224(392), 250–252, 255 Carini, J. P., A:165(140), 184(140), 190(140), 243, 247 Carmichael, R., B:309(42), 435 Carr, J. A., B:19(37), 89 Carroll, P. J., B:513(119), 585 Carus, T. L., B:440(1), 492 Carusotto, J., A:446(130), 473 Casalini, R., A:178(224), 188(224), 200(282), 201(287), 202(282,317,321), 208(339), 209(321), 233–234(224), 250–253, B:499(10–12,14,16), 502(48), 503(12,62,71–74,80–81,84,87, 89,92,94,96–97,99,101–102), 504(12,72), 505(84,80–81), 507(87,89,92,96), 508(113), 510(62,114–115), 512(62), 514(133), 517(114–115,141), 519(141,165–166), 527(207), 529(101–102), 530(80), 531–532(101–102), 541(253), 548(279), 551(80), 553(101–102), 556(80,290–291), 560(114–115), 561(207), 565(96,207), 575(207, 290–291,369), 577(114), 579–580(48), 582–590, 592 Casati, F. G., A:441–442(119), 473 Caspi, A., B:443(26), 493

Cassettari, M., A:16(83), 18(83), 20–21(83), 23–24(83), 119 Casteleijn, G., A:32(142), 121 Castin, Y., A:446(130), 473 Castro, F., A:29(136), 120 Cattaneo, G., B:481(95), 495 Cavaille, J. Y, A:201(297), 252, B:527(194), 549(282), 556(282,286), 574(194), 587, 590 Cerrada, M. L., B:556(289), 590 Cetinbas, M., A:446(131), 473 Chaikin, C. E., B:481(103), 495 Chakravarty, S., A:261(24), 321 Challa, G., B:627(58), 632 Chamberlin, R. V., A:11(43), 118, 185(267), 251, B:501(29), 529(29), 578(29), 583 Chandrasekhar, S., B:440(9), 492 Chang, I., A:171(193,198), 249, B:526(183), 587 Chauvet, G. A., B:86(109), 92 Chay, I. C., A:170(182), 248 Chay, I.-C., B:565(308), 575(352), 591–592 Chechkin, A. V., B:445(43), 450(60), 453(60,63–64), 456(60,64), 458(64), 468(77), 474(86), 475(43,90), 481(100), 483(60,64), 484(64), 492(113), 493–496 Chemla, D. S., A:358(1), 469 Chen, H. S., B:529(219–221), 588 Chen, S.-H., A:32(144), 33(151), 121, 156(115), 165(115), 247 Chen, X. K., A:142(55), 177(55), 182(55), 245 Chen, Y., B:14(27), 88 Cheng, P. Y., B:14(27), 88 Chigirinskays, Y., B:46(62), 49(62), 90 Chirikov, B. V., A:441–442(119), 473 Cho, M., A:147(68), 245 Choi, B., B:209(156), 282 Chong, S.-H., A:156(108), 184(265), 225(265), 247, 251 Christensen, R., B:205(132), 206–218(132), 230(132), 232(132), 281 Christensen, T., A:170(174), 186(278), 190(278), 205(327), 248, 251, 253 Christianson, H., B:187(124), 281 Chryssikos, G., A:16(79), 18(79), 20–21(79), 23(79), 119 Chukbar, K. B., B:253(238), 257(238), 284 Chung, I., A:328(3), 331(3), 344(3), 354 Cicerone, M. T., A:171(195), 249, B:522(168–169), 587

author index Cichos, F., A:328(6), 331(6), 333(6), 349(43), 351(6), 353(6), 354, 356 Ciliberto, S., A:259(8–9), 303(8), 307(8), 321, B:546(267), 590 Clark, W. G., A:23–24(121), 120 Clarke, R. N., A:3(4), 16–18(4), 23(117–118), 24(131), 117, 120 Clarke, T., B:274(243), 284 Clerc, J. P., B:132–133(31), 278 Cluzel, P., A:334(23), 355 Cocho, G., B:445–446(41), 493 Cochrane, J., B:527(198), 588 Codastefano, P., A:32(144), 33(151), 121 Coddens, G., A:215(354), 224(387), 253, 255, B:548(271), 590 Coffey, W. T., A:10(26), 12(26), 117, 180(239–240), 250, B:253(250), 255(250), 260(244), 267(250), 271(244–246), 284, 287–288, 289(8,14), 290(8), 292(8,18,21–22,24–26), 293(8,18,21,23,27–28), 294(8), 296–297(8), 299(22), 309(8), 312(24,43), 313(8,24), 314(8), 319(8), 322–324(8), 329(8), 330(8,25), 331–333(62), (334(8,62), 336(8,62), 338(8,67–68), 339(8), 341(8,67), 343–344(8,67–68), 347(8,67,78,81), 348(8), 350(8), 353(8), 355–358(8), 364(8,24,43,67), 366–368(8), 376(24), 378(8), 380(8), 384(8,43), 388(90,92), 389(92), 393–394(8), 398(26,96–97), 399(8), 401(8,26), 403(8), 407(24), 417(8), 418(25), 420(101), 424(22,101), 426(8,67), 434(67), 434–436, 442–443(13), 493 Cohen, E. G. D., A:452(153), 474 Cohen, M., B:174–175(96–97), 280 Cohen, M. H., A:13(70), 97(70), 118, 156(101–102), 246 Cohen-Solal, G., A:231(407), 255 Colby, R. H., B:569(332–333), 591 Cole, K. S. A:9(16), 106(16), 117 B;286(3), 291(3), 300(3), 424(3), 434 Cole, R. H., A: 9(16–17), 10(28), 16(79), 18(79), 20(79), 21(79,104), 23(79,116), 47(78), 48(17,78), 88(245), 96–98(78), 103–105(78), 106(16), 117, 119–120, 124, 136(42), 137(48), 165(139), 245, 247, B:286(3–4), 291(3), 300(3), 322(4), 424(3–4), 434 Collin, R. E., A:17(97), 119

639

Collins, J. J., B:22(39), 89 Colmenero, J., A:11(41), 106(263), 118, 124, 180(234), 190(277), 201(277,302–304), 202(277,303,318), 204(277), 236(277,303), 250–252, B:502(50), 503(69), 518(142–148,151), 536(237), 548(143,271,273), 558(294), 567(322), 572(340), 579(144–148,151), 584, 586, 589–592 Colombini, D., B:224(174), 282 Colomer, P., B:575(373), 593 Comes, R., A:44(171), 121 Comez, L., A:169(160), 207(160), 216(359), 231(160), 248, 254, B:514(125–126), 526(184), 586–587 Compiani, M., A:452(151), 474 Compte, A., B:235(195), 283, 446(53), 494 Coniglio, A., B:144(60), 279 Cook, R. J., A:451(145), 474 Coppens, M.-D., B:443(29), 493 Cordero, F., A:89(256), 102(256), 124 Corezzi, S., B:499(10–11), 503(82), 505(82), 514(126), 556(290), 575(290,371,374–375), 582, 585–586, 590, 592–593 Corry, R. A., B:617–618(36), 620(36), 631 Cortes, P., B:575(360), 592 Corti, M., A:89(256), 102(256), 124 Cory, R. A., B:566(313), 591 Cox, D. R., A:423(94), 472 Cox, H. E., B:14(28), 88 Cregg, P. J., B:271(245), 284 Crimmins, T. F., A:241(430), 256 Crisanti, A., A:259–260(4), 276(4), 321 Crissman, M., 255 Crocker, J. C., B:529(213), 530(213), 550(213), 578(213), 588 Crossley, J., A:147(66), 170(66), 245 Crothers, D. S. F., A: 180(240), 250, B:292(21), 293(21,27), 347(78), 420(101), 424(101), 435–437 Cudkowicz, M. E., B:18(35), 88 Cugliandolo, L. S., A:259(3,5–6). 260(2–3), 276(2–3,40,46–47), 278(40), 289(46–47), 304(57), 305(5–6), 307(5–6), 309–312(5–6), 320–323 Cui, H. M., A:23–24(123), 120 Cukierman, M., A:172(202), 249 Culicchi, C., B:575(370), 592 Cullis, A. G., A:41(165), 121

640

author index

Cummins, H. Z., A:130(5), 141(53), 142(53,55), 145(61–62), 156(109), 159(109), 163(5,134), 177(5,55,133–134, 221), 179(5,134,227,231), 182(5,55,133–134,244), 183(231), 185(270), 193(280), 193(231), 216(357), 223(133,372–376), 224(372,374,381–384), 230(231), 243, 245, 247, 250–251, 254 Currat, R., A:44(171), 121 D’Anna, G., A:259(14), 321 da Luz, M. G. E., B:445(40), 493 Dahan, M., A: 328(5), 331(5), 336(5), 342(33), 344(5), 351(5), 353(5), 354, 356, 425(98), 432–433(98), 456(98), 472 Dalnoki-Veress, K., B:601(8–10), 609(9), 617–618(37), 630–631 Dammers, A. J., B:443(29), 493 Darcy, H., A:76(233), 123 Das, S. P., A:130(21), 156(21), 162(131), 221(21), 244, 247 Dauxois, T., B:546(267), 590 David, F., A:68(221), 123 David, L., A:201(296), 252, B:556(286–288), 590 Davidson, D. W., A:9(17), 48(17), 117, 165(139), 247, B:286(4), 322(4), 424(4), 434 Davis, E. A., A:9(21), 117 Davis, H., B:481(102), 495 Davis, M., A:6–9(8), 16(8), 117 Dawes, G. S., B:14(28), 88 Dawson, C. A., B:4(7), 13(24), 87–88 De Gennes, P. G., B:601(11–12), 630 De Jagher, P. C., B:481(96), 495 De Lucca, C. J., B:22(39), 89 De Oliveira, C. E. M., A:40(179), 45(179), 46(257), 47–48(179), 93–94(179), 96(179), 101(179,257), 102(257), 122, 124 Dean, D. S., A:304(57), 323 Dean, P., B:132(45), 278 Debenedetti, P. G., A:157(127–128), 247, B:546(255), 589 Debierre, J.-M., B:137(51), 279 DeBolt, M. A., A:89(247), 124 Debus, O., A:180(233), 216(233), 250 Debye, P., A:154(95), 246, B:286–289(1), 292(1), 305(1), 316(1), 484 Decena, J. A., B:14(27), 88 Deegan, R. D., A:202(313), 236(313), 252 Deering, W., B:4(5), 22–24(5), 58(5), 87 DeGennes, P. G., A:113(277), 124

Degiorgio, V., A:11(38), 67(38), 118 Dehmelt, H., A:448(140), 450(144), 466(140), 473–474 De´jardin, J. L., B:338(68–69), 436 De´jardin, P. M., B:338(69), 436 Delasanti, M., B:518(158), 547(158), 586 Den Nijs, M. P., B:135(40), 278 Denoyer, F., A:44(171), 121 Deppe, D. D., A:171(196), 249 Derrida, B., B:132–133(32), 278 Desbiolles, P., A: 328(5), 331(5), 336(5), 342(33), 344(5), 351(5), 353(5), 354, 356, 425(98), 432–433(98), 456(98), 472 Descamps, M., A:173(210), 249, B:548(277), 590 Destexhe, A., B:25–26(43), 89 Deutscher, G., B:174–175(105), 280 Dhar, A., A:346(38), 356 Dhinojwala, A., A:171(196), 249 Di Leonardo, R., A:156(116), 165(116), 247 Dianoux, A. J., A:216(362), 254 DiBiasio, A., A:33(151), 121 Diehl, R., A:171(188), 249 Diezemann, G., A:130(11), 132(28), 148(11), 150(11,349), 152–154(11), 171(189,199), 188(331), 200(11), 205(331), 209(331), 213(189,331), 214(11,349), 235(11), 241(427), 244, 249, 253, 256, B:501(29), 529(29), 530(226), 549–550(226,280), 578(29), 583, 588, 590 Dill, J. F., A:89(247), 124 Ding, M., A:460(160), 474 Ding, Y., A:173(213), 208(335), 249, 253 Dissado, L. A., A:61(216), 123 Ditlevsen, P. D., B:474(87), 475(91), 495 Dittrich, T., A:373(25–26), 470 Dixon, P. K., A:165(140), 169(165), 184(140), 190(140,279), 192(279), 247–248, 251, B:508(112), 585 Dlubek, G., B:499(9), 582 Doliwa, B., A:157(129), 240(426), 247, 256, B:526(188), 550(188), 587 Dom, B. E., A:89(247), 124 Domb, C., B:132(42), 278 Donth, E., A:48(183), 84(183), 106(262), 115(282), 122, 124–125, 130(13–14), 200(276), 204(325), 244, 251, 253, B:556(290), 572(338), 575(290), 590, 592 Doob, J., B:78(99), 91 Doolittle, A. K., A:13(69), 97(69), 118, B:499(7), 582

author index Doolittle, D. B., B:499(7), 582 Dorfman, E., A:45(181), 122 Dorfmuller, T., B:513(116–118,121), 585–586 Dorfmu¨ller, T., A:184(252–254), 194(251–254), 251 Dorget, M., B:224(172), 282 Dornic, I., A:346(29), 355 Dorsey, A. T., A:261(24), 321 Doß, A., A:169(154–155), 188(331), 205(154–155,331), 209(154–155,331), 210(154–155), 213(331), 236(154–155), 248, 253 Dosseh, G., B:518(149), 579(149), 586 Doster, W., B:513(123), 586 Douglas, J. F., B:235(199), 283 Dozier, W. D., A:61(217), 123 Drake, J. M., A:61(217), 123 Drake, P. W., A:89(247), 124 Dressel, M., A:48(188), 51(188), 122 Dreyfus, C., A;141–142(53), 177(221), 179(229), 216(357), 224(388), 245, 250, 254–255 Drezin, Y. A., B:183(117), 281 Driebe, D. J., A:449(142), 473 Dries, T., A:170(177), 210(177), 235(177), 248 Drndic, M., A:335(28), 355 Du Bois Reymond, P. J., B:98(19), 102(19), 278 Du Prez, F. E., B:507(111), 585 Du, W. M., A:130(5), 141(53), 142(53,55), 147(68), 163(5,133–134), 177(5,55,133–134), 179(5,133–134), 182(5,55,133–134), 193(280), 223(372–303), 224(372,381–384), 243, 245, 247, 251, 254 Dubinskii, A. A., A:231(403), 255 Duchne, A. M., B:163(87), 280 Duer, M. J., A:148(79), 246 Dunlap, D. H., A:374(33), 470 Dunn, A. G., B:135(47), 278 Duong, L.-T., B:525(180), 550(180), 587 Duplantier, B., A:68(221), 123, B:144(58), 279 Durand, D., B:518(158), 547(158), 575(368), 586, 592 Dutcher, J. R., B:596(2), 601(8,10), 617–618(37), 630–631 Duval, E., A:165(145), 182(145), 216(362), 247, 254 Duvvuri, K., A:202(311), 208(311), 252, B:525(180), 550(180), 573(344), 587, 592 Dux, H., A:194(251), 251 Dvinskikh, S., A:212(346), 253

641

Dwyer, G., B:14(27), 88 Dykhne, A. M., B:183(117), 186(119), 281 Dyre, J. C., A:9(20), 11(20), 117, 170(174–175), 186(278), 190(278), 205(327), 241(428), 248, 251, 253, 256, B:174–175(99), 280, 363(83), 400(83), 436, 540(248), 589 Dzuba, S. A., A:231(406), 255 Earle, K. A., A:218(367), 254 Easteal, A. J., A:89(247), 124 Eberhard, M., A:10–11(25), 41(25), 117 Eckert, H., A:148(78), 246 Ecolivet, C., A:223(376), 254 Ediger, M., B:522(168–171), 523(178–179), 527(206), 560(206), 565(206), 569(335), 571(336), 587–588, 591–592 Ediger, M. D., A:130(4), 171(195,197,200), 214(200,352), 243, 249, 253 Edwards, B. F., B:137(57), 279 Edwards, D. A., B:502(52), 584 Efremov, M. Y., B:620(45–46), 631 Efros, A. L., A:331(8), 333(8), 354 Egorov, V. M., A:201(294), 252 Ehlich, D., A:171(191), 249 Eiermann, P., A:169(167), 177(167), 197(167), 204(324), 210–212(324), 236(167), 248, 253 Einstein, A., A:76(235), 110(235), 123, B:286–288(2), 294(2), 306(2), 311(2), 434, 440(4), 488(4), 492 Eisenberg, D., A:13(52), 98(52), 112(52), 118 Eisler, H.-J., A:328(2), 354 El Goresy, T., A:188(331), 205(331), 209(331), 213(331), 253 Elbaum, M., B:443(26), 493 Eliazar, I., B:456(65), 494 Ellerby, D. J., B:19(37), 89 Ellison, C. J., B:617–618(41), 631 Elterman, P. B., A:89(247), 124 Emel’yanov, V. I., B:183(115), 280 Emelyanov, Y. A., A:201(294), 252 Emery, J. R., B:575(368), 592 Emonet, T., A:334(23), 355 Empedocles, S. A., A:328(7), 331(7), 353(7), 354, 358(3–4), 432(4), 470 Endou, H., B:506(106), 585 Erde´lyi, A., A:298(53), 323, B:437 Ermolina, I., A:3(5), 16(86), 18–25(86), 117, 119 Eska, G., A:139(50), 163(50), 232–235(50), 245

642

author index

Esquinazi, P., A:234(417), 255 Essam, J. W., B:132(43–44), 135(47), 278 Etienne, S., B:529(217–218), 588 Etros, A., B:144(67), 187(67), 194(67), 279 Evans, E., B:194(130), 281 Evans, G. J., B:312(43), 364(43), 384(43), 435 Evans, K. E., B:209(145–146,157), 281–282 Evans, M. W., B:312(43), 364(43), 384(43), 397(93), 412(98), 426(93), 435, 437 Evans, R. S., B:209(156), 282 Evans, W. M., A:132–133(27), 180(27), 244 Even, U., A:60(214–215), 123 Eyring, H., A:12(49–51), 106(264), 118, 124 Ezquerra, T. A., B:502(42–43), 554(43), 583 Fabian, L., A:132(23), 156(23,115), 165(115), 244, 247 Facchi, P., A:373(28), 470 Faetti, S., A:413(88), 472 Failla, R., A:374(36), 470 Faivre, A., A:201(296), 252, B:556(286–288), 590 Fajans, J., B:474(88), 495 Fakhraai, Z., B:564(306), 591 Falconer, K., B:43(57), 89 Family, F., B:144(60), 279 Fannin, P. C., B:324(54), 347(54,73), 436 Farago, B., A:182(241,243), 190(277), 201(277,299,301–302), 202(277), 204(277), 215(241,243). 236(277), 250–252, B:518(144–145,147–148), 536(237), 579(144–145,147–148), 586, 589 Fatuzzo, E., A:136(43), 245 Faust, R., A:201(302), 252, B:518(144–145), 579(144–145), 582, 592 Fayer, M. D., A:147(70), 182(70), 216(356), 224(70,390–391), 225(395–397), 226(70), 245, 254–255 Feder, E., A:56(213), 66(213), 68(213), 109–110(213), 123, B:96(4), 160(4), 277 Feder, J., A:459(158), 474, B:43(58), 89, 97(9), 277 Fedotov, V. D., A:19(103), 21(103), 23(103), 33(147), 119–121 Fejfar, A., A:42(168), 121 Felderhof, B. U., B:338(70), 436 Feldman Y., A:3(2,5), 6(9), 12(46–47), 14(78), 16(86), 19(103,113), 20(86), 21(2,86,103,106), 22(86), 23(86,103,113), 24(86,113,126), 25(86), 26–27(132),

30(132), 32(113,143,146), 33(143,147,149), 34(143), 35(47), 36–37(2), 38(2,153–161), 39(2,156), 40(2,143,153–156,179), 41(2,153–154,156,160–161), 42(156–161), 43(2,153), 44(160), 45(179), 46(257), 47(78,179), 48(78,179,186), 49–51(186), 52(186,208), 53–54(208), 55(154), 56(47,154), 58(153–156), 59(2), 60(156), 61(2,154), 62(154), 63(2,143,153–161), 64(154,156–160), 65(2), 68(225), 72(47), 73(2), 82–85(208), 86(244), 89(244), 90(208,244), 91(244), 93–94(179), 96(78,155,179), 97(78,155), 98(78,153,155), 99(153,155), 100(153,155), 101(179,257), 102(257), 103(78,153,156), 104–105(78), 108(2), 111(2,46), 113(275–276), 114(2), 115(275), 116–124 Feldman, Y., B:235(204), 283 Feldmann, J., A:349(45), 356 Feller, W., A:345(35), 356, 403(72), 472, B:46–47(61), 90, 241(218), 259(218), 262(218), 283, 467(69), 472(69), 487(69), 494 Feng, S., B:155(78), 212(164,166), 279, 282 Ferrari, C., B:575(364–366), 592 Ferretti, M., A:89(256), 102(256), 124 Ferry, J. D., B:499(6), 582 Fetters, L., B:607(18), 631 Fetters, L. J., A:201(299,301), 252 Fick, A., B:440(6), 492 Findley, L., B:80(105), 91 Finkel, P., A:223(376), 254 Finn, P. A., A:216(360), 254 Fioretto, D., A:169(160), 207(160), 216(359), 231(160,409), 248, 254–255, B:499(11), 503(82), 505(82), 514(125–126), 526(184), 541(253), 556(290), 575(290,369,374–375), 582, 585–587, 589–590, 592–593 Firtion, R., B:18(35), 88 Fischer, E. W., A:11(42), 16(90), 48(193), 106(262), 118–119, 122, 124, 141–142(54), 165(141), 169(160,170–172), 170(170–172), 182(245–246), 184(255), 185(141,268), 194(255), 196(170–171), 200(171), 207(160), 223(377), 231(160), 245, 247–248, 250–251, 254, B:502(47), 514(130–131), 518(159), 526(184), 531(230), 562(297), 569(330–331), 579(159), 583, 586–588, 590–591 Fisher, B. R., A:328(2), 354

author index Fisher, M. P. A., A:261(24), 321 Fishman, S., A:444(124), 447(124), 473 Fitz, B., B:575(372), 593 Flat, A. Y., B:124(36), 132(36), 278 Fleischer, G., A: 153–154(89), 171(192), 246, 249, B:522(167), 587, 607(17), 631 Fleming, G. R., A:147(68), 245 Fleurov, V. N., A:406(76), 472 Flinn, P. A., A:216(361), 254 Flores, J., A:373(27,29), 374(32,34), 470 Floriani, E., A:389–390(49), 405(49), 406(49,79), 414(49), 444(126), 471–473 Flory, P. J., A:13(66–68), 118, B:624(55), 626(55), 632 Floudas, G., B:503(61,63,65–66,70), 507(111), 527(199,201), 565(307), 584–585, 588, 591, 607(17), 631 Fogedby, H. C., B:248(223), 253(223), 284, 446(54–55), 448(57), 449–450(54), 456(57), 479(54), 483(54), 494 Folgerø, K., A:16(89), 23–24(122), 119–120 Fonseca, T., A:452(151), 474 Fontana, M. D., A:40(173–174), 122, 184(256), 194(256), 208(256), 251 Fontanella, J. J., B:499(12), 502(35), 503–504(12), 528–529(35), 556(35), 582–583 Ford, G. W., A:261(21,27), 321–322 Ford, J., A:441–442(119), 473 Førdedal, H., A:23–24(119), 120 Fornazero, J., A:201(296), 252, B:556(287), 590 Forrest, J. A., B:564(306), 566(314), 591, 601(8,10), 617–618(37,39–40), 630–631 Fourier, J. B. J., A:76(234), 123 Fox, C., B:237(216), 265(216), 283, 305(38), 308(38), 313(38), 435 Fox, R. F., A:367(18), 470 Fox, T. G., A:13(66–68), 118, B:624(55), 626(55), 632 Fradkin, L., A:334(17), 355 Frank, D. J., B:174–175(93), 280 Frank, M., A:179–180(227), 250 Franosch, T., A:156(106), 223(374), 224(106,374), 246 Frantsuzov, P. A., A:351(50), 356 Franz, S., A:276(45), 322 Frederick, H. J., B:5(14), 15–17(14), 87 Frederix, P. L. T. M., A:349(46), 356 Freed, J. H., A:218(367), 254

643

Freeman, B. D., B:513(120), 586 Frey, E., B:466(67), 488(107), 494–495 Frey, U., B:4(7), 13(24), 87–88 Friberg, S. E., A:31(138,141), 120 Frick, B., A:106(263), 124, 182(243), 201(299,301,303–304), 215(243), 250, 252, B:518(143,149), 548(143,271), 562(300–301), 579(149), 586, 590–591 Fricke, R., A:99(252–253), 124 Friisø, T., A:16(89), 119 Frisch, U., A:415(92), 472, B:119–120(22), 122(22), 144(22), 278, 467(75), 471(75), 494 Fro¨hlich, H., A:136(44), 245, B:238(221), 284, 289(10), 290(20), 404(10), 434 Fro¨jdh, P., A:68(224), 123 Fro¨lich, H., A:4(6), 6–8(6), 105(6), 117 Fromm, D. P., A:331(9–10), 349(41–42), 354, 356, 358(5), 470 Fronzoni, L., A:413(88), 472, B:546(264), 589 Frunza, L., A:99(252–253), 124 Frunza, S., A:99(252–253), 124 Frusawa, H., A:205(328), 253, B:536(234,236), 589 Fryer, D. S., B:613(42), 617–618(42), 631 Fuchs, M., A:184(263), 223(372,374), 224(372,374,381–383), 251, 254 Fuentes, A. F., B:580(380), 593 Fujara, F., A:151(83), 153(87–89), 154(89), 170(177–178), 171(186,188,192–193,201), 172(201), 180(233,235), 182(242), 210(177–178), 212(87–88), 213(87), 215(201,242), 216(233), 224(387), 235(177–178), 246, 248–250, 255, B:522(167), 587 Fujima, T., A:205(328), 253, B:536(234,236), 589 Fujimori, H., A:201(298), 252, B:563(304), 591 Fujiwara, S., A:109(269), 124, B:235(197), 283 Fukao, K., B:596(4), 618(43), 630–631 Fulcher, G. S., A:13(61), 118, B:499(4), 582 Fulinski, A., A:395(62), 471 Fuse, Y., B:572(341), 592 Fytas, G., A:184(252–254), 194(252–254), 208(336), 251, 253, B:503(66), 513(116–118,121), 526(182), 527(199–201), 569(332), 584–588, 591 Gadenne, P., B:174–175(105,108), 280 Gaimes, N., A:231(407), 255

644

author index

Gainaru, C., A:139(50), 163(50), 164(137), 167–168(137), 179(230), 188(137,230,275,331), 189(230,275), 190(137,275), 191(230,275), 193(275), 194(137,230), 195–200(275), 202(230), 203(137,230), 205(230,331), 206(137), 208(323), 209(137,230,331), 213(331), 216(137), 222(137), 223(230), 232(50,137), 233–234(50), 235(50,323), 236(323), 245, 247, 250–251, 253 Gakhov, F. D., B:474(82), 495 Galaubic-Latka, M., B:46(60), 69(90), 71–72(90), 89, 91 Gallagher, A., A:331(9–10), 349(41–42), 354, 356, 358(5), 470 Gallet, F., A:259–260(12), 303(12), 307(12), 315(12), 317(12), 319(12), 321 Gallone, G., B:575(371), 592 Gammaitoni, L., B:325(55), 436 Gangal, A. V., B:237(211), 283 Gapinski, J., A:145(60), 169(160–161), 184(161), 194(161), 207(160–161), 231(160–161), 245, 248, B:514(129,131,134–135), 526(184), 586–587 Garanin, D. A., B:289(14), 347(77,79), 361(79), 434, 436 Garcı´a-Barriocanal, J., B:580(380), 593 Garcimartı´n, A., A:259(9), 321 Garg, A., A:261(24), 321 Garrington, D. C., A:147(66), 170(66), 245 Garti, N., A:32(143,146), 33(143,147), 34(143), 40(143), 63(143), 121 Garwe, E., A:204(324), 253 Gaspard, P., A:450(143), 474 Gaunt, D. S., B:132(46), 135(46), 137(46), 278 Gawlinski, E. T., B:135(52), 279 Ge, S., B:620(49), 631 Gedde, U. W., B:614(34), 631 Geil, B., A:153(87), 170(179), 171(192–193,201), 172(201), 212–213(87), 215(201), 246, 248–249, B:501(29), 522(167), 529(29), 578(29), 583, 587 Geisel, T., A:396(63), 471, B:39–40(51), 89 Gelfand, L. M., A:392(57), 471 Gemkow, M. J., A:26(133), 120 Georges, A., A:405(75), 472, B:253(222), 284 Gerharz, B., A:184(255), 194(255), 251, B:531(230), 588 Gernstein, G. L., B:24(41), 89 Gerritsen, H. C., A:349(46), 356

Gervais, F., A:40(173), 122 Gestblom, B., A:16(85), 18(85), 20(85), 21(85,104), 22(85), 23–24(85,119), 119–120 Gestblom, P., A:21(104), 119 Ghirardi, G. A., A:439(115), 446(115), 449(115), 473 Giacobino, E., A:342(33), 356 Giannelis, E. P., B:564(305), 591 Giannitsis, A. T., B:324(54), 347(54), 436 Gibbs, J. H., A:156(100), 246, B:499(8), 582 Giese, K., A:16–17(91), 119 Gilath, I., A:38(157), 42(157), 63–64(157), 121 Giller, C., B:70(91), 91 Gilroy, K. S., A:235(421), 256, B:361(82), 436 Gimel, J. C., A:179(232), 187(232), 250 Giovanetti, V., A:446(132), 473 Girardi, F., A:466(169), 474 Giri, M. R., B:208–211(140), 281 Girling, L. G., B:17(29), 88 Girshberg, Y., A:44(170), 121 Gitterman, M., B:467(72), 494 Giuntoli, M., A:440(117), 462(117), 473 Givant, A., A:38(160–161), 4142(160–161), 63(160–161), 64(160), 121 Glarum, S., A:136(41), 245 Gla¨ser, H., A:182(246), 250 Gleason, M. J., A:241(430), 256 Glenny, R. W., B:17(30), 88 Glinchuk, M. D., A;40(176), 94(176), 122 Glo¨ckle, W. G., B:235–236(194), 237(194), 283, 291(16), 435, 469(79), 495 Glorieux, C., A:216(356), 254 Glotzer, S. C., A:240(426), 256 Glozman, A., A:334(17), 355 Gmeiner, J., A:203(344), 214(344), 253 Gnedenko, B. V., A:387(45), 471, B:48–49(63), 75(63), 90, 444(34), 493, Gneiting, T., B:34(48), 89 Gobet, F., B:546(267), 590 Gochiyaev, V. Z., A:180(236), 250 Godre`che, C., A:276(44), 322–323, 334(25), 345(25), 356(29), 355, 431(103), 472 Goering, H., B:562(300–301), 591 Goetze, W., B:235–236(191), 283 Goitianda, L., B:558(294), 590 Goldberger, A. L., B:4(6), 5(10,12), 13(25), 17(31), 18(35), 19(19), 65(25), 69(89), 78–79(25), 80(103,105–106), 84(25), 87–88, 91 Goldstein, M., A:52(207), 123, 167(151), 170(151), 201(151), 205(151,332),

author index 237(151,332), 248, 253, B:499(15), 501(17), 503(17,104), 529(17), 583, 585 Goldwater, D., B:80(103), 91 Gomez, D., A:202(318), 252, B:503(69), 584 Gomez, J., B:536(237), 589 Gomez-Ribelles, J. L., B:575(359), 592 Gomi, S., B:235(198), 283 Gonchar, V. Y., B:445(43), 450(60), 453(60,63–64), 456(60,64), 458(64), 468(77), 474(86), 475(43,90), 483(60,64), 484(64), 492(113), 493–496 Goncharov, Y. G., A:48(188), 51(188), 122, 140(51), 245 Gong, J., A:373(30), 470 Goodman, J. W., A:275(38), 322 Gorbatschow, W., B:596(3), 630 Gorenﬂo, R., A:298(54), 323, B:75(94), 91, 253(234), 257(234), 284, 488–489(111–112), 496 Gottke, S. D., A:147(70), 182(70), 224(70,390–391), 226(70), 245, 255 Go¨ttmann, O., A:23(114), 120 Go¨tze, W. A:14(74–75), 48(74), 119, 130(16–17,19), 156(16–17,19,104,106,108), 158(19), 159(16–17,19), 161(130), 162(17,132), 177(19), 179(17), 182(17), 184(265), 221(19), 223(374), 224(106,374,380–383), 225(265,380,398–400), 227(380), 230(19), 244, 246–247, 251, 254–255 Gowrek, T., B:499(9), 582 Goychuk, I., A:334(21–22), 355 Grabert, H., A:297–298(49–50), 322 Graessley, W. W., A:113(278), 125 Graham, M. R., B:17(29), 88 Graham, R., A:373(25–26), 470 Granek, R., B:443(26), 493 Grant, E. R., A:17(96), 119 Grant, J. P., A:23(117–118), 120 Grassberger, P., B:144(61), 279 Gravalides, C., B:503(65), 584 Gray, L. J., B;257(242), 274(242), 284 Gray, R. W., B:527(197), 588 Greaves, G. N., B:580(378), 593 Green, P. F., A:177(222), 233(222), 250 Grest, G., B:174–175(96–97), 280 Grest, G. S., A:33(148,150), 70(150), 121, 156(102), 246 Gribbon, P., A:115(281), 125 Griezer, F., B:183(114), 280

645

Grifﬁn, L., B:5(11,13–14), 15–17(14), 18(11,36), 19(36), 21(36,38), 22(38), 27(36), 28(38), 44(36), 45–46(11,36), 87–89 Grigera, T. S., A:234(415), 255, 259(7), 321 Grigolini, P., A:337(30), 355, 363(8), 368(19), 371(19), 373(19,31), 374(35–36), 389(49–50), 390(49–51), 391(50,52,55), 392(50,59), 394(60), 396(59–60), 397(60), 398(50,65), 401(67,71), 403(59,71), 404(71), 405(49,74), 406(49,79), 407(80), 409(52,82,84), 410(84), 411(51,84), 413(51,80,87–88), 414(49), 415(87,93), 419(93), 424(96–97), 425(97,99–100), 426(97,100), 429(96,102). 430(96), 431(96,102), 432(105), 434(105), 433(96), 435(108), 436(110), 438(108), 439(59,112–114), 440(117), 441(59), 444(31,122–123,125–126), 445(31,112,127), 446(112–114,132), 447(31,122), 448(139), 449(59,71,114,141), 450(139), 451(139,146), 452(150–151), 453(155), 454(96–97), 455(156), 456(157), 461(161), 462(117), 463(161,165), 464(165–167), 466(99–100,105,169), 467(96,105,155), 468(59–60,93,100,155), 470–474, B:5(13), 34(47), 35(49), 39(52), 40(53), 47(47), 50(47), 55(53), 62(53), 64–65(53), 76(98), 77(53), 87(53), 87, 89, 91, 292–293(20), 313(20), 324(20), 435 Grimau, M., A:11(40), 118 Grinberg, O. Y., A:218(366), 254 Groothues, H., B:562(299), 590 Gross, E. P., B:305(39), 312(39), 368(39), 376(39), 418(39,100), 435, 437 Grulke, E. A., B:599(7), 630 Gsin, N., A:446(134), 473 Gua`rdia, E., A:84(241), 123 Guissani, Y., A:177(221), 250 Guitter, E., A:68(221), 123 Gundersen, H. J. G., B:80(101), 91 Gupta, P. K., A:89(247), 124 Gutina, A., A:26–27(132), 30(132), 38(153–157), 39(156), 40(153–156), 41(153,156), 42(153–154,156–157), 48–52(186), 55–56(154), 58(153–156), 60(156), 61–62(154), 63(153–157), 64(154,156–157), 96–97(155), 98–100(153,155), 103(153,156), 121–122 Guyon, E., B:132–133(31), 278

646

author index

Ha, T., A:349(44), 356, 358(1), 469 Haase, M., A:334(19), 355 Habasaki, J., B:528(212), 548(276), 580(276), 588, 590 Habib, S., A:439(116), 473 Hack, T., A:11(42), 118, B:569(331), 591 Hadjichristidis, N., B:503(70), 584 Hager, N. E., A:22(122), 120 Hakim, V., A:261(26), 322 Halalay, I. C., A:182(248), 216(248), 250 Haley, J. C., B:569(335), 591 Hall, C. K., B:513(120), 547(269), 586, 590 Hall, D. B., A:171(196), 249 Hallbruker, A., A:103–104(259), 124 Halley, J. M., B:486(106), 495 Hamann, H. F., A:331(9), 349(41), 354, 356, 358(5), 470 Hamilton, K. E., A:171(196), 249 Hamilton, P., A:451(146), 474 Hanada, F., B:627(57), 632 Hanai, S., B:506(107), 585 Ha¨nggi, P., A:334(21–22), 355, B:325(55), 327(59), 380(59), 427(59), 436, 474–475(84,89), 476(89), 488(108), 495–496 Hanneken, J. W., B:274(243), 284 Hansen, C., A:169–170(172), 185(268), 231–232(412), 248, 251, 255 Hansen, F. N., B:518(156), 547(156), 586 Hansen, J. P., A:132–134(26), 156(110), 158–159(26), 244, 247 Harden, J. L., A:303(55), 307(55), 323 Hardingham, T. E., A:115(281), 125 Harm, S. H., B:17(29), 88 Harris, A. B., B:135(48), 278 Harrison, G., B:527(197–198), 588 Harrowell, P., B:235–236(192), 283 Hartley, S., B:486(106), 495 Hartmann, K., A:151(83), 246 Hartmann, L., B:596(3), 630 Haruvy, Y., A:38(157), 42(157), 63–64(157), 121 Hashin, Z., B:164(88), 204–205(133–134), 225(133–134), 230(133), 280–281 Hashitsume, N., A:266(31), 269(31), 304(31), 306(31), 322, 394(61), 400(61), 405–406(61), 454(61), 471 Hasted, J. B., A:110(270), 112(270), 124 Hatakeyama, H., A:91(249), 124 Hatakeyama, T., A:91(249), 124

Hatta, A., A:11(37), 118 Hausdorff, J. M., B:4(6), 5(10,12), 13(25), 17(31), 18(35), 19(10), 65(25), 78–79(25), 84(25), 87–88 Hauwede, J., B:596(3), 630 Havlin, S., A:66(220), 109(220), 123, B:4(6), 13(25), 17(32), 65(25), 69(89), 78–79(25), 84(25), 87–88, 91, 443(15), 445(40), 493 Havriliak, S., A:9(15), 117, B:286(5), 294(5), 434 Hayashi, Y., A:3(5), 48–51(186), 52(186,208), 53–54(208), 82–85(208), 86(244), 88(246), 89(244,246,248), 90(208,244), 91(244), 117, 122–124 Hayes, R. R., A;40(175), 122 He, Y., B:569(335), 571(336), 591–592 Heammersley, J. H., B:131(27), 278 Heard, S. H., B:183(114), 280 Heinrich, W., B:503(88), 507(88), 585 Helfand, E., B:547(269), 590 Hemmere, P. C., B;273(248), 284 Hempel, E., B:562(299), 572(338), 590, 592 Hendricks, J., B:548(272), 590 Heneghan, C., B:24(42), 89 Heng, B. C., A:115(281), 125 Henry, H. T., B:19(37), 89 Hensel-Bielowka, S., A:52(202), 122, 200(282), 202(319–320,322), 208(320), 251–253, B:502(40,44), 503(40,44,85–86,93,62,75,77), 504(40), 505(77,85–86), 507(93), 508(44,113), 510(62), 512(62), 531(44,77), 532(44), 537(40), 551(40), 553(44,77), 583–585 Hentschel, H. G. E., B:119–120(23), 122(23), 144(23), 278 He´risson, D., A:259(10–11), 321 Hermier, J. P., A: 328(5), 331(5), 336(5), 342(33), 344(5), 351(5), 353(5), 354, 356, 425(98), 432–433(98), 456(98), 472 Hernandez, J., A:130(5), 163(5,134), 177(5,134), 179(5,134), 182(5,134), 223(376), 224(383,387), 243, 247, 254–255 Herring, C., B:183(116), 188(116), 280 Herrmann, A., A:334(19), 355 Herrmann, H. J., B:132(39), 137(39,53–54), 278–279 Hess, K. U., A:169–170(169), 197(169), 248 Hesse, W., A:13(62), 118, B:499(5), 582 Hettich, R., B:614(33), 631 Hetzenauer, H., A:112(272), 124

author index Heuberger. G., A:171(193–194), 249 Heuer, A., A:157(129), 240(426), 247, 256, B:501(28), 526(188), 529(225), 550(188), 583, 587–588 Heyes, C. D., A:349(47), 356 Hildebrand, A:224(381), 254 Hilfer, R., A:12(45), 42(166), 75(45), 78(45), 118, 121, 179(226), 250, B:55(77), 90, 237(208), 240(208), 283, 298–299(33), 422–423(102), 435, 437 Hilhorst, H. J., A:309–310(58), 323 Hilke, M., A:374(34), 470 Hill, N. E., A:6–9(8), 16(8), 117 Hilland, J., A:16(89), 119 Hines, W. A., A:23–24(121), 120 Hinze, G., A:130(11), 147(70), 148(11), 150(11,349), 152(11), 153(11,86), 154(11), 169(154–155,161), 171(186,189,199), 177(220), 182(70), 184(161), 188(331), 194(161), 200(11), 205(154–155,331), 207(161), 209(154–155,331), 210(154–155,220,345), 213(86,189,215,331), 214(11,349), 216(356), 224(70,390), 226(70), 231(161), 234(220), 235(11), 236(154– 155,345,424), 244–246, 248–250, 253–256, B:501(29), 529(29), 530(226), 549(226,280), 550(226,280,284), 578(29), 583, 588, 590 Hirayama, F., B:235–236(181), 282 Hiwatari, Y., B:528(212), 588 Hobb, P. V., A:51(200), 122 Hofacker, I., A:184(263), 251 Hoffman, A., A:165(141), 185(141), 247 Hoﬁneister, R., A:40(180), 122 Hofmann, A., A:48(193), 122 Hohlbein, J., A:334(18), 355 Hohng, S., A:349(44), 356 Holland, D., B:292–293(21), 420(101), 424(101), 435, 437 Hong, D. C., B:137(54), 279 Hong, K. M., A:332(11,16), 355 Hori, H., B:572(341), 592 Ho¨ring, S., B:572(338), 592 Hoshi, M., A:23–24(124), 120 Hosking, J. T. M., B:31(46), 89 Howard, I. A., A:85(243), 124 Howard, M., A:68(224), 123 Howells, W. S., A:215(355), 254 Hsu, W. Y., B:208–211(140), 281 Hu, B. L., A:121(442), 473 Hu¨bner, C. G., A:334(18–19), 355

647

Huffman, D. R., B:183(112), 280 Hughes, B. D., A:452(152), 474, B:443–445(14), 467–468(14), 493 Hui, P., B:174–175(91–92,102), 181(91–92), 280 Hult, A., B:614(34), 631 Hunklinger, S., A:201–202(290), 252, B:503(67–68), 537(67), 584 Hurt, W. D., A:23–24(125), 120 Hurtz, J., A;224(386), 254 Hushur, A., A:102(254), 124 Hutchinson, J. M., B:575(360), 592 Huth, H., A:190(274), 201(274), 251, B:556(290), 575(290), 590, 624(54), 626(54), 632 Huxley, J. S., B:5(16), 88 Hwang, Y. H., A:130(5), 163(5), 177(5), 179(5), 182(5), 193(280–281), 223(373,375–376), 224(384), 243, 251, 254 Hwang, Y.-H., B:513(122), 586 Idiyatullin, Z., A:33(147), 121 Idrissi, H., A:40(174), 122 Ignaccolo, M., A:374(36), 391(55), 451(146), 470–471, 474 Ikeda, R. H., B:208–211(140), 281 Imada, M., A:68(223), 123 Immergut, E. H., B:599(7), 630 Imry, Y., B:174–175(95), 280 Ingenhouse, J., B:440(2), 492 Inokuti, M., B:235–236(181), 282 Inoue, A., B:527(220–221), 588 Iomin, A., A:444(124), 447(124), 473 Isaqeson, J., B:97(13), 277 Ischenko, V. V., B:347(77), 436 Isichenko, M. B., 132(35), 278 Israeloff, N. E., A:135(38–39), 234(415), 244–245, 255, 259(7), 321 Issac, A., A:328(6), 331(6), 333(6), 351(6), 353(6), 354 Ito, K., A:205(328), 253, B:536(234,236), 589 Ivanov, E. N., A: 153(90–91), 246 Ivanov, P. Ch., B:4(7), 13(24), 69(89), 87–88, 91 Izrailev, F. M., A:441–442(118–119), 473 Ja¨ckle, J., A:235(420), 255 Jackson, C. L., B:566(311), 591 Jackson, W. B., A:106(261), 124 Jacobsen, P., B:574(346), 592 Jacobsson, P., A:169(156), 205(156), 209(156), 248, B:503(103), 585

648

author index

Jakobsen, T., A:23(120,122), 24(122), 120 Jal, J. F., A:201(296), 216(362), 252, 254, B:556(287), 590 Jang, M. S., A:102(254), 124 Jarosz, M., A:335(28), 355 Jenkins, S., A:24(131), 120 Jeschke, G., A:218(368), 231(404), 254–255 Jespersen, S., B:448(57), 456(57), 494 Jian, T., B:527(211), 588 Jiang, G. Q., A:23–24(121), 120 Joabsson, F., A:115(279), 125 Joessang, T., B:97(9), 277 Johari, G. P., A:48(189), 51(189), 52(207), 103–104(259), 122–124, 130(1), 134(1), 167(151), 170(151), 201(1,151,295), 205(151), 207(1,333), 232(1), 237(1,151,333), 240(333,425), 243, 248, 252–253, 256, B:324(52),436, 499(13), 501(17–20), 502(58), 503(17–20,104), 506(20), 527(194), 529(17–19,215,217–218), 538(243), 540(250–252), 549(282), 550(285), 556(282), 574(194), 575(348–349,353–358,361–367), 582–585, 587–590, 592 Johnsen, R., A:179(232), 187(232), 250 Johnson, S. T., A:331(10), 354 Jones, A. A., B:571(336), 592 Jones, R. A. L., B:566(313), 591, 617–618(36,38), 620(36), 631 Jones, R. B., B:338(70), 436 Jonscher, A. K., A:9–10(18–19), 117, B:174–175(103), 235–236(184–185), 237–238(185), 244(184), 250(185), 280–282, 546(259), 589 Joos, A., A:364(10), 470 Joosten, J. G. H., A:32(142), 121 Jordan, C., B:527(194), 574(194), 587 Jo¨rg, T., A:213(347), 236(424), 253, 256, B:550(284), 590 Jortner, J., A:60(214–215), 123, B:144(68–69), 172(68–69), 188(68), 279 Joseph, D. D., B:257(240), 274(240), 284 Joung, A. P., B:129(26), 224(26), 233(26), 278 Jovin, T. M., A:26(133), 120 Jue, P. K., A:208(337), 253 Julin, C., B:174–175(105), 280 Jung, P., B:325(55), 436 Jung, W., B:569(328), 591 Jung, Y., A:334(27), 346(37), 355–356, 433–434(107), 472

Justl, A., A:200(431), 202(317), 252, 256, B:556(291), 575(291), 590 Kaatze, U., A:16(88,91), 17(91), 23(114), 52(204,208), 53–54(208), 82–85(208), 90(208), 110(271,273), 119–120, 122–124 Kac, M., A:261(21,27), 321–322 Kaganova, I. M., B:188(129), 281 Kahle, S., A:169(161), 184(161), 194(161), 201(304), 207(1616), 231(161), 248, 252, B:572(338), 592 Kai, Y., A:224(394), 255 Kakalios, J., A:106(261), 124 Kakutani, H., B:502(56), 584 Kallimanis, A. S., B:486(106), 495 Kalmykov, Y. P., A:180(238,240), 250, B:253(250), 255(250), 267(250), 284, 287–290(8), 292(8,22,24–26), 293(8,24,27), 294(8), 296–297(8), 299(22), 309(8), 312(24), 313(8,24), 314(8), 319(8), 322–324(8), 329(8), 330(8,25), 331(62), 332–333(62–63), 334(8,62–63), 336(8,62), 338(8,67–69,72), 339(8), 341(8,67), 343(8,67), 344(8,63,67), 347(8,67,78,81), 348(8), 350(8), 353(8), 355–358(8), 364(8,24,67), 366–368(8), 376(24), 378(8), 380(8), 384(8), 388(90,92), 389(92), 393–394(8), 398(26,96–97), 399(8), 401(8,26), 403(8), 407(24), 417(8), 418(25), 424(22), 425(103), 426(8,67), 434(67), 434–437, 442–443(13), 493 Kamath, S., B:569(333), 591 Kaminska, E., B:503(78–79), 505(78–79), 531–532(229), 584, 588 Ka¨mmerer, S., A:133(30), 165–166(30), 244 Kantelhardt, J. W., B:17(32), 88 Kantor, Y., B:188(121), 208(139), 212(165,167), 213(165), 279, 281–282 Kaplan, E., B:24(42), 89 Kaplan, T., B:257(242), 274(242), 284 Kappler, E., 441(8), 492 Karagiorgis, M., A:449(141), 473 Karatasos, K., B:564(305), 591 Ka¨rger, J., B:443(30), 493 Kariniemi, V., B:80(102), 91 Kaspar, H., B:499(9), 582 Kasper, G., A:201–202(290), 252, B:503(67–68), 584 Kastner, M., A:335(28), 355 Katana, G., A:11(42), 118, B:569(331), 591 Katori, H., B:445(44), 493

author index Kaufmann, S., A:209(341), 212(341), 253 Kauzmann, W., A:13(52,64), 98(52), 112(52), 118, 156(98), 246 Kawana, S., B:617–618(38), 631 Kawasaki, K., A:156(103), 246 Kawashima, N., A:68(222), 123 Kay, B. D., A:103–104(260), 124 Keddie, J. L., B:566(313), 591, 617–618(36), 620(36), 631 Keiding, S. R., A:52(205), 112(274), 122, 124 Keller, J. B., B:186(118), 281 Kenkre, V. M., A:369–370(22), 376(37), 470 Kenny, J. M., B:575(375), 593 Kerr, G. P., B:627(56), 692 Kerstein, A. R., B:137(57), 279 Kertesz, J., B:97(7), 132–133(30), 186(7), 277–278 Kessairi, K., B:573(343), 592 Khinchin, A. I., A:462(163), 474 Kiebel, M., A:170(177), 171–172(201), 180(233), 210(177), 215(172), 216(233), 235(177), 248–250 Kilbas, A. A., B:236(206), 240(206), 283, 452(62), 457(62), 494 Kim, B., A:184(264), 251 Kim, Y. H., B:613(21), 620(47), 631 Kimble, H. J., A:451(145), 474 Kimmich, R., A:11(35–36), 118, 148(74–75), 246 King, E., A:16(81), 18(81), 20–21(81), 23(81), 119 Kinoshita, S., A:224(394), 255 Kircher, O., A:210(343), 253 Kirilina, E. P., A:231(406), 255 Kirkpatrick, S., B:188(70), 279 Kirkpatrick, T. R., A:156(122), 247 Kisliuk, A., A:145(60), 164(138), 165(145–146), 173(146), 177(138), 180(237), 182(145–146), 215(146), 223(217), 245, 247–250 Kister, S. S., B:97(11), 277 Kitai, K. A:201(288), 251 B:536(235), 573(235), 589 Kivelson, D., A:133(29), 142(29), 156(123), 170(176), 179(228), 185(123,269), 244, 247–248, 250–251 Kivelson, S., A:157(123), 185(123), 247 Klafter, J., A:3(1),10(31–32), 11(31,33), 12(31), 54(210–211), 62(217), 75–76(31), 106–108(31), 116–117, 123, 334(24), 355, 384(42–43), 385(42,45), 403(73),

649

406(77–78), 432(104), 450(44), 463(43), 466(43), 467(104), 471–472, B:38(50), 49(65), 54(41), 55(78), 76(78,97), 89–91, 235–236(187–189), 237(189,209–210), 253(226–227,236–237), 257(226,236–237), 264(237), 265(227), 283–284, 287(7), 291–292(7), 296(7), 297–299(7,31), 300(37), 302(37), 303(7,37), 304(31), 305(7,31), 307(7,31), 308–310(31), 313–314(7), 325(7,31), 326(7,57), 327(31), 335(7), 365(7,31,86), 366(7,86), 384(7), 394(37), 434–437, 443(18,20,24–25,28), 445(20,35–36), 446(45–47), 447(49,52), 448(58), 450(60), 453(60,63–64), 456(60,64–65), 458(64), 467(76), 468(77), 470(76), 471(45–46,80–81), 474(76,85–86), 479(20,46), 481(45,93,97–98,100), 483(20,46,60,64), 484(64), 486(20,46), 488(36,109), 493–496 Klauzer, A., A:12(44), 118 Kleiman, R. N., A:14(76), 119 Klein, R., B:97(10), 277 Klian, H.-G., B:235–236(194), 237(194), 283 Klik, I., B:347(80), 436 Klimontovich, Y. L., B:479(92), 482(105), 495 Klonowski, W., B:22(40), 89 Klupper, M., B:224(171), 282 Knaak, W., A:182(241), 215(241), 250 Knaebel, A., A:303(55), 307(55), 323 Knauss, L. A., A:223–224(372), 254 Ko, J. H., A:102(254–255), 124 Kob, W., A:133(30), 156(111–114), 165(30,112–114), 166(30), 221(111–113), 244, 247, B:568(327), 591, 608(19), 631 Kobelev, V., B:64(85), 91 Kobitski, A. Yu, A:349(47), 356 Koch, F., A:10–11(25), 38(163), 41(25), 42(163), 63(163), 117, 121 Kocka, J., A:42(168), 121 Koda, S., A:146(63), 245 Kogut, J., A:78–79(239), 123 Kogut, J. B., B:132(34), 278 Kohlrausch, R., A:10(29), 117, B:235–236(179), 249(179), 282, B:501(30), 583 Kojima, S., A:102(254–255), 124 Kolmogorov, A. N., A:387(45), 447(135), 471, 473, B:48–49(63), 75(63), 90, B:444(34), 493 Kolokoltsov, V., B:253(230), 284 Kolpakov, A. G., B:209(142), 281 Koltulski, M., B:253(229), 284

650

author index

Kolwankar, K. H., B:237(211), 283 Kolwankar, K. M., B:54(70), 90 Koper, G. J. M., A:309–310(58), 323 Koplin, C., A:201(306), 203(306), 208(323), 209(306), 231(306), 235(323), 236(306,323), 237(306), 238(306,429), 252–253, 256, B:530(227), 549–550(227), 588 Ko¨plinger, J., B:503(67), 537(67), 584 Korelov, V., B:253(230), 284 Korn, G. A., A:29(135), 67(135), 107(135), 120 Korn, T. M., A:29(135), 67(135), 107(135), 120 Kornyshev, A. A., B:174–175(94), 280 Korobkova, E., A:334(23), 355 Koroteev, I. I., B:183(115), 280 Korus, J., B:572(338), 592 Korzh, S. A., B:188(127), 281 Kosevich, A. G., B:209(141), 215(141), 281 Koshino, K., A:364(14), 470 Koss, R., B:174–175(101), 280 Kosslick, H., A:99(252–253), 124 Kotaka, T., B:601(15), 631 Kotulski, M., B:299(35), 435 Koutalas, G., B:503(70), 584 Kowalski, S. E., B:17(29), 88 Kox, D. R., B:253(232), 257(232), 284 Kozlov, G. V., A:140(51), 245 Kozlovich, N., A:6(9), 12(47), 32(143,146), 33(143,147,149), 34(143), 35(47), 38(153–154,157), 40(143,153–154), 41(153), 42(153–154,157), 55(154), 56(47,154), 58(153–154), 60(143), 61–62(154), 63(153–154,157), 64(154,157), 68(225), 72(47), 98–100(153), 103(153), 113(275), 115(275), 117–118, 120–121, 123–124, B:235(204), 283 Krakoviak, V., A:224(387–379), 254 Kramarenko, V. Y., B:502(42), 583 Kramers, H. A., B:289(11), 327(58), 434, 436, 474(83), 495 Krauss, T. D., A:332(15), 355 Krauth, W., A:276(42), 322 Krauzmann, M., A:224(387), 254 Kemer, F., A:3(3), 9(14),11(42), 16(3,87,90), 17(3), 18(3,87,98), 26(14), 48(3,193), 51(3), 106(87), 117–119, 122, A:130(12), 134(37), 137–138(37), 148(12), 154(12), 165(141), 185(141), 209(12), 213–214(12), 244, 247, B:324(53), 436, 562(299), 569(328,331),

590–591, 596(1,3,5–6), 601(16), 607(18), 608(19), 616(1), 623–624(1), 562(5), 630–631 Krenz, G. S., B:4(7), 13(24), 87–88 Kriegs, H., B:514(125–126,132), 586 Kristiak, J., A:216(363), 254 Krook, M., B:418(100), 437 Kroy, K., B: 488(107), 495 Kubo, R., A:134(32–33), 135(32), 244, 266(30–31), 269(30–31,34), 270(34), 273(34), 304(30–31), 305(30), 306(31), 322, 394(61), 400(61), 405–406(61), 432(106), 454(61), 471–472, B:327(60), 399(60), 436 Kudlik, A., A:130(6), 163–165(6), 167(6), 168(6,153), 170(153,180), 177(220), 180(237), 184(259–260), 188(6), 190–191(6), 192(259–260), 200(6), 201(6,153,285,308–309), 202(6,153,285,315), 203(6,315), 204(6), 205(6,153,285), 206(6), 209(6), 210(220), 212(346), 230(401), 231–232(6), 234(6,220), 236(6,153,285,315), 243, 248, 250–252, 255 Kuebler, S. C., B:501(29), 529(29), 578(29), 583 Kulik, V., A:231(406), 255 Kumar, S. K., B:569(332–333), 591 Kunin, W. E., B:486(106), 495 Kuno, M., A:331(9–10), 349(41–42), 354, 356, 358(2,5), 384(2), 416(2), 432(2), 451(2), 469–470 Kurchan, J., A:259(5–6,13), 276(40), 278(40), 304(57), 305(5–6), 307(5–6), 309–312(5–6), 321–323 Kurochkin, V. I., A:218(366), 254 Kuwabara, S., A:16(82), 18(82), 20–21(82), 23–24(82), 119 Lacasta, A. M., A:461(162), 474 Ladin, Z., B:5(10), 19(10), 87 Lagrkov, A. N., B:174–175(108), 280 Lairez, D., B:575(368), 592 Lakatos-Lindenberg, K., A:378(38), 460(38), 470 Lakes, R., B:209(144,150–151), 225(175–178), 228(175–177), 281–282 Lamb, J., B:527(197–198), 588 Lambert, M., A:44(171), 121 Lamperti, J., A:345(36), 356 Landau, L. D., A:74–76(227), 123, B:174(90), 209(141), 215(141), 275(219), 280–281, 284, 481(104), 495

author index Landauer, R., B:160(85), 280 Landolﬁ, M., B:546(263), 589 Lane, J. W., A:172(202), 249 Langevin, D., A:31(140), 120 Langevin, P., B:27(44), 89 Langoff, L., A:334(17), 355 Laredo, E., A:11(40), 118 Larralde, H., B:445–446(41), 493 Larsen, U. D., B:209(153–154), 282 Latka, D., B:46(60), 69(90), 71–72(90), 89, 91 Latka, M., B:5(9), 46(60), 69(90), 71–72(90), 87, 89, 91 Latora, V., A:447(137), 473 Latz, A., A:132(23), 156(23), 184(263), 224(381–383), 244, 251, 254 Laughlin, R. B., A:469(170), 474 Laughlin, W. T., A:169(164), 170(164), 172(164), 248 Lauritsen, K. B., A:68(224), 123 Lauritzen, J. I., B:331(61), 436 Lauthie, M. F., B:532(231), 553(231), 589 Laux, A., A:373(31), 444–445(31), 447(31), 470 Lax, M., A:63(219), 76(237–238), 123, B:235–236(182), 282, 443(22), 493 Leatherdale, C. A., A:328(7), 331(7), 353(7), 354, 358(3), 470 Lebedev, S. P., A:140(51), 245 Lebedev, Y. L., A:218(366), 254 Lebon, M. J., A:177(221), 216(357), 250, 254 Leduc, M. B., B:14(28), 88 Lee, A., B:575(350–351), 592 Lee, C., B:17(31), 88 Lee, S. B., B:135(50), 279 Lee, S. S., A:102(254), 124 Lee, T., B:225(175), 228(175), 282 Lefevre, G. R., B:17(29), 88 Lefschetz, S., B:481(103), 495 Leggett, A. J., A:261(23–24), 321 Legrand, F., A:180(235), 250 Leheny, R. L., A:165(143–144), 184(143,261), 190(143), 192(143–144), 247, 251 Lei, X. L., B:174–175(100), 280 Lellinger, D., B:575(354), 592 LeMehaute, A., A:10(23), 106–107(23), 117 Lenk, R., A:184(259), 251 Lennon, J. J., B:486(106), 495 Leon, C., B:502–503(39), 519(161), 537–538(39), 548(276,278), 551(39), 574(278), 580(276,380), 583, 587, 590, 593

651

Leo´n, C., A:178(22), 185(257–258), 191–193(258), 201(258), 208(258), 233(223), 250–251 Leporini, D., A:219–220(369), 231(404–405), 254–255 Lepri, S., B:546(268), 590 Lequeux, F., A:303(55), 307(55), 323 Leung, K., A:93(251), 124 Leutheusser, E., A:156(105), 246 Levandowsky, M., B:445(37), 493 Levelut, C., A:231(407), 255 Levin, V. V., A:21(106), 119 Levine, B. D., B:4(8), 70(91,93), 87, 91 Levine, Y. K., A:32(142), 121 Levinshtein, M. E., B:144(71), 279 Levita, G., B:575(369–371), 592 Levitov, L., A:335(28), 355 Levitt, A., B:529(213), 530(213), 550(213), 578(213), 588 Levitz, P., B:443(31), 493 Le´vy, P., A:388(47), 471, B:440(10), 444(33), 492–493 Lewis, G. N., B:558(293), 590 Lewis, L. J., A:216(358), 254, B:548(274–275), 590 Leyser, H., B:513(123), 586 Leyva, V., A:40(177), 122 Li, G., A:130(5), 141(53), 142(53,55), 145(61), 163(5,134), 177(5,55,133–134), 179(5,134), 182(5,55,133–134,244), 193(280), 223(133,372–374), 224(374,381–384,387), 243, 245, 247, 251, 254–255 Li, H.-M., B:523(173), 587 Li, J., A:225(396), 255 Li, L., A:218(365), 254 Li, Y., B:17(31), 88 Lichtenberg, A. L., B:546(254), 589 Lieberman, M. A., B:546(254), 589 Liebovitch, L. S., B:8(21), 23(21), 42(21), 67(21), 88 Lifshitz, E. M., A:74–76(227), 123, 334(17), 355, B:174(90), 209(141), 215(141), 275(219), 280–281, 284, 481(104), 495 Liggins, E. C., B:14(28), 88 Lilge, D., A:184(252), 194(252), 251 Lin, G., B:571(336), 592 Lin, M. Y., B:97(10), 277 Linberg, R., A:31(141), 120 Lindblad, G., A:446(133), 469(133), 473 Lindenberg, K., A:461(162), 474, B:66(86), 91

652

author index

Lindman, B., A:31(139), 120 Lindsay, H. M., B:97(10), 277 Lindsay, S. M., A:163(135), 225(135), 247 Linsay, C. P., B:527(210), 574(210), 588 Lipsitz, L. A., B:17(31), 88 Liu, F., B:614(34), 631 Liu, S. H., B:257(241), 274(241), 284 Liu, X., A:235(419), 255 Liu, Y., B:620(50), 631 Livi, A., B:541(253), 575(369–370), 589, 592 Livi, R., B:546(268), 590 Lobb, C. J., B:174–175(93), 280 Lobo, R., A:142(56), 245 Lochhass, K. H., B:499(9), 582 Lockwenz, H., A:204(324), 253 Lodge, T. P., B:568(326), 569(334–335), 591 Loﬁnk, M., A:172(203), 249 Loidl, A., A:13(59), 16(92), 48(188,195), 50(195), 51(188,195), 52(195,201), 118–119, 122, 130(9–10), 136(9–10), 140(52), 163(9–10), 164(9), 165(9–10), 169(158–159), 175(9), 175(9), 177(9–10), 178(9), 180(9), 182(9–10), 183(9) 184(9–10,262), 185(10,258), 191(9,258), 192(9–10,258), 193(9,258), 201(9,258), 202(316), 208(158–159,258), 218(9), 223–224(9–10), 231(402), 236(316), 244–245, 248, 251–252, 255, B:502(39), 503(39,100), 529(216), 537(39), 538(39,100,244,246), 550(100), 551(39), 558–559(244), 583, 585, 588–589 Lonneckegabel, V., A:16(88), 119 Loreto, V., A:259(14), 321 Lorthioir, C., B:572(340), 592 Losert, W., A:193(280), 223(373), 251, 254 Lovejoy, S., B:46(62), 49(62), 90 Lovesey, S. W., A:270(37), 322 Lovicu, G., A:201(287), 251 Lowen, S. B., B:24(42), 89 Lozano, G., A:276(46–47), 289(46–47), 322 Lu, I. R., B:529(222), 588 Lu, S. T., A:23–24(125), 120 Lu, Y. J., A:23–24(123), 120 Lubensky, T. S., B:97(13), 277 Lubianiker, Y., A:42(167), 121 Lucchesi, M., A:177(219), 201(287), 202(317), 224(392), 250–252, 255, B:499(10–11,14,16), 503(80), 505(80), 530(80), 537–538(240), 541(253), 548(240), 551(80,240), 556(80,240,291), 573(343), 575(291), 582–584, 589–590, 592

Luchko, Yu., B:451(61), 474(61), 494 Luck, J. M., A:276(44), 334(25), 345(25), 322–323, 431(103), 472 Lung, C. W., B:312(45–46), 435 Lunkenheimer, P., A:13(59), 16(92), 48(188,195), 50(195), 51(188,195), 52(195,201), 118–119,122, 130(9–10), 136(9–10), 137(45), 140(52), 163(9–10), 164(9), 165(9–10), 169(158–159), 175(9), 175(9), 177(9–10), 178(9), 180(9), 182(9–10), 183(9) 184(9–10,262), 185(10,258), 191(9,258), 192(9–10,258), 193(9,258), 201(9), 202(316), 208(158–159,258), 218(9), 223–224(9–10), 231(402), 236(316), 244–245, 248, 251–252, 255, B:502(39), 503(39,100), 529(216,223), 537(39), 538(39,100,244,246), 550(100), 551(39), 558–559(244), 583, 585, 588–589 Lupascu, V., B:620(51), 631 Lupton, J. M., A:349(45), 356 Lusceac, S. A., A:203(344), 208(323), 214(344), 235(323,423), 236(323), 253,256 Luterova´, K., A:42(168), 121 Lutz, E., A:298(52), 322, B:67(87), 91, 312(47), 414–417(47), 435 Lutz, T. R., B:571(356), 592 Lyamshev, I. M., B:124(37), 132(37), 278 Ma, S. K., A:74–75(228), 79(228), 123 MacCulloch, M. J., B:80(104), 91 Macdonald, D. W., B:445(38), 493 Macedo, P. B., A:89(247), 124, 169(162–163), 248 Maciejewsk, A., B:5(9), 87 Madden, P. A., A:132(24), 133(24,29), 141(24), 142(29), 171(24), 244 Magill, J. H., B:519(160), 523(160,173–177), 579(160), 587 Maglione, M., A:16(92), 119, 201(296–297), 223(376), 252, 254, B:556(287), 590 Magrid, N. M., B:80(105), 91 Mahesh Kumar, M., A:102(258), 124 Mai, C., B:529(217–218), 588 Mai, J., B:481(94), 495 Maier, G., B:289(15), 434 Mainardi, F., A:298(54), 323, B:75(94), 91, 253(234), 257(234), 284, 451(61), 474(61), 488–489(112), 494, 496 Majumdar, A., B:4(7), 13(24), 87–88 Majumdar, S. N., A:346(38–39), 356

author index Makse, H. A., A:259(13), 321 Malinovsky, V. K., A:173(212), 180(236), 249–250 Malmstro¨m, E., B:614(34), 631 Mandel, L., A:275(39), 322 Mandelbrot, B. B., A:56(212), 123, 401(69), 466(69), 471, B:2(1), 3(1,3), 7(1,3), 24(41), 55(76), 87, 89–90, 97(14–15), 113(14–15), 278 Mandell, A. J., B:80(105–106), 91 Mangel, T., A:171(193), 249 Mangion, M. B. M., B:574(348), 592 Manias, E., B:564(305), 591 Mannella, R., A:405(74), 409–411(84), 444(125–126), 472–473, B:546(264), 589 Manneville, P., A:381(40), 471 Manor, N., A:60(214–215), 123 Mantegazza, F., A:11(38), 67(38), 118 Mantegna, R. N., B:481(99), 486(99), 495 Maranas, J. K., B:518(150), 548(150), 579(150), 586 Marchesoni, F., B:325(55), 436, 488(108), 496 Marcus, R. A., A:351(49–50), 356 Marczuk, K., A:38(162), 121 Maresch, G. G., A:231(403), 255 Margolin, G., A:334(26), 338(31), 340(31), 349(26,40), 350(26), 353(26), 355–356, B:443(18), 471(80), 493, 495 Margolina, A., B:132(39), 137(39), 278 Marianer, S., B:132–133(28), 144(28), 278 Marichev, O. I., B:265(217), 283, 313(48), 423–424(48), 435, 452(62), 457(62), 494 Marom, G., A:113(275–276), 115(275), 124 Marsan, D., B:46(62), 49(62), 90 Marsh, R. L., B:19(37), 89 Martin, A. J., B:289(15), 434 Martin, G. J., B:80(105), 91 Martin, J., A:349(43), 356 Martin, J. E., A:11(39), 118 Martin, S. W., A:48(185), 122, 130(7), 165(7), 172(7), 231(7), 243, B:499(1), 580(1), 582 Martinelli, M., A:219–220(369), 231(405), 254–255 Martsenyuk, M. A., B:338(65), 341(65), 436 Martz, E. O., B:209(150), 281 Marychev, O. I., B:236(206), 240(206), 283 Masciovecchio, C., A:231(409), 255 Mashimo, S., A:16(79,82), 18(79,82), 20–21(79,82), 23(79,82,123–124), 24(82,123–124), 88(246), 89(246,248),

653

106(265), 110(265), 119–120, 124, B:526(182), 587 Maslanka, S., A:201(292), 252, B:531(228), 574(345), 588, 592 Mason, P. R., A:136(43), 245 Mason, S. M., A:451(147), 474 Mason, T. G., A:316(59), 323 Massa, C. A., A:219–220(369), 231(405), 254–255 Massawe, E. S., B:331–334(62), 336(62), 436 Massot, W., A:179(229), 182(229), 223(182), 250 Masuko, M., B:506(107–109), 585 Masumoto, T., B:527(220–221), 588 Mateos, J. L., B:445–446(41), 493 Mathai, A. M., B:469(78), 495 Matheson, R. R., A:139(49), 245 Matsumoto, K., B:506(106), 585 Matsuoka, T., A:146(63), 245 Matsushita, M., B:506(105), 585 Matthews, P. C., B:546(265), 589 Mattsson, J., A:147(65), 169(65,156), 170(65), 184–185(65), 194(65), 204(65), 205(156), 207–208(65), 209(156), 216–218(65), 231(65), 245, 248, B:574(346), 592 Mauger, A., A:260(15–16,18), 269(36), 321–322 Maunuksela, J., A:460(159), 474 Maurer, F. H. J., B:224(174), 282 Maxwell, J. C., B:160(82), 163(82), 279 Maxwell-Garnett, J. C., B:142(83), 160(83), 279 May, R. J., A:184(253), 194(253), 251 Mayer, E., A:103–104(259), 124 Mayor, P., A:259(14), 321 Mazenko, G. F., A:162(131), 184(264), 247, 251 Mazo, R. M., B:253(251), 267(251), 284, 287–288(9), 388(9), 414(9), 417(9), 434 Mazur, P., A:261(21), 321 Mazza, A., A:368(19), 371(19), 373(19), 470 Mazzacurati, V., A:143(58), 245 McBrierty, V. J., A:148(71), 245 McCarthy, D. J., B:289(14), 434, 575(360), 592 McClintock, P. V. E., B:474–476(89), 495 McConnell, J. R., B:305(41), 312(41), 380–381(41), 388(41), 435 McCrum, N. G., A:201(293), 252, B:529(214), 588 McDonald, I. R., A:132–134(26), 158–159(26), 244 McGoldnick, S. G., B:260(244), 271(244–246), 284

654

author index

McKenna, G. B., A:48(185), 122, 130(7), 165(7), 172(7), 231(7), 243, B:499(1), 526(181), 527(195), 556(289), 566(311,316–317), 567(317), 575(350-351), 580(1), 582, 587, 590–592 McKinney, S., B:17(30), 88 McLeish, T. C. B., B:569(334), 591 McMahon, R. J., A:171(197), 249 McMahon, R. M., B:523(178), 587 McMillan, P. F., A:48(185), 122, 130(7), 165(7), 172(7), 231(7), 243 B:499(1), 580(1), 582 Meakin, P., B:54(75), 90, 96(6), 97(10), 186(6), 277 Medick, P., A:130(15), 148(15), 154(94), 188–191(275), 193(275), 195–200(275), 201(15,306), 203(306,344), 208(323), 209(15,306), 214(94,344), 231(306), 235(15,323), 236(15,306,323), 237–238(306), 244, 246, 251–253, B:530(227), 549–550(227), 588 Medvedev, F. A., B:97(16), 278 Meerschart, M. M., B:445(42), 493 Meier, G., A:16(90), 119, 169(161), 182(245–246), 184(161,255), 194(161,255), 207(1616), 223(377), 231(161), 248, 250–251, 254, B:513(116–118), 514(134), 531(230), 585–586, 588 Meijerink, A., A:349(46), 356 Meir, Y., B:135(48), 278 Me´lin, R., A:276(41), 278(41), 322 Menconi, G., A:451(146), 474 Mendoz-Suarez, G., B:580(380), 593 Menon, N., A:184(261), 185(266), 190(279), 197(266), 251 Merikoski, J., A:460(159), 474 Mermet, A., A:216(362), 254 Merzouki, A., A:21(107), 120 Meseguer, J. M., B:575(359), 592 Messin, G., A: 328(5), 331(5), 336(5), 342(33), 344(5), 351(5), 353(5), 354, 356, 425(98), 432–433(98), 456(98), 472 Metrat, G., A:40(173), 122 Metus, J., B:17(31), 88 Metzler, R., A:10(31), 11(31,33), 12(31), 75–76(31), 106–108(31), 117, 383(41,43), 396(64), 463(43), 466(43,168), 471,474, B:41(54–55), 61–62(83), 76(97), 89–91, 235–236(194), 237(194,209–210,212), 253(226–227,235–237), 257(226,235–237), 264(237), 265(226), 283–284, 287(7),

291–292(7), 296(7), 297–299(7,31), 300(37), 302(37), 303(7,37), 304(31), 305(7,31), 307(7,31), 308–310(31), 313–314(7), 325(7,31), 326(7,57), 327(31), 335(7), 365(7,31,85–86), 366(7,85–86), 374(89), 384(7), 394(37), 415(89), 422(89), 434–437, 443(18,20,24-25,28), 445(20), 446(46,48), 447(50,52), 448(57–58), 450(60), 453(60,63–64), 456(57,60,64), 458(64), 468(77), 471(46,80–81), 474(85–86), 479(20,46), 481(46), 483(20,46,60,64), 484(64), 486(20,46), 493–495 Meunier, J., A:303(56), 307(56), 323 Mews, A., A:328(1), 354 Mezard, M., A:156(120–121), 247 Me´zard, M., A:276(42), 322 Mezei, F., A:145(60), 182(241), 215(241,353), 245, 250, 253 Micic, O. I., A:349(42), 356 Mijangos, C., B:548(271), 590 Mijovic, J., B:569(335), 575(372), 591, 593 Mikkelsen, K. V., A:52(205), 122 Mikulecky, D. C., A:451(149), 474 Milgotin, B., A:16(86), 18–25(86), 119 Miller, E., A:21(108), 120 Miller, K., A:75(232), 123 Miller, K. S., 91 Minakov, A., B:624(54), 626(54), 632 Minoguchi, A., A:201(288), 251, B:536(235), 573(235), 589 Miramontes, O., B:445–446(41), 493 Mirollo, R., B:546(265), 589 Mishima, M., B:4(7), 13(24), 87–88 Mishima, O., A:48(182), 122 Misra, B., A:364(13), 470 Mistus, J., B:4(6), 13(25), 65(25), 79–79(25), 84(25), 87–88 Mitchell, S. L., B:18(35), 88 Mitescu, C. D., B:132–133(31), 278 Mitropolsky, Y. A., B:481–482(101), 495 Mittal, R., A:85(243), 124 Miura, N., A:21(110), 24(110), 88(246), 89(246,248), 120, 124 Miyamoto, Y., B:596(4), 618(43), 630–631 Mizukami, M., B:563(304), 591 Moeller, R. P., A:89(247), 124 Mohanty, U., A:241(427), 256 Mo¨ller, F., A:10–11(25), 38(163), 41(25), 42(163), 63(163), 117, 121 Monaco, G., A:173(209), 231(409), 249, 255

author index Monkenbusch, M., A:201(302–304), 202(303), 236(303), 252, B:518(144), 579(144), 586 Monnerie, L., B:527(194), 574(194), 587 Monserrat, S., B:575(359–360,373), 592–593 Montagnini, A., A:391(52), 409(52), 471 Montroll, E. W., A:76(236), 123, 366(17), 376(37), 387(46), 409(83), 470–472, B:48–49(64), 73(64), 90, 235(183,190), 236(183), 253(231), 257(231), 282–284, 291–292(17), 295(17,29), 299(36), 306(17), 364(17), 435, 443(21,23), 467(71), 479(71), 493–494 Moon, R. E., B:5(14), 15–17(14), 87 Mopsic, F. I., A:18(99), 119, 137(46), 245 Morabito, G., A:439(112), 445–446(112), 473 Moreno, K. J., B:580(380), 593 Moretti, D., B:488–489(112), 496 Mori, H., A:106(266–267), 124 Morineau, D., B:532(231–232), 553(231–232), 589 Morita, A., B:338(66), 388(91), 436–437 Morito, N., B:529(219), 558 Moss, F., B:474–476(89), 495 Mossa, S., A:156(116–117), 165(116–117), 247 Mott, J. F., A:9(21), 117 Mouray, T., B:614(32), 631 Moynihan, C. T., A:89(247), 124, B:580(378), 593 Mpoukouvalas, K., B:507(111), 585 Mroczkowski, T. J., B:474–476(89), 495 Mukamel, S., A:147(68), 245 Mu¨llen, K., A:334(19), 355 Mu¨ller, J., A:349(45), 356 Munch, J. P., B:518(158), 547(158), 586 Munch, J.-P., A:303(55), 307(55), 323 Munk, T., B:466(67), 494 Munteanu, E.-L., B:443(19), 493 Murat, M., B:132–133(28), 144(28), 278 Murdock, M. D., A:24(128), 120 Murphy, E. J., B:445(39), 493 Murray, C. A., B:596(2), 630 Murthy, S. S. N., 201(291,307), 252 Mutch, W. A. C., B:17(29), 85–86(107), 88, 92 Muzy, J. F., 46(59), 89 Myers, G. A., B:80(105), 91 Myllis, M., A:460(159), 474 Nabet, B., A:29(136), 120 Nadler, W., A:332(13–14), 334(13), 355 Nagatani, T., B:137(55), 279

655

Nagel, S. R., A:130(4), 165(140,143–144), 184(140,143,261), 185(166), 190(140,143,279), 192(143–144,279), 197(166), 202(312–313), 236(312–323), 243, 247, 251–252 Naito, S., A:23–24(124), 120 Nakajima, A., B:627(57), 632 Nakamura, N., A:234(414), 255 Nakayama, T., B:506(108), 585 Naoi, S., A:91(249), 124 Naoki, M., B:506(105–106), 585 Napolitano, A., A:169(162), 248 Narahari Achar, B. N., B:274(243), 284 Nashitsume, N., B:327(60), 399(60), 436 Nealey, P. F., B:613(42), 617–618(42), 631 Ne´el, L., B:289(12), 394(12), 434 Neelakantan, A., B:518(150), 548(150), 579(150), 586 Negami, S., A:9(15), 117, B:286(5), 294(5), 434 Nelson, K. A., A:182(248–250), 216(248–250,356), 241(430), 250, 254, 256 Nelson, P., A:130(3), 221(3), 243 Nemilov, S. V., A:170(173), 248 Nenashev, B. G., A:173(212), 249 Nesbitt, D. J., A:331(9–10), 349(41–42), 354, 356 Neubauer, B., B:80(101), 91 Neuhauser, R. G., A:328(7), 331(7), 353(7), 354, 358(3–4), 432(4), 470 Neumeister, S., B:527(199), 588 Ngai, K. L., A:13(57), 48(184–185,187), 51(187), 118, 122, 130(7–8), 165(7), 169(152), 172(7), 173(214), 178(224–225), 185(257–258), 188(224–225), 191–193(258), 200(152), 201(152,258), 202(152,314), 205(152,329–330), 208(152,258), 209(152), 216(364), 231(7), 233(224–225), 234(8, 224–225), 243, 248–254, B:235–236(184), 244(184), 282, 499(1–2), 500(2), 501(21–26), 502(32–41,44–46,48–49,50), 503(38–40,44,76,78–80,99), 504(40), 505(38,76,78–79), 508(44), 516(2,22,45), 517(36,46,136–138,140), 518(24,142,155,157,160), 519(161–164), 522(172), 523(160), 525(172), 526(172,281–182,187), 527(189–192,200–208,211), 528(35–36,212), 529(35,38), 530(32,36,38), 531(38,44,76,229), 532(44,229), 536(239),

656

author index

Ngai, K. L. (continued) 537(39–40,239–240), 538(39,240), 543(140), 546(256–262), 547(24,155,157,260–262,270), 548(24,26,38,41,240,275–279), 549(38), 550(36), 551(32–34,36,39–41,240), 553(44,76), 556(34–35,240), 560(25,191,202,205–206,208,295), 561(207), 562(298,302–303), 564(302–303), 565(25,189,202,206–207,295,307–308), 566(303,310,315), 567(318–324), 568(172,302,325–326), 569(329), 571(329,337), 572(339,342), 573(136,342), 574(32,36,278,345), 575(201), 579(48,160,162), 580(1–2,22,25–26,48,155, 157,162,276,377–379), 582–593 Nicolai, T., A:179(232), 187(232), 250 Nienhaus, G. U., A:349(47), 356 Nierwetberg, J., A:396(63), 471 Nigmatullin, R. R., A:10(22–23), 12(22,47), 35(47), 54(209), 56(47), 61(216), 72(47), 106–107(22–23), 117–118, 123, B:235(200–204), 236–237(200–203), 240(200–203), 283, 293(28), 299(28), 312–315(28), 324(28), 435 Nikkuni, T., B:506(108), 585 Nilgens, H., A:141–142(54), 245 Niquet, G., A:201(296), 252, B:556(287), 590 Nir, I., A:32(143,146), 33(143,147), 34(143), 40(143), 63(143), 121 Nivanen, L., A:10(23), 106–107(23), 117 Nogales, A., B:502(43), 554(43), 583 Noh, T. W., B:174–175(98), 280 Nomura, H., A:146(63), 245 Nonnenmacher, T. F., A:11(35), 118, 383(41), 396(64), 471, B:41(55), 61–62(83), 89–90, 235–236(194), 237(194,212), 253(235), 257(235), 283–284, B:291(16), 435 469(79), 495 Noolandi, H., A:332(11,16), 355 Noreland, E., A:16(85), 18(85), 20–24(85), 119 Nori, F., A:259(14), 321 Novikov, V. N., A:52(203), 62(203), 122, 145(59), 164(136,138), 169(168–169), 170(169,183), 172(208), 173(213), 175(13 6,183), 177(136,138,168), 179(136,183,230), 180(236), 181–182(183), 183(136), 188–189(136,230), 190(272), 191(230), 194(230), 197(168–169), 201(272), 202–203(230), 205(230), 208(335), 209(230), 216(183), 222(183), 223(136,230),

225(395–397), 230(183,401,408,411), 232(183), 245, 247–251, 253, 255, 335(28), 355, B:125(24), 136(24), 138(24), 149(24), 160(49), 167(79), 170(49), 209(160), 212(163), 221(170), 231(160), 235–236(196), 237(196,215), 278–279, 282–283, B:292–293(23), 299(23), 305(23), 324(23), 400(23), 435 Nozaki, R., A:16(84), 18(84), 20(84), 21(84,109), 23–24(84,115), 119–120, 201(288), 251, B:536(235), 573(235), 574(345), 589, 592 Nozik, A. J., A:349(42), 356 Nuriel, H., A:113(275), 115(275), 124 Nuyken, O., B:614(33), 631 Nyden, M., A:115(279), 125 Nyrstro¨m, B., B:518(156), 547(156), 586 O’Reilly, J. M., B:567(324), 591 Ocio, M., A:259(10–11), 321 Odagaki, T., A:63(219), 123 Oddershede, L., B:443(19), 493 Odelevski, V. I, B:163(86), 280 Oguni, M., A:201(298), 252, B:563(304), 591 Ohlemacher, A., A:201(308), 252 Oka, A., A:23–24(115), 120 Okabe, H., B:506(107–108), 585 Okumura, H., B:527(220–221), 588 Oldekop, W., A:172(204), 249 Oldham, K., A:75(231), 123, B:236(205), 240(205), 283, 298(32), 305(32), 318(32), 435 Olemskoi, A. I., B:124(36), 132(36), 278 Oliveria, M., B:97(8), 186–187(8), 277 Olsen, N. B., A:170(174–175), 186(278), 190(278), 205(326–327), 241(428), 248, 251, 253, 256, B:363(83), 400(83), 436, 538(245,247), 540(248), 589 Ondar, M. A., A:218(366), 254 Onsager, L., A:429(101), 472 Oppenheim, I., A:241(427), 256 Oranskii, L. G., A:218(366), 254 Orbach, R., A:10(24), 117 Ornstein, L. S., B:253(247), 284, 370(87), 373(87), 437 Orrit, M., A:328(4), 331(4), 333(4), 335(4), 339(32), 340(4), 354 Osler, T. J., B:237(214), 283 Oswald, J., A:42(168), 121 Ota, T., A:16(82), 18(82), 20–21(82), 23–24(82), 119 Ott, E., B:54(68), 90

author index Øye, G., A:38(159), 42(159), 63–64(159), 121 Ozaki, T., B:24(42), 89 Pablo, J. J., B:613(42), 617–618(42), 631 Packer, K. J., A:148(71), 245 Padro´, J. A., A:84(241), 123 Pagnini, G., B:451(61), 474(61), 488–489(112), 494, 496 Paiu, J. M., B:224(172), 282 Pakula, T., A:169(160), 207(160), 231(160), 248 Pala, M. G., 448(139), 450–451(139), 453(155), 467–468(155), 473–474 Palade, L. I., B:527(196), 588 Palatella, L., A:337(30), 355, 415(93), 419(93), 424(96), 425–426(100), 429(96,102), 430(96), 431(96,102), 432(105), 433(96), 434(105), 448(139), 450(139), 451(139,146), 453(155), 454(96), 463–464(165), 466(100,105), 467(96,105,155), 468(93,100,155), 472–474 Palimec, L. F., B:224(172), 282 Palmieri, L., A:169(160), 207(160), 231(160), 248, B:526(184), 587 Paluch, M., A:52(202), 122, 169(152,154–155), 200(152,282), 201(152,292), 202(152,282,319–320,322), 205(152,154–155,329), 208(152,320,339), 209(152,154–155), 210(154–155), 236(154–155), 248, 251–253, B:499(11–12), 502(38,40,44), 503(12,38,40,44,62, 71–79,82–86,90–95,97–99), 504(12,40,72), 505(38,72,75–79,82–86), 507(90–95), 508(44,113), 510(62), 512(62), 514(129–133,135), 516–517(98), 519(165–166), 529–530(38), 531(38,44,75–77,228–229), 532(44,229), 536(239), 537(40,239), 540(249), 548–549(38), 551(40), 553(44,76–77), 565(94), 574(345), 582–589, 592 Panagiotopoulos, P. D., B:209(154), 282 Panina, L. V., B:347(77), 436 Paolucci, D. M., A:182(250), 216(250), 250 Papadopoulos, P., B:503(70), 584 Paptheodorou, G., A:173(211), 249 Paradisi, P., A:425(99), 456(157), 466(99), 472, 474, B:488–489(112), 496 Pardi, L. A., A:219–220(369), 231(405), 254–255 Parisi, G., A:156(120–121), 247, 269(35), 276(40), 278(40), 322 B:119–120(22), 122(22), 144(22), 278

657

Park, J. B., B:209(150), 281 Parker, T. J., A:16(93), 119 Parthun, M. G., B:575(355–358,362), 592 Pascazio, S., A:373(28), 470 Pathmanathan, K., B:529(215), 588 Pathria, P. K., A:74–76(226), 79(226), 91(226), 95(226), 110(226), 123 Patkowski, A., A:141–142(54), 145(60), 169(160–161), 179(229), 182(229,245–246), 184(161), 194(161), 207(160–161), 223(229,377), 231(160–161), 245, 248, 250, 254, B:513(116–118), 514(129–135), 526(187), 562(297), 585–587, 590 Patterson, G. D., A:208(337), 253, B:513(119), 527(210), 574(210), 585, 588 Pattnaik, R., A:215(354), 253 Paul, W., B:292–293(19), 297–299(19), 303(19), 305(19), 435 Pauli, W., A:362(6), 470 Pawelzik, U., A:201(308–309), 252 Pawlus, S., A:202(322), 205(329), 253, B:503(72,76–77,79,90–91,95), 504(72), 505(76–77,79), 507(76,90–91,95), 531(76–77), 536–537(239), 553(76–77), 584–585, 589 Paz, J. P., A:364(11), 439(116), 442(120), 470, 473 Pearle, P., A:439(115), 446(115), 449(115), 473 Pearson R. B., B:135(41), 278 Pecora, R., A:132–133(25), 140(25), 141–142(54), 171(25), 184(25), 244–245 Peigen, H.-O., B:101(21), 110(21), 113(21), 144(21), 278 Pelant, I., A:42(168), 121 Peliti, L., A:259(5), 305(5), 307(5), 309–312(5), 321 Pelletier, J. M., B:529(222), 588 Pelligrini, Y.-P., B:188(128), 281 Pelous, J., A:231(407), 255 Peng, C. K., B:4(6), 5(10,12), 13(25), 17(31), 18(35), 19(10), 65(25), 79–79(25), 84(25), 87–88 Penzel, T., B:17(32), 88 Percival, I. C., A:446(134), 473 Perez, J., A:201(297), 252, B:527(194), 529(217–218), 549(282), 556(282,286), 574(194), 587–588, 590 Peristeraki, D., B:503(70), 584 Perrin, J., B:53(67), 90, 440(5), 492 Perry, C. H., A;40(175), 122

658

author index

Peseckis, F., B:446(56), 494 Pesin, Y. B., A:447(136), 473 Pethrick, R. A., B:575(368), 592 Petong, P., A:23(114), 120 Petrenko, F., A:51(199), 122 Petrie, S. E. B., B:574(347), 592 Petrosky, T., A:448–449(138), 473 Petrov, Y. I., B:183(113), 280 Petrucci, S., A:106(264), 124 Petry, W., A:167(147), 171–172(201), 177(147), 180(147,233,235), 182(147,242), 215(201,242), 216(147,233), 224(387), 248–250, 255, B:513(123), 586 Pettini, M., B:546(263), 589 Petychakis, L., B:607(17), 631 Pﬁster, G., B:443(17), 493 Phalippou, J., B:97(12), 277 Philips, P., A:374(33), 470 Philips, W. A., B:361(82), 436 Phillies, G. D. J., B:518(154–155), 547(154–155), 580(155), 586 Phillips, D. W., B:527(198), 588 Phillips, J. C., B:235–236(193), 283 Phillips, W. A., A:234(416), 235(421), 255–256 Piazza, F., B:546(268), 590 Piazza, R., A:11(38), 67(38), 118 Pick, R. M., A;141–142(53), 147(69), 177(221), 179(229,231), 182(69), 183(231), 193(231), 224(69,388–389), 230(231), 245, 250, 255 Pietronero, L., B:170(89), 280 Pimenov, A., A:48(188), 51(188), 122, 140(52), 245 Pines, D., A:469(170), 474 Pionteck, J., B:499(9), 582 Pischorn, U., A:209(341), 212(341), 253 Pisignano, D., A:177(219), 201(287), 202(317), 250–252, B:556(291), 575(291), 590 Pitaevski, L. P., B:174(90), 209(141), 215(141), 280–281 Plazek, D. J., A:13(57), 48(184), 118, 122, 170(182), 173(214), 248–249, B:501(25), 502(45), 516(45), 517(137), 518(160), 523(160,174–177), 526(181), 527(189–193,199,203–206), 560(25,191,205–206,295), 565(25,189,193,206,295,307–308), 575(352), 579(160), 580(25), 583, 586–588,590–592 Podeszwa, R., A:85(242), 123

Podlubny, I., B:236(207), 240(207), 283, B:488–489(110), 496 Pohl, R. O., A:235(419), 255 Pollard, R. D., A:24(130), 120 Pollock, E. L., A:142(57), 245 Polygalov, E., A:16(86), 18(86), 19(86,103), 20(86), 21(86,103), 22(86), 23(86,103), 24–25(86), 119 Pomeau, Y., A:381(39), 471 Potapova, I., A:328(1), 354 Pottel, R., A:16(88), 52(204), 119, 122 Pottier, N., A:260(15–19), 261(25), 266(25), 236(36), 286–287(25), 297–299(51), 321–322 Pound, R. V., A:150(82), 246 Power, G., B:324(52), 436, 540(250–252), 589 Prall, D., B:209(151), 281 Prassas, M., B:97(12), 277 Pratesi, G., A:182(247), 223(247), 250 Prevosto, D., A:177(219), 201(287), 224(392), 250–251, 355, B:499(14,16), 503(79–81), 505(79–81), 530(80), 537–538(240), 548(240), 551(80,240), 556(80,240), 573(343), 582–584, 589, 592 Preziosi, L., B:257(240), 274(240), 284 Price, A. H., A:6–9(8), 16(8), 117 Prigogine, I., A:448–449(138), 473 Prince, P. A., B:445(39), 493 Privalko, V. P., B:160(49), 167(79), 170(49), 235–237(196), 279, 283, 292–293(23), 299(23), 305(23), 324(23), 400(23), 435, 502(42), 583 Procaccia, I., B:119–120(23), 122(23), 144(23), 278 Prokhorov, A. M., A:140(51), 245 Prudnikov, A. P., B:265(217), 283, 313(48), 423–424(48), 435 Psurek, T., B:503(83–86,95), 505(83–86), 507(95), 574(345), 585, 592 Pugachev, A. M., A:173(212,216), 174(216), 249 Puglia, D., B:575(375), 593 Puquet, A., A:415(92), 472 Purcell, E. M., A:150(82), 246 Pusey, P. N., B:518(152), 547(152), 586 Putselyk, S., A:139(50), 163(50), 232–235(50), 245 Puzenko, A., A:3(2), 6(9), 14(78), 21(2), 26–27(132), 30(132), 33(149), 36–37(2), 38(2,153–154), 40(153–154), 41(2,153), 42(153–154), 43(2), 47–48(78), 52–54(208), 55–56(154), 58(153–154), 59(2), 61(2,154),

author index 62(154), 63(2,153–154), 64(154), 65(2), 73(2), 78(240), 82–85(208), 86(244), 89(244), 90(208,244), 91(244), 96–97(78), 98(78,153), 99–100(153), 103(78,153), 104–105(78), 108(2), 111(2), 114(2), 116–117, 119–121, 123–124 Qi, F., A:188(331), 205(331), 209(331), 213(331), 236(424), 253, 256 Qiu, X. H., A:214(352), 253, B:527(206), 560(206), 565(206), 588 Qui, F., B:550(284), 590 Quinlan, C. A., B:518(154), 547(154), 586 Quinn, K. P., B:271(245–246), 284 Quitmann, D., A:165(145–146), 173(146), 180(237), 182(145–146), 215(146), 223(217), 224(386), 247–250, 254 Quittet, A. M., A:44(171), 121 Qzhang, J., B:174–175(100), 280 Rabotnov, Y. N., B:54(73), 90 Rademacher, C. M., B:527(209), 560(209), 588 Rademann, K., A:60(214–215), 123 Radloff, U., A:201(309), 252 Rafaelli, G., A:451(146), 463–464(165), 474 Rahman, M., A:401(68), 471 Raicu, V., B:174–175(104), 280 Raikher, Y. L., B:338(65), 341(65), 436 Rajagopal, A. K., B:568(325–326), 591 Rajagopalon, B., B:68(88), 91 Rammal, R., B:129(25), 233(25), 278, 284 Ramos-Fernandez, G., B:445–446(41), 493 Ramsay, D. J., A:208(337), 253 Randall, M., B:558(293), 590 Raposo, M. P., B:445(40), 493 Raskovich, E. Y., A:23–24(121), 120 Read, B. E., A:201(293), 252, B:529(214), 588 Redner, S., B:467–468(70), 471(70), 472(70), 494 Reichl, L. E., B:80(100), 91 Reid, C. J., B:397(93), 398(95), 412(98), 426(93), 437 Reinecke, H., B:548(271), 590 Reinsberg, S. A., A:214(352), 253, B:526(188), 550(188), 587 Reiser, A., A:201–202(290), 252, B:503(68), 584 Reisfeld, R., A:38(158), 42(158), 60(214–215), 63–64(158), 121, 123 Reiter, G., B:566(312), 591 Reitsma, F., A:451(148), 474 Ren, S. Z., B:518(153–154), 547(153–154), 586

659

Rendell, R. W., A:48(184,187), 51(187), 122, B:501(24–25), 518(24), 547(24,270), 548(24), 560(25), 565(25), 580(25), 583, 590 Resinger, T., B:503(63,65–66), 584 Reuther, E., A:334(19), 355 Reynolds, P. J., B:132–133(33), 140(33), 278 Rhodes, C. J., B:445(38), 493 Riccardi, J., A:389–392(50), 398(50), 413(87), 415(87), 471–472 Ricci, M., A:167(150), 177(219), 182(247), 223(247), 224(150,388–389,392), 248, 250, 255 Richards, R. T., B:14(28), 88 Richardson, C., B:518(154), 547(154), 586 Richardson, L. F., B:57(79), 90 Richert, R., A:13(58), 14(77), 18(100–102), 48(197), 118–119, 122, 137(47), 169–170(170–172), 185(268), 190(277), 196(170–171), 200(171), 201(277,286), 202(277,310–311), 204(277), 207(334), 208(310–311), 214(351), 231–232(412), 236(277), 240(286), 245, 248, 251–253, 255, B:501(29), 502(47), 518(159), 525(180), 529(29), 536(233), 540(233), 550(180,283), 573(344), 578(29), 579(159), 583, 587, 589–590, 592 Richter, D., A:182(243), 201(299,301–304), 202(303), 215(243), 236(303), 250, 252, B:518(143–145,147–148), 527(195), 536(237), 548(143,272), 579(144-145,147–148), 586–587, 589–590 Riegger, T., A:169(159), 208(159), 248, B:538(246), 589 Rigamonti, A., A:89(256), 102(256), 124 Righini, R., A:182(247), 223(247), 224(393), 250, 255 Rimini, A., A:439(115), 446(115), 449(115), 473 Risken, H., B:271(244–246), 283, 313(49), 398(94), 418(49), 435, 437, B:441–442(12), 492 Ritchie, D. S., B:135(47), 278 Ritort, F., A:259–260(4), 276(4,43,45), 321–322 Rivera, A., A:139(50), 163(50), 178(223), 232(50), 233(50,223), 234–235(50), 245, 250, B:548(276), 580(276), 590 Rizos, A. K., B:527(192,211), 565(307), 587–588, 591 Roberts, P. L., B:271(245), 284

660

author index

Robertson, C. G., B:527(209), 548(279), 560(209), 588, 590 Robinson, J. E., A:142(56), 245 Robyr, P., A:115(280), 125 Rocard, M. Y., B:312(44)k 369(44), 385(44), 435 Rocco, A., A:372(23), 398(65), 447(23), 452(150), 469(23), 470–471, 474, B:237(213), 283 Rodrigues, S., A:142(56), 245 Roe, R. J., B:526(185–186), 587 Rogach, A. L., A:349(45), 356 Roggatz, I., A:203(344), 214(344), 253 Roland, C. M., A:178(224), 185(257), 188(224), 200(282), 201(292), 202(282,319,321–322), 205(329), 208(339), 209(321), 233–234(224), 250–253, B:499(12), 502(35,46,48–50), 503(12,62,71–75,77,84,87,89–92,94–99, 101–102), 504(12,72), 505(75,77,84), 507(87,89–92,94–99), 508(113), 510(62,114–115), 512(62), 514(129), 517(46,114–115,138–139,141), 519(141,165–166,205–207), 528(35), 529(35,101–102), 531(75,77,101–102,228), 532(101–102), 536(49,239), 537(239), 548(275,278–279), 553(77,101–102), 556(35), 560(114–115,205–206), 561(207), 565(94,96,206–207,308), 567(318–324), 569(329), 571(329,337), 574(278), 575(207), 577(114), 579–580(48), 582–592 Rolla, P. A., A:169(160), 200(431), 201(287) 202(317), 207(160), 224(392), 231(160), 248, 251–252, 255–256, B:499(10–11,14,16), 503(79–82), 505(79–82), 526(184), 537–538(240), 541(253), 548(240), 551(80,240), 556(80,240,291), 573(343), 575(291,369–371), 582–585, 587, 589–590, 592 Roman, F., B:575(373), 593 Romano, G., A:201(287), 251 Romanov, E., B:64(85), 91 Romanychev, G., A:16(86), 18–25(86), 119 Romero, A. H., A:461(162), 474 Roncaglia, R., A:373(31), 436(110), 444(31,122), 445(31), 447(31,122), 470, 473 Rønne, C., A:52(205), 112(274), 122, 124 Rosa, A., A:424(96), 425–426(100), 429–431(96), 433(96), 449(141), 454(96), 466(100), 467(96), 468(100), 472–473 Rosen, B. W., B:164(88), 280

Rosen, M., A:331(8), 333(8), 354 Rosenblum, M. G., B:69(89), 91 Rosenquist, E., B:97(9), 277 Ro¨sner, P., B:529(223), 588 Ross, B., A:75(232), 91, 123 Rossitti, S., B:70(92), 91 Ro¨ssler, E. A., A:48(194), 52(203), 62(203), 122, 130(2,6,11,15), 139(50), 145(59), 146(64–65), 148(2,11,15), 149–150(11), 151(84), 152–153(11), 154(11,93–94,96–97) 163(6,50,64), 164(6,64,136,146), 165(6,142), 166(64), 167(6), 168(6,142,153), 169(65,157,166–169), 170(65,153,169,177,181,183,185), 171(64,187,190), 173(146,216), 174(216), 175(136,183), 176(185), 177(64,136,167–168,218,220), 178(64), 179(64,136,142,183,185,230), 180(64,237), 181(183), 182(64,146,183,185), 183(64,136), 184(62,142,259–260), 185(65), 186–187(142), 188(6,136,142,230,273,275,331), 189(136,142,230,275), 190(6,142,275), 191(6,142,230,275), 192(259–260), 193(275), 194(64–65,142,230), 195(275), 196(64,275), 197(167–169,275),198–199(275), 200(6,11,275,431), 201(6,15,96–97,153,190,285,300,305–306), 202(6,142,153,157,230,285,315,317), 203(6,230,306,315,344), 204(6,65,324), 205(6,142,153,157,230,285,331), 206(6,142), 207(65,96–97), 208(65,323), 209(6,15,142,157,230,306,331,340–341), 210(177,220,324,342), 211(324), 212(324,341,346), 213(331), 214(11,94,344), 215(146), 216(64–65,183), 217–218(65), 219–220(360), 222(183), 223(136,217–218,230,370), 230(183,401), 231(6,65,96–97,305–306, 410–411), 232(6,50,183), 233(50), 234(6,50,220), 235(11,15,50,177,323,410,422), 236(6,15,153,167,285,306,315,323), 237(93,96,306), 238(93,306,429), 239(96–97), 243–256 B:537(241), 549–550(280–281), 556(291), 572(241), 575(291), 589–590 Rostig, S., B:17(32), 58–59(82), 88, 90 Roszka, S. J., B:503(98), 516–517(98), 585 Rouch, J., A:32(144–145), 121 Roux, J. N., A:156(110), 247 Ruby, S. L., A:216(361), 254

author index Rueb, C. J., B:224(173), 282 Ruelle, D., B:54(71), 90 Runt, J., B:503(89), 507(89), 585 Ruocco, G., A:143(58), 156(116–117), 165(116–117), 173(209), 231(409), 245, 247, 249, 255 Russ, S., B:443(30), 493 Russell, E. V., A:135(38–39), 244–245 Russina, M., A:145(60), 215(353), 245, 253 Ruths, T., B:562(296–297), 590 Ryabov, Y., B:235(204), 283, 293(28), 299(28), 312–315(28), 324(28), 435 Ryabov, Ya., A:3(2), 10(22), 12(22,46–47), 14(78), 21(2), 35(47), 36–41(2), 38(155), 40(155,179), 43(2), 45(179), 46(257), 47(78,179), 48(78,179,186), 49–51(186), 52(186,208), 53–54(208), 56(47), 58(155), 59(2), 61(2), 63(2,155), 65(2), 72(47), 73(2), 82–85(208), 78(240), 86(244), 89(244), 90(208,244), 91(244), 93–94(179), 96–98(78,155), 99–100(155), 101(179,257), 102(257), 103–105(78), 106–107(22), 108(2), 111(2,46), 113(276), 114(2), 116–124 Rysiakiewicz-Pasek, E., A:38(156,162), 39–42(156), 58(156), 60(156), 63–64(156), 103(156), 121 Rzoska, S. J., B:503–504(72), 584 Sa’ar, A., A:38(160–161), 4142(160–161), 63(160–161), 64(160), 121 Sack, R. A., B:305(40), 312(40), 368–369(40), 376(40), 380–381(40), 384–385(40), 394(40), 401(40), 418(40), 435 Safran, S. A., A:33(148,150), 70(150), 121 Sahimi, M., B:96–97(2–3), 147(76), 154(3), 155(78), 160(2–3), 208(135,137–138), 212(166,168–169), 277, 279, 281–282 Saint-James, D., A:261(25), 266(25), 286–287(25), 297–299(51), 321–322 Saito, S., B:502(55–56,59), 584 Saiz, L., A:84(241), 123 Sakai, A., A:177(133), 182(133), 223(133), 247 Saletan, G. E., A:392(57), 471 Saleur, H., B:144(58), 279 Salnikov, E. S., A:231(406), 255 Salomon, M., A:106(264), 124 Salvetti, G., A:16(83), 18(83), 20–21(83), 23–24(83), 119, B:575(364–367), 592 Sambrooks, J. E., B:80(104), 91

661

Samko, S. G., B:236(206), 240(206), 283, 452(62), 457(62), 494 Sampoli, M., A:156(116–117), 165(116–117), 247 Samwer, K., B:529(223), 588 Sancho, J. M., A:461(162), 474 Sander, L., B:96(5), 277 Sandercock, J. R., A:163(135), 225(135), 247 Sanders, A. W., B:4(8), 70(93), 87, 91 Sanders, J. V., B:183(114), 280 Sanny, J., A:23–24(121), 120 Sano, H., A:332(12), 355 Santamara, J., A:178(223), 233(223), 250 Santamaria, J., B:580(380), 593 Santangelo, P., B:502(50), 503(92), 507(92), 527(206), 560(206), 565(206), 584–585, 588 Sanz, A., B:502(43), 554(43), 583 Sanz, J., A:178(223), 233(223), 250 Saraidarov, T., A:38(158), 42(158), 63–64(158), 121 Sarychev, A. K., B:141(65), 174–175(107–109), 279–280 Sasabe, H., A:89(247), 124, B:502(55–56,59), 584 Sastry, S., A:223(371), 254 Sauer, J. A., A: 255 Saupe, A., B:289(15), 434 Saupe, D., B:101(21), 110(21), 113(21), 144(21), 278 Saxena, R. K., B:469(78), 495 Scafetta, N., A:451(146), 464(167), 474, B:5(13), 34(47), 47(47), 50(47,66), 51–52(66), 87, 89–90 Scaife, B. K. P., A:9(12), 11(12), 117 Scardicchio, A., A:373(28), 470 Scarpini, F., A:216(359), 254 Scha¨dler, V., A:231(404), 255 Schaefer, D., A:209(341), 212(341), 253 Scha¨fer, H., A:9(14), 26(14), 117 Scheidler, P., B:568(327), 591 Scheidsteger, T., A:156(107), 246 Scher, H., A:76(237–238), 123, B:235–236(182–183), 282 443(17–18,22–23), 471(80), 493, 495 Scherer, N. F., A:147(68), 245 Schertzer, D., B:46(62), 49(62), 90 Schick, C., B:562(300–301), 591, 624(54), 626(54), 632 Schick, W., B:235–236(194), 237(194), 283 Schiener, B., B:501(29), 529(29), 578(29), 583

662

author index

Schiessel, H., B:237(212), 283 Schillcock, J. C., B:209(148), 281 Schilling, R., A:130(18,20), 132(23,30), 156(18,20,23,107), 157(20), 165–166(30), 244, 246 Schirmacher, W., A:10–11(25), 41(25), 117 Schlather, M., B:34(48), 89 Schlegel, G., A:328(1), 354 Schlesinger, M. F., B:235(190), 283 Schlipf, S., B:445(44), 493 Schlosser, E., A:16(87), 48(193), 106(87), 119, 122, B:527(202–203), 560(202), 565(202), 588 Schmaljohann, D., B:614(33), 631 Schmidt, A., B:474(88), 495 Schmidt-Rohr, K., A:148(72–73), 150(72), 152(172), 154–155(72), 201(308–309), 209(72), 214(72), 246, 252, B:501(27), 529(224), 578(27), 583, 588 Schmitt, F., B:46(62), 49(62), 90 Schnauss, W., A:151(83), 170(178), 171–172(201), 184(262), 185(258), 191–193(258), 201(258), 208(258), 210(178), 215(201), 235(178), 246, 248–249, 251 Schneider, G., B:530(227), 549–550(227), 588 Schneider, U., A:52(201), 122, 130(9), 135(9), 140(52), 163–165(9), 169(158), 175(9), 177–178(9), 180(9), 182–184(9), 191–193(9), 201(9,306), 203(306), 208(158), 209(306), 218(9), 223–224(9), 231(306,402), 235(422), 236–238(306), 244–245, 248, 252, 255–256, B:502(39), 503(39,100), 537(39), 538(39,100,244), 550(100), 551(39), 558–559(244), 583, 585, 589 Schneider, W. R., A:10(24), 117, B:250(224), 253(224), 284 Schober, H., A:145(60), 245, B:548(272), 590 Schoﬁeld, A., B:529(213), 530(213), 550(213), 578(213), 588 Scho¨nhals, A., A:3(3), 16–17(3), 18(3,87), 48(3,193), 51(3), 99(252–253), 106(87), 117, 119, 122, 124, 134(37), 137–138(37), 165(141), 185(141), 204(325), 244, 247, 253, B:324(53), 436, 527(202–203), 560(202), 565(202,300–301, 309), 588, 591, 596(1), 616(1), 623–624(1), 630 Schramm, P., A:297–298(49–50), 322 Schroder, T. B., B:174–175(99), 280 Schrøder, T. B., A:9(20), 11(20), 117 Schro¨dinger, E., B:3(4), 87, 467(68), 494

Schroeder, M. J., B:502(35), 528–529(35), 556(35), 583 Schroers, J., A:218(365), 254 Schro¨ter, K., A:190(274), 200(276), 201(274), 204(325), 251, 253, B:556(290), 572(338), 575(290), 590, 592 Schulte, A., B:513(123), 586 Schultze, H., B:569(328), 591 Schumer, R., B:445(42), 493 Schuster, F. L., B:445(37), 493 Schuster, H. G., B:101(20), 110(20), 114(20), 144(20), 278 Schu¨tz, G. M., A:276(48), 322 Schwan, H. P., A:16(94), 18(94), 119 Schweiger, A., A:218(368), 254 Schwettman, A., A:374(36), 470 Sciortino, F., A:32(145), 120, 132(23), 156(23,115), 165(115), 244, 247 Scopigno, T., A:173(209), 249 Sebzda, T., B:5(9), 87 Sedita, J. S., B:567(324), 591 Sedlacik, R., A:42(168), 121 Segal, I., A:45(181), 122 Segre, P. N., B:518(152), 547(152), 586 Seifert, U., B:209(148), 281 Sekula, M., A:202(322), 253, B:503(76–78,92), 505(76–78), 507(92), 531(76–77,229), 532(229), 553(76–77), 584–585, 588 Sen, P. N., B;212(164), 282 Senatra, D., A:182(247), 223(247), 250 Senitzky, I. R., A:261(20), 321 Senker, J., A:177(220), 210(220), 212(346), 234(220), 250, 253 Serghei, A., B:596(5–6), 608(19), 614(31), 617(31), 652(5), 630–631 Serra, R., A:452(151), 474 Servoin, J. L., A:40(173), 122 Seshadri, V., A:392(58), 411(86), 412(58), 471–472, B:75(95), 76(95–96), 77–78(95), 91, 449(59), 485(59), 494 Sette, F., A:173(209), 231(409), 249, 255 Seyfried, M. S., A:24(128), 120 Sgardelis, S. P., B:486(106), 495 Shacklette, J. M., B:209(155), 282 Shahin, Md., 201(291), 252 Shalaev, V. M., B:174–175(105,1081–09), 280 Shannon, R. D., A:93(250), 124 Shappir, J., A:38(160–161), 4142(160–161), 63(160–161), 64(160), 121 Shaw, C. D., B:54(69), 90

author index Shen, G. Q., A:130(5), 145(61), 163(5), 177(5), 179(5,227,231), 180(227), 182(5), 183(231), 193(231,280–281), 223(373,375–376), 224(384), 230(231), 243, 245, 250–251, 254, B:513(122), 586 Sheppard, R. J., A:17(96), 119 Sherlock, B. G., A:30(137), 120 Shermergor, T. D., B:204–205(31), 216–220(31), 230(31), 232(31), 281 Shi, W. F., B:518(153), 547(153), 586 Shibata, T., A:146(63), 245 Shimizu, K. T., A:328(7), 331(7), 334(23), 353(7), 354, 358(3–4), 364(14), 432(4), 470 Shimizu, T., B:627(57), 632 Shimomura, M., A:48(190), 51(190), 55(190), 86(190), 88(190), 122, 201(289), 252 Shinyashiki, N., A:12(46), 16(82), 18(82), 20(82), 21(82,110), 23(82), 24(82,110), 48(190), 51(190), 55(190), 86(190), 88(190,246), 89(246,248), 106(265), 110(265), 111(46), 118–120, 122, 124, 201(289), 252 Shiokawa, K., A:121(442), 473 Shiotsubo, M., A:21(110), 24(110), 120 Shiozaki, Y., A:23–24(115), 120 Shiryaev, A. N., B:257(239), 262(239), 284 Shklovskii, B. I, B:141(64,66–67), 187(67), 194(67), 279 Shlesinger, M. F., A:3(1), 14(71–73), 54(211), 116, 118, 123, 376(37), 387(46), 388(48), 409(83), 452(152), 470–472, 474, B:38(50), 49(65), 54(72), 58(80), 70(80), 89–90, 295(29), 299(36), 435, 445(35–36), 446(45,47), 471(45), 481(45,98), 488(36), 493–495 Shliomis, M. I., B:338(65), 341(65), 436 Shuler, K. E., A:378(38), 460(38), 470 Shuster, R. H., B:224(171), 282 Sibbald, C. L., A:24(129), 120 Sidebottom, D. L., A:177(222), 208(338), 233(222), 250, 253 B:514(127), 586 Sidorenko, A., B:613(28), 631 Sigmund, O., B:209(152–153), 282 Silbey, R. J., A:334(27), 346(37), 355–356 Silbey, R. S., B:297(30), 372–374(30), 379–380(30), 394(30), 398(30), 400(30), 413(30), 416(30), 421(30), 435, 446(49), 494 Sillescu, H., A:48(196), 122, 130(2), 132(28), 148(2), 150(349–350), 151(83,85), 153(87),

663

169(154–155), 170(177–178), 171(186,188–189,191–194,198–199,201), 172(201,203), 180(233,235), 188(331), 205(154–155,331), 209(154–155,331,340–341), 210(154–155,177–178,342,345), 212(87,341), 213(87,189,331,347), 214(349–350), 215(201), 216(233), 224(387), 235(177–178), 236(154–155,345,424), 243–244, 246, 248–250, 253, 255–256, B:501(29), 522(167), 526(183), 529(29), 530(22), 549(226), 550(226,284), 578(29), 583, 587–588, 590 Simmons, J. H., A:169(163), 248 Simon, G., B:614(35), 631 Singh, A. P., A:184(265), 223–224(374), 225(265,400), 251, 254–255 Sixou, B., B:556(288), 590 Sjo¨blom, J., A:16(85), 18(85), 19(113), 20–22(85). 23(85,113,119–120), 24(85,113,119,127), 31(141), 32(113), 38(159), 42(159), 63–64(159), 119–121 Sjo¨gren, L., A:14(74), 16(82), 18(82), 20–21(82), 23–24(82) 48(74), 119, 130(17), 156(17), 159(17), 162(17,132), 179(17), 182(17), 244, 247, B:235–236(191), 283 Sjo¨lander, A., A:14(75), 119, 156(104), 246 Skal, A. S., B:141(64), 279 Skodvin, T., A:19(113), 23(113,120), 24(113,127), 32(113), 120 Smith, G. D., B:548(272), 590 Smith, R. S., A:103–104(260), 124 Smith, S. W., B:513(120), 586 Smith, V. L., B:253(232), 257(232), 284 Smoluchowski, M., A:364(9), 470 Snaar, J. E. M., A:115(280), 125 Sokolov, A. P., A:48(194), 122, 145(59–60), 164(138), 165(145–146), 169(167–168), 173(146,208,213), 177(138,167–168,218), 180(236–237), 182(145–146), 190(272), 197(167–168), 201(272), 208(335), 215(146), 223(217–218), 224(386) 231(408), 236(197), 245, 247–251, 253–255, 432(104), 467(104), 472 Sokolov, I. M., A:106(268), 124, 461(162), 466(168), 474, B:55(78), 76(78), 90, 157(81), 279, 374(89), 415(89), 422(89), 437, 446(48), 481(94,100), 488(109), 494–496 Soles, C. L., A:216(364), 254, B:519(164), 587 Song, P. H., B:174–175(98), 280

664

author index

Sorensen, C. M., B:514(127), 518(153), 547(153), 586 Sornette, D., B:58(81), 90 South, G. P., A:17(96), 119 Soutougin, N. N., A:61(216), 123 Spanier, J., A:75(231), 123, B:236(205), 240(205), 283, 298(32), 305(32), 318(32), 435 Sparre Andersen, E., B:467(73–74), 494 Sperl, M., A:225(398–399), 255 Spiess, H. W., A:148(72–73), 150(72,80), 151(85), 152(172), 153(88), 154–155(72), 201(308–309), 209(72,340–341), 212(88,341), 214(72,352), 231(403–404), 246, 252–253, 255, B:501(27–29), 526(188), 529(29,224–225), 550(188), 578(27,29), 583, 587–588 Spyrou, N. M., A:23(117–118), 120 Stafstrom, S., B:209(155), 282 Stanislavsky, A. A., A:399(66), 471 Stanley, H. E., A:48(182), 122, B:4(6–7), 13(24–25), 17(31), 54(74), 65(25), 69(89), 78–79(25), 84(25), 87–88, 90–91, 132(33,38), 133(33), 135(52), 137(53–54), 140(33), 144(72), 278–279, 445(39–40), 481(99), 486(99), 493, 495 Stannarius, R., A:9(14), 26(14), 117, B:562(299), 590 Starkweather, H. W., A:139(49), 245 Stauber, J., A:13(54), 112(54), 118 Stauffer, D., A:38(152), 62(152), 65–66(152), 72(152), 80(152), 121, B:96–97(1), 132(1,39), 133(1), 135(1), 137(39), 138(1), 144(59), 147(1), 156(1), 160(1), 208(136), 277–279, 281 Stavroulakis, G. E., B:209(154), 282 Stebbins, J. F., A:148(76), 246 Stefanich, M., A:444(123), 473 Steffen, W., A:48(194), 106(262), 122, 124, 141–142(54), 145(60), 169(160), 177(218), 179(229), 182(229,245–246), 207(160), 223(218,229,377), 231(160), 245, 248, 250, 254, B:514(125–126), 526(184), 586–587 Stegun, I., B:316(51), 319(51), 323(51), 332(51), 334(51), 341(51), 367–369(51), 381–382(51), 385(51), 389–390(51), 403(51), 421(51), 414–426(51), 436 Stein, D. L., A:332(13–14), 334(13), 355 Stepanov, A., A:201(294), 252 Stephensen, H., B:70(92), 91 Sternin, E., A:9(14), 26(14), 117

Stevens, J. R., A:208(337), 253, B:513(119), 585, 601(8), 630 Stickel, F., A:169–170(170–172), 196(170–171), 200(171), 248 B:502(47), 518(159), 579(159), 583, 587 Stillinger, F. H., A:157(126–128), 240(126), 247, B:546(255), 589 Sto¨cker, M., A:38(159), 42(159), 63–64(159), 121 Stockmayer, W. H., B:602(13), 631 Stoll, B., B:503(88), 507(88), 585 Stott6, N. E., A:328(2), 354 Straley, J. P., B:132(29), 144(29,63), 278–279 Straud, D., B:174–175(91–92,101–102), 181(91–92), 280 Street, R. A., A:106(261), 124, 332(16), 355 Strelniker, Y. M., B:187(126), 281 Strenski, P. N., B:137(51), 279 Strikman, S., B:204–205(133), 225(133), 230(133), 281 Strogatz, S. H., B:546(265), 589 Stroud, D. G., B:187(123), 188(122–123), 281 Strube, B., A:231(408), 255 Struik, L. C. E., A:259(1), 276(1), 320 Struzik, Z. R., B:69(89), 91 Stuchly, S. S., A:24(129), 120 Sudarshan, E. C. G., A:364(13), 470 Su¨dland, N., A:11(35), 118 Sudo, S., A:48(190), 51(190), 55(190), 86(190), 88(190), 122, 201(289), 252 Suki, B., B:4(7), 13(24), 87–88 Sulkowski, W. W., A:201(292), 252, B:531(228), 588 Surovtsev, N., A:145(59), 170(185), 173(212), 176(185), 179(185), 182(185), 216(362), 223(370), 230(401), 231(411), 245, 248, 254–255 Suzuki, A., B:506(107–109), 585 Svanberg, C., B:503(103), 585 Swallen, S. F., A:171(197), 249, B:523(178), 587 Swierczynski, Z., B:5(9), 87 Swinney, H. L., B:443(27), 493 Sykes, M. F., B:132(42–44,46), 135(46), 137(46), 278 Symm, G. T., A:23(117–118), 120 Szeto, H. H., B:14(27), 88 Szydlowsky, M., B:445(43), 475(43), 493 Taborek, P., A:14(76), 119 Tachiya, M., A:332(12), 355 Talkner, P., B:327(59), 380(59), 427(59), 436, 474–475(84), 495

author index Tammann, G., A:13(62), 118 Tammann, V. G., B:499(5), 582 Tanatarov, L. V., B:450(60), 453(60,63–64), 456(60,64), 458(64), 468(77), 483(60,64), 484(64), 494–495 Tang, J., A:351(49), 356 Tao, N. J., A:130(5), 163(5,134), 177(5,134), 179(5,134), 182(5,134,244), 224(381–383), 243, 247, 250, 254 Tarboton, D. G., B:68(88), 91 Tarjus, G., A:170(176), 179(228), 185(269), 248, 250–251 Tartaglia, P., A:32(144–145), 33(151), 121, 132(23), 156(23), 244 Taschin, A., A:167(150), 177(219), 224(150,392), 248, 250, 255 Taupitz, M., A:151(84), 246 Taus, J., B:235(186), 282 Taylor, L. R., B:5(17), 6(17–18), 7(20), 88 Taylor, R. A. J., B:6(18), 88 Tegmark, M., A:446(128), 468(128), 473 Teich, M. C., B:24(42), 89 Teichroeb, J. H., B:617–618(39), 631 Teitler, S., B:568(325), 591 Telleria, I., B:558(294), 590 Temkin, S. I., B:364(84), 436 Terki, F., A:231(407), 255 Tesi, G., A:446(132), 473 Tessieri, L., A:439(112–114), 445(112), 446(112–114,131), 473 Texter, J., A:68(225), 123 The-Ha, Ho, A:42(168), 121 Theis, C., A:132(23), 156(23), 244 Theocaris, P. S., B:209(154), 282 Theodorakopoulos, N., A:235(420), 255 Thirumalai, D., A:156(122), 247 Thomas, W. R., B:194(130), 281 Thompson, E., A:235(419), 255 Thon, G., B:443(19), 493 Thrane, L., A:52(205), 122 Thuresson, K., A:115(279), 125 Tikhonov, A. N., A:9(13), 117 Timoken, J., A:460(159), 474 Titov, S. V., A:180(238,240), 250, B:292(21–22,24–26), 293(21,27), 299(22), 312–313(24), 330(25), 338(68), 347(81), 364(24), 376(24), 388(90,92), 389(92), 398(26,96–97), 401(26), 407(24), 418(25),420(101), 424(22,101), 435–437 Tjomsland, T., A:16(89), 119

665

Toda, M., A:266(31), 269(31), 304(31), 306(31), 322, 394(61), 400(61), 405–406(61), 454(61), 471, B:327(60), 399(60), 436 Tolic-Nørrelykke, I. M., B:443(19), 493 To¨lle, A., A:167(148), 182(148), 215–216(148), 221(148), 248, B:14(124), 586 Tombari, E., A:16(79,83), 18(79,83), 20–21(79,83), 23(79,83), 24(83), 119, B:575(364–367), 592 Topp, K. A., A:235(418), 255 Torell, L. M., A:184(256), 194(256), 208(256,338), 251, 253 Torkelson, J. M., A:171(196), 249, B:617–618(41), 631 Tornberg, N. E., A;40(175), 122 Torquato, S., B:209(152), 282 Torre, R., A:147(69), 167(150), 177(219), 182(69,247), 223(247), 224(69,150,388–389,392–393), 245, 248, 250, 255 Tosatti, E., B:170(89), 280 Toulouse, G., B:129(25), 233(25), 278, 284 Toulouse, J., A:215(354), 216(357), 223(372,376), 224(372), 253–254 Tracht, U., B:501(29), 529(29), 578(29), 583 Tran-Cong, Q., B:572(341), 592 Trefa´n, G., A:389–390(49), 405(49), 406(49,79), 414(49), 471–472 Tretiakov, K. V., B:209(159), 282 Triebwasser, S., A:44(169), 121 Trimper, S., A:276(48), 322 Tsallis, C., A:391(53), 409(53), 471 Tsang, K. Y., B:501(23), 546–547(260–262), 583, 589 Tschirwitz, C., B:530(227), 549–550(227), 588 Tschirwitz, T., A:52(203), 62(203), 122, 130(6), 163(6), 164(6,136), 165(6,142), 167(6), 168(6,142,153), 170(153,183), 175(136), 177(136), 179(136,142,183,230), 181–182(183), 183(136), 184(142), 186–187(142), 188(6,136,142,230,273), 189(136,142,230), 190(6,142), 191(6,142,230), 194(142,230), 200(6), 201(6,153,285,306), 202(6,142,153,230,285), 203(6,230,306), 204(6), 205(6,142,153,230,285), 206(6,142), 209(6,142,230,306), 216(183), 222(183), 223(136,230), 230(183,401), 231(6,306), 232(6,183), 234(6), 236(6,153,285,306), 237–239(306), 243, 247- 248, 250–252, 255

666

author index

Tsironis, G. P., B:546(266), 589 Tucker, J. C., A:172(205), 249 Turnbull, D., A:13(70), 97(70), 118, 156(101), 246 Turner, S. R., B:613(22), 614(32), 631 Tweer, H., A:169(163), 248 Uchaikin, V. V., B:253(224,230), 271(225), 284 Uhlenbeck, G. E., B:253(247), 284, 370(87), 373(87), 437 Uhlmann, D. R., A:169(164), 170(164), 172(164,202), 248–249 Ullersma, P., A:261(22), 321 Ullman, R., B:338(64), 436 Umehara, T., A:16(82), 18(82), 20–21(82), 23–24(82), 119 Unger, R., B:572(338), 592 Urakawa, O., B:572(341), 592 Vacher, R., A:231(407), 255 Valiev, K. A., A: 153(91), 246 Van de Moortele, B., B:529(222), 588 Van der Hulst, H. C., B:183(111), 280 Van der Waerden, B. L., B:98(18), 100(18), 278 Van Dijk, M. A., A:32(142), 121 Van Hook, A., A48(191), 122 Van Hove, L., A:362(7), 470 Van Kampen, N. G., A:415(91), 472, B:338(71), 436, 440–442(11), 492, Van Koningsveld, A., A48(192), 122 Van Megen, W., A:156(119), 247 Van Oijen, A. M., A:328(4), 331(4), 333(4), 335(4), 340(4), 354 Van Sark, W. G. J. H. M., A:349(46), 356 Van Turnhout, J., A:27(134), 120 Van Vleck, J. H., B:372(88), 418(88), 437 Vannimeus, J., B:132–133(32), 278 Va´rez, A., A:178(223), 233(223), 250 Varnik, F., B:612(20), 631 Vasquez, A., A:216(360), 254 Verberk, R., A:328(4), 331(4), 333(4), 335(4), 339(32), 340(4), 354–355 Verdini, L., A:169(160), 207(160), 231(160), 248, B:526(184), 587 Verdonck, B., B:507(111), 585 Verney, V., B:527(196), 588 Veronesi, S., A:16(83), 18(83), 20–21(83), 23–24(83), 119 Verveer, P. J., A:26(133), 120 Viasnoff, V., A:303(55), 307(55), 323

Vicsek, T., B:97(7), 132–133(30), 186(7), 277–278 Vidal, C., A:381(39), 471 Vigier, G., B:556(286,288), 590 Vij, J. K., B:324(52), 436, 540(250–252), 589 Vilar, J. M. G., A:334(23), 355 Vinogradov, A. P., B:141(65), 279 Viot, R., A: 185(269), 251 Virasoro, M. A., B:129(25), 233(25), 278, 284 Virgilio, M., A:451(146), 474 Virnik, K., A:38(156), 39–42(156), 58(156), 60(156), 63–64(156), 103(156), 121 Viswanathan, G. M., B:445(39–40), 493 Vitali, D., A:439(112–114), 445(112), 446(112–114,132), 473 Vlachos, G., B:564(305), 591 Vladimirov, V. S., B:313(50), 436 Vogel, H., A:13(60), 118, B:499(3), 530(227), 549–550(227,281), 582, 588, 590 Vogel, M., A:130(15), 148(15), 154(93–94,96–97), 201(15,96–97,305–306), 203(306), 207(96–97), 208(323), 209(15,306), 212(346), 214(94), 231(96–97,305–306), 235(15,323), 236(15,306,323), 237(93,96,306), 238(93,306), 239(96–97,306), 240(426), 244, 246, 252–253, 256 Vo¨gtle, F., B:613(23), 631 Voigtmann, T., A:224(380), 225(380,400), 227(380), 254–255 Voit, B., B:613(22,24), 614(32–33), 631 Volkov, A. A., A:140(51), 245 Vollmer, H. D., B:398(94), 437 Volmari, A., A:7(10), 117 Von Borczyskowski, C., A:328(6), 331(6), 333(6), 349(43), 351(6), 353(6), 354, 356 Von Meerwall, E. D., B:569(335), 591 Voulgarakis, N. K., B:546(266), 589 Vugmeister, B. E., A:40(176), 94(176), 122 Wadayama, T., A:11(37), 118 Waddington, J. L., B:80(104), 91 Wagner, H., A:137(47), 201(286), 207(334), 240(286), 245, 251, 253, B:536(233), 540(233), 589 Wahnstro¨m, G., A:216(358), 254, B:548(274), 590 Walderhaug, H., B:518(156), 547(156), 586 Waldron, J. T., B:253(250), 255(250), 267(250), 284, 287–290(8), 292–294(8), 296–297(8),

author index 309(8), 313–314(8), 319(8), 322–324(8), 329–330(8), 331–333(62), 334(8,62), 336(8,62), 338(8,67), 339(8), 341(8,67), 343–344(8,67), 347(8,67,78), 348(8), 350(8), 353(8), 355–358(8), 364(8,67), 366–368(8), 378(8), 380(8), 384(8), 393–394(8), 399(8), 401(8), 403(8), 417(8), 426(8,67), 434(67), 434, 436, 442–443(13), 493 Wallqvist, A., A:52(205), 122 Walter, F., B:614(32), 631 Walther, H., B:445(44), 493 Walther, L. E., A:135(38), 244 Wang, C. H., A:184(252–254), 194(252–254), 251, B:513(121), 586 Wang, C.-Y., B:522(170), 587 Wang, K. G., B:312(45–46), 435 Wang, X. J., A:450(143), 474 Wang, Y. C., B:225(175,178), 228(175), 282 Ward, D. W., A:241(430), 256 Warham, A. G. P., A:24(131), 120 Warschewski, U., A:169(167), 177(167), 197(167), 236(167), 248 Wasylshyn, D. A., B:575(361–363), 592 Watanabe, H., B:338(66), 436 Watts, D. C., A:10(30), 117, 201(283), 204(283), 207(283), 251, B:235–236(180), 249(180), 282, 299(34), 435, 501(31), 502(31,57), 583–584 Wawryszczuk, J., B:499(9), 582 Weast, R. C., A:50–51(198), 76(236), 122–123 Webber, C. L., B:18(33), 88 Weber, M., A:231(403), 255 Weber, R., B:620(48), 631 Weberskirch, R., B:614(33), 631 Webman, I., A:33(148,150), 63(218), 70(150), 121, 123, B:144(68–69), 172(68–69), 174–175(96–97), 188(68), 212–213(165), 279–280, 282 Weeks, E. R., B:443(27), 493, 529(213), 530(213), 550(213), 578(213), 588 Wegner, G., B:503(65–66), 584 Wehn, R., A:231(402), 255, B:538(246), 589 Wei, J. Y., B:5(10), 18(35), 19(10), 87–88 Weinga¨rtner, H., A:7(10), 117 Weiss, G. H., A:10(33), 117, 366(17), 470, B:253(231), 257(231), 267(249), 284, B:291–292(17), 295(17), 306(17), 364(17), 435, 443(21,32), 493 Weiss, S., A:358(1), 469

667

Weiss, U., A:262(28), 296–298(28), 322 Weisskopf, V. F., B:372(88), 418(88), 437 Weitz, D. A., A:316(59), 323, B:97(8,10), 186–187(8), 277, 529(213), 530(213), 550(213), 578(213), 588 Welton, T. A., A:134–135(31), 244 Weron, K., A:12(44), 118, B:299(35), 435 West, B. J., A:374(35), 389–390(49), 392(58–59), 394(60), 396(59–60), 397(60), 398(65), 403(59), 405(49,74), 406(49,79), 409(82,84), 410(84), 411(84,86), 412(58), 414(49,89), 415(93), 419(93), 424(97), 425–426(97,100), 429(102), 431(102), 436(110), 439(59), 440(59,117), 441(59), 444(122–123,125), 447(122), 454(97), 462(117), 466(100), 468(59–60,93,100), 470–473 B:4(5,8), 5(9,11,13–14), 8(21), 13(23), 14(26), 15–17(14), 18(11,36), 19(36), 21(36,38), 22(5,38), 23(5,21), 24(5), 27(36), 28(38), 31(32), 35(49), 39(52), 40(23,53), 42(21,56), 44(36), 45(11,36), 46(11,36,60), 48(64), 49(64–65), 50–52(66), 54(23,52), 55(53), 58(5,23,80,82), 59(23,82), 62(53), 64–65(53), 66(86), 67(21), 69(11,90), 70(80,93), 71–72(90), 73(64), 75(95), 76(95–96), 77(53,95), 78(95), 80(106), 87(53,56), 87–91, 237(213), 283, 292–293(20), 313(20), 324(20), 435, 449(59), 467(71), 479(71), 485(59), 494 West, D. J., B:5(11), 18(11), 45–46(11), 69(11), 87 Westgren, A., 441(7), 492 Whalley, E., A:48(189), 51(189), 122, B:502(58), 584 Wheatcraft, S. W., B:445(42), 493 Wheeler, J. A., A:446(128), 468(128), 473 When, R., A:169(159), 208(159), 248 White, B. S., B:235–236(187), 283, 445(37), 493 White, C. T., B:235–236(184), 244(184), 282, 546(258–259), 589 Whitworth, R., A:51(199), 122 Wiebel, S., A:167(149–150), 179–180(227), 224(149–150,385), 248, 250, 254 Wiedersich, J., A:145(59), 170(183,185), 175(183), 176(185), 179(183,185), 181(183), 182(183,185), 212(346), 216(183), 222(183), 223(370), 230(183,401), 231(410–411), 232(183), 235(410), 245, 248–249, 253–255 Wiesner, U., A:231(404), 255

668

author index

Wilcoxon, J. P., A:11(39), 118 Wilder, J. A., A:89(247), 124 Wilhelm, M., A:214(352), 253, B:501(28–29), 529(29,225), 578(29), 583, 588 Wilkie, J., A:446(131), 473 Williams, B. D., A:165(140), 184(140), 190(140), 243, 247 Williams, G., A:10(27,30), 117, 134(35–36), 136(40), 147(35,66), 170(66), 171(35), 201(283,293), 204(283), 207(36,283), 236(283), 244–245, 251–252, B:235–236(180), 249(180), 282, 299(34), 435, 501(31), 502(31,51–54,57,60), 506(110), 529(214), 536(238), 583–585, 588–589 Willis, J. C., B:7(19), 88 Willner, L., A:201(304), 252 Wilner, L., B:527(195), 587 Wilson, K. G., A:78–79(239), 123, B: 132(34), 278 Witten, T. A., B:96(5), 277 Woiwod, P., B:7(20), 88 Wojciechowski, K. W., B:167(79), 209(143, 147,149,158–160), 211(162), 212(163), 213(143), 221(170), 231(160), 237(215), 279, 281–283 Wojcik, K., A:40(174), 122 Wolf, E., A:275(39), 322 Wong, W. H., A:23–24(121), 120 Wong, Y. M., B:446(47), 481(98), 494–495 Woo, W. K., A:328(7), 331(7), 353(7), 354, 358(3–4), 432(4), 470 Woodward, A. E., A:255 Wren, R., B:538(244), 558–559(244), 589 Wu, H.-L., A:374(33), 470 Wu, L., A:165(140), 184(140), 190(140), 201(284), 202(312), 209(284), 236(284,312), 247, 251–252 Wu, N., A:408(81), 472 Wu, Y., A:218(365), 254, B:14(27), 88 Wu¨bbenhorst, H., A:27(134), 120 Wu¨bbenhorst, M., B:596(2), 630 Wurm, B., A:13(53), 118 Wuttke, J., A:167(147,149), 171–172(201), 177(147), 179(227), 180(147,227,235), 182(147), 215(201), 216(147), 224(149,387), 248–250, 255 Wyss, W., B:250(224), 253(224), 284 Xu, M. Z., A:106(264), 124 Xu, Z., A:102(258), 124

Yacoby, Y., A:44(170,172), 94(172), 121 Yagi, S., A:40(180), 122 Yagi, T., A:224(394), 255 Yagihara, S., A:12(46), 16(82), 18(82), 20(82), 21(82,110), 23(82), 24(82,110), 48(190), 51(190), 55(190), 86(190), 88(190), 89(248), 106(265), 110(265), 111(46), 118–120, 122, 124, 201(289), 252 Yagil, Y., B:174–175(105), 280 Yamaguchi, M., A:224(394), 255 Yamamoto, S., A:11(37), 118 Yamamuro, O., B:517(149), 543(149), 586 Yang, J., A:451(146), 474 Yang, W., A:460(160), 474 Yang, Y., A:182(248–249), 216(248–249), 250 Yannopoulos, S., A:173(211), 249 Yao, Y. D., B:347(80), 436 Yariv, A., A:40(177,180), 122 Yaughan, E. F., A:6–9(8), 16(8), 117 Ye, Z.-G., A:102(258), 124 Yee, A. F., A:216(362,364), 254, B:519(164), 587 Yip, S., A:130(3), 221(3), 243 Yoffe, O., A:45(181), 122 Yonezawa, F., B:235(197–198), 283 Yonezawara, F., A:109(269), 124 Yoo, P. J., B:620–621(53), 632 Yoon, D. Y., B:548(272), 590 Yu, J., A:23–24(123), 120 Zabolitzky, J. G., B:208(136), 281 Zabransky, B. J., A:216(361), 254 Zacherl, A., A:396(63), 471 Zakhidov, A. A., B:209(155), 282 Zaslavsky, G. M., A:3(1), 116, 388(48), 391(54), 443(54), 444(124), 447(124), 463–464(54), 471, 473, B:253(228,233), 257(233), 284, 445(35), 493 Zavada, T., A:11(35), 118 Zelezny, V., A:42(168), 121 Zemany, L., B:5(12), 87 Zeng, X., B:174–175(91–92), 181(91–92), 280 Zeng, X. C., A:179(228), 250 Zenitani, H., A:201(288), 251, B:536(235), 573(235), 589 Zetsche, A., B:569(328,330), 591 Zhang, H. P., A:179(231), 183(231), 193(231), 230(231), 250 Zhang, R., B:4(8), 70(91,93), 87, 91

author index Zhang, S. H., B:503(89), 507(89), 585 Zhang, W. B., B:518(153), 547(153), 586 Zhao, M., B:613(26), 631 Zheng, S., B:614(35), 631 Zheng, X. D., B:527(190), 587 Zhou, Y., B:613(27), 631 Zhu, P. W., B:614(35), 631 Ziman, J., B:157(80), 279 Zimmerman, H., B:501(28), 526(188), 529(225), 550(188), 583, 587–588 Zimmermann, H., A:188(331), 205(331), 209(331), 213(331,347), 253 Ziolo, J., A:202(322), 205(329), 253, B:503(72,82–83,85–86,92–93,75–77), 504(72), 505(75–77,82–83,85–86), 507(92–93), 531(72,75–77), 536–537(239), 553(76–77), 584–585, 589 Zirkel, A., B:548(272), 590 Zoppi, M., A:132(22), 134(22), 244 Zorn, B., A:201(299,301), 252 Zorn, R., B:518(143), 527(195), 548(143), 562(300–301), 586–587, 591

669

Zosimov, V. V., B:124(37), 132(37), 278 Zqruki, J., B:97(12), 277 Zschiegner, S., B:443(30), 493 Zubarev, D. N., A:74–75(229), 79(229), 123 Zuckerman, J. H., B:4(8), 70(91,93), 87, 91 Zuev, Y., A:16(86), 18(86), 19(86,103), 20(86), 21(86,103), 22(86), 23(86,103), 24–25(86), 33(147), 119–121 Zukoski, C. F., B:224(173), 282 Zumofen, G., A:10(32), 117, 334(18,24), 355, 384(42), 385(42,44), 388(48), 401(70), 403(70,73), 450(44), 471–472, 38(50), 89, 235–236(187–189), 237(189), 283, B:445(36), 467(76), 474(67), 481(93,97), 488(36), 493–495 Zurek, W. H., A:364(11–12), 373(24), 439(116), 442(120), 470, 473 Zwanzig, R., B:331(61), 436 Zwanzig, R. W., A:74–75(230), 77(230), 79(230), 123, 262(29), 305(29), 322, 364–365(15–16), 470 Zwerger, W., A:261(24), 321

SUBJECT INDEX Letter in boldface indicates the volume. Accelerating state, Zwanzig projection method, Anderson localization, A:373–374 Ac conductivity, porous glasses relaxation response, A:40–41 Action potentials, fractal-based time series analysis, B:23–26 Adam-Gibbs theory conﬁned system relaxation kinetics, A:103–104 entropy model, vitriﬁcation of liquids, B:499–502 ferroelectric crystals, liquid-like behavior, A:94–95 temperature/pressure superpositioning, B:510–511 Admittance broadband dielectric spectroscopy, A:16–18 time-domain spectroscopy, A:20–21 Aftereffect function dielectric relaxation Cole-Davidson and Havriliak-Negami behavior, B:314–316 fractional Smoluchowski equation solution, B:320–325 inverse Fourier transform calculations, B:424 Agglomerates of particles, fractal viscoelastic properties, B:224–235 Aging effects anomalous diffusion, non-Ohmic models, A:302–303 noise and friction, A:296–297 one- and two-time dynamics, A:297–303 Mittag-Lefﬂer relaxation, A:298–299 particle coordinate and displacement, A:299–300 time-dependent quantum diffusion coefﬁcient, A:300–302

velocity correlation function, A:299 quantum aging, A:303 blinking quantum dots, A:337–342 on-off correlation function, A:338 on-off mean intensity, A:337–338 Brownian motion Langevin model, A:279–284 displacement response and correlation function, A:280 ﬂuctuation-dissipation ratio, A:280–281 temperature effects, ﬂuctuationdissipation ratio, A:283–284 time-dependent diffusion coefﬁcient, A:281–283 velocity correlation function, A:279–280 overdamped classical motion, A:277–279 displacement response and correlation functions, A:277–278 ﬂuctuation-dissipation ratio, A:278–279 overview, A:276–277 quantum motion, A:284–296 displacement response correlation function, A:288–289 time-dependent diffusion coefﬁcient, A:289–291 effective temperature determination, A:292 modiﬁed ﬂuctuation-dissipation theorem, A:291–292 Ohmic model temperature, A:292–295 time-dependent diffusion coefﬁcient, A:286–291 velocity correlation function, A:285–286 continuous time random walk applications, A:429–431 formal approach, A:423–425 non-Poisson processes, A:421–423

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

671

672

subject index

Aging effects (continued) Le´vy ﬂight processes, theoretical background, B:443–448 non-Poisson dichotomous noise, four-time correlation function, A:421 out-of-equilibrium linear response theory, A:309–311 vitriﬁcation of liquids coupling model above glass transition temperature, B:558–561 structural-Johari-Goldstein relaxation correspondence, B:552–555 Johari-Goldstein secondary relaxation, B:538–540 Airy function, supercooled liquids and glasses, depolarized light scattering, A:143–148 Algebraic decay, fractal structures, non-Debye relaxation, B:236–253 Allometric aggregation data analysis breathing rate variability, B:15–18 gait analysis, B:21–22 heart beat fractals, B:11–14 research problems of, B:83–84 scaling behavior, B:5–10 a-scaling, supercooled liquids and glasses dynamic susceptibility, A:230–231 formation above glass transition temperature, A:180–182 mode-coupling theory, A:160–162 Alpha relaxation hyperbranched polymer ﬁlm conﬁnement effects, B:614–620 polystyrene ﬁlms, conﬁnement effects, B:623–628 Amorphous polymers fractal structure, B:96–98 vitriﬁcation structural relaxation-dispersion coordination, B:517–528, B:526–528 temperature/pressure dependence, B:507–509 Analytic signal, zero-temperature FDT, A:275–276 Anderson localization non-Ohmic bath, A:459 Zwanzig projection method, A:368–374 Angular frequency, non-Ohmic dissipation, aging effects, noise and friction, A:297

Angular velocity correlation function (AVCF), dielectric relaxation, inertial effects Barkai/Silbey fractional Klein-Kramers equation, B:378–379 fractional Langevin equation, B:416–419 Metzler/Klafter fractional Klein-Kramers equation, B:370–372 symmetric top molecules, B:389–398 Anomalous continuity equation, ‘‘strange kinetic’’ percolation, A:76–81 Anomalous dielectric relaxation. See Dielectric relaxation Anomalous diffusion. See also Fractional rotational diffusion aging phenomena, non-Ohmic models classical aging effects, A:302–303 noise and friction, A:296–297 one- and two-time dynamics, A:297–303 Mittag-Lefﬂer relaxation, A:298–299 particle coordinate and displacement, A:299–300 time-dependent quantum diffusion coefﬁcient, A:300–302 velocity correlation function, A:299 quantum aging effects, A:303 in double-well periodic potential, B:331–338 fractal structures, B:253–264 Brownian motion, B:253–255 Smoluchowski equation for Brownian particles, B:255–257 subordinate processes, B:259–264 generalized master equation, perturbation response, A:405–407 Le´vy ﬂight processes, multimodality, B:466–467 non-Poisson renewal processes, quantum to classicial transition, A:442–447 out-of-equilibrium physics Kubo generalized susceptibilities, A:312–315 effective temperature, A:312–313 friction coefﬁcient, A:314–315 mobility modiﬁcation, A:313–314 linearly coupled particle equation of motion, A:307–308 linear response theory, A:309–312 age- and frequency-dependent response functions, A:309–311 quasi-stationary regime, effective temperature, A:311–312

subject index overview, A:303–304 power-law behaviors, A:317–319 friction coefﬁcient, A:317–318 mean-square displacement, A:318 temperature determination, A:318–319 stationary medium, A:308–309 Stokes-Einstein relation, A:315–317 thermal bath, A:304–307 ﬂuctuation-dissipation theorems, A:304–306 regression theorem, A:306–307 scaling dynamics, simple random walks, B:30–31 Anti-Fourier transform, continuous time random walk Gaussian stochastic process, A:400–401 generalized central limit theorem, A:387–388 Antisymmetric equation, continuous time random walk, correlation function, A:427–429 Arrhenius law conﬁned system relaxation kinetics, A:97–98 doped ferroelectric crystals, A:101–102 dielectric spectroscopy ferroelectric crystals, A:47–48 hydrogen-bonding liquids, A:50–55 fractal viscoelasticity, shear-stress relaxation, B:232–235 fractional rotational diffusion bistable potential with nonequivalent wells, B:348–364 potential phenomena, B:325–331 hydrogen-bonding liquids, scaling phenomena, A:88–93 non-Poisson renewal processes, modulation theories, A:452–456 relaxation kinetics, A:12–15 supercooled liquids and glasses, formation above glass transition temperature, b-peak glass formation, A:201–209 vitriﬁcation of liquids, coupling model, B:548–551 activation enthalpies, B:555–556 Asymmetric double-well potential, fractional rotational diffusion, B:347–364 Asymptotic behavior chaotic fractal percolation clusters, B:133–137 dielectric relaxation, Cole-Cole behavior, B:303–305

673

fractal structure, B:97–98 anomalous diffusion, B:262–264 Brownian particles with memory, B:266–267 dielectric relaxation, B:247–253 elasticity, B:206–209 shear-stress relaxation, B:233–235 fractional rotational diffusion bistable potential with nonequivalent wells, bimodal approximation, B:357–364 direct current (DC) electric ﬁeld, B:344–347 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:334–338 Le´vy ﬂight processes dissipative nonlinearity, B:483–484 power-law asymptotics of stationary solutions, B:455–456 Sparre Anderson universality, B:471–472 theoretical background, B:445–448 multifractal spectrum, diffusion entropy analysis, B:50–52 power law blinking quantum dots, case studies, A:339–342 dipole correlation function, A:10–12 ionic microemulsion dipole correlation function, A:67 non-Ohmic bath, A:457–459 ‘‘strange kinetic’’ percolation, A:73–81 supercooled liquids and glasses mode coupling tests, A:221–225 mode-coupling theory, A:162 Atomic force microscopy (AFM) polystyrene ﬁlms, conﬁnement effects, B:622–628 thin polymer ﬁlm preparation, B:597–601 Atomic polarization, static electric ﬁelds, A:5 Attractor, Julia fractal sets, B:112–113 Autocorrelation function dielectric relaxation, fractional Smoluchowski equation solution, B:322–325 fractional Brownian motion, B:60–61 scaling dynamics fractional random walks, B:33–34 inverse power-law, B:34–35 simple random walks, B:30–31 supercooled liquids and glasses, molecular reorientation, A:131–134 dielectric spectroscopy, A:136–140

674

subject index

Average cluster dimension, chaotic fractal percolation clusters, B:135–137 Average coordination number chaotic fractal structure models, B:154–157 fractal structures, negative Poisson’s ratio, B:215 Avogadro’s number fractal anomalous diffusion, Brownian motion, B:255 vitriﬁcation of liquids, coupling model principles, B:546–551 Axial tension, fractal structure, negative Poisson’s ratio, B:211–215 Backward-wave oscillator, supercooled liquids and glasses, molecular reorientation, A:140 Ballistic interval, fractal structures, Brownian particle inertial effects, diffusion equation, B:271–274 Barkai-Silbey format, fractional Klein-Kramers equation, B:372–379 fractional Langevin equation, B:416–419 linear and symmetrical top molecules, B:380–388 periodic potentials, B:400–414 symmetric top molecules, B:394–398 Barrier crossings fractal structures, dielectric relaxation, B:238–253 fractional rotational diffusion double-well periodic potential, anomalous diffusion and dielectric relaxation, B:331–338 in potentials, B:330–331 Le´vy ﬂight processes, B:474–481 Brownian motion, B:475 Cauchy analytical approximation, B:478–480 numerical solution, B:475–478 starting equations, B:474–475 Bernoulli map, non-Poisson renewal processes, trajectory and density entropies, A:449–451 Bessel function continued fraction solution, longitudinal and transverse responses, B:426–427 dielectric relaxation, inertial effects, periodic potentials, B:403–414

fractal structures, Brownian particles with memory, B:273–274 fractional rotational diffusion direct current (DC) electric ﬁeld, B:341–347 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:332–338 b-peak glass formation above melting point low-temperature regime, A:185–201 relaxation properties, A:201–209 below melting point, A:235–241 Bhatnagar-Gross-Krook model, dielectric relaxation, B:418–419 Bias ﬁeld effects, bistable potential with nonequivalent wells, B:348–364 Bifurcation time, Le´vy ﬂight processes, B:461–464 Bimodal approximation fractional rotational diffusion bistable potential with nonequivalent wells, B:355–364 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:334–338 Le´vy ﬂight processes nonunimodality, B:458–459 trimodal transient state, B:462–463 Binary mixtures, vitriﬁcation of liquids, component dynamics, B:567–574 Binomial coefﬁcients dielectric relaxation, Cole-Davidson and Havriliak-Negami behavior, B:316 fractal functions, B:276–277 Binomial random walk on a line, blinking quantum dot nonergodicity, A:345–346 Biological phenomena fractals, B:3–4 fractional dynamics, B:65–73 Bistable potential, fractional rotational diffusion, nonequivalent wells, B:347–364 bimodal approximation, B:355–364 matrix continued fraction solution, B:350–354 Blinking quantum dots. See also Quantum dots aging phenomena, A:337–342 on-off correlation function, A:338 on-off mean intensity, A:337–338

subject index basic principles, A:328 continuous time random walk aging phenomena, A:423–425 correlation function, A:426–429 Le´vy dynamic derivation, A:392 Nutting law, A:383–384 overview, A:358–362 diffusion model, A:331–334 experimental data, A:350–353 nonergodicity, A:342–350 time-averaged correlation function, A:347–350 time-averaged intensity distribution, A:344–346 non-Poisson renewal processes decoherence theory, A:440–441 modulation theories, A:451–456 quantum-to-classical transition, A:441–447 physical models, A:328–334, A:466–467 stochastic models, A:334–336 Bloch equation, generalized master equation, A:433–435 Bloembergen-Purcell-Pound (BPP) expression, spin-lattice relaxation and lineshape analysis, A:150–152 Boltzmann constant canonical equilibrium, A:410 conﬁned system relaxation kinetics, A:98 static electric ﬁelds, A:7 Boltzmann distribution dielectric relaxation Cole-Cole behavior, B:302–305 continuous-time random walk model, B:299 fractional Smoluchowski equation solution, B:320–325 inertial effects fractionalized Klein-Kramers equation, B:371–372 periodic potentials, B:399–414 fractal anomalous diffusion, Brownian motion, B:255 fractal viscoelasticity, shear-stress relaxation, B:232–235 Bolzano function, fractal sets, B:98–101 Bonding properties chaotic fractal percolation clusters, B:132–137 lattice structural models, B:149–157 Scala-Shklovsky model, B:141–147

675

fractal structures dielectric properties, B:175–183 negative Poisson’s ratio, B:212–215 Booster dynamic system, canonical equilibrium, A:410 B-operation, Bolzano function, fractal sets, B:99–101 Born model, chaotic fractal structures, B:144–147 Bose distribution function, supercooled liquids and glasses, depolarized light scattering, A:140–148 Box counting dimension, multifractal spectrum, B:43–46 Breathing rate variability (BRV) fractal analysis, B:14–18 multifractal analysis, B:46 Broadband dielectric spectroscopy (BDS) basic principles, A:16–18 hydrogen-bonding liquids, A:52–55 scaling phenomena, water-rich mixtures, A:86–93 hyperbranched polymer ﬁlm conﬁnement effects, B:613–620 cis-1,4-polyisoprene ﬁlm preparation, B:602–607 polystyrene ﬁlms, conﬁnement effects, B:628 thin polymer ﬁlm preparation, B:596–601 Bromwich integral, fractal anomalous diffusion, B:262–264 Brownian motion aging phenomena Langevin model, A:279–284 displacement response and correlation function, A:280 ﬂuctuation-dissipation ratio, A:280–281 temperature effects, ﬂuctuationdissipation ratio, A:283–284 time-dependent diffusion coefﬁcient, A:281–283 velocity correlation function, A:279–280 overdamped classical motion, A:277–279 displacement response and correlation functions, A:277–27 ﬂuctuation-dissipation ratio, A:278–279 overview, A:276–277 quantum motion, A:284–296 displacement response and correlation function, A:288–289

676

subject index

Brownian motion (continued) displacement response and time-dependent diffusion coefﬁcient, A:289–291 effective temperature determination, A:292 modiﬁed ﬂuctuation-dissipation theorem, A:291–292 Ohmic model temperature, A:292–295 time-dependent diffusion coefﬁcient, A:286–288 velocity correlation function, A:285–286 continuous time random walk, phenomenological modeling, A:364 dielectric relaxation Cole-Cole behavior, B:303–305 continuous-time random walk model, B:294–299 Debye noninertial rotational diffusion, B:305–312 disordered systems, B:287–293 inertial effects, B:364–365 Barkai-Silbey fractional Klein-Kramers equation, B:373–379 fractional Langevin equation, B:414–419 linear and symmetrical top molecules, B:380–388 Metzler/Klafter fractional KleinKramers equation, B:365–372 periodic potentials, B:398–414 symmetric top molecules, B:394–398 dielectric relaxation, microemulsions, A:32–38 fractal structures anomalous diffusion, B:253–255 distribution function, memory particles, B:264–267 inertial effects, B:267–274 diffusion equation with fractional derivatives, B:269–274 one-dimensional lattice random walks, B:267–269 fractional diffusion equations, B:73–76 fractional dynamics generalized Weierstrass function, B:59–61 probability density, B:83–84 fractional rotational diffusion bistable potential with nonequivalent wells, B:347–364

bimodal approximation, B:357–364 direct current (DC) electric ﬁeld, B:338–347 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:335–338 in potentials, B:327–331 generalized master equation, A:364 Le´vy ﬂight processes barrier crossing, B:475–478 dissipative nonlinearity, nonlinear friction, B:482 power-law asymptotics of stationary solutions, B:457–457 theoretical background, B:440–448 multifractal spectrum, diffusion entropy analysis, B:47–52 physiological time series, scaling behavior, B:9–10 scaling dynamics fractional random walks, B:33–34 simple random walks, B:30–31 ‘‘strange kinetic’’ percolation, A:76–81 Bulk modulus, fractal elastic properties, B:204–209 negative Poisson’s ratio, B:212–215 ‘‘Cage level’’ dynamics, hydrogen-bonding liquids, scaling phenomena, glycerol-rich mixtures, A:84–86 Calculus of fractional dynamics, B:55–61 physiologic phenomena, B:84 Caldeira-Leggett model non-Ohmic dissipation, aging effects, one-and two-time dynamics, A:297–303 statistical mechanics, A:262–264 Canonical equilibrium, dynamic approach to, A:409–410 Cantor set Cayley tree fractal structures, B:128–131 fractal construction, B:103–104 Haussdorff-Besicovitch dimension, B:116–119 fractal functions, B:276–277 Capacitance conductivity fractal dielectric properties, B:181–183 thin polymer ﬁlm preparation, B:599–601 Cauchy principal value dc conductivity computation, Hilbert transform, A:28–30

subject index dielectric relaxation Cole-Cole behavior, B:303–305 continuous-time random walk model, B:298–299 fractal anomalous diffusion, B:260–264 fractional calculus, B:57–61 Le´vy ﬂight processes barrier crossing, B:478–481 space-fractional Fokker-Planck equation, B:451 symmetrized correlation function, A:270–273 Zwanzig projection method, Anderson localization, A:369–374 Cayley tree fractal sets, B:128–131 fractal structures dielectric relaxation, B:247–253 shear-stress relaxation, B:233–235 Central limit theorem (CLT) continuous time random walk, A:387–388 multifractal spectrum, diffusion entropy analysis, B:47–52 noncanonical equilbrium, A:413 scaling dynamics, simple random walks, B:29–31 Cerebral blood ﬂow (CBF), fractional dynamics, B:69–73 Chain condition chaotic fractal structures, linear elasticity, B:145–147 fractional diffusion equations, B:73–76 Le´vy ﬂight processes, ﬁrst arrival time problems, B:468–470 ‘‘Chain conﬁnement’’ concept, cis-1,4-polyisoprene, B:607–614, B:630 Chaotic fractal structures, B:131–160 conductivity, B:142–144 iterative averaging method, B:173–174 variational method, B:164 dielectric properties, frequency dependence of, B:174–183 ﬁnite lattice properties, B:155–157 linear elasticity, B:144–147 models, B:147–160 nucleating cell probability functions, B:157–160 percolation systems, B:131–147 cluster structures, B:131–132 critical indices, B:132–137 physical properties, B:141–147

677

renormalization-group transformations, B:137–141 Chapman-Kolmogorov equation, dielectric relaxation, inertial effects, fractional Klein-Kramers equation, B:365–372, B:374 Charge carrier mobility hydrogen-bonding liquids, scaling phenomena, glycerol-rich mixtures, A:84–86 ‘‘strange kinetic’’ percolation, A:80–81 Chemical reaction rate theory, relaxation kinetics, A:12–15 Chemical shift anisotropy (CSA), supercooled liquids and glasses nuclear magnetic resonance, A:149–150 spin-lattice relaxation and line-shape analysis, A:151–152 stimulated echo experiments, two-dimensional NMR, A:153–155 Classical Brownian motion, aging phenomena non-Ohmic dissipation, A:302–303 overdamped aging effects, A:277–279 Classical limit, ﬂuctuation-dissipation theorem, A:273–274 Cluster fractal structure chaotic fractals conductivity, B:144–147 lattice structural models, B:149–157 percolation threshold values, B:131–137 renormalization group transformations, B:138–141 Scala-Shklovsky model, B:141–147 dielectric relaxation, B:248–253 ring-shaped structures, dielectric properties, B:180–183 scalar dependence, B:97–98 Coarse graining, ‘‘strange kinetic’’ percolation, A:78–81 Coaxial reﬂection and transmission, supercooled liquids and glasses, molecular reorientation, A:139 Cole-Cole (CC) equation dielectric relaxation Debye noninertial rotational diffusion, B:310–312 disordered systems, B:290–293 fractal structures, B:239–253 fractional diffusion equation, B:300–305 fractional Smoluchowski equation solution, B:319–325

678

subject index

Cole-Cole (CC) equation (continued) inertial effects, B:364–365 Barkai/Silbey fractionalized Klein-Kramers equation, B:376–379 Kramers-Moyal expansion, B:419–420 Metzler/Klafter fractionalized Klein-Kramers equation, B:369–372 symmetric top molecules, B:394–398 top molecule rotations, B:385–388 microscopic models, B:293–325 dielectric response, A:9–12 fractional rotational diffusion bistable potential with nonequivalent wells, B:355–364 direct current (DC) electric ﬁeld, B:342–347 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:334–338 in potentials, B:330–331 inverse Fourier transform calculations, B:424 polymer-water mixtures, dielectric relaxation, A:110–113 supercooled glass formation above glass transition temperature, b-peak glass formation, A:202–209 symmetric relaxation, peak broadening in complex systems, A:106–110 vitriﬁcation of liquids coupling model, B:550–551 Johari-Goldstein secondary relaxation, non-Arrhenius temperature dependence, B:536–538 primitive relaxation, B:529–530 Cole-Davidson equation dielectric relaxation Debye noninertial rotational diffusion, B:312 disordered systems, B:290–293 fractional diffusion equation, B:313–316 fractional Smoluchowski equation solution, B:319–325 microscopic models, B:293–325 dielectric response, A:9–12 hydrogen-bonding liquids, A:51–55 fractional rotational diffusion, double-well periodic potential, anomalous diffusion and dielectric relaxation, B:337–338 inverse Fourier transform calculations, B:424

supercooled liquids and glasses formation above glass transition temperature, high-temperature regime, A:179–182 mode coupling tests, A:221–225 spin-lattice relaxation and line-shape analysis, A:151–152 vitriﬁcation of liquids, Johari-Goldstein secondary relaxation, nonArrhenius temperature dependence, B:536–538 Collision rate dielectric relaxation continuous-time random walk model, B:294–299 inertial effects, Barkai-Silbey fractional Klein-Kramers equation, B:373–379 dielectric relaxation, inertial effects, fractional Langevin equation, B:418–419 Colloidal systems, fractal dielectric properties, B:183 Complex systems (CS) blinking quantum dots, A:456–461 non-Ohmic bath, A:456–459 recurrence properties, A:459–461 control of, B:82–83 dielectric relaxation band dielectric spectroscopy methods, A:16–18 data ﬁtting problems, A:25–30 continuous parameter estimation, A:25–27 dc-conductivity problems, A:26–27 Hilbert transform dc-conductivity computation, A:28–30 software tools for, A:30 frequency and time domains, A:8–12 kinetic mechanisms, A:12–15 overview, A:2–3 spectroscopic and data analysis principles, A:15–30 static electric ﬁelds, A:3–7 time-dependent electric ﬁeld, A:7–12 time-domain spectroscopy, A:18–25 data processing, A:25 hardware tools, A:21–22 nonuniform sampling, A:22–23 sample holders, A:23–24

subject index dielectric relaxation, inertial effects Barkai/Silbey fractionalized Klein-Kramers equation, B:376–379 Metzler/Klafter fractionalized Klein-Kramers equation, B:366–372 periodic potentials, B:401–414 fractional rotational diffusion in potentials bistable potential, nonequivalent wells, B:347–364 bimodal approximation, B:355–364 matrix continued fraction solution, B:350–354 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:331–338 theoretical principles, B:325–331 uniform DC external ﬁeld, B:338–347 scaling dynamics, B:28–35 theoretical principles, A:461–463 Compliance modulus tensor, fractal viscoelasticity frequency dependence, B:216–235 Component dynamics, vitriﬁcation of liquids, binary mixtures, B:567–574 Conditional probability, non-Poisson dichotomous noise, four-time correlation function, A:417–421 Conductivity chaotic fractal structures, B:142–144 fractal physical properties, B:160–174 effective medium model, B:163 galvanomagnetic properties, B:185–204 iterative averaging method for, B:164–174 Maxwell model, B:163 variational model, B:164 vitriﬁcation of liquids, coupling model, B:548–551 Conductivity tensor, fractal galvanomagnetic properties, B:183–204 Conﬁgurational entropy, vitriﬁcation of liquids, B:499–502 Conﬁned systems, dielectric relaxation kinetics, A:95–105 doped ferroelectric crystal relaxation, A:100–102 glassy water model, A:103–104 models for, A:96–98 static properties and dynamics, A:104–105 temperature dependence of models, A:102–104

679

water dielectric relaxation, A:98–100 Conﬁnement factor, relaxation kinetics, A:14–15 Conﬁnement-induced mode (CIM) hyperbranched polymers, B:613–620 Le´vy ﬂight processes, B:453–467 bifurcation time, B:461–464 consequences of, B:464–467 Langevin equation formal solution, B:459–461 n-modal state phase diagrams, B:463–464 nonunimodality of stationary solutions, B:456–459 power-law asymptotics of stationary solutions, B:455–456 stationary quartic Cauchy oscillator, B:453–454 theoretical background, B:448 trimodal transient state, B:462–463 cis-1,4-polyisoprene thin ﬁlm preparation, B:601–607 molecular assignment, B:607 simulations, B:608–614 polystyrene ﬁlms, B:620–628 Coniglio-Sarychev-Vinogradov model, chaotic fractal percolation clusters, B:141–147 Connecting set (CS) formation chaotic fractal percolation clusters, lattice structural models, B:149–157 fractal square lattice construction, B:126–128 Constant ﬁeld effect, fractional rotational diffusion, bistable potential with nonequivalent wells, B:348–364 Constant ﬂux approximation, Le´vy ﬂight processes, Cauchy case, B:479–481 Constitutive relations equations, fractal conductivity, B:161–163 Continued fraction solution dielectric relaxation, inertial effects Barkai-Silbey fractional Klein-Kramers equation, B:374–379 Metzler/Klafter fractionalized Klein-Kramers equation, B:368–372 periodic potentials, B:401–414 symmetric top molecules, B:391–398 top molecule rotations, B:383–388 fractional rotational diffusion, bistable potential with nonequivalent wells, B:350–354 bimodal approximation, B:357–364

680

subject index

Continued fraction solution (continued) longitudinal and transverse responses, B:425–427 spherical top molecules, B:430–434 Continuity equations, ‘‘strange kinetic’’ percolation, A:76–81 Continuous parameter estimation dielectric spectroscopy, A:27 frequency-domain analysis, A:25–26 Continuous time random walk (CTRW) dichotomous ﬂuctuations, B:38–41 dielectric relaxation Cole-Cole behavior, B:300–305 disordered systems, B:291–293 microscopic models, B:293–299 fractal structures, anomalous diffusion, B:257–264 generalized master equations, A:375–378 aging phenomena applications, A:429–431 formal approach, A:423–425 non-Poisson processes, A:421–423 anomalous diffusion, perturbation response, A:405–407 blinking quantum dot complexity, A:456–461 non-Ohmic bath, A:456–459 physics parameters, A:466–467 recurrence variables, A:459–461 canonical equilibrium, dynamic approach, A:409–410 comparisons of, A:464–465 complexity theory, A:461–463 intermittent dynamic model, A:378–384 experimental vs. theoretical laminar regions, A:379–381 Manneville map, A:381–383 Nutting law, A:383–384 limitations of, A:431–435 noncanonical equilibrium dynamic approach, A:410–413 information approach, A:407–409 non-Poisson processes decoherence theory, A:435–451 dichotomous noise/higher-order correlation functions, A:414–421 DF property, A:415–416 four-time correlation function, A:416–421 modulation theories, A:450–456 nonordinary environment, A:439–441 quantum-to-classical transition, A:441–447

trajectory and density entropies, A:447–450 overview, A:358–362 phenomenological approach, A:362–364 quantum-like formalism, correlation functions, A:425–429 quantum measurement processes, A:467–469 scaling property differences, A:463–464 superdiffusion condition, A:384–404 central limit theorem, A:385–388 Gaussian case, A:399–401 generalized master equation vs., A:392–399 Le´vy derivation, A:388–392 multiscaling, A:403–404 trajectories approach, A:401–403 Zwanzig projection method, A:364–375 Anderson localization, A:368–374 Le´vy ﬂight processes, theoretical background, B:445–448 overview, A:358–362 ‘‘strange kinetic’’ percolation, A:76–81 Convolution relations continuous time random walk, trajectory approach, A:402–403 Fourier transformation, A:324–325 Convolution theorem, symmetric relaxation, peak broadening in complex systems, A:107–110 Cooperative dynamic/scaling phenomena, disordered systems, A:55–116 conﬁned relaxation kinetics, A:95–105 dielectric spectrum broadening, A:105–116 porous materials, static percolation, A:55–65 ‘‘strange kinetic’’ percolation, A:73–81 Cooperative relaxation phenomenon hydrogen-bonding liquids, A:51–55 microemulsion structure and dynamics, A:33– 38 Copper ions, ferroelectric crystals conﬁned system relaxation kinetics, A:100– 102 dielectric relaxation, A:44–48 liquid-like behavior, A:93–95 Correlation function blinking quantum dots aging phenomena, A:338 nonergodicity, time-averaged correlation function, A:347–350 Brownian motion aging phenomena Langevin model

subject index displacement response, A:280 velocity correlation function, A:279–280 quantum motion displacement response, A:289–290 velocity correlation function, A:285–286 continuous time random walk quantum-like formalism, A:425–429 vs. generalized master equation, A:393–399 dichotomous ﬂuctuations, B:35–41 noncanonical equilibrium, dynamic approach to, A:411–413 non-Ohmic bath, A:458–459 non-Ohmic dissipation, aging effects, MittagLefﬂer relaxation, A:298–299 non-Poisson dichotomous noise, higher-order functions, A:414–421 out-of-equilibrium physics, A:259–261 overdamped classical Brownian motion aging, A:277–278 supercooled liquids and glasses b-peak glass formation, A:203–209 formation above glass transition temperature, dynamic susceptibility, A:165–169 molecular reorientation, A:131–134 Zwanzig projection method, Anderson localization, A:373–374 Correlation length, chaotic fractal percolation clusters, B:132–137 lattice structural models, B:149–157 renormalization-group transformations, B:138–141 Correspondence principle, Zwanzig projection method, Anderson localization, A:374 Coulomb interaction blinking quantum dots, diffusion model, A:332–334 supercooled liquids and glasses, depolarized light scattering, A:142–148 Coupled particle equation of motion, statistical mechanics, Caldeira-Leggett model, A:262–264 Coupling model (CM), vitriﬁcation of liquids, B:501–502 basic properties, B:501–502 binary component dynamics, B:567–574 dispersion invariance to temperature and pressure combinations, B:561

681

enthalpy activation in glassy state, B:555–556 many-molecule dynamics, B:577–580 nanometric relaxation, B:562–567 polymer systems primary/secondary relaxation interrelations, B:574–577 pressure-temperature dependence above glass transition temperature, B:556–561 primitive relaxation, B:529–530 structural relaxation and Johari-Goldstein relaxation correspondence, B:551–555 theoretical background, B:546–551 Coupling spectral density, phenomenological dissipation modeling, A:265–266 Critical exponents, supercooled liquids and glasses, mode-coupling theory, A:159–162 Critical indices chaotic fractal structure Coniglio-Sarychev-Vinogradov and ScalaShklovsky models, B:141–144 lattice structural models, B:150–157 linear elasticity, B:144–147 percolation clusters, B:131–137 fractal elasticity, B:207–209 Le´vy ﬂight processes power-law asymptotics of stationary solutions, B:455–456 stationary quartic Cauchy oscillator, B:453–454 superharmonicity, B:464–467 multifractals, B:122–125 Critical slowing down effect, microemulsion structure and dynamics, A:38 Critical temperature, supercooled liquids and glasses dynamic susceptibility, A:230 mode coupling tests, A:221–225 nonergodicity parameter, A:158–162 Cross-correlation function, supercooled liquids and glasses, molecular reorientation, dielectric spectroscopy, A:136–140 Crossover temperature Brownian motion aging effects, quantum model, A:287–288 Ohmic temperature, A:294–295 supercooled liquids and glasses mode-coupling theory, A:159–162

682

subject index

Crossover temperature (continued) nonergodicity parameter, A:216–220 vitriﬁcation of liquids, coupling model, B:546–551 Cube cell fractal structures conductivity, iterative averaging method, B:169–174 dielectric properties, B:176–183 galvanomagnetic properties, B:190–204 Cumulant operator expansion, dielectric relaxation, Cole-Davidson and Havriliak-Negami behavior, B:315–316 Cylindrical capacitor, time-domain spectroscopy, A:23–24 Data analysis allometric aggregation, scaling behavior, B:5–10 dielectric spectroscopy, software for, A:30 multifractals, B:42–43 diffusion entropy analysis, B:46–52 spectrum of dimensions, B:42–46 time-domain spectroscopy, A:25 Debye-Fro¨lich model, dielectric relaxation Cole-Davidson and Havriliak-Negami behavior, B:316 continuous-time random walk model, B:299 disordered systems, B:289–293 fractional Smoluchowski equation solution, B:324–325 Debye relaxation dielectric polarization, time-dependent electric ﬁelds, A:8 dielectric relaxation Cole-Cole behavior, B:304–305 Cole-Davidson and Havriliak-Negami behavior, B:313–316 fractional Smoluchowski equation solution, B:316–325 inertial effects fractional Klein-Kramers equation, B:366–372 linear and symmetrical top molecules, B:381–388 periodic potentials, B:400–414 noninertial rotational diffusion, B:305–312 failure in disordered systems, B:286–293 fractal structures mean relaxation time, B:239–253 shear-stress relaxation, B:234–235

fractional rotational diffusion, direct current (DC) electric ﬁeld, B:347 vitriﬁcation of liquids, coupling model, B:548–551 Debye-Stokes-Einstein (DES) equation, vitriﬁcation of liquids binary mixtures, B:570–574 structural relaxation-dispersion coordination, B:520–528 Debye-Waller factor, supercooled liquids and glasses below glass transition temperature, A:233–241 nonergodicity parameter, A:157–162 nonergodicity parameter, temperature dependence, A:215–220 Decoherence theory continuous time random walk, phenomenological modeling, A:364 generalized master equation, A:364 non-Poisson renewal processes, A:435–439 nonordinary environment, A:439–441 quantum to classical transition, A:441–447 trajectory and density entropies, A:447–451 quantum mechanics measurements, A:468–469 Zwanzig projection method, Anderson localization, A:373–374 Decoupling phenomenon, supercooled liquids and glasses, formation above glass transition temperature, A:169–173 Deformation polarization, static electric ﬁelds, A:5 Degrees of freedom, ‘‘strange kinetic’’ percolation, A:78–81 Delta function, fractional dynamics, B:64–73 Delta trick procedure, continuous time random walk, vs. generalized master equation, A:396–399 Density entropy, non-Poisson renewal processes, A:447–451 Dephasing term, non-Poisson renewal processes, decoherence theory, A:436–439 Depolarized light scattering, supercooled liquids and glasses, molecular reorientation, A:140–148 Derivatives fractal function, B:57–59 fractal structure analysis, B:95–96

subject index conductivity galvanomagnetic properties, B:192–204 iterative averaging method, B:167–174 non-Debye relaxation, B:236–253 fractional calculus, B:55–61 fractional diffusion equations, B:75–76 Detailed balance principle, dielectric relaxation, Cole-Cole behavior, B:302–305 ‘‘Devil’s Stairs’’ construction, fractal sets, B:104–105 Dichotomous Factorization (DF) assumption continuous time random walk correlation function, A:428–429 vs. generalized master equation, A:395–399 non-Poisson statistics, higher-order correlation functions, A:414–421 DF property, A:415–416 four-time correlation function, A:416–421 Dichotomous ﬂuctuations, fractal physiology, B:35–41 early time behavior, B:40–41 exact solution, B:39–40 Dielectric frequency dependence, fractal physical properties, B:174–183 iterative averaging method, B:175–183 Dielectric loss hyperbranched polymer ﬁlm conﬁnement effects, B:615–620 cis-1,4-polyisoprene ﬁlm preparation, B:602–607 supercooled glass formation below glass transition temperature, A:233–241 thin polymer ﬁlm preparation, B:597–601 Dielectric permittivity broadband dielectric spectroscopy, A:17–18 dc conductivity, A:26–27 fractal relaxation, B:239–253 microemulsion structure and dynamics, A:32–38 porous glasses, A:38–41 static electric ﬁelds, A:4–7 supercooled liquids and glasses, molecular reorientation, dielectric spectroscopy, A:135–140 Dielectric relaxation complex systems broadband dielectric spectroscopy methods, A:16–18 data ﬁtting problems, A:25–30 continuous parameter estimation, A:25–27

683

dc-conductivity problems, A:26–27 Hilbert transform dc-conductivity computation, A:28–30 software tools for, A:30 frequency and time domains, A:8–12 kinetic mechanisms, A:12–15 overview, A:2–3 spectroscopic and data analysis principles, A:15–30 static electric ﬁelds, A:3–7 time-dependent electric ﬁeld, A:7–12 time-domain spectroscopy, A:18–25 data processing, A:25 hardware tools, A:21–22 nonuniform sampling, A:22–23 sample holders, A:23–24 conﬁned system relaxation kinetics, A:104–105 disordered systems, A:30–55 conﬁned relaxation kinetics, A:95–105 cooperative dynamic/scaling phenomena, A:55–116 porous materials, static percolation, A:55–65 Debye noninertial rotational diffusion model, B:305–312 dielectric spectrum broadening, A:105–116 ferroelectric crystals, A:44–48 liquid-like behavior, A:93–95 H-bonding liquids, A:48–55 scaling behavior, A:81–93 microemulsions, A:31–38 dynamic percolation, A:65–73 microscopic models, B:293–325 continuous-time random walk model, B:294–299 fractional diffusion equations Cole-Cole model, B:300–305 Cole-Davidson/Havriliak-Negami behavior, B:313–316 fractional Smoluchowski equation, B:316–325 porous materials, A:38–44 glasses, A:38–41 silicon, A:41–44 ‘‘strange kinetic’’ percolation, A:73–81 theoretical principles, B:286–293 in double-well periodic potential, anomalous diffusion and, B:331–338 fractal structures, B:237–253

684

subject index

Dielectric relaxation (continued) fractional rotational diffusion, in potentials, B:329–331 inertial effects fractional Klein-Kramers equation Barkai-Silbey format, B:372–379 linear and symmetrical top molecules, B:380–398 Metzler/Klafter format, B:365–372 periodic potentials, B:398–414 fractional Langevin equation, B:414–419 linear and symmetrical top molecules, B:380–398 fractional rotational diffusion, B:388–398 rotators in space, B:380–388 periodic potentials, B:398–414 theoretical principles, B:364–365 vitriﬁcation of liquids amorphous polymers, B:507–509 molecular glass-forming liquids, B:503–507 structural relaxation time, B:509–511 Dielectric spectroscopy (DS) basic principles, A:3, A:18–30 continuous parameter estimation, A:27–30 disordered materials, A:105–116 microcomposite materials, A:113–116 symmetric relaxation peak broadening, A:106–110 polymer-water mixtures, A:110–113 frequency band, A:15–16 microemulsion structure and dynamics, A:32–38 supercooled liquids and glasses formation above glass transition temperature b-peak glass formation, A:201–209 dynamic susceptibility, A:163–169 high-temperature regime, A:173–182 low-temperature regime, A:182–201 future research issues, A:241–243 mode coupling tests, A:221–225 dynamic susceptibility, A:228–230 two-correlator models, A:225–228 molecular reorientation, A:134–140 coaxial reﬂection and transmission techniques, A:139 frequency response analysis, A:139 quasi-optical and FTIR spectroscopy, A:140

time-domain analysis, A:137–139 nonergodicity parameter, temperature dependence, A:216–220 Differential-recurrence equation continued fraction solution, longitudinal and transverse responses, B:425–427, B:426–427 dielectric relaxation, inertial effects, Barkai/ Silbey fractionalized KleinKramers equation, B:375–379 fractional rotational diffusion bistable potential with nonequivalent wells, B:351–354 direct current (DC) electric ﬁeld, B:344–347 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:331–338 Kerr-effect resposne, B:432–434 Differential scanning calorimetry (DSC) ferroelectric crystals, liquid-like behavior, A:94–95 hydrogen-bonding liquids, scaling phenomena, water-rich mixtures, A:86–93 hyperbranched polymer ﬁlm conﬁnement effects, B:614–620 Diffusing particles, ﬂuctuation-dissipation theorem, A:259–261 Diffusion anomalous diffusion aging phenomena, non-Ohmic models classical aging effects, A:302–303 noise and friction, A:296–297 one- and two-time dynamics, A:297–303 Mittag-Lefﬂer relaxation, A:298–299 particle coordinate and displacement, A:299–300 time-dependent quantum diffusion coefﬁcient, A:300–302 velocity correlation function, A:299 quantum aging effects, A:303 out-of-equilibrium physics Kubo generalized susceptibilities, A:312–315 effective temperature, A:312–313 friction coefﬁcient, A:314–315 mobility modiﬁcation, A:313–314 linearly coupled particle equation of motion, A:307–308

subject index linear response theory, A:309–312 age- and frequency-dependent response functions, A:309–311 quasi-stationary regime, effective temperature, A:311–312 overview, A:303–304 power-law behaviors, A:317–319 friction coefﬁcient, A:317–318 mean-square displacement, A:318 temperature determination, A:318–319 stationary medium, A:308–309 Stokes-Einstein relation, A:315–317 thermal bath, A:304–307 ﬂuctuation-dissipation theorems, A:304–306 regression theorem, A:306–307 blinking quantum dots, A:331–334 nonergodicity, A:345–346 fractal structures, B:95–96, B:235– 277 anomalous diffusion, B:253–264 Brownian motion, B:253–264 controlled aggregation, B:97–98 dielectric relaxation, B:237–246 distribution function of Brownian particles, B:264–267 fractional derivatives diffusion equation, B:269–274 inertial effects of Brownian particle, B:267–277 non-Debye relaxation, B:235–253 one-dimensional lattice random walks, B:267–269 shear-stress relaxation, B:232–235 Smoluchowski equation for Brownian particles, B:255–257 fractional diffusion equations, B:73–76 scaling dynamics fractional random walks, B:31–34 inverse power-law autocorrelation functions, B:34–35 simple random walks, B:28–31 in time series, B:27–35 supercooled liquids and glasses formation above glass transition temperature, time constants and decoupling phenomenon, A:171–173 mode coupling tests, A:222–225 symmetric relaxation, peak broadening in complex systems, A:110

685

Diffusion entropy analysis (DEA) anomalous diffusion, B:84 multifractal spectrum, B:46–52 scaling dynamics fractional random walks, B:34 inverse power-law autocorrelation functions, B:35 Le´vy random walks, B:84 Digital sampling oscilloscopes, time-domain spectroscopy, A:21–22 Diglycidyl ether of bisphenol A (DGEBA), supercooled glass formation, above glass transition temperature, lowtemperature regime, A:200–201 Dilatometry, hyperbranched polymer ﬁlm conﬁnement effects, B:618–620 Dimensionless variables, Le´vy ﬂight processes, barrier crossing, B:474–475 Dipolarons, supercooled liquids and glasses, depolarized light scattering, A:142–148 Dipole correlation function (DCF) dielectric response, A:10–12 microemulsion structure and dynamics, A:33–38 dynamic percolation, ionic medium, A:65–67 static percolation, porous materials, A:55–65 glasses, A:58–63 silicons, A:63–65 ‘‘strange kinetic’’ percolation, A:79–81 Dipole-induced-dipole (DID), supercooled liquids and glasses, depolarized light scattering, A:141–148 Dipole systems dielectric relaxation Cole-Cole behavior, B:300–305 Debye noninertial rotational diffusion, B:305–312 fractional Smoluchowski equation solution, B:316–325 inertial effects, periodic potentials, B:399–414 overview, B:286–293 fractional rotational diffusion bistable potential with nonequivalent wells, B:347–364 direct current (DC) electric ﬁeld, B:338–347 in potentials, B:327–331

686

subject index

Dirac delta function chaotic fractal percolation clusters, renormalization-group transformations, B:140–141 dielectric relaxation Cole-Davidson and Havriliak-Negami behavior, B:314–316 disordered systems, B:290–293 fractal structures conductivity, iterative averaging method, B:168–174 dielectric properties, B:176–183 Hall’s coefﬁcient, B:189–204 shear-stress relaxation, B:232–235 Le´vy ﬂight processes, theoretical background, B:443–448 non-Ohmic bath, A:458–459 Direct current (DC) conductivity dielectric spectroscopy, A:26–27 ferroelectric crystals, A:47–48 Hilbert transform, A:28–30 porous silicon, A:42–44 fractional rotational diffusion, B:338–347 hydrogen-bonding liquids, scaling phenomena, glycerol-rich mixtures, A:84–86 Dirichlet function fractal sets, B:98 Le´vy ﬂight processes, method of images inconsistency, B:472–474 Discrete-time random walk, dielectric relaxation, B:294–299 Disease mechanisms, fractional dynamics and, B:86–87 Disordered media physics, fractal theory and, B:96–98 Disordered systems cooperative dynamic/scaling phenomena, A:55–116 conﬁned relaxation kinetics, A:95–105 dielectric spectrum broadening, A:105–116 porous materials, static percolation, A:55–65 ‘‘strange kinetic’’ percolation, A:73–81 dielectric relaxation, A:30–55 Debye noninertial rotational diffusion model, B:305–312 ferroelectric crystals, A:44–48 liquid-like behavior, A:93–95 H-bonding liquids, A:48–55 scaling behavior, A:81–93

microemulsions, A:31–38 dynamic percolation, A:65–73 microscopic models, B:293–325 continuous-time random walk model, B:294–299 fractional diffusion equations Cole-Cole model, B:300–305 Cole-Davidson/Havriliak-Negami behavior, B:313–316 fractional Smoluchowski equation, B:316–325 porous materials, A:38–44 glasses, A:38–41 silicon, A:41–44 theoretical principles, B:286–293 time-domain spectroscopy, A:22–23 Dispersion mechanisms fractal dielectric relaxation, B:244–253 vitriﬁcation of liquids basic principles, B:498–502 coupling model, temperature/pressure dependence, B:561 structural relaxation properties, B:516–528 time and pressure invariance, B:502–516 amorphous polymers, B:507–509 molecular glass-formers, B:503–507 photon correlation spectroscopy analysis, B:512–516 structural relaxation time, B:509–516 Displacement response Brownian motion aging phenomena Langevin model, A:280 overdamped classical Brownian motion, A:277–278 quantum motion, A:288–289 time-dependent diffusion coefﬁcient and, A:289–291 non-Ohmic dissipation, aging effects, particle coordinates, A:299–300 Dissipative coupling, fractional rotational diffusion, potential phenomena, B:325–331 Dissipative nonlinearity, Le´vy ﬂight processes, B:481–487 asymptotic behavior, B:483–484 dynamical equation, B:482–483 nonlinear friction term, B:481–482 probability density function, B:485–486 quadratic/quartic numerical solutions, B:484–485

subject index Dissipative systems fractional dynamics, physical/physiological models, B:66–73 statistical mechanics Caldeira-Leggett model, A:262–264 Ohmic dissipation, A:267–268 phenomenological modeling, A:265–268 Distribution function continuous time random walk, Manneville map, A:382–383 dielectric relaxation, inertial effects, symmetric top molecules, B:389–398 fractal structures Brownian particles with memory, B:264–267 conductivity Hall’s coefﬁcient, B:189–204 iterative averaging method, B:168–174 viscoelasticity, shear-stress relaxation, B:234–235 fractional rotational diffusion bistable potential with nonequivalent wells, B:349–364 direct current (DC) electric ﬁeld, B:340–347 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:331–338 in potentials, B:326–331 Le´vy ﬂight processes, Sparre Anderson universality, B:471–472 Doorway variable canonical equilibrium, A:410 noncanonical equilibrium, A:411–413 Double-well periodic potential anomalous diffusion and dielectric relaxation, B:331–338 dielectric relaxation, inertial effects, B:401–414 fractional rotational diffusion, bistable potential with nonequivalent wells, B:347–364 Le´vy ﬂight processes, barrier crossing, B:474–475 tunneling models, A:235 Drude dielectric function, fractal properties, frequency dependence, B:174–183 Dynamical equations, Le´vy ﬂight processes, dissipative nonlinearity, B:482–483 Dynamical fractals, deﬁned, B:4

687

Dynamic Kerr-effect reponse, linear molecules, B:427–429 dielectric relaxation, inertial effects, B:384–388 symmetric top molecules, B:394–398 Dynamic susceptibility, supercooled liquids and glasses basic properties, A:129–131 depolarized light scattering, A:140–148 formation above glass transition temperature, A:163–169, A:228–231 Dyre-Olsen minimal model, fractional rotational diffusion, bistable potential with nonequivalent wells, B:363–364 Effective-medium model, fractal conductivity, B:160, B:163 galvanomagnetic properties, B:188–204 negative Poisson’s ratio, B:212–215 Einstein diffusion coefﬁcient, Brownian motion aging effects Langevin model, A:281–283 quantum model, A:287–288 Einstein relation anomalous diffusion dynamics, perturbation response, A:405–407 dielectric relaxation, Cole-Cole behavior, B:300–305 Le´vy ﬂight processes, theoretical background, B:442–448 out-of-equilibrium physics, effective temperature and generalized susceptibilities, A:312–315 scaling dynamics, simple random walks, B:29–31 thermal bath diffusion, A:304–305 Einstein-Smoluchowski equation dielectric relaxation, disordered systems, B:287–293 fractal structures, Brownian particle inertial effects, B:271–274 Einstein-Stokes relation, Le´vy ﬂight processes, theoretical background, B:442–448 Elasticity, fractal structures, B:204–209 negative Poisson’s ratio, B:211–215 viscoelasticity frequency dependence, B:216–235 Elastic neutron scattering, vitriﬁcation of liquids, coupling model, structuralJohari-Goldstein relaxation correspondence, B:553–555

688

subject index

Electrical phenomena fractal-based time series analysis, neuron ﬁring, B:23–26 fractional rotational diffusion, direct current (DC) electric ﬁeld, B:338–347 Electric polarizability tensor, supercooled liquids and glasses, depolarized light scattering, A:141–148 Electroencephalography, physiological time series, neuron action potentials, B:25–26 Electromagnetic wavelength, broadband dielectric spectroscopy, A:16–18 Electron-hole pairing, blinking quantum dots, diffusion model, A:332–334 Electron paramagnetic resonance (EPR), supercooled liquids and glasses dynamic susceptibility, A:230 nonergodicity parameter, A:218–220 Electron polarization, static electric ﬁelds, A:5 End-to-end distances, cis-1,4-polyisoprene thin ﬁlm preparation, B:608–614 Enhanced diffusion, dielectric relaxation, inertial effects, Barkai-Silbey fractional Klein-Kramers equation, B:373–379 Ensemble-averaged correlation function, blinking quantum dots aging phenomena, A:337–342 on-off aging correlation function, A:338 stochastic models, A:336 Ensemble averaging, Le´vy ﬂight processes, theoretical background, B:444–448 Enthalpy Johari-Goldstein secondary relaxation, polymer vitriﬁcation, B:540–541 vitriﬁcation of liquids, coupling model, structural-Johari-Goldstein relaxation correspondence, B:555–556 Entropy-temperature dependence, vitriﬁcation of liquids basic principles, B:501–502 Johari-Goldstein secondary relaxation, B:540–541 molecular mobility and, B:541–545 Epoxy resins, structural and Johari-Goldstein relaxation interrelations, B:574–577 Equations of motion dichotomous ﬂuctuations, B:38–41

fractals, B:2–3 non-Poisson dichotomous noise, A:415–416 out-of-equilibrium environment, linearly coupled particles, A:307–308 statistical mechanics, Caldeira-Leggett model, A:262–264 Equivalent static percolation cluster (ESPC), hyperscaling relationship, A:68–71 Ergodic hypothesis, blinking quantum dots, stochastic models, A:336 Euclidean dimension, polymer-water mixtures, dielectric relaxation, A:110–113 Euclidean space, fractal sets Haussdorff-Besicovitch dimension, B:116–119 square lattice construction, B:126–128 Euler-Lagrange equations, fractals, B:2 Evenness property, allometric aggregation data analysis, scaling behavior, B:6–10 Excess wing (EW) function hydrogen-bonding liquids dielectric response, A:52–55 scaling behavior, glycerol-rich mixtures, A:81–86 supercooled liquids and glasses b-peak glass formation, A:205–209 formation above glass transition temperature, dynamic susceptibility, A:163–169 molecular glass formation above melting point, low-temperature regime, A:182–201 vitriﬁcation of liquids coupling model, structural-JohariGoldstein relaxation correspondence, B:552–555 Johari-Goldstein secondary relaxation, B:531–535 Excitation transfer, static percolation, porous materials, A:56–65 glasses, A:58–63 Experimental laminar regions, continuous time random walk, intermittent dynamic model, A:379–381 Exponential logarithmic laws, fractal structures, non-Debye relaxation, B:236–253 Extreme quantum case, ﬂuctuation-dissipation theorem, A:274–276 Eyring law, relaxation kinetics, A:12–15

subject index Fabry-Perot interferometer, supercooled liquids and glasses, A:130–131 depolarized light scattering, A:143–148 Fast Fourier Transform, dc conductivity computation, Hilbert transform, A:29–30 Fast relaxation, microemulsion structure and dynamics, A:33–38 Feigenbaum number, fractal sets, Verhulst dynamics, B:114–115 Feller pseudodifferential operator, fractional diffusion equations, B:75–76 Feller’s subordination process, fractional dynamics, B:68–73 Ferroelectric crystals conﬁned system relaxation kinetics, A:100–102 dielectric relaxation, A:44–48 liquid-like behavior, A:93–95 Fick’s law fractal anomalous diffusion Brownian motion, B:256–257 Brownian particle inertial effects, B:271–274 Le´vy ﬂight processes, theoretical background, B:442–448 Finite-dimensional calculations, fractal structural models, B:155–157 Finite lattice properties, fractal structural models, B:147–157 Finite variance statistical methods (FVSM) heart beat fractals, B:11–14 multifractal spectrum, B:52 physiological time series, scaling behavior, B:10 scaling dynamics, fractional random walks, B:33–34 First-order approximation fractional Fokker-Planck equation, Gru¨nwaldLetnikov solution, B:488–492 Le´vy ﬂight processes bifurcation time, B:461–464 differential equations, B:460–461 First passage time density (FPTD), Le´vy ﬂight processes, B:467–474 ﬁrst arrival time, B:467–470 method of images inconsistency, B:472–474 Sparre Anderson universality, B:470–472 Fitting equation, multifractal analysis, B:45–46

689

Fitting function dielectric spectra, continuous parameter estimation, A:25–26 hydrogen-bonding liquids, A:51–55 Fixed axis rotation model dielectric relaxation Debye noninertial rotational diffusion, B:306–312 inertial effects Barkai-Silbey fractional Klein-Kramers equation, B:373–379 linear and symmetrical top molecules, B:380–388 Metzler/Klafter fractional KleinKramers equation, B:365–372 top molecule rotations, B:385–388 fractional rotational diffusion, bistable potential with nonequivalent wells, bimodal approximation, B:357–364 Fluctuation-dissipation theorems (FDTs) Brownian motion aging effects Langevin model, A:280–281 temperature effects, A:283–284 modiﬁed quantum theorem, A:291–292 overdamped classical Brownian motion, A:278–279 quantum displacement response and timediffusion coefﬁcient, A:289–291 dielectric relaxation, inertial effects, fractional Langevin equation, B:414–419 fractional dynamics, physical/physiological models, B:66–73 noncanonical equilibrium, A:412–413 non-Ohmic dissipation, aging effects classical model, A:302–303 noise and friction, A:296–297 time-dependent diffusion coefﬁcient, A:300–302 non-Poisson renewal processes, decoherence theory, A:439 out-of-equilibrium physics and, A:259–261 thermal bath diffusion, A:304–307 phenomenological dissipation modeling, A:266 supercooled liquids and glasses, molecular reorientation, A:134 dielectric spectroscopy, A:135–140 time-domain formulation, A:268–276 classical limit, A:273–274 dissipative response function, A:271–273

690

subject index

Fluctuation-dissipation theorems (continued) extreme quantum case, A:274–276 symmetrized correlation function, A:269–273 Zwanzig projection method, Anderson localization, A:372–374 Fluctuation power spectrum, supercooled liquids and glasses, molecular reorientation, A:134 Fokker-Planck equation. See also Fractional Fokker-Planck equation canonical equilibrium, A:410 dielectric relaxation Cole-Cole behavior, B:304–305 Cole-Davidson and Havriliak-Negami behavior, B:313–316 continuous-time random walk model, B:299 disordered systems, B:288–293 fractional Smoluchowski equation solution, B:316–325 inertial effects Barkai-Silbey fractional Klein-Kramers equation, B:374–379 basic principles, B:364–365 fractional Langevin equation, B:417–419 linear and symmetrical top molecules, B:381–388 Metzler/Klafter fractional Klein-Kramers equation, B:366–372 symmetric top molecules, B:388–398 dielectric response, A:11–12 fractal systems Brownian motion, B:255–257 Brownian particles with memory, B:264–267 fractional rotational diffusion bistable potential with nonequivalent wells, B:348–364 bimodal approximation, B:355–364 direct current (DC) electric ﬁeld, B:338–347 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:331–338 in potentials, basic principles, B:325–331 Le´vy ﬂight processes conﬁnement and multimodality, B:453–467 differential equations, B:461 space-fractional Fokker-Planck properties, B:450–451

theoretical background, B:441–448 Zwanzig projection method, A:367 Fourier transform (FT) broadband dielectric spectroscopy, A:18 Brownian motion aging phenomena Langevin classical model, velocity correlation function, A:280 quantum model, velocity correlation function, A:285–286 continued fraction solution, longitudinal and transverse responses, B:425–427 continuous time random walk generalized central limit theorem, A:386–388 Le´vy dynamic derivation, A:389–392 noncanonical equilibrium, A:408–409 trajectory approach, A:402–403 vs. generalized master equation, A:395–399 dc conductivity computation, Hilbert transform, A:29–30 dichotomous ﬂuctuations, B:39–40 dielectric relaxation Debye noninertial rotational diffusion, B:311 fractional Smoluchowski equation solution, B:317–325 inertial effects Barkai/Silbey factional Klein-Kramers equation, B:375–379 Metzler/Klafter fractionalized Klein-Kramers equation, B:366–372 periodic potentials, B:401–414 equations, A:323–324 ﬂuctuation-dissipation theorem analytic signal, A:275–276 response function, A:273 fractal structures anomalous diffusion, B:260–264 Brownian particles with memory, B:265–267, B:272–274 dielectric relaxation, B:242–253 viscoelasticity frequency dependence, B:217–235 fractional Brownian motion, B:59–61 fractional diffusion equations, B:74–76 fractional rotational diffusion bistable potential with nonequivalent wells, matrix continued fraction solution, B:353–354 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:331–338

subject index Le´vy ﬂight processes frequency domain induction equations, B:452–453 Langevin equation with, B:77–80 stationary quartic Cauchy oscillator, B:453–454 theoretical background, B:446–448 multifractal spectrum, diffusion entropy analysis, B:48–52 noncanonical equilibrium, A:412–413 out-of-equilibrium environment age- and frequency-dependent response functions, A:310–311 Stokes-Einstein relation, A:316–317 scaling dynamics fractional random walks, B:32–34 simple random walks, B:31 supercooled liquids and glasses, molecular reorientation, A:133–134 dielectric spectroscopy, A:135–140 time-domain spectroscopy, A:137–139 time-domain spectroscopy, A:20–21 Fourier transform infrared (FTIR) spectroscopy, supercooled liquids and glasses, molecular reorientation, A:140 Four-time correlation function, non-Poisson dichotomous noise, A:416–421 Fox functions dielectric relaxation Cole-Cole behavior, B:305 fractional Smoluchowski equation solution, B:319–325 fractal structures, dielectric relaxation, B:242–253 inverse Fourier transform calculations, B:422–424 Le´vy ﬂight processes, ﬁrst arrival time problems, B:469–470 Fractals Bolzano function, B:98–101 Cayley tree, B:128–131 chaotic structures, B:131–160 conductivity, B:142–144 ﬁnite lattice properties, B:155–157 linear elasticity, B:144–147 models, B:147–160 nucleating cell probability functions, B:157–160

691

percolation systems, B:131–147 cluster structures, B:131–132 critical indices, B:132–137 physical properties, B:141–147 renormalization-group transformations, B:137–141 disordered media physics, B:96–98 dielectric relaxation, continuous-time random walk model, B:295–299 dynamic percolation, ionic microemulsions, static/dynamic dimensions, A:71–73 Hausdorff-Besicovich dimensions, B:115–119 life science applications, B:4 multifractals lognormal distribution, B:123–125 structural properties, B:119–125 nowhere differential functions, B:98–101 physical properties conductivity, B:142–144, B:160–174 effective medium model, B:163 iterative averaging method for, B:164–174 Maxwell model, B:163 variational model, B:164 dielectric frequency dependence, B:174–183 iterative averaging method, B:175–183 elastic properties, B:204–209 galvanomagnetic properties, B:183–204 cube cell properties, B:198–199 Hall’s coefﬁcient, iterative averaging method for, B:189–198 layered structures, B:199–204 linear elasticity, B:144–147 negative Poisson ratio, B:209–215 relaxation and diffusion, B:235– 277 anomalous diffusion, B:253–264 Brownian motion, B:253–264 dielectric relaxation, B:237–246 distribution function of Brownian particles, B:264–267 fractional derivatives diffusion equation, B:269–274 inertial effects of Brownian particle, B:267–277 non-Debye relaxation, B:235–253 one-dimensional lattice random walks, B:267–269 Smoluchowski equation for Brownian particles, B:255–257 viscoelastic frequency dependence, B:215–235

692

subject index

Fractals (continued) iterative averaging method, B:221–222 negative shear modulus, B:225–231 shear stress relaxation, B:232–235 two-phase calculations, B:222–225 physiology and complexity dichotomous ﬂuctuations with memory, B:35–41 early time behavior, B:40–41 exact solution, B:39–40 dynamic scaling models, B:26–52 fractional random walks, B:31–34 inverse power-law autocorrelation functions, B:34–35 simple random walks, B:28–31 in time series, B:27–35 multifractals and data processing, B:42–52 diffusion entropy analysis, B:46–52 spectrum of dimensions, B:42–46 theoretical background, B:2–4 time series scaling, B:4–26 allometric aggregation data analysis, B:5–10 breathing function, B:14–18 dynamic models, B:27–35 gait cycle, B:18–22 heartbeats, B:10–14 neurons, B:22–26 polymer-water mixtures, dielectric relaxation, A:110–113 set functions, B:101–115 Cantor set, B:103–104 conductivity, iterative averaging method, B:169–174 ‘‘Devil’s stairs,’’ B:104–105 Julia sets, B:110–113 Koch’s ‘‘snowﬂake,’’ B:102–103 Peano function, B:105–110 Sierpinski carpet, B:103 square lattice construction, B:125–128 Van der Waerden function, B:101–102 Verhulst dynamics, B:113–115 static percolation, porous materials, A:55–65 glasses, A:58–63 silicon, A:63–65 ‘‘strange kinetic’’ percolation, A:73–81 structural properties, B:94–96 symmetric relaxation, peak broadening in complex systems, A:108–110 Fractal set mass equation, chaotic fractal percolation clusters, B:136–137

Fractal time random walk, dielectric relaxation Cole-Cole behavior, B:305 continuous-time random walk model, B:299 disordered systems, B:292–293 inertial effects, fractionalized Klein-Kramers equation, B:368–372 Fractional Brownian motion (FBM) continuous time random walk, A:401 non-Ohmic bath, A:459 recurrence properties, A:459–461 Fractional derivative technique dielectric relaxation Cole-Davidson and Havriliak-Negami behavior, B:313–316 continuous-time random walk model, B:297–299 Smoluchowski equation solution, B:316–325 fractal structures anomalous diffusion, B:261–264 Brownian particle inertial effects, diffusion equation, B:269–274 Brownian particles with memory, B:265–267 classical set, B:275–277 non-Debye relaxation, B:236–253 Le´vy ﬂight processes, space-fractional Fokker-Planck equation, B:450–451 Fractional dynamics calculus, B:55–61 Brownian motion, B:59–60 differential equation, B:84 functional derivatives, B:57–59 dielectric relaxation, inertial effects, B:364–365 fractionalized Klein-Kramers equation, B:369–372 Langevin equations, B:61–73 diffusion equations, B:73–76 Le´vy statistics, B:76–80 physical/physiological models, B:65–73 scaling dynamics, random walks, B:31–34 theoretical background, B:52–55 Fractional Fokker-Planck equation Gru¨nwald-Letnikov solution, B:488–492 Le´vy ﬂight processes bifurcation time, B:462–464 Cauchy case, B:478–481 convergence corroboration, B:467 dissipative nonlinearity, B:483–487

subject index Fractional Klein-Kramers equation (FKKE), dielectric relaxation, inertial effects Barkai-Silbey format, B:372–379 fractional Langevin equation, B:416–419 linear and symmetrical top molecules, B:380–398 Metzler/Klafter format, B:365–372 periodic potentials, B:398–414 Fractional Langevin equation, dielectric relaxation, inertial effects, B:414–419 Fractional rotational diffusion basic principles and equations, B:73–76 dielectric relaxation Cole-Cole behavior, B:300–305 Cole-Davidson and Havriliak-Negami behavior, B:313–316 Debye noninertial rotational diffusion, B:305–312 disordered systems, B:292–293 inertial effects linear and symmetrical top molecules, rotators in space, B:380–388 symmetric top molecules, B:388–398 in potentials bistable potential, nonequivalent wells, B:347–364 bimodal approximation, B:355–364 matrix continued fraction solution, B:350–354 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:331–338 theoretical principles, B:325–331 uniform DC external ﬁeld, B:338–347 Fractional stress relaxation, fractional dynamics, B:63–73 Fractional time evolution, ‘‘strange kinetic’’ percolation, A:75–81 Fragility index, vitriﬁcation of liquids dispersion correlation with structural relaxation, B:516–517 Johari-Goldstein secondary relaxation, molecular mobility and, B:543–545 Frank-Lobb algorithm, fractal dielectric properties, frequency dependence, B:175 Free induction decay (FID), supercooled liquids and glasses nuclear magnetic resonance, molecular reorientation, A:211–214

693

spin-lattice relaxation and line-shape analysis, A:151–152 time windows and spin-lattice interactions, A:150 Free spectral range (FSR), supercooled liquids and glasses, depolarized light scattering, A:143–148 Free volume models relaxation kinetics, A:13–15 vitriﬁcation of liquids, temperature/pressure superpositioning, B:510–511 Frequency dependence dielectric relaxation, inertial effects fractionalized Klein-Kramers equation, B:366–372 periodic potentials, B:407–414 fractal structures dielectric properties, B:174–183 viscoelastic properties, B:215–235 iterative averaging method, B:221–222 negative shear modulus, B:225–231 shear stress relaxation, B:232–235 two-phase calculations, B:222–225 fractional rotational diffusion double-well periodic potential, anomalous diffusion and dielectric relaxation, B:336–338 in potentials, B:327–331 Le´vy ﬂight processes, induction equations, B:452–453 Frequency-dependent permittivity, time-domain spectroscopy, A:19–21 Frequency-dependent response functions, outof-equilibrium linear response theory, A:309–311 Frequency domain broadband dielectric spectroscopy, A:17–18, A:25 continuous parameter estimation, A:25–26 dielectric polarization, time-dependent electric ﬁelds, A:8–12 supercooled liquids and glasses depolarized light scattering, A:145–148 formation above glass transition temperature, low-temperature regime, A:184–201 molecular reorientation, dielectric spectroscopy, A:135–140 supercooled liquids and glasses, molecular reorientation, A:139

694

subject index

Frequency-temperature superposition (FTS) supercooled glass formation above glass transition temperature b-peak glass formation, A:201–209 dynamic susceptibility, A:167–169 high-temperature regime, A:176–182 low-temperature regime, A:192–201 mode coupling tests, A:220–228 two-correlator models, A:225–228 supercooled liquids and glasses, future research issues, A:242–243 Friction coefﬁcient non-Ohmic dissipation, aging effects, A:296–297 out-of-equilibrium physics Kubo formula, A:314–315 power-law behaviors, A:317–318 phenomenological dissipation modeling, coupling spectral density, A:265–266 Frobenius-Perron equation, non-Poisson processes, trajectory and density entropies, A:448–451 Fro¨hlich’s relaxation model dielectric relaxation, inertial effects, periodic potentials, B:398–414 fractal structures, dielectric relaxation, B:237–253 fractional rotational diffusion double-well periodic potential, anomalous diffusion and dielectric relaxation, B:335–338 potential phenomena, B:325–331 Frustration limited domains, supercooled liquids and glasses, glass transition phenomenon, A:157 Gait cycle, fractal-based time series analysis, B:18–22 Galvanomagnetic properties, fractal structures, B:183–204 cube cell properties, B:198–199 Hall’s coefﬁcient, iterative averaging method for, B:189–198 layered structures, B:199–204 Gamma function dielectric relaxation, fractional Smoluchowski equation solution, B:319–325 fractal structures, non-Debye relaxation, B:236–253

fractional calculus, B:56–61 inverse Fourier transform calculations, B:423–424 noncanonical equilbrium, A:413 non-Ohmic dissipation, aging effects, timedependent diffusion coefﬁcient, A:300–302 scaling dynamics, fractional random walks, B:31–34 ‘‘strange kinetic’’ percolation, A:75–81 Gaussian distribution continuous time random walk, A:399–401 dielectric relaxation Cole-Cole behavior, B:300–305 Debye noninertial rotational diffusion, B:312 inertial effects, fractional Langevin equation, B:415–419 fractal anomalous diffusion Brownian motion, B:256–257 distribution function of Brownian particles, B:267 fractional dynamics, B:64–73 Le´vy ﬂight processes ﬁrst arrival time problems, B:468–470 method of images inconsistency, B:473–474 theoretical background, B:443–448 multifractal spectrum, diffusion entropy analysis, B:47–52, B:50–52 noncanonical equilibrium, A:413 information approach, A:407–409 physiological time series, B:8–10 gait analysis, B:19–22 neuron action potentials, B:23–26 scaling dynamics fractional random walks, B:33–34 inverse power-law autocorrelation functions, B:34–35 simple random walks, B:29–31 statistical properties, Caldeira-Leggett model, A:264 Generalized central limit theorem (GCLT) continuous time random walk, A:385–388 Le´vy dynamic derivation, A:390–392 Le´vy ﬂight processes, theoretical background, B:444–448 multifractal spectrum, diffusion entropy analysis, B:48–52 non-Poisson renewal processes, decoherence theory, A:440–441

subject index Generalized gamma distribution (GGE), supercooled liquids and glasses b-peak glass formation, A:205–209 formation above glass transition temperature, low-temperature regime, A:187–201 Generalized Master Equation (GME) aging phenomena applications, A:429–431 formal approach, A:423–425 non-Poisson processes, A:421–423 anomalous diffusion, perturbation response, A:405–407 blinking quantum dot complexity, A:456–461 non-Ohmic bath, A:456–459 physics parameters, A:466–467 recurrence variables, A:459–461 canonical equilibrium, dynamic approach, A:409–410 complexity theory, A:461–463 continuous-time random walk, A:375–378 comparisons of, A:464–465 quantum measurement processes, A:467–469 intermittent dynamic model, A:378–384 experimental vs. theoretical laminar regions, A:379–381 Manneville map, A:381–383 Nutting law, A:383–384 limitations of, A:431–435 noncanonical equilibrium dynamic approach, A:410–413 information approach, A:407–409 non-Poisson renewal processes decoherence theory, A:435–451 dichotomous noise/higher-order correlation functions, A:414–421 DF property, A:415–416 four-time correlation function, A:416–421 modulation theories, A:450–456 nonordinary environment, A:439–441 quantum-to-classical transition, A:441–447 trajectory and density entropies, A:447–450 overview, A:358–362 phenomenological approach, A:362–364 quantum-like formalism, correlation functions, A:425–429 scaling property differences, A:463–464

695

superdiffusion condition, A:384–404 central limit theorem, A:385–388 continuous-time random walk vs., A:392–399 Gaussian case, A:399–401 Le´vy derivation, A:388–392 multiscaling, A:403–404 trajectories approach, A:401–403 Zwanzig projection method, A:364–375 Anderson localization, A:368–374 Generalized Weierstrass function (GWF) fractal function derivatives, B:57–59 fractional Brownian motion, B:59–61 Geometrical fractals, deﬁned, B:4 Gibbs ensemble anomalous diffusion dynamics, perturbation response, A:405–407 continuous time random walk, Gaussian stochastic process, A:400–401 generalized master equation, A:431–435 non-Poisson processes aging and, A:421–423 trajectory and density entropies, A:447–451 Zwanzig projection method, Anderson localization, A:368–374 Gibbs phase exponent, ‘‘strange kinetic’’ percolation, A:79–81 Gilroy-Philips model, fractional rotational diffusion, bistable potential with nonequivalent wells, B:361–364 Glass transition temperature. See also Vitriﬁcation of liquids basic principles of, B:498–502 deﬁned, A:128–131 hydrogen-bonding liquids, scaling phenomena, A:89–93 hyperbranched polymer ﬁlm conﬁnement effects, B:617–620 microcomposite material dielectric relaxation, A:113–116 polystyrene ﬁlms, conﬁnement effects, B:620–628 supercooled liquids and glasses basic properties, A:128–131 mode coupling theory, A:157–162 molecular glass formation above melting point, A:162–231 dynamic susceptibility evolution, A:163–169, A:228–231

696

subject index

Glass transition temperature (continued) high-temperature regime, A:173–182 low-temperature regime, A:182–214 b-peak glass formation, A:201–209 excess wing evolution, A:182–201 mode coupling theory tests, A:220–228 asymptotic scaling laws, A:221–225 two-correlator model, A:225–228 molecular reorientation mechanism, A:209–214 nonergodicity parameter temperature dependence, A:214–220 time constants and decoupling phenomenon, A:169–173 molecular glass formation below melting point, A:231–241 b-glass formation, A:235–241 low-temperature regime, constant loss phenomena, A:231–235 molecular reorientation dynamics, A:131–155 correlation function, spectrum, and susceptibility, A:131–134 depolarized light scattering, A:140–148 dielectric spectroscopy, A:134–140 coaxial reﬂection and transmission techniques, A:139 frequency response analysis, A:139 quasi-optical and FTIR spectroscopy, A:140 time-domain analysis, A:137–139 nuclear magnetic resonance, A:148–155 spin-lattice relaxation and line-shape analysis, A:150–152 stimulated echo experiments and twodimensional NMR, A:152–155 time windows and spin-lattice interactions, A:148–150 theoretical principles, A:155–162 thin polymer ﬁlms, B:595–596 vitriﬁcation of liquids coupling model, nanometric relaxation, B:565–567 Johari-Goldstein secondary relaxation, non-Arrhenius temperature dependence, B:536–538 many-molecule dynamics, B:577–580 Global characteristic times, dielectric relaxation, continuous-time random walk model, B:295–299

Glycerol dielectric relaxation, A:48–55 hydrogen-bonding liquids, scaling behavior, A:81–86 water-rich mixtures, A:86–93 Glycerol mixtures, sorbitol in, structural and Johari-Goldstein relaxations in, B:573–574 Gordon’s sum rule, dielectric relaxation Debye noninertial rotational diffusion, B:312 inertial effects, B:364–365 Barkai/Sibley fractionalized KleinKramers equation, B:378–379 Metzler/Klafter fractionalized KleinKramers equation, B:369–372 Green-Kubo method anomalous diffusion dynamics, perturbation response, A:405–407 noncanonical equilibrium, dynamic approach to, A:411–413 non-Poisson dichotomous noise, A:415–416 Green’s function dielectric relaxation, fractional Smoluchowski equation solution, B:319–325 fractional diffusion equations, B:75–76 Gross-Sack solution, dielectric relaxation, inertial effects Barkai/Silbey fractionalized Klein-Kramers equation, B:376–379 fractional Langevin equation, B:418–419 Metzler/Klafter fractionalized Klein-Kramers equation, B:368–372 Gru¨nwald-Letnikov method fractional Fokker-Planck solution, B:488–492 Le´vy ﬂight processes, B:467 method of images inconsistency, B:474 Guided-wave propagation, broadband dielectric spectroscopy, A:17–18 Hahn-echo sequence, supercooled liquids and glasses, spin-lattice relaxation and line-shape analysis, A:151–152 Hall-Helfand function, vitriﬁcation of liquids, coupling model, B:547–551 Hall parameters, fractal galvanomagnetic properties, B:185–204 Hamiltonian equations canonical equilibrium, A:410 dielectric relaxation, inertial effects, B:420–421

subject index statistical mechanics, Caldeira-Leggett model, A:262 ‘‘strange kinetic’’ percolation, A:74–81 Hardware tools, time-domain spectroscopy, A:21–22 Harmonic analysis fractal function derivatives, B:58–59 Le´vy ﬂight processes conﬁnement and multimodality, B:453–467 dissipative nonlinearity, B:481–487 physiological time series, neuron action potentials, B:25–26 Hashin-Strikman model fractal elastic properties, B:204–209 negative shear modulus, B:225–235 fractal structures, B:95–96 Haussdorff-Besicovitch measure fractional dimensions, B:115–119 Julia fractal sets, B:113 Havriliak-Negami (HN) law dielectric relaxation, A:9–12 disordered systems, B:290–293 fractional diffusion equationdielectric relaxation, B:313–316 fractional Smoluchowski equation solution, B:322–325 microscopic models, B:293–325 fractal dielectric relaxation, B:244–253 fractional rotational diffusion, double-well periodic potential, anomalous diffusion and dielectric relaxation, B:337–338 hyperbranched polymer ﬁlm conﬁnement effects, B:617–620 cis-1,4-polyisoprene ﬁlm preparation, B:605–607 symmetric relaxation, peak broadening in complex systems, A:106–110 vitriﬁcation of liquids coupling model, B:550–551 Johari-Goldstein secondary relaxation, non-Arrhenius temperature dependence, B:536–538 Heart rate variability (HRV) fractal-based time series, B:10–14 breathing rate variability and, B:14–18 fractional dynamics, B:70–73 Le´vy statistics, Langevin equation with, B:78–80 multifractal analysis, B:46 statistical analysis, limitations of, B:81–82

697

Heaviside unit function dielectric relaxation, Cole-Davidson and Havriliak-Negami behavior, B:313–316 fractal structures, Brownian particles with memory, B:272–274 fractal viscoelasticity, shear-stress relaxation, B:232–235 Hermite polynomials, dielectric relaxation, inertial effects Barkai/Silbey fractionalized Klein-Kramers equation, B:375–379 Metzler/Klafter fractionalized Klein-Kramers equation, B:366–372 symmetric top molecules, B:389–398 top molecule rotations, B:381–388 Heterogeneity length-scale, vitriﬁcation of liquids, coupling model, B:550–551 Hidden phase transition, supercooled liquids and glasses, formation above glass transition temperature, lowtemperature regime, A:185–201 Hierarchical ‘‘blob’’ model, fractal construction dielectric relaxation, B:250–253 viscoelasticity, B:231 Hierarchical coordination, fractal structures dielectric relaxation, B:248–253 shear-stress relaxation, B:232–235 Higher-order correlation function, non-Poisson dichotomous noise, A:414–421 High-frequency modes dielectric relaxation, inertial effects, periodic potentials, B:412–414 fractional rotational diffusion, in potentials, B:330–331 High-resolution alpha analyzer, polystyrene ﬁlms, conﬁnement effects, B:621–628 High-temperature regime, supercooled liquids and glasses b-peak glass formation, A:203–209 formation above glass transition temperature, A:173–182 Hilbert transform dielectric spectra, dc conductivity, A:27–30 ﬂuctuation-dissipation theorem, extreme quantum case, zero-temperature FDT, A:274–276 Le´vy ﬂight processes, space-fractional Fokker-Planck equation, B:451

698

subject index

Histogram analysis, blinking quantum dots, nonergodicity, A:350–353 Hoelder index, fractal functions, B:275–277 Ho¨lder experiment, multifractal spectrum, B:43–46 Holtsmark analysis, Le´vy ﬂight processes, B:478–481 Homogeneity, fractional diffusion equations, B:74–76 Homopolymer dynamics, in binary mixtures, B:567–574 Hooke’s law fractal viscoelasticity frequency dependence, B:215–235 Le´vy ﬂight processes, theoretical background, B:448 Hopping transport, supercooled liquids and glasses, nonergodicity parameter, A:158–162 Hurst exponent fractional Brownian motion, B:60–61 fractional dynamics, physical/physiological models, B:67–73 multifractal spectrum, diffusion entropy analysis, B:47–52 physiological time series, scaling behavior, B:8–10 scaling behavior, B:83 scaling dynamics, inverse power-law autocorrelation functions, B:35 Hybrid function, supercooled liquids and glasses, formation above glass transition temperature, high-temperature regime, A:180–182 Hydrogen-bonding liquids dielectric relaxation, disordered systems, A:48–55 scaling behavior, A:81–93 universal scaling behavior, A:81–93 glycerol-rich mixtures, A:81–86 water-rich mixtures, A:86–93 vitriﬁcation amorphous polymers, B:508–509 Johari-Goldstein secondary relaxation, B:532–535 Hydrogen nuclear magnetic resonance, supercooled liquids and glasses, time windows and spin-lattice interactions, A:148–150

Hydrophilic interactions, polymer-water mixtures, dielectric relaxation, A:110–113 Hydrophobic interactions, polymer-water mixtures, dielectric relaxation, A:110–113 Hyperbranched polymers, conﬁnement effects, B:613–620 Hypergeometric function dielectric relaxation, inertial effects Barkai/Silbey fractionalized KleinKramers equation, B:376–379 Metzler/Klafter fractionalized KleinKramers equation, B:368–372 fractional dynamics, B:65–73 Hyperscaling relationship (HSR), dynamic percolation ionic microemulsions, A:68–71 Impedance, broadband dielectric spectroscopy, A:16–18 Induction equations, Le´vy ﬂight processes, frequency domain, B:452–453 Inductively coupled plasma optical emission spectrometry (ICP-OES), ferroelectric crystals, dielectric relaxation, A:45–48 Inequalities, fractal construction, dielectric relaxation, B:246–253 Inertial effects dielectric relaxation disordered systems, B:292–293 fractional Klein-Kramers equation Barkai-Silbey format, B:372–379 linear and symmetrical top molecules, B:380–398 Metzler/Klafter format, B:365–372 periodic potentials, B:398–414 fractional Langevin equation, B:414–419 linear and symmetrical top molecules, B:380–398 fractional rotational diffusion, B:388–398 rotators in space, B:380–388 periodic potentials, B:398–414 theoretical principles, B:364–365 fractal structures, Brownian particles, B:267–274 diffusion equation with fractional derivatives, B:269–274

subject index one-dimensional lattice random walks, B:267–269 Inﬁnite cluster density, chaotic fractal percolation clusters, B:133–137 Inﬁnitely short memory limit, Ohmic dissipation, A:268 Information approach, noncanonical equilibrium, A:407–409 Information dimension, multifractals, B:121–125 Infrared (IR) spectrosopy, polystyrene ﬁlms, conﬁnement effects, B:625–628 Inhomogeneous media, fractal dielectric relaxation, B:250–253 Inphasing coupling/inphasing strength, nonPoisson renewal processes, decoherence theory, A:436–439 Integration theorem continued fraction solution, longitudinal and transverse responses, B:425–427 dielectric relaxation Debye noninertial rotational diffusion, B:309–312 fractional Smoluchowski equation solution, B:318–325 inertial effects periodic potentials, B:401–414 top molecule rotations, B:383–388 fractional dynamics inertial effects Barkai/Silbey fractional Klein-Kramers equation, B:375–379 physical/physiological models, B:66–73 fractional rotational diffusion, double-well periodic potential, anomalous diffusion and dielectric relaxation, B:332–338 Integrodifferential equation, statistical mechanics, Caldeira-Leggett model, A:263–264 Interaction-induced mechanisms, supercooled liquids and glasses, depolarized light scattering, A:141–148 Interferometric techniques, supercooled liquids and glasses, depolarized light scattering, A:143–148 Intermediate interval, fractal structures, Brownian particle inertial effects, diffusion equation, B:271–274 Intermittent dynamic model, continuous time random walk, A:378–384

699

experimental vs. theoretical laminar regions, A:379–381 Manneville map, A:381–383 Nutting law, A:383–384 Intermolecular interactions, vitriﬁcation of liquids, B:502 coupling model principles, B:546–551 structural and Johari-Goldstein relaxations, B:574–577 Intrawell relaxation modes, fractional rotational diffusion direct current (DC) electric ﬁeld, B:344–347 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:334–338 potential phenomena, B:325–331 Inverse Fourier transform calculation of, B:421–424 dichotomous ﬂuctuations, B:39–40 dielectric relaxation Debye noninertial rotational diffusion, B:311 fractional Smoluchowski equation solution, B:317–325 fractal structures anomalous diffusion, B:260–264 Brownian particles with memory, B:265–267, B:272–274 dielectric relaxation, B:242–253 viscoelasticity frequency dependence, B:217–235 fractional Brownian motion, B:59–61 fractional diffusion equations, B:74–76 fractional rotational diffusion bistable potential with nonequivalent wells, matrix continued fraction solution, B:353–354 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:331–338 Le´vy ﬂight processes frequency domain induction equations, B:452–453 Langevin equation with, B:77–80 multifractal spectrum, diffusion entropy analysis, B:48–52 scaling dynamics fractional random walks, B:32–34 simple random walks, B:31 Inverse Laplace transform, fractional dynamics, B:62–73

700

subject index

Inverse power-law spectrum multifractal spectrum, diffusion entropy analysis, B:48–52 scaling dynamics autocorrelation functions, B:34–35 fractional random walks, B:31–34 Investigation diffusion equation, fractal anomalous diffusion, Brownian motion, B:256–257 Ionic polarization, static electric ﬁelds, A:5 ‘‘Islands of mobility’’ theory b-peak glass formation, below melting point, A:237–241 supercooled liquids and glasses, b-peak glass formation, A:206–209 Isothermal spectra, hydrogen-bonding liquids, dielectric relaxation, A:50–55 Isotropic bodies, fractal structures, negative Poisson’s ratio, B:211–215 Isserlis’s theorem, dielectric relaxation inertial effects fractional Langevin equation, B:417–419 Kramers-Moyal expansion, B:420 microscopic models, B:294 Iteration function, Julia fractal sets, B:110–113 Iterative averaging, fractal structures conductivity methods, B:164–174 dielectric properties, B:175–183 elastic properties, B:205–209 frequency dependence of dielectric properties, B:175–183 Hall’s coefﬁcient, B:189–204 viscoelasticity properties, B:221–235 Johari-Goldstein (JG) b-relaxation dielectric response, hydrogen-bonding liquids, A:52–55 molecular glass-forming liquids, temperature/ pressure dependence, B:503–507 structural relaxation time, B:530–545 aging phenomena, B:538–540 enthalpy, entropy, and volume mimicry, B:540–541 molecular mobility dependences, B:541–545 non-Arrhenius temperature dependence, B:536–538 pressure dependence, B:530–535 temperature-pressure invariance, B:535–536

vitriﬁcation of liquids basic principles, B:501–502 coupling model, B:501–502 basic properties, B:501–502 binary component dynamics, B:567–574 dispersion invariance to temperature and pressure combinations, B:561 enthalpy activation in glassy state, B:555–556 many-molecule dynamics, B:577–580 nanometric relaxation, B:562–567 polymer systems primary/secondary relaxation interrelations, B:574–577 pressure-temperature dependence above glass transition temperature, B:556–561 primitive relaxation, B:529–530 structural relaxation correspondence with, B:551–555 theoretical background, B:546–551 many-molecule dynamics, B:577–580 primitive relaxation and, B:528–530 Jonscher’s power law wings dielectric response, A:9–12 static percolation, porous materials, silicons, A:63–65 Julia sets, fractal construction, B:110–113 Kauzmann temperature, supercooled liquids and glasses, glass transition phenomenon, A:156–162 Kelvin-Voigt model, fractal viscoelasticity, B:218–235 Kerr-effect relaxation differential-recurrence relations, B:432–434 inertial effects, B:384–388 symmetric top molecules, B:394–398 linear molecules, B:427–429 Kicked rotor condition, non-Poisson renewal processes, A:443–447 Kinetic equations conﬁned systems, A:95–105 doped ferroelectric crystal relaxation, A:100–102 glassy water model, A:103–104 models for, A:96–98 static properties and dynamics, A:104–105 temperature dependence of models, A:102–104 water dielectric relaxation, A:98–100

subject index dielectric relaxation, A:12–15 Cole-Davidson and Havriliak-Negami behavior, B:314–316 fractional Smoluchowski equation solution, B:324–325 inertial effects, Barkai/Silbey fractional Klein-Kramers equation, B:379 Le´vy ﬂight processes, nonunimodality, B:456–459 Kirkwood correlation factor static electric ﬁelds, A:6–7 supercooled liquids and glasses, formation above glass transition temperature, time constants and decoupling phenomenon, A:171–173 Kirkwood-Fro¨hlich equation static electric ﬁelds, A:7 supercooled liquids and glasses, molecular reorientation, dielectric spectroscopy, A:136–140 Klein-Kramers equation, dielectric relaxation continuous-time random walk model, B:294–299 inertial effects, B:364–365 fractionalized Metzler/Klafter format, B:365–372 Klimontovich theory, Le´vy ﬂight processes, dissipative nonlinearity, B:482 Koch’s ‘‘snowﬂake’’ construction Cayley tree fractal structures, B:128–131 fractal sets, B:102–103 Haussdorff-Besicovitch dimension, B:115–119 Kohlrausch functions, supercooled glass formation above glass transition temperature, low-temperature regime, A:193–201 Kohlrausch stretched exponential function, vitriﬁcation of liquids binary mixtures, B:568–574 coupling model, B:546–551 structural-Johari-Goldstein relaxation correspondence, B:551–555 many-molecule dynamics, B:578–580 Kohlrausch-Williams-Watts (KWW) law dielectric relaxation, Cole-Davidson and Havriliak-Negami behavior, B:313–316

701

dielectric response, A:10–12 hydrogen-bonding liquids, A:51–55 microemulsion structure and dynamics, A:38 fractal dielectric relaxation, B:249–253 fractional dynamics, B:62–73 static percolation, porous materials, A:57–65 glasses, A:58–63 silicons, A:63–65 vitriﬁcation of liquids amorphous polymers, B:508–509 Coupling Model, B:501–502 coupling model principles, B:546–551 photon correlation spectroscopy studies, B:513–516 primitive relaxation and, B:529–530 probe rotational time correlation functions, B:521–528 Rouse modes, B:527–528 structural relaxation time, B:510–511 Kolmogorov-Arnold-Moser (KAM) theory, fractional dynamics, B:53–55 Kolmogorov-Sinai (KS) entropy, non-Poisson renewal processes, A:447–451 Kolmogorov turbulence, fractal functions, B:275–277 Kramers escape rate fractional rotational diffusion bistable potential with nonequivalent wells, B:355–364 in potentials, B:327–331 Le´vy ﬂight processes, barrier crossing, B:474–481 Kramers-Kronig relations dc conductivity, A:27 Hilbert transform, A:28–30 ﬂuctuation-dissipation theorem, extreme quantum case, zero-temperature FDT, A:275–276 thermal bath diffusion, A:305–306 Kramers-Moyal expansion, dielectric relaxation fractional Smoluchowski equation solution, B:324–325 inertial effects Cole-Cole relaxation, B:419–420 fractional Langevin equation, B:417–419 microscopic models, B:293–325 Kronecker’s delta, dielectric relaxation, inertial effects, fractionalized Klein-Kramers equation, B:367–372

702

subject index

Kubo formula ﬂuctuation-dissipation theorem phenomenological dissipation modeling, A:266 response function, A:270–273 out-of-equilibrium physics effective temperature and generalized susceptibilities, A:312–315 quasi-stationary regime, A:311–312 Stokes-Einstein relation, A:315–317 thermal bath diffusion, A:304–306 Kubo-Martin-Schwinger condition, ﬂuctuationdissipation theorem, symmetrized correlation function, A:269–273 Kummer function, dielectric relaxation, inertial effects, fractionalized KleinKramers equation, B:368–372 Lagrange multiplier, noncanonical equilibrium, A:408–409 Laminar regions continuous time random walk, intermittent dynamic model, A:379–381 non-Poisson dichotomous noise, four-time correlation function, A:417–421 Langevin equation. See also Fractional Langevin equation dielectric relaxation Debye noninertial rotational diffusion, B:312 inertial effects, fractional Klein-Kramers equation, B:365–372 microscopic models, B:293–325 fractal anomalous diffusion, Brownian motion, B:253–255 fractional dynamics basic equations, B:62–73 biological phenomena, B:61–73 physical/physiological models, B:65–73 random walk model, B:83–84 fractional rotational diffusion, direct current (DC) electric ﬁeld, B:341–347 Le´vy ﬂight processes barrier crossing, B:474–481 dissipative nonlinearity, B:482–483 noise properties, B:449–450 power-law asymptotics of stationary solutions, B:456 statistics, B:76–80 theoretical background, B:442–448 noncanonical equilibrium, A:412–413

non-Ohmic dissipation, aging effects noise and friction, A:296–297 one-and two-time dynamics, A:297–303 numerical solutions to, B:492 Ohmic dissipation, A:268 out-of-equilibrium linear response theory, quasi-stationary regime, A:311–312 phenomenological dissipation modeling, classical/quantal formats, A:266–267 quantum Brownian motion aging, A:284–296 random walk model vs., B:83–84 stationary medium, particle diffusion in, A:308–309 statistical mechanics, Caldeira-Leggett model, A:264 Zwanzig projection method, A:367 Anderson localization, A:371–374 Langevin model, Brownian motion aging phenomena, A:279–284 displacement response and correlation function, A:280 ﬂuctuation-dissipation ratio, A:280–281 temperature effects, ﬂuctuation-dissipation ratio, A:283–284 time-dependent diffusion coefﬁcient, A:281–283 velocity correlation function, A:279–280 Laplace transform blinking quantum dots case studies, A:339–342 on-off aging correlation function, A:338 on-off mean intensity, A:337–338 stochastic models, A:335–336 canonical equilibrium, A:410 continued fraction solution, longitudinal and transverse responses, B:425–427 continuous time random walk, A:376–378 laminar regions, A:380–381 Le´vy dynamic derivation, A:389–392 recurrence properties, A:460–461 trajectory approach, A:402–403 vs. generalized master equation, A:395–399 dichotomous ﬂuctuations, B:39–41 dielectric polarization, time-dependent electric ﬁelds, A:8 dielectric relaxation Cole-Cole behavior, B:301–305 Debye noninertial rotational diffusion, B:309–312 disordered systems, B:288–293

subject index fractional Smoluchowski equation solution, B:318–325 inertial effects Barkai/Silbey fractionalized Klein-Kramers equation, B:375–379 fractional Langevin equation, B:415–419 Metzler/Klafter fractionalized Klein-Kramers equation, B:366–372 periodic potentials, B:401–414 symmetric top molecules, B:391–398 top molecule rotations, B:383–388 fractal structures anomalous diffusion, B:261–264 Brownian particle inertial effects, B:271–274 dielectric relaxation, B:240–253 fractional dynamics, B:62–73 physical/physiological models, B:68–73 fractional rotational diffusion bistable potential with nonequivalent wells, B:349–364 matrix continued fraction solution, B:351–354 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:332–338 in potentials, B:328–331 generalized master equation, A:429–431 inverse Fourier transform calculations, B:421–424 Kerr-effect response, B:432–434 Le´vy ﬂight processes bifurcation time, B:462–464 ﬁrst arrival time problems, B:468–470 method of images inconsistency, B:473–474 stationary probability density function, B:460–461 theoretical background, B:446–448 non-Ohmic bath, A:458–459 out-of-equilibrium physics, Stokes-Einstein relation, A:315–317 supercooled glass formation above glass transition temperature, lowtemperature regime, A:188–201 symmetric relaxation, peak broadening in complex systems, A:107–110 Larmor frequency b-peak glass formation, below melting point, A:236–241

703

supercooled liquids and glasses nuclear magnetic resonance, A:149–150 molecular reorientation, A:210–214 spin-lattice relaxation and line-shape analysis, A:151–152 Lattice fractal structures conductivity, iterative averaging method, B:164–174 one-dimensional lattice, Brownian particle inertial effects, B:267–269 Layered structures, fractal galvanomagnetic properties, B:199–204 Legendre functions continued fraction solution, longitudinal and transverse responses, B:425–427 dielectric relaxation fractional Smoluchowski equation solution, B:323–325 inertial effects linear and symmetrical top molecules, B:381–388 symmetric top molecules, B:389–398 fractional rotational diffusion, bistable potential with nonequivalent wells, B:351–354 multifractal spectrum, B:43–46 supercooled liquids and glasses, molecular reorientation, A:133–134 dielectric spectroscopy, A:137–140 Length-scale measurements, vitriﬁcation of liquids, structural relaxationdispersion coordination, B:526–528 Lennard-Jones potential supercooled liquids and glasses, mode coupling tests, A:223–225 vitriﬁcation of liquids, coupling model, B:548–551 Le´vy-Feller diffusion, fractional diffusion equations, B:75–76 Le´vy ﬂight processes barrier crossing, B:474–481 Brownian motion, B:475 Cauchy analytical approximation, B:478–480 numerical solution, B:475–478 starting equations, B:474–475 conﬁnement and multimodality, B:453–467 bifurcation time, B:461–464 consequences of, B:464–467 Langevin equation formal solution, B:459–461

704

subject index

Le´vy ﬂight processes (continued) n-modal state phase diagrams, B:463–464 nonunimodality of stationary solutions, B:456–459 power-law asymptotics of stationary solutions, B:455–456 stationary quartic Cauchy oscillator, B:453–454 trimodal transient state, B:462–463 deﬁnition and basic properties, B:449–453 fractional Fokker-Planck equation, B:450–451 frequency domain induction equations, B:452–453 Langevin equation, B:449–450 dielectric relaxation basic properties, B:296–299 Cole-Cole behavior, B:303–305 continuous-time random walk model, B:295–299 dissipative nonlinearity, B:481–487 asymptotic behavior, B:483–484 dynamical equation, B:482–483 nonlinear friction term, B:481–482 probability density function, B:485–486 quadratic/quartic numerical solutions, B:484–485 ﬁrst passage and arrival time problems, B:467–474 ﬁrst arrival time, B:467–470 method of images inconsistency, B:472–474 Sparre Anderson universality, B:470–472 fractional dynamics, B:84–85 physical/physiological models, B:68–73 fractional rotational diffusion, double-well periodic potential, anomalous diffusion and dielectric relaxation, B:336–338 Langevin equation with, B:76–80 multifractal spectrum, diffusion entropy analysis, B:47–52 numerical solution methods, B:488–492 Gru¨nwald-Letnikov fractional FokkerPlanck equation solution, B:488–492 Langevin equation, B:492 physiological time series, scaling behavior, B:10 theoretical background, B:440–448

Le´vy ‘‘sneaking,’’ dielectric relaxation, inertial effects, fractional Klein-Kramers equation, B:372–379 Le´vy statistics canonical equilibrium, A:409–410 complex systems, A:461–463 continuous time random walk, A:358–362 dynamic derivation of, A:388–392 generalized central limit theorem, A:388 Liouville formalism, A:358–362 multiscaling, A:403–404 Nutting law, A:383–384 trajectory approach, A:401–403 noncanonical equilibrium, A:408–409 dynamic approach to, A:410–413 non-Poisson dichotomous noise, A:415–416 non-Poisson renewal processes, decoherence theory, A:439–441 Le´vy-walk diffusion relation dielectric relaxation, inertial effects, BarkaiSilbey fractional Klein-Kramers equation, B:374–379, B:420–421 multifractal spectrum, B:51–52 Life support devices, fractional dynamics, B:84–85 future research issues, B:84–85 Light scattering (LS) mechanisms, supercooled liquids and glasses, A:141–148 formation above glass transition temperature dynamic susceptibility, A:163–169 high-temperature regime, A:173–182 low-temperature regime, A:183–201 future research issues, A:241–243 mode coupling tests, A:221–225 dynamic susceptibility, A:228–230 nonergodicity parameter, temperature dependence, A:216–220 Lindblad master equation non-Poisson renewal processes, quantum to classicial transition, A:446–447 quantum mechanics measurements, A:469 Linear elasticity, chaotic fractal structures, B:144–147 Linear molecules dielectric relaxation, inertial effects, B:393–398 dynamic Kerr-effect reponse, B:427–429 Linear response theory fractional rotational diffusion, direct current (DC) electric ﬁeld, B:339–347

subject index out-of-equilibrium environment anomalous diffusion, A:309–312 age- and frequency-dependent response functions, A:309–311 quasi-stationary regime, effective temperature, A:311–312 particle equations of motion, A:307–308 thermal bath diffusion, A:305–306 time-domain spectroscopy, A:21 Linear top molecules, inertial effects, anomalous dielectric relaxation, B:380–398 rotators in space, B:380–388 Line-shape analysis, supercooled liquids and glasses, spin-lattice relaxation, A:150–152 Liouville equation canonical equilibrium, A:410 continuous time random walk, A:358–362 anomalous diffusion dynamics, A:406–407 correlation function, A:427–429, A:428–429 generalized master equation vs., A:464–465 Le´vy dynamic derivation, A:389–392 overview, A:358–362 phenomenological modeling, A:363–364 vs. generalized master equation, A:398–399 dichotomous ﬂuctuations, B:38–41 dielectric relaxation, inertial effects fractionalized Klein-Kramers equation, B:371–372 periodic potentials, B:413–414 symmetric top molecules, B:394–398 generalized master equation vs. continuous time random walk, A:392–399 non-Poisson processes, trajectory and density entropies, A:448–451 non-Poisson renewal processes, modulation theories, A:455–456 ‘‘strange kinetic’’ percolation, A:74–81 Zwanzig projection method, A:364–375 Locally preferred structures, supercooled liquids and glasses, glass transition phenomenon, A:157 Logarithmic derivatives fractal elasticity, B:206–209 viscoelastic properties, B:223–235 fractal galvanomagnetic properties, B:190–204 vitriﬁcation of liquids, coupling model, structural-Johari-Goldstein relaxation correspondence, B:552–555

705

Logarithm of standard deviation, heart beat fractals, B:12–14 Lognormal distribution, multifractals, B:123–125 Longitudinal response continued fraction solution, B:425–427 dielectric relaxation, inertial effects, periodic potentials, B:399–414 fractional rotational diffusion bistable potential with nonequivalent wells, B:349–364 bimodal approximation, B:356–364 direct current (DC) electric ﬁeld, B:339–347 Long-jump model, dielectric relaxation, B:296–299 Lorentzian line Brownian motion aging phenomena, quantum model, velocity correlation function, A:285–286 Ohmic dissipation, A:267–268 supercooled liquids and glasses, nuclear magnetic resonance, molecular reorientation, A:211–214 Loss of spectral reserve dielectric relaxation, inertial effects Barkai/Silbey fractionalized KleinKramers equation, B:376–379 periodic potentials, B:404–414 symmetric top molecules, B:394–398 top molecule rotations, B:385–388 fractional rotational diffusion, direct current (DC) electric ﬁeld, B:344–347 Le´vy statistics, Langevin equation with, B:80 molecular glass-forming liquids, temperature/ pressure dependence, B:503–507 vitriﬁcation of liquids coupling model, structural-JohariGoldstein relaxation correspondence, B:551–555 Johari-Goldstein secondary relaxation, B:532–535 structural relaxation-dispersion coordination, B:521–528 Low-frequency ranges broadband dielectric spectroscopy, A:16–18 dielectric relaxation, inertial effects, periodic potentials, B:412–414 fractional rotational diffusion double-well periodic potential, anomalous diffusion and dielectric relaxation, B:336–338 in potentials, B:327–331

706

subject index

Low-temperature regime, supercooled liquids and glasses molecular glass formation above melting point, A:182–214 b-peak glass formation, A:201–209 excess wing evolution, A:182–201 molecular glass formation below melting point, A:232–241 Lumped-Impedance Methods, broadband dielectric spectroscopy, A:16–18 Lyapunov coefﬁcient, non-Poisson renewal processes, A:447–451 Mach-Zehnder interferometer, supercooled liquids and glasses, molecular reorientation, A:140 Macroscopic correlation function ionic microemulsion dipole correlation function, A:66–67 static percolation, porous materials, A:56–65 ‘‘strange kinetic’’ percolation, A:74–81 supercooled liquids and glasses, molecular reorientation, dielectric spectroscopy, A:135–140 symmetric relaxation, peak broadening in complex systems, A:109–110 Macroscopic dipole moment, dielectric polarization, static electric ﬁelds, A:3–7 Macroscopic ﬂuctuations chaotic fractals, renormalization-group transformations, B:137–141 fractals, B:3–4 fractal sets, square lattice construction, B:125–128 Magic correlation time, supercooled glass formation above glass transition temperature, low-temperature regime, A:190–201 Magnetic dipole moment values, fractional rotational diffusion, direct current (DC) electric ﬁeld, B:347 Magnetic ﬁeld fractal conductivity, galvanomagnetic properties, B:194–204 fractional rotational diffusion, direct current (DC) electric ﬁeld, B:338–347 Maier-Saupe model, dielectric relaxation, disordered systems, B:289–293

Manneville map continuous time random walk, A:381–383 non-Poisson dichotomous noise, four-time correlation function, A:416–421 non-Poisson renewal processes, trajectory and density entropies, A:449–451 Many-molecule relaxation, vitriﬁcation of liquids, B:501–502 coupling model principles, B:546–551 above glass transition temperature, B:559–561 glass transition temperatures and, B:577–580 primitive relaxation and, B:528–530 Markov approximation dichotomous ﬂuctuations, B:38–41 fractal anomalous diffusion, B:259–264 fractional diffusion equations, B:73–76 Le´vy ﬂight processes, theoretical background, B:443–448 Le´vy statistics, Langevin equation with, B:76–80 multifractal spectrum, diffusion entropy analysis, B:49–52 noncanonical equilibrium, A:412–413 Zwanzig projection method, A:365–375 Anderson localization, A:371–374 Pauli master equation, A:366–367 Markov master equation continuous time random walk Le´vy dynamic derivation, A:390–392 Poisson statistics, A:378 vs. generalized master equation, A:396–399 non-Poisson renewal processes, quantum to classicial transition, A:446–447 Mass exponent fractional dynamics, physical/physiological models, B:68–73 multifractal spectrum, B:44–46 MATLAB software, dielectric spectroscopy data analysis and modeling, A:29–30 Matrix continued fraction solution. See Continued fraction solution Maximum entropy method, noncanonical equilibrium, A:408–409 Maxwell-Boltzmann distribution dielectric relaxation, inertial effects fractionalized Klein-Kramers equation, B:366–372

subject index fractional Langevin equation, B:416–419 symmetric top molecules, B:391–398 top molecule rotations, B:383–388 Kerr-effect relaxation, B:428–429 Maxwell-Cattaneo equation, fractal anomalous diffusion, Brownian particle inertial effects, B:269–274 Maxwell equation dichotomous ﬂuctuations, B:38–41 dielectric polarization, static electric ﬁelds, A:4–7 dielectric relaxation, inertial effects, fractionalized Klein-Kramers equation, B:370–372 Maxwell ﬁelds, supercooled liquids and glasses, molecular reorientation, dielectric spectroscopy, A:135–140 Maxwell model, fractal structures conductivity, B:163 distribution function of Brownian particles, B:267 viscoelasticity, B:218–235 Mean crossing time (MCT), Le´vy ﬂight processes barrier crossing, B:475 Cauchy case, B:479–481 Mean dipole moment, dielectric relaxation, Debye noninertial rotational diffusion, B:308–312 Mean intensity, blinking quantum dots, aging phenomena, A:337–342 Mean relaxation time, fractal dielectric relaxation, B:239–253 Mean-square displacement dielectric relaxation, continuous-time random walk model, B:294–299 fractal anomalous diffusion, Brownian motion, B:255 Le´vy ﬂight processes, theoretical background, B:442–448 out-of-equilibrium physics power-law behaviors, A:318 Stokes-Einstein relation, A:315–317 vitriﬁcation of liquids, structural relaxation-dispersion coordination, B:525–528 Mellin-Barnes integrals, fractal structures, dielectric relaxation, B:242–253

707

Memory kernel continuous time random walk, A:378 Le´vy dynamic derivation, A:389–392 continuous time random walk, vs. generalized master equation, A:397–399 dichotomous ﬂuctuations, B:35–41 fractal structures, Brownian particle inertial effects, B:269–274 generalized master equation, A:429–431 symmetric relaxation, peak broadening in complex systems, A:106–110 Mesoscopic clusters hydrogen-bonding liquids, scaling phenomena, A:90–93 ionic microemulsion dipole correlation function, A:66–67 microcomposite material dielectric relaxation, A:115–116 Metal-insulator transition, fractal dielectric properties, B:177–183 Method of images, Le´vy ﬂight processes, B:472–474 Metzler/Klafter fractional Klein-Kramers equation, dielectric relaxation, inertial effects, B:365–372 fractional Langevin equation, B:417–419 Microcomposite materials, dielectric relaxation, A:113–116 Microemulsions, dielectric relaxation, A:31–38 Microscopic correlation function, supercooled liquids and glasses, molecular reorientation, dielectric spectroscopy, A:136–140 Microscopic ﬂuctuations chaotic fractals, renormalization-group transformations, B:137–141 fractals, B:3–4 fractal theory and, B:96–98 dielectric relaxation, B:251–253 Microscopic models, dielectric relaxation, disordered systems, B:293–325 continuous-time random walk model, B:294–299 fractional diffusion equations Cole-Cole model, B:300–305 Cole-Davidson/Havriliak-Negami behavior, B:313–316 fractional Smoluchowski equation, B:316–325

708

subject index

Mie theory, fractal dielectric properties, B:183 Minimum scaling property, supercooled liquids and glasses, mode-coupling theory, A:160–162 Mittag-Lefﬂer function dielectric relaxation Cole-Cole behavior, B:305 Cole-Davidson and Havriliak-Negami behavior, B:313–316 continuous-time random walk model, B:298–299 Debye noninertial rotational diffusion, B:307–312 fractional Smoluchowski equation solution, B:319–325 inertial effects fractionalized Klein-Kramers equation, B:366–372 fractional Langevin equation, B:415–419 fractal structures Brownian particles with memory, B:265–267, B:274 dielectric relaxation, B:241–253 fractional dynamics, B:62–73 fractional rotational diffusion direct current (DC) electric ﬁeld, B:343–347 in potentials, B:327–331 inverse Fourier transform calculations, B:423–424, B:424 non-Ohmic dissipation, aging effects, A:298–299 Mobility, out-of-equilibrium physics Kubo formula, A:313–315 power-law behaviors, A:317–318 Mode-Coupling Theory (MCT) relaxation kinetics, A:14–15 supercooled liquids and glasses basic principles, A:130–131 formation above glass transition temperature, A:220–228 asymptotic scaling laws, A:221–225 dynamic susceptibility, A:167–169 high-temperature regime, A:178–182 low-temperature regime, A:185–201 time constants and decoupling phenomenon, A:170–173 two-correlator model, A:225–228 future research issues, A:242–243

glass transition phenomenon, A:156–162 nonergodicity parameter, A:157–162 temperature dependence, A:215–220 Modulation theories non-Poisson renewal processes, A:451–456 scaling properties, A:463–464 Molecular assignment, cis-1,4-polyisoprene thin ﬁlm preparation, B:607 Molecular dynamics dielectric response, A:10–12 glass-forming liquids coupling model, temperature/pressure dependence, B:561 dispersion mechanisms photon correlation spectroscopy studies, B:513–516 temperature/pressure dependence, B:503–507 Johari-Goldstein secondary relaxation aging effects, B:538–540 mobility dependences on temperature, pressure, volume, and entropy, B:541–545 primitive relaxation, B:529–530 structural relaxation-dispersion coordination, B:526–528 supercooled liquids and glasses, A:131–155 correlation function, spectrum, and susceptibility, A:131–134 depolarized light scattering, A:140–148 dielectric spectroscopy, A:134–140 coaxial reﬂection and transmission techniques, A:139 frequency response analysis, A:139 quasi-optical and FTIR spectroscopy, A:140 time-domain analysis, A:137–139 formation above glass transition temperature, A:165–169 mode coupling tests, A:224–225 nonergodicity parameter, A:158–162 nuclear magnetic resonance, A:148–155 mechanisms of, A:209–214 spin-lattice relaxation and line-shape analysis, A:150–152 stimulated echo experiments and twodimensional NMR, A:152–155 time windows and spin-lattice interactions, A:148–150

subject index vitriﬁcation of liquids, coupling model nanometric relaxation, B:562–527 structural-Johari-Goldstein relaxation correspondence, B:552–555 polymer ﬁlms conﬁnement effects cis-1,4-polyisoprene, B:601–607 molecular assignment, B:607 simulations, B:608–614 hyperbranched polymers, B:613–620 polystyrene ﬁlms, B:620–628 glass-forming systems, B:595–596 thin ﬁlm preparation, B:596–601 Molecular weight dependence, cis-1,4polyisoprene ﬁlm preparation, B:603–607 Monochromatic waves, supercooled liquids and glasses, molecular reorientation, A:140 Mono-fractal time series allometric aggregation, B:8–10 multifractal spectrum, B:43–46 Monotonous relaxation kinetics, dielectric response, A:14–15 Montroll-Weiss problem, fractal structures, anomalous diffusion, B:260–264 Mo¨ssbauer-Lamb factor, supercooled liquids and glasses, nonergodicity parameter, temperature dependence, A:215–220 Multifractals data analysis, B:42–52 diffusion entropy analysis, B:46–52 spectrum of dimensions, B:42–46 fractional dynamics, physical/physiological models, B:66–73 structural analysis, B:119–125 Multimodality, Le´vy ﬂight processes, B:453–467 bifurcation time, B:461–464 consequences of, B:464–467 Langevin equation formal solution, B:459–461 n-modal state phase diagrams, B:463–464 nonunimodality of stationary solutions, B:456–459 power-law asymptotics of stationary solutions, B:455–456

709

stationary quartic Cauchy oscillator, B:453–454 theoretical background, B:448 trimodal transient state, B:462–463 Multiple organ dysfunction syndrome (MODS), fractional dynamics, B:85–86 Multiscaling, continuous time random walk, A:403–404 Multi-window time sale, time-domain spectroscopy, A:22–23 Nagel scaling, supercooled glass formation above glass transition temperature, low-temperature regime, A:184–201 Nanocrystal (NC) structure, blinking quantum dots, A:328–331 diffusion model, A:331–334 nonergodicity, A:342–350 on-off mean intensity, A:337–338 stochastic models, A:334–336 Nano-geometry, porous silicon dielectric relaxation, A:41–44 Nanometric relaxation, vitriﬁcation of liquids, coupling model, B:562–567 Nearly constant loss (NCL) phenomenon, molecular glass formation, below melting point, A:231–241 Ne´el-Brown model, dielectric relaxation, disordered systems, B:289–293 Negative Poisson’s ratio, fractal structures, B:209–215 Negative shear modulus, fractal viscoelastic properties, B:225–235 Neuron analysis, fractal-based time series analysis, B:22–26 Neutron scattering (NS) studies supercooled liquids and glasses, A:130–131 formation above glass transition temperature, dynamic susceptibility, A:165–169 molecular reorientation, A:132–134 nonergodicity parameter, temperature dependence, A:215–220 vitriﬁcation coupling model, structuralJohari-Goldstein relaxation, B:553–555

710

subject index

Newtonian equation dielectric relaxation, inertial effects periodic potentials, B:413–414 symmetric top molecules, B:397–398 fractal structures dielectric relaxation, B:240–253 viscoelasticity frequency dependence, B:216–235 Noise properties, Le´vy ﬂight processes, Langevin equation, B:449–450 Noise spectral density, non-Ohmic dissipation, aging effects, A:296–297 Non-Arrhenius temperature dependence, vitriﬁcation of liquids coupling model, above glass transition temperature, B:556–561 Johari-Goldstein secondary relaxation, B:536–538 Noncanonical equilibrium, dynamic vs. thermodynamic approach, A:410–413 information approach, A:407–409 Non-Debye electric response dynamic percolation, ionic microemulsions, static/dynamic dimensions, A:72–73 models for, A:8–12 porous materials, A:38–44 static percolation, porous materials, A:55–65 symmetric relaxation, peak broadening in complex systems, A:106–110 Non-Debye relaxation, fractal structures physical principles, B:235–253 shear-stress relaxation, B:235 Nonequilibrium state, fractal structures, dielectric relaxation, B:240–253 Nonequivalent wells, bistable potential with, fractional rotational diffusion, B:347–364 bimodal approximation, B:355–364 matrix continued fraction solution, B:350–354 Nonergodicity parameter blinking quantum dots, A:342–350 time-averaged correlation function, A:347–350 time-averaged intensity distribution, A:344–346 supercooled liquids and glasses formation above glass transition temperature, high-temperature regime, A:180–182

mode coupling theory, A:157–162 dynamic susceptibility, A:229–230 temperature dependence, A:214–220 Nonhomogeneity, Le´vy ﬂight processes, differential equations, B:460–461 Noninertial rotational diffusion dielectric relaxation, Debye model, B:305–312 direct current (DC) electric ﬁeld, B:339–347 Nonlinear friction, Le´vy ﬂight processes, B:481–482 Non-Markov property non-Poisson dichotomous noise, four-time correlation function, A:416–421 Zwanzig projection method, A:366–367 Non-Ohmic dissipation models aging phenomena classical effects, A:302–303 noise and friction, A:296–297 one- and two-time dynamics, A:297–303 Mittag-Lefﬂer relaxation, A:298–299 particle coordinate and displacement, A:299–300 time-dependent quantum diffusion coefﬁcient, A:300–302 velocity correlation function, A:299 quantum effects, A:303 oscillator baths, A:456–459 Nonordinary environment, non-Poisson renewal processes, decoherence theory, A:439–441 Non-Poisson statistics aging phenomena, A:421–423 continuous time random walk generalized master equation vs., A:393–399 intermittent dynamic model, A:378–384 overview, A:358–362 trajectory approach, A:403 dichotomous noise/higher-order correlation functions, A:414–421 DF property, A:415–416 four-time correlation function, A:416–421 generalized master equation, A:429–435 renewal processes decoherence theory, A:435–439 nonordinary environment, A:439–441 quantum to classical transition, A:441–447 trajectory and density entropies, A:447–451 modulation theories, A:451–456

subject index Nonpolar dielectric, static electric ﬁelds, A:5–7 Nonuniform sampling, time-domain spectroscopy, A:22–23 Nonunimodality, Le´vy ﬂight processes, B:456–459 Nonvanishing eigenvalues, fractional rotational diffusion bistable potential with nonequivalent wells, bimodal approximation, B:355–364 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:333–338 Normal diffusion mechanisms, fractional rotational diffusion double-well periodic potential, anomalous diffusion and dielectric relaxation, B:332–338 in potentials, B:326–331 Nowhere differential functions, fractal sets, B:98–101 Bolzano function, B:98–101 Cantor set, B:103–104 ‘‘Devil’s Stairs’’ construction, B:104–105 Julia sets, B:110–113 Koch’s ‘‘snowﬂake,’’ B:102–103 Peano function, B:105–110 Sierpinski carpet, B:103 van der Waerden function, B:101–102 Nuclear magnetic resonance (NMR) b-peak glass formation, below melting point, A:236–241 supercooled liquids and glasses, A:130–313 molecular reorientation, A:133–134, A:148–155, A:209–214 spin-lattice relaxation and line-shape analysis, A:150–152 stimulated echo experiments and twodimensional NMR, A:152–155 time windows and spin-lattice interactions, A:148–150 vitriﬁcation of liquids coupling model, B:549–551 primitive relaxation, B:529–530 Nucleation ﬁeld, fractional rotational diffusion, bistable potential with nonequivalent wells, B:359–364 Nutting law continuous time random walk, A:383–384 fractional dynamics, B:62–73

711

non-Poisson dichotomous noise, four-time correlation function, A:416–421 Nyquist equation out-of-equilibrium physics, effective temperature, A:315 thermal bath diffusion, A:305 Ohmic dissipation. See also Non-Ohmic dissipation models Brownian motion aging phenomena, quantum model effective temperature, A:292–295 velocity correlation function, A:285–286 statistical mechanics, A:267–268 Ohm’s law, fractal galvanomagnetic properties, B:185–204 One-dimensional lattice, fractal anomalous diffusion, Brownian particle inertial effects, B:267–269 On-off times, power-law distribution, blinking quantum dots aging correlation function, A:338 diffusion model, A:332–334 mean intensity, A:337–338 probability density function, A:339–342 Onsager theory blinking quantum dots, diffusion model, A:332–334 generalized master equation, A:429–435 Open-ended coaxial line sensor, time-domain spectroscopy, A:23–24 Optical heterodyne techniques, supercooled liquids and glasses, depolarized light scattering, A:145–148 Optical Kerr effect (OKE) studies, supercooled liquids and glasses, A:130–131 depolarized light scattering, A:147–148 future research issues, A:241–243 mode coupling tests, A:224–225 Orientation properties dielectric relaxation Cole-Cole behavior, B:300–305 inertial effects, top molecule rotations, B:382–388 fractal galvanomagnetic properties, B:199–204 fractional rotational diffusion, direct current (DC) electric ﬁeld, B:338–347 static electric ﬁelds, A:5–7

712

subject index

Orientation properties (continued) supercooled liquids and glasses, stimulated echo experiments, two-dimensional NMR, A:154–155 Ornstein-Uhlenbeck process dielectric relaxation, inertial effects Barkai-Silbey fractional Klein-Kramers equation, B:373–379 Metzler/Klafter fractional Klein-Kramers equation, B:370–372 fractal structures, Brownian particles with memory, B:273–274 fractional dynamics, B:64–73 Le´vy ﬂight processes dissipative nonlinearity, B:483 theoretical background, B:447–448 Le´vy statistics, Langevin equation with, B:77–80 Orthogonality dielectric relaxation, inertial effects fractionalized Klein-Kramers equation, B:367–372 symmetric top molecules, B:390–398 top molecule rotations, B:381–388 fractional rotational diffusion, double-well periodic potential, anomalous diffusion and dielectric relaxation, B:332–338 Oscillation function aging effects, time-dependent diffusion coefﬁcient, A:300–302 non-Ohmic bath, A:456–459 Out-of-equilibrium physics anomalous diffusion Kubo generalized susceptibilities, A:312–315 effective temperature, A:312–313 friction coefﬁcient, A:314–315 mobility modiﬁcation, A:313–314 linearly coupled particle equation of motion, A:307–308 linear response theory, A:309–312 age- and frequency-dependent response functions, A:309–311 quasi-stationary regime, effective temperature, A:311–312 overview, A:303–304 power-law behaviors, A:317–319 friction coefﬁcient, A:317–318 mean-square displacement, A:318 temperature determination, A:318–319

stationary medium, A:308–309 Stokes-Einstein relation, A:315–317 thermal bath, A:304–307 ﬂuctuation-dissipation theorems, A:304–306 regression theorem, A:306–307 recent developments in, A:259–261 Packing arrangement, fractal dielectric relaxation, B:250–253 Pake spectrum, supercooled liquids and glasses nuclear magnetic resonance, molecular reorientation, A:211–214 time windows and spin-lattice interactions, A:150 Paraelectric phase, ferroelectric crystals, dielectric relaxation, A:46–48 Parallel layer structures, fractal conductivity, B:162–163 Parallel plate capacitor, time-domain spectroscopy, A:23–24 Parallel realization, time-domain spectroscopy, A:22–23 Particle coordinate and displacement non-Ohmic dissipation, aging effects, A:299–300 out-of-equilibrium environment, equations of motion, A:307–308 Partition function dielectric relaxation, inertial effects, periodic potentials, B:399–414 fractional rotational diffusion bistable potential with nonequivalent wells, B:350 bimodal approximation, B:356–364 in potentials, B:326–331 multifractal spectrum, B:42–46 Pauli master equation continuous time random walk intermittent dynamic model, A:378–384 laminar regions, A:381 non-Poisson renewal processes, A:450–451 phenomenological approach, A:362–364 Zwanzig projection method, A:366–367 Anderson localization, A:368–374 Peano function, fractal sets, B:105–110 Pearson random walk, Le´vy ﬂight processes, theoretical background, B:442–448 Percolation chaotic fractal clusters, B:131–147

subject index basic properties, B:131–132 critical indices, B:132–137 fractal cluster conductivity properties, B:142–144 fractal structure, scalar dependence, B:97–98 linear elasticity, B:144–147 microemulsions dielectric relaxation, A:32–38 dynamic percolation, ionic mediums, A:65–73 dipole correlation function, A:65–67 fractal dimensions, A:71–3 hyperscaling, A:68–71 static percolation, porous materials, A:55–65 glasses, A:58–63 silicon, A:63–65 ‘‘strange kinetic’’ phenomena, A:73–81 threshold values chaotic fractal structures, B:153–160 percolation clusters, B:131–137 fractal structure conductivity, B:160–163 galvanomagnetic properties, B:186–204 iterative averaging method, B:167–174 negative Poisson’s ratio, B:209–215 viscoelastic properties, B:222–235 Percus-Yevick approximation, supercooled liquids and glasses, mode-coupling theory, A:159–162 Periodic potentials fractional rotational diffusion in bistable potential, nonequivalent wells, B:347–364 bimodal approximation, B:355–364 matrix continued fraction solution, B:350–354 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:331–338 theoretical principles, B:325–331 uniform DC external ﬁeld, B:338–347 inertial effects, anomalous dielectric relaxation, B:398–414 Perovskite ferroelectrics, dielectric relaxation, A:44–48 Perry’s linear relation, ferroelectric crystals, dielectric relaxation, A:44–48

713

Perturbation response, anomalous diffusion dynamics, A:405–407 Perturbation theory, fractal conductivity, galvanomagnetic properties, B:185–204 Phase-space distribution function, dichotomous ﬂuctuations, B:39–41 Phase transition, ferroelectric crystals, A:44–48 Phenomenological modeling continuous time random walk, A:362–364 dissipative systems, A:265–267 fractals, B:3–4 fractional rotational diffusion, double-well periodic potential, anomalous diffusion and dielectric relaxation, B:337–338 physiological time series, heart rate variability, B:13–14 Phenyl glycidyl ether (PGE), supercooled glass formation, above glass transition temperature, low-temperature regime, A:200–201 Photoluminescence (PL), porous silicon dielectric relaxation, A:41–44 Photon correlation spectroscopy (PCS), supercooled liquids and glasses b-peak glass formation, A:207–209 depolarized light scattering, A:146–148 formation above glass transition temperature, low-temperature regime, A:184–201 future research issues, A:242–243 Photon correlation spectroscopy (PCS), vitriﬁcation of liquids, temperature/ pressure dependence, B:512–516 Physical/physiological models, fractional dynamics, B:65–73 Physiological time series complex systems, B:81–82 scaling behavior, B:4–26 allometric aggregation data analysis, B:5–10 breathing function, B:14–18 clinical applications of, B:82–83 dynamic models, B:27–35 gait cycle, B:18–22 heartbeats, B:10–14 neurons, B:22–26 Picoline mixtures, structural and JohariGoldstein relaxations in, B:572–573

714

subject index

Planar square lattices, fractal sets, B:126–128 Planck’s constant, relaxation kinetics, A:13–15 Plasma frequencies, fractal dielectric properties, B:177–183 Pochhammer symbol, dielectric relaxation, Cole-Davidson and HavriliakNegami behavior, B:316 Poisson statistics allometric aggregation data analysis, scaling behavior, B:6–10 continuous time random walk generalized master equation vs., 465–466, A:396–399 intermittent dynamic model, A:378–384 Markov master equation, A:378 overview, A:358–362 trajectory approach, A:402–403 dielectric relaxation Cole-Cole behavior, B:305 disordered systems, B:289–293 fractal-based time series analysis, neuron action potentials, B:23–26 fractal structures, negative Poisson’s ratio, B:209–215 multifractal spectrum, diffusion entropy analysis, B:50–52 non-Poisson renewal processes, modulation theories, A:452–456 quantum-to-classical transition, A:442–447 ‘‘strange kinetic’’ percolation, A:74–81 Polarization fractal dielectric relaxation, B:245–253 fractional rotational diffusion bistable potential with nonequivalent wells, B:351–354 direct current (DC) electric ﬁeld, B:338–347 static electric ﬁelds, A:3–7 Poley absorption, dielectric relaxation, inertial effects, B:398 periodic potentials, B:407–414 cis-1,4-Polyisoprene (PIP), conﬁnementinduced thin ﬁlm preparation, B:601–607 molecular assignment, B:607 simulations, B:608–614 Polymer ﬁlms conﬁnement effects cis-1,4-polyisoprene, B:601–607 molecular assignment, B:607

simulations, B:608–614 hyperbranched polymers, B:613–620 polystyrene ﬁlms, B:620–628 glass-forming systems, B:595–596 thin ﬁlm preparation, B:596–601 Polymeric composites fractal viscoelastic properties, B:224–235 vitriﬁcation aging effects, B:538–540 coupling model, B:550–551 above glass transition temperature, B:560–561 nanometric relaxation, B:562–567 structural-Johari-Goldstein relaxation correspondence, B:554–555 Johari-Goldstein secondary relaxation, B:531–535 photon correlation spectroscopy studies, B:512–516 polystyrene compounds, B:572 structural and Johari-Goldstein relaxations, 574–577, B:572 temperature/pressure dependence, B:507–509 Polymer-water mixtures, dielectric spectroscopy, A:110–113 Polystyrene ﬁlms, conﬁnement effects, B:620–628 Porosity, static percolation, porous materials, A:55–65 glasses, A:58–63 silicon, A:63–65 Porous materials dielectric relaxation, A:38–44 conﬁned system kinetics, A:96–98, A:103–104 glasses, A:38–41 silicon, A:41–44 static percolation, A:55–65 glasses, A:58–63 silicon, A:63–65 Positron annihilation lifetime spectroscopy (PALS), A:216 Potential-energy landscape (PEL), supercooled liquids and glasses, A:157 Power curve, allometric aggregation data analysis, scaling behavior, B:6–10 Power-law principles allometric aggregation data analysis, scaling behavior, B:5–10 blinking quantum dots

subject index aging phenomena, A:337–342 on-off correlation function, A:338 on-off mean intensity, A:337–338 basic principles, A:328, A:329 diffusion model, A:331–334 experimental data, A:350–353 nonergodicity, A:342–350 time-averaged correlation function, A:347–350 time-averaged intensity distribution, A:344–346 physical models, A:328–334 stochastic models, A:334–336 continuous time random walk, aging phenomena, A:424–425 dielectric relaxation Cole-Cole behavior, B:304–305 fractional Smoluchowski equation solution, B:320–325 inertial effects, Barkai-Silbey fractional Klein-Kramers equation, B:374–379 fractal structures, dielectric relaxation, B:242–253 Le´vy ﬂight processes asymptotic behavior, B:455–456 barrier crossing, B:477–478 dissipative nonlinearity, B:485–487 Le´vy statistics, Langevin equation with, B:78–80 out-of-equilibrium physics, A:317–319 friction coefﬁcient, A:317–318 mean-square displacement, A:318 temperature determination, A:318–319 scaling dynamics, simple random walks, B:31 Pre-percolation cluster mobility, ‘‘strange kinetic’’ percolation, A:80–81 Pressure dependence, vitriﬁcation of liquids coupling model above glass transition temperature, B:556–561 structural-Johari-Goldstein relaxation correspondence, B:553–555 dispersion mechanisms, B:502–516 amorphous polymers, B:507–509 molecular glass-formers, B:503–507 photon correlation spectroscopy analysis, B:512–516 structural relaxation time, B:509–516 Johari-Goldstein b-relaxation

715

basic principles, B:530–535 molecular mobility and, B:541–545 structural relaxation-dispersion coordination, B:517–528 Probability density function (PDF). See also Stationary probability density function blinking quantum dots case studies, A:339–342 nonergodicity, A:345–346 stochastic models, A:334–336 chaotic fractal structure models, B:153–157 nucleating cell derivations, B:157–160 continuous time random walk Gaussian stochastic process, A:400–401 generalized central limit theorem, A:385–388 multiscaling, A:403–404 dichotomous ﬂuctuations, B:38–41 dielectric relaxation Cole-Cole behavior, B:300–305 continuous-time random walk model, B:295–299 fractional Smoluchowski equation solution, B:317–325 inertial effects, symmetric top molecules, B:388–398 fractal anomalous diffusion, B:258–264 Brownian particle distribution function, B:264–267 Brownian particle inertial effects, B:267–269 fractal conductivity, iterative averaging method, B:164–174 fractional diffusion equations, B:73–76 fractional Fokker-Planck equation, Gru¨nwaldLetnikov solution, B:489–492 Le´vy ﬂight process barrier crossing, B:475 bifurcation time, B:462–464 conﬁnement and multimodality, B:453–467 dissipative nonlinearity, B:481–483 ﬁrst arrival time problems, B:468–470 frequency domain induction equations, B:452–453 indices, B:464–467 Langevin equation, statistics, B:77–80 method of images inconsistency, B:472–474 stationary quartic Cauchy oscillator, B:453–454

716

subject index

Probability density function (continued) theoretical background, B:441–448 trimodal transient state, B:462–463 multifractal dimensions, B:119–125 diffusion entropy analysis, B:47–52 noncanonical equilibrium, A:411–413 non-Poisson dichotomous noise, higher-order correlation functions, A:414–421 non-Poisson renewal processes, modulation theories, A:455–456 scaling dynamics, fractional random walks, B:33–34 Probability distribution dielectric relaxation continuous-time random walk model, B:295–299 inertial effects, periodic potentials, B:413–414 fractal structures anomalous diffusion, B:257–259 dielectric relaxation, B:249–253 Probe rotational time correlation functions, vitriﬁcation of liquids binary mixtures, B:568–574 structural relaxation-dispersion coordination, B:521–528 Projection operators continuous time random walk, vs. generalized master equation, A:393–399 Zwanzig projection method Anderson localization, A:369–374 generalized master equation, A:365–367 Propagator representation, dielectric response, A:11–12 Proportionality coefﬁcient, fractal elastic properties, negative shear modulus, B:226–235 Pseudo-Poissonian process, fractal anomalous diffusion, B:259–264 Quadratic function fractional dynamics, physical/physiological models, B:72–73 Le´vy ﬂight processes, numerical solution, B:484–485 Quadrupolar interaction, supercooled liquids and glasses nuclear magnetic resonance, A:149–150 stimulated echo experiments, twodimensional NMR, A:153–155

Quantum Brownian motion, aging phenomena, A:284–296 displacement response and correlation function, A:288–289 displacement response and time-dependent diffusion coefﬁcient, A:289–291 effective temperature determination, A:292 modiﬁed ﬂuctuation-dissipation theorem, A:291–292 non-Ohmic dissipation, A:303 Ohmic model temperature, A:292–295 time-dependent diffusion coefﬁcient, A:286–288 velocity correlation function, A:285–286 Quantum dots. See also Blinking quantum dots aging phenomena, A:337–342 on-off correlation function, A:338 on-off mean intensity, A:337–338 basic principles, A:328 diffusion model, A:331–334 experimental data, A:350–353 nonergodicity, A:342–350 time-averaged correlation function, A:347–350 time-averaged intensity distribution, A:344–346 physical models, A:328–334 stochastic models, A:334–336 Quantum mechanics continuous time random walk correlation function, A:425–429 measurement processes, A:467–469 phenomenological modeling, A:362–364 dichotomous ﬂuctuations, B:35–41 generalized master equation, A:362–364 non-Poisson renewal processes, A:441–447 Quantum Zeno effect generalized master equation, A:364 non-Poisson renewal processes, A:450–451 Quartic potential, Le´vy ﬂight processes dissipative nonlinearity, B:483 numerical solution, B:484–485 Quasielastic neutron scattering, vitriﬁcation of liquids, structural relaxation-dispersion coordination, B:518–528 Quasi-homogeneous medium, fractal conductivity, B:160–163 Quasi-optical methods broadband dielectric spectroscopy, A:16–18

subject index supercooled liquids and glasses, molecular reorientation, A:140 Quasi-stationary approximation fractal dielectric properties, B:176–183 out-of-equilibrium linear response theory, effective temperatures, A:311–312 Rabi frequency, generalized master equation, A:433–435 Radio-frequency ranges, broadband dielectric spectroscopy, A:16–18 Random activation energy models, dielectric relaxation disordered systems, B:293 fractional Smoluchowski equation solution, B:324–325 Random copolymers, structural and JohariGoldstein relaxations in, B:572 Random force Le´vy ﬂight processes, theoretical background, B:442–448 non-Ohmic dissipation, aging effects, noise and friction, A:296–297 overdamped aging effects, classical Brownian motion, A:277–279 statistical properties, Caldeira-Leggett model, A:264 thermal bath diffusion, A:306 Random fractal model, static percolation, porous materials, glasses, A:61–63 Random jump orientation, supercooled liquids and glasses, stimulated echo experiments, two-dimensional NMR, A:154–155 Random trap model, blinking quantum dots, A:331 Random walks dichotomous ﬂuctuations, B:35–41 early time behavior, B:40 exact solution, B:39–40 late time behavior, B:40–41 dielectric relaxation disordered systems, B:287–293 inertial effects, fractional Klein-Kramers equation, B:365–372 fractal anomalous diffusion, B:257–259 Brownian particle inertial effects, B:267–269 fractional dynamics, physical/physiological models, B:65–73

717

fractional rotational diffusion, bistable potential with nonequivalent wells, B:363–364 Le´vy ﬂight processes ﬁrst arrival time problems, B:468–470 multimodality, B:466–467 theoretical background, B:440–448 multifractal spectrum, B:42–46 cis-1,4-polyisoprene thin ﬁlm preparation, B:608–614 scaling dynamics fractional random walks, B:31–34 simple models, B:28–31 trajectories b-peak glass formation, below melting point, A:238–241 blinking quantum dot nonergodicity, binomial random walk on a line, A:345–346 supercooled liquids and glasses, molecular reorientation, A:133–134 Rank squares, Peano function, fractal sets, B:105–110 Ratio of factorials, fractional calculus, B:56–61 Recurrence relation continued fraction solution longitudinal and transverse responses, B:425–427 spherical molecules, B:430–431 dielectric relaxation, inertial effects Barkai/Silbey fractionalized Klein-Kramers equation, B:375–379 Metzler/Klafter fractionalized Klein-Kramers equation, B:368–372 periodic potentials, B:402–414 symmetric top molecules, B:389–398 top molecule rotations, B:382–388 fractional Brownian motion, A:460–461 Reduced Model Theory (RMT), non-Poisson renewal processes, modulation theories, A:455–456 Reﬂectometer instrumentation, time-domain spectroscopy, A:21–22 Regression theorem, thermal bath diffusion, A:306–307 Regular fractal model, static percolation, porous materials, glasses, A:61–63 Regularization technique, dielectric spectra, continuous parameter estimation, A:26

718

subject index

Reisz fractional derivative continuous time random walk, vs. generalized master equation, A:398–399 fractional diffusion equations, B:76 Relatively dual model, chaotic fractal structure, B:152–157 Relaxation. See also Dielectric relaxation; Structural relaxation models fractal structures, B:95–96, B:235– 277 anomalous diffusion, B:253–264 Brownian motion, B:253–264 dielectric relaxation, B:237–246 distribution function of Brownian particles, B:264–267 fractional derivatives diffusion equation, B:269–274 inertial effects of Brownian particle, B:267–277 non-Debye relaxation, B:235–253 one-dimensional lattice random walks, B:267–269 shear-stress relaxation, B:232–235 Smoluchowski equation for Brownian particles, B:255–257 fractal viscoelasticity, B:220–235 fractional rotational diffusion direct current (DC) electric ﬁeld, B:340–347 in potentials, B:328–331 cis-1,4-polyisoprene ﬁlm preparation, B:605–607 vitriﬁcation of liquids, Johari-Goldstein (JG) b-relaxation, B:501–502 Relaxation stretching, supercooled liquids and glasses, A:129–131 low-temperature regime, A:193–201 Relaxation time distribution dielectric relaxation, disordered systems, B:290–293 fractal structures, dielectric relaxation, B:237–253 fractional rotational diffusion direct current (DC) electric ﬁeld, B:340–347 in potentials, B:328–331 polymer ﬁlm molecular dynamics hyperbranched polymer ﬁlm conﬁnement effects, B:615–620 polystyrene ﬁlms, conﬁnement effects, B:625–628

Renewal processes, non-Poisson statistics, decoherence theory, A:435–439 nonordinary environment, A:439–441 quantum to classical transition, A:441–447 trajectory and density entropies, A:447–451 Rennie entropy, multifractal dimensions, B:120–125 Renormalization group model chaotic fractal percolation clusters, B:137–141 continuous time random walk, A:462–463 Rescaling techniques, Le´vy ﬂight processes, space-fractional Fokker-Planck equation, B:451 Resistivity, fractal conductivity, B:161–163 Response function ﬂuctuation-dissipation theorem classical limit, A:273–274 symmetrized correlation function and, A:269–273 out-of-equilibrium linear response theory, age- and frequency-dependent response functions, A:309–311 Riccati equation, Le´vy ﬂight processes, dissipative nonlinearity, B:481–482 Riemann-Liouville (RL) formula dielectric relaxation Cole-Cole behavior, B:304–305 continuous-time random walk model, B:298–299 Debye noninertial rotational diffusion, B:312 fractal structures anomalous diffusion, B:263–264 non-Debye relaxation, B:236–253 viscoelasticity, B:219–235 fractional Brownian motion, B:59–61 fractional calculus, B:57–61 Le´vy ﬂight processes power-law asymptotics of stationary solutions, B:455–456 space-fractional Fokker-Planck equation, B:451 ‘‘strange kinetic’’ percolation, A:75–81 symmetric relaxation, peak broadening in complex systems, A:107–110 Riesz fractional derivative fractional Fokker-Planck equation, Gru¨nwald-Letnikov solution, B:489–492

subject index Le´vy ﬂight processes convergence corroboration, B:467 nonunimodality, B:457–459 space-fractional Fokker-Planck equation, B:450–451 Riesz-Weyl properties, Le´vy ﬂight processes, theoretical background, B:447–448 Ring-shaped clusters, fractal dielectric properties, B:180–183 Rocard equation, dielectric relaxation, inertial effects Barkai/Silbey fractional Klein-Kramers equation, B:376–379 Metzler/Klafter fractional Klein-Kramers equation, B:369–372 symmetric top molecules, B:394–398 top molecule rotations, B:385–388 Root-mean-square parameters, polystyrene ﬁlms, conﬁnement effects, B:622–628 Rotational diffusion equation dielectric relaxation Debye noninertial rotational diffusion, B:305–312 disordered systems, B:288–293 fractional Smoluchowski equation solution, B:319–325 vitriﬁcation of liquids, structural relaxationdispersion coordination, B:520–528 Rough approximation, fractal galvanomagnetic properties, B:196–204 Rouse modes cis-1,4-polyisoprene ﬁlm preparation, B:601–607 vitriﬁcation of liquids coupling model, above glass transition temperature, B:451 structural relaxation-dispersion coordination, B:527–528 Sack’s parameter, dielectric relaxation, inertial effects Barkai/Silbey fractionalized Klein-Kramers equation, B:375–379 Metzler/Klafter fractionalized Klein-Kramers equation, B:367–372 symmetric top molecules, B:394–398 Saddle-like relaxation, doped ferroelectric crystals, A:100–102 Saddle-point method, fractal structures, dielectric relaxation, B:249–253

719

Salol systems, structural relaxation time studies of, B:514–516 Sample holders, time-domain spectroscopy, A:23–24 Sawyer-Tower setup, supercooled liquids and glasses, molecular reorientation, A:137–149 Scala-Shklovsky model, chaotic fractal percolation clusters, B:141–147 Scaling phenomena chaotic fractal percolation clusters, B:131–137 dichotomous ﬂuctuations, B:40–41 dielectric relaxation, continuous-time random walk model, B:295–299 disordered systems, A:55–116 conﬁned relaxation kinetics, A:95–105 dielectric spectrum broadening, A:105–116 doped ferroelectric crystals, liquid-like behavior, A:93–95 dynamic percolation, ionic microemulsions, A:65–73 porous materials, static percolation, A:55–65 glasses, A:58–63 silicon, A:63–65 ‘‘strange kinetic’’ percolation, A:73–81 dynamic models, B:26–52 fractional random walks, B:31–34 inverse power-law autocorrelation functions, B:34–35 simple random walks, B:28–31 in time series, B:27–35 fractal function derivatives, B:57–59 fractal structure, B:97–98 dielectric properties, frequency dependence of, B:174–183 hydrogen-bonding liquids, A:81–93 glycerol-rich mixtures, A:81–86 water-rich mixtures, A:86–93 Julia fractal sets, invariance property, B:113 multifractal spectrum, B:42–46 diffusion entropy analysis, B:46–52 physiological time series, B:4–26 allometric aggregation data analysis, B:5–10 breathing function, B:14–18 dynamic models, B:27–35 gait cycle, B:18–22 heartbeats, B:10–14

720

subject index

Scaling phenomena (continued) neuron action potentials, B:23–26 neurons, B:22–26 supercooled liquids and glasses formation above glass transition temperature, low-temperature regime, A:184–201 mode coupling tests, A:221–225 mode-coupling theory, A:159–162 Scanning grating monochromators, supercooled liquids and glasses, depolarized light scattering, A:143–148 Scattering parameters broadband dielectric spectroscopy, A:17–18 fractal viscoelasticity frequency dependence, B:217–235 Schro¨dinger equation fractals, B:3–4 non-Poisson renewal processes decoherence theory, A:437–439 quantum to classicial transition, A:446–447 Secondary relaxation molecular glass-forming liquids, temperature/ pressure dependence, B:505–507 structural relaxation time, B:530–545 aging phenomena, B:538–540 enthalpy, entropy, and volume mimicry, B:540–541 molecular mobility dependences, B:541–545 non-Arrhenius temperature dependence, B:536–538 pressure dependence, B:530–535 temperature-pressure invariance, B:535–536 vitriﬁcation of liquids, primitive relaxation and, B:528–530 ‘‘Self-concentration’’ effect, vitriﬁcation of liquids, binary mixtures, B:569–574 Self-diffusion mechanisms, vitriﬁcation of liquids, structural relaxationdispersion coordination, B:522–528 Self-similarity chaotic fractal structures, conductivity properties, B:142–144 dielectric relaxation Cole-Cole behavior, B:305 continuous-time random walk model, B:299 fractal construction dielectric relaxation, B:246–253 Haussdorff-Besicovitch dimension, B:118–119 lattice structural models, B:147–157

negative shear modulus, B:230–235 shear-stress relaxation, B:233–235 square lattices, B:126–128 fractal dielectric properties, B:178–183 Julia fractal sets, B:113 Semi-inﬁnite intervals, Le´vy ﬂight processes, method of images inconsistency, B:473–474 Series realization, time-domain spectroscopy, A:22–23 Shannon entropy anomalous diffusion, B:84 multifractal spectrum diffusion entropy analysis, B:46–52 dimensional analysis, B:119–125 noncanonical equilibrium, A:408–409 Shear modulus, fractal elastic properties, B:204–209 negative Poisson’s ratio, B:212–215 viscoelasticity frequency dependence, B:218–235 negative shear modulus, B:225–235 Shear-stress relaxation, fractal viscoelasticity, B:232–235 Short-memory limit, Brownian motion aging effects, quantum model, A:287–288 Sierpinski carpet, fractal sets, B:103 Haussdorff-Besicovitch dimension, B:119 Silica glasses dielectric relaxation, A:38–41 static percolation, A:55–63 Silicon materials dielectric relaxation, A:41–44 static percolation, A:63–65 Similarity transformation, fractal construction, Haussdorff-Besicovitch dimension, B:118–119 Simulations, cis-1,4-polyisoprene thin ﬁlm preparation, B:608–614 Single droplet components, dielectric relaxation, microemulsions, A:32–38 Single-mode approximation, fractional rotational diffusion, direct current (DC) electric ﬁeld, B:344–347 Singularity spectrum fractional dynamics, B:69–73 physical/physiological models, B:72–73 multifractal analysis, B:44–46 Sleep patterns, fractal-based time series analysis breathing function, B:17–18 neuron action potentials, B:25–26

subject index Slow relaxation ﬂuctuation-dissipation theorem out-of-equilibrium physics and, A:259–261 phenomenological dissipation modeling, A:266 time-domain formulation, A:268–276 classical limit, A:273–274 dissipative response function, A:271–273 extreme quantum case, A:274–276 symmetrized correlation function, A:269–273 statistical mechanics, dissipative systems Caldeira-Leggett model, A:262–264 Ohmic dissipation, A:267–268 phenomenological modeling, A:265–267 ‘‘Smearing out’’ phenomenon, fractal structures, dielectric relaxation, B:237–253 Smirnov-Levy process, fractal anomalous diffusion, B:260–264 Smoluchowski equation continuous time random walk, phenomenological modeling, A:364 dielectric relaxation continuous-time random walk model, B:294–299 disordered systems, B:287–293 fractional equation solution, B:316–325 inertial effects, fractional Klein-Kramers equation, B:366–372 fractal anomalous diffusion, Brownian motion, B:255–257 fractional rotational diffusion, direct current (DC) electric ﬁeld, B:338–347 generalized master equation, A:364 Software, dielectric spectroscopy data analysis and modeling, A:30 Space-fractional Fokker-Planck equation, Le´vy ﬂight processes frequency domain induction equations, B:452–453 properties of, B:450–451 theoretical background, B:447–448 Sparre Anderson universality, Le´vy ﬂight processes ﬁrst passage time density, basic principles, B:467–468 random walk analysis, B:470–472 Spatial inhomogeneity fractal structures, non-Debye relaxation, B:236–253

721

vitriﬁcation of liquids, structural relaxationdispersion coordination, B:523–528 Spectral density Ohmic dissipation, A:267–268 phenomenological dissipation modeling, A:265–266 supercooled liquids and glasses, depolarized light scattering, A:140–148 Spherical top molecules continued fraction solution, B:430–434 dielectric relaxation, inertial effects, B:393–398 Spin-lattice relaxation b-peak glass formation, A:235–241 supercooled liquids and glasses, nuclear magnetic resonance, A:148–150 line-shape analysis, A:150–152 Sputtering regimes, fractal dielectric properties, B:183 Square lattice construction chaotic fractal percolation clusters, B:132–137 dielectric relaxation, disordered systems, B:287–293 fractal sets, B:125–128 structural models, B:147–157 Square-root cusp singularity, supercooled liquids and glasses, mode-coupling theory, A:161–162 Standing wave methods, broadband dielectric spectroscopy, A:17–18 Static electric ﬁelds, dielectric polarization, A:3–7 Static lattice site percolation (SLSP) model hyperscaling relationship, A:68–71 ionic microemulsion dipole correlation function, A:65–67 Static percolation, porous materials, A:55–65 glasses, A:58–63 silicon, A:63–65 Stationary medium, particle diffusion equations in, A:308–309 Stationary probability density function, Le´vy ﬂight processes differential equations, B:460–461 nonlinear friction, B:481–482 nonunimodality proof, B:456–459 power-law asymptotics, B:455–456 quadratic and quartic nonlinearity, B:484–485 Stationary quartic Cauchy oscillator, Le´vy ﬂight processes, B:453–454

722

subject index

Statistical density matrix, Zwanzig projection method, Anderson localization, A:369–374 Statistical fractals, deﬁned, B:4 Statistical mechanics dielectric polarization, A:6–7 dissipative systems Caldeira-Leggett model, A:262–264 Ohmic dissipation, A:267–268 phenomenological modeling, A:265–267 ‘‘strange kinetic’’ percolation, A:74–81 Steepness index, vitriﬁcation of liquids dispersion correlation with structural relaxation, B:516–517 Johari-Goldstein secondary relaxation, molecular mobility and, B:543–545 Stickel analysis, supercooled liquids and glasses, formation above glass transition temperature, time constants and decoupling phenomenon, A:169– 173 Stimulated echo experiments b-peak glass formation, below melting point, A:239–241 supercooled liquids and glasses nuclear magnetic resonance, molecular reorientation, A:212–214 two-dimensional nuclear magnetic resonance, A:152–155 Stirling’s approximation, scaling dynamics, fractional random walks, B:32–34 Stochastic analysis blinking quantum dots, A:334–336 dichotomous ﬂuctuations, B:35–41 fractional Brownian motion, B:60–61 fractional diffusion equations, B:73–76 fractional rotational diffusion, bistable potential with nonequivalent wells, B:363–364 Le´vy ﬂight processes barrier crossing, B:474–475 overview of, B:488 scaling behavior, B:82–83 correlation function, B:82–83 Stokes-Einstein (SE) relation out-of-equilibrium environment, A:315–317 supercooled liquids and glasses, formation above glass transition temperature, time constants and decoupling phenomenon, A:171–173

vitriﬁcation of liquids binary mixtures, B:570–574 structural relaxation-dispersion coordination, B:520–528 Stokes equation, Le´vy ﬂight processes, dissipative nonlinearity, B:481–482 Strain forces, fractal viscoelasticity frequency dependence, B:215–235 ‘‘Strange kinetics’’ dielectric relaxation, basic principles, A:3 percolation phenomena, A:73–81 Stress forces, fractal viscoelasticity frequency dependence, B:215–235 Stretched exponental laws, fractal structures, non-Debye relaxation, B:236–253 Stride rate variability (SRV) fractal-based time series analysis, B:19–22 fractional dynamics, B:68–73 multifractal spectrum analysis, B:45–46 Structural relaxation time fractional rotational diffusion, bistable potential with nonequivalent wells, B:361–364 Johari-Goldstein relaxation and, B:530–545 aging phenomena, B:538–540 enthalpy, entropy, and volume mimicry, B:540–541 molecular mobility dependences, B:541–545 non-Arrhenius temperature dependence, B:536–538 pressure dependence, B:530–535 temperature-pressure invariance, B:535–536 molecular glass-forming liquids, temperature/ pressure dependence, B:503–507 vitriﬁcation of liquids basic principles, B:498–502 coupling model, B:501–502 basic properties, B:501–502 binary component dynamics, B:567–574 dispersion invariance to temperature and pressure combinations, B:561 enthalpy activation in glassy state, B:555–556 Johari-Goldstein relaxation correspondence, B:551–555 many-molecule dynamics, B:577–580 nanometric relaxation, B:562–567 polymer systems primary/secondary relaxation interrelations, B:574–577

subject index pressure-temperature dependence above glass transition temperature, B:556–561 primitive relaxation, B:529–530 theoretical background, B:546–551 dispersion correlation with, B:516–528 primitive relaxation and, B:528–530 temperature/pressure superpositioning, B:509–511 Sturm-Liouville representation dielectric relaxation, inertial effects, fractionalized Klein-Kramers equation, B:366–372 fractional rotational diffusion bistable potential with nonequivalent wells, B:349–364 direct current (DC) electric ﬁeld, B:340–347 double-well periodic potential, anomalous diffusion and dielectric relaxation, B:333–338 in potentials, B:326–331 Subdiffusion dielectric relaxation continuous-time random walk model, B:295–299 Debye noninertial rotational diffusion, B:306–312 inertial effects Barkai-Silbey fractional Klein-Kramers equation, B:373–379 periodic potentials, B:407–414 Le´vy ﬂight processes barrier crossing, B:474–481 theoretical background, B:443–448 Subjective collapse, non-Poisson renewal processes, quantum to classicial transition, A:446–447 Subordinate processes, fractal anomalous diffusion, B:259–264 Supercooled liquids and glasses basic properties, A:128–131 glass transition phenomenon, A:155–162 mode coupling theory, A:157–162 molecular glass formation above melting point, A:162–231 dynamic susceptibility evolution, A:163–169, A:228–231 high-temperature regime, A:173–182

723

low-temperature regime, A:182–214 b-peak glass formation, A:201–209 excess wing evolution, A:182–201 mode coupling theory tests, A:220–228 asymptotic scaling laws, A:221–225 two-correlator model, A:225–228 molecular reorientation mechanism, A:209–214 nonergodicity parameter temperature dependence, A:214–220 time constants and decoupling phenomenon, A:169–173 molecular glass formation below melting point, A:231–241 b-glass formation, A:235–241 low-temperature regime, constant loss phenomena, A:231–235 molecular reorientation dynamics, A:131–155 correlation function, spectrum, and susceptibility, A:131–134 depolarized light scattering, A:140–148 dielectric spectroscopy, A:134–140 coaxial reﬂection and transmission techniques, A:139 frequency response analysis, A:139 quasi-optical and FTIR spectroscopy, A:140 time-domain analysis, A:137–139 nuclear magnetic resonance, A:148–155 spin-lattice relaxation and line-shape analysis, A:150–152 stimulated echo experiments and twodimensional NMR, A:152–155 time windows and spin-lattice interactions, A:148–150 Superdiffusion dielectric relaxation, continuous-time random walk model, B:295–299 generalized Master Equation, A:384–404 central limit theorem, A:385–388 continuous-time random walk vs., A:392–399 Gaussian case, A:399–401 Le´vy derivation, A:388–392 multiscaling, A:403–404 trajectories approach, A:401–403 Superharmonicity, Le´vy ﬂight processes multimodal states, B:463–464 nonunimodality, B:457–459 power-law asymptotics, B:455–456

724

subject index

Surface-to-volume ratio polystyrene ﬁlms, conﬁnement effects, B:620–628 thin polymer ﬁlm preparation, B:598–601 Survival probability blinking quantum dots, diffusion model, A:332–334 continuous time random walk, laminar regions, A:380–381 Susceptibility minimum, supercooled liquids and glasses, mode coupling tests, A:221–225 Symmetrical top molecules, inertial effects, anomalous dielectric relaxation, B:380–398 rotators in space, B:380–388 Symmetric relaxation continuous time random walk, correlation function, A:427–429 peak broadening in complex systems, A:106–110 Symmetrized correlation function ﬂuctuation-dissipation theorem, A:269–273 classical limit, A:273–274 quantum Brownian motion aging, A:284–296 Tauberian theorem fractal structure, anomalous diffusion, B:262–264 scaling dynamics, simple random walks, B:31 Taylor series chaotic fractal percolation clusters, renormalization-group transformations, B:139–141 fractal functions, B:277 fractional rotational diffusion, bistable potential with nonequivalent wells, bimodal approximation, B:357–364 Julia fractal sets, B:110–113 Le´vy ﬂight processes, theoretical background, B:442–448 noncanonical equilibrium, A:407–409 physiological time series heart rate variability, B:12–14 scaling behavior, B:6–10 scaling dynamics, inverse power-law autocorrelation functions, B:34–35 Temperature dependence Brownian motion aging effects Langevin model, ﬂuctuation-dissipation ratio and, A:283–284

quantum motion determination, A:292 Ohmic model, A:292–295 hydrogen-bonding liquids, scaling phenomena, glycerol-rich mixtures, A:82–86 microemulsion structure and dynamics, A:33–38 out-of-equilibrium environment Kubo formula for generalized susceptibilities, A:312–315 power-law behaviors, A:318–319 quasi-stationary regime, A:311–312 Stokes-Einstein relation, A:315–317 cis-1,4-polyisoprene ﬁlm preparation, B:604–607 porous glasses relaxation response, A:40–41 porous silicon dielectric relaxation, A:41–44 supercooled glass formation above glass transition temperature b-peak glass formation, A:202–209 nonergodicity parameter, A:214–220 supercooled glass formation below glass transition temperature, A:232–241 vitriﬁcation of liquids basic principles, B:499–502 coupling model above glass transition temperature, B:560–561 nanometric relaxation, B:562–567 structural-Johari-Goldstein relaxation correspondence, B:553–555 dispersion mechanisms, B:502–516 amorphous polymers, B:507–509 molecular glass-formers, B:503–507 photon correlation spectroscopy analysis, B:512–516 structural relaxation time, B:509–516 Johari-Goldstein secondary relaxation, B:530–536 molecular mobility and, B:541–545 non-Arrhenius temperature dependence, B:536–538 many-molecule dynamics, B:577–580 structural and Johari-Goldstein relaxation interrelations, B:575–577 structural relaxation-dispersion coordination, B:518–528

subject index Tensor components fractal viscoelasticity frequency dependence, B:216–235 fractional rotational diffusion, bistable potential with nonequivalent wells, B:348–364 Theoretical laminar regions, continuous time random walk, intermittent dynamic model, A:379–381 Thermal bath diffusion, out-of-equilibrium physics, A:304–307 ﬂuctuation-dissipation theorems, A:304–306 particle diffusion equations, A:309 regression theorem, A:306–307 Thermodynamic noncanonical equilibrium dynamic approach to, A:410–413 information approach to, A:407–409 Tikhonov regularization procedure, hyperbranched polymer ﬁlm conﬁnement effects, B:617–620 Time average correlation function, blinking quantum dots, stochastic models, A:336 Time-averaged correlation function, blinking quantum dot nonergodicity, A:347–350 Time-averaged intensity distribution, blinking quantum dot nonergodicity, A:344–346 Time behavior dichotomous ﬂuctuations, B:40–41 fractal structures, dielectric relaxation, B:249–253 Time constants supercooled glass formation above glass transition temperature, b-peak glass formation, A:202–209 supercooled liquids and glasses, formation above glass transition temperature, A:169–173 Time correlation function, supercooled liquids and glasses, molecular reorientation, A:132–134 Time-dependent diffusion coefﬁcient Brownian motion aging effects Langevin model, A:281–283 quantum model, A:286–288 displacement response and, A:289–291

725

non-Ohmic dissipation, aging effects, A:300–302 Time-dependent electric ﬁelds dielectric polarization, A:7–12 supercooled liquids and glasses, molecular reorientation, dielectric spectroscopy, A:135–140 Time-dependent microscopic density, supercooled liquids and glasses, molecular reorientation, A:131–134 Time-dependent rate, vitriﬁcation of liquids, coupling model, B:547–551 Time domain analysis dielectric relaxation, inertial effects, fractionalized Klein-Kramers equation, B:372 dielectric response, A:8–12 ﬂuctuation-dissipation theorem (FDT), A:268–276 classical limit, A:273–274 dissipative response function, A:271–273 extreme quantum case, A:274–276 symmetrized correlation function, A:269–273 fractional rotational diffusion, direct current (DC) electric ﬁeld, B:343–347 Time-domain measurement system (TDMS), time-domain spectroscopy, A:22 Time-domain reﬂectometry (TDR), deﬁned, A:18 Time-domain spectroscopy (TDS) basic principles, A:18–25 data processing, A:25 deﬁned, A:16 hardware tools, A:21–22 microemulsion structure and dynamics, A:33–38 nonuniform sampling, A:22–23 sample holders, A:23–24 supercooled liquids and glasses depolarized light scattering, A:147–148 formation above glass transition temperature, high-temperature regime, A:179–182 molecular reorientation, A:137–139 Time-independent projection operators, Zwanzig projection method, A:365–367

726

subject index

Time series analysis of disease states, B:86–87 physiological time series complex systems, B:81–82 scaling behavior, B:4–26 allometric aggregation data analysis, B:5–10 breathing function, B:14–18 dynamic models, B:27–35 gait cycle, B:18–22 heartbeats, B:10–14 neurons, B:22–26 scaling dynamics, B:27–35 Time-temperature superposition principle, supercooled liquids and glasses, mode-coupling theory, A:160–162 Time windows, supercooled liquids and glasses, nuclear magnetic resonance, A:148–150 Top molecules inertial effects, anomalous dielectric relaxation, linear and symmetrical molecules, B:380–398 fractional rotational diffusion, B:388–398 rotators in space, B:380–388 spherical molecules, continued fraction solution, B:430–434 Trajectories continuous time random walk, A:401–403 Le´vy ﬂight processes, barrier crossing, B:475–478 non-Poisson renewal processes, A:447–451 Transcranial Doppler ultrasonography (TCD), fractional dynamics, B:70–73 Transition probability, dielectric relaxation, continuous-time random walk model, B:299 Translational diffusion coefﬁcient, vitriﬁcation of liquids, structural relaxationdispersion coordination, B:520–528 Transmission line theory, time-domain spectroscopy, A:18–19 Transverse response continued fraction solution, B:425–427 dielectric relaxation, inertial effects, periodic potentials, B:399–414 fractional rotational diffusion bistable potential with nonequivalent wells, B:348–350 bimodal approximation, B:356–364

direct current (DC) electric ﬁeld, B:339–347 Trapping events dielectric relaxation, inertial effects fractional Klein-Kramers equation, B:365–372 linear and symmetrical top molecules, B:381–388 periodic potentials, B:400–414 Le´vy ﬂight processes, theoretical background, B:443–448 Traveling-wave methods, broadband dielectric spectroscopy, A:17–18 Triangular lattices, chaotic fractal percolation clusters, renormalization-group transformations, B:138–141 Trimodal transient state, Le´vy ﬂight processes, B:462–463 Tri-styrene, structural and Johari-Goldstein relaxations, B:572–577 Tsallis entropy, noncanonical equilibrium, A:409 Two-correlator model, supercooled liquids and glasses, mode-coupling tests, A:225–227 Two-dimensional nuclear magnetic resonance, supercooled liquids and glasses molecular reorientation, A:213–214 stimulated echo experiments, A:152–155 Two-rotational-degree-of-freedom model, dielectric relaxation, inertial effects, top molecule rotations, B:387–388 Ultrametric space Cayley tree fractal structures, B:128–131 fractal shear-stress relaxation, B:234–235 Uniform compression modulus, fractal structure, negative Poisson’s ratio, B:211–215 Universality, Le´vy ﬂight processes ﬁrst passage time density, B:467–474 Sparre Anderson universality, B:470–472 Van der Waerden function, fractal sets, B:101–102 Van Vleck-Weisskopf model, dielectric relaxation, inertial effects fractionalized Klein-Kramers equation, B:372 fractional Langevin equation, B:418–419 Variational approach, fractal conductivity, B:164 Velocity correlation function

subject index Brownian motion aging phenomena Langevin model, A:279–280 quantum Brownian motion, A:285–286 non-Ohmic dissipation, aging effects Mittag-Lefﬂer relaxation, A:298–299 Wiener-Khintchine theorem, A:299 out-of-equilibrium physics, Kubo formula, A:313–314 Velocity distribution, Le´vy ﬂight processes, dissipative nonlinearity, B:481–487 Ventilation mechanics, fractal-based time series analysis, B:17–18 Verhulst dynamics, fractal sets, B:113–115 Viscoelastic frequency dependence fractal structures, B:215–235 iterative averaging method, B:221–222 negative shear modulus, B:225–231 shear stress relaxation, B:232–235 two-phase calculations, B:222–225 vitriﬁcation of liquids coupling model above glass transition temperature, B:560–561 nanometric relaxation, B:562–567 structural relaxation-dispersion coordination, B:525–528 Viscosity coefﬁcients, vitriﬁcation of liquids, coupling model, B:548–551 Viscous drag coefﬁcient, fractal anomalous diffusion, Brownian motion, B:253–255 Viscous freezing. See Glass transition temperature Vitriﬁcation of liquids basic principles, B:498–502 coupling model basic properties, B:501–502 binary component dynamics, B:567–574 dispersion invariance to temperature and pressure combinations, B:561 enthalpy activation in glassy state, B:555–556 many-molecule dynamics, B:577–580 nanometric relaxation, B:562–567 polymer systems primary/secondary relaxation interrelations, B:574–577 pressure-temperature dependence above glass transition temperature, B:556–561

727

primitive relaxation, B:529–530 structural relaxation and Johari-Goldstein relaxation correspondence, B:551–555 theoretical background, B:546–551 dispersion mechanisms basic principles, B:498–502 structural relaxation coordination with, B:516–528 time and pressure invariance, B:502–516 amorphous polymers, B:507–509 molecular glass-formers, B:503–507 photon correlation spectroscopy analysis, B:512–516 structural relaxation time, B:509–516 Johari-Goldstein secondary relaxation primitive relaxation and, B:528–530 structural relaxation and, B:530–545 aging phenomena, B:538–540 enthalpy, entropy, and volume mimicry, B:540–541 molecular mobility dependences, B:541–545 non-Arrhenius temperature dependence, B:536–538 pressure dependence, B:530–535 temperature-pressure invariance, B:535–536 Vogel-Fulcher-Tammann-Hesse (VFTH) equation, vitriﬁcation of liquids, B:499–502 many-molecule dynamics, B:577–580 temperature dependence, B:518–528 Vogel-Fulcher-Tammann (VFT) equation conﬁned system relaxation kinetics, A:102–104 ferroelectric crystals dielectric spectroscopy, A:47–48 liquid-like behavior, A:94–95 hydrogen-bonding liquids, A:52–55 scaling phenomena glycerol-rich mixtures, A:82–86 water-rich mixtures, A:86–93 relaxation kinetics, A:13–15 supercooled liquids and glasses, formation above glass transition temperature low-temperature regime, A:196–201 time constants and decoupling phenomenon, A:169–173 Voigt model, fractal viscoelasticity, B:218–235

728

subject index

Voltage step, time-domain spectroscopy, A:18–19 Volume elastic modulus fractal viscoelasticity frequency dependence, B:216–235 vitriﬁcation of liquids, Johari-Goldstein secondary relaxation, B:540–541 molecular mobility and, B:541–545 Volume-pressure dependence, vitriﬁcation of liquids, basic principles, B:501–502 Von Neumann entropy non-Poisson processes, trajectory and density entropies, A:448–451 non-Poisson renewal processes, decoherence theory, A:435–439 Von Schweidler component, supercooled liquids and glasses formation above glass transition temperature, A:173 high-temperature regime, A:179–182 mode coupling tests, A:221–225 Waiting-time probability density function, dielectric relaxation, inertial effects, fractional Klein-Kramers equation, B:365–372 Water/oil-continuous (W/O) microemulsion, dielectric relaxation, A:31–38 Water-rich mixtures conﬁned system relaxation kinetics, A:98–100 hydrogen-bonding liquids, scaling phenomena, A:86–93 Wave function, non-Poisson renewal processes, decoherence theory, A:439 Wave functions, broadband dielectric spectroscopy, A:16–18 Weierstrass function dielectric relaxation, continuous-time random walk model, B:297–299 fractional dynamics, B:53–55 West-Seshadri (WS) noncanonical equilibrium, A:412–413 non-Poisson dichotomous noise, A:415–416 Weyl fractional operator, Le´vy ﬂight processes, space-fractional Fokker-Planck equation, B:450–451

Wick’s theorem, dielectric relaxation, microscopic models, B:294 Wiener-Hopf equation, Le´vy ﬂight processes, method of images inconsistency, B:473–474 Wiener-Khintchine theorem Brownian motion aging phenomena Langevin model, velocity correlation function, A:280 quantum model, velocity correlation function, A:285–286 dielectric relaxation, fractional Smoluchowski equation solution, B:325 non-Ohmic dissipation, aging effects, velocity correlation function, A:299 supercooled liquids and glasses, molecular reorientation, A:134 Wiener process dielectric relaxation, disordered systems, B:292–293 fractal functions, B:275–277 fractional Brownian motion, B:60–61 fractional dynamics, physical/physiological models, B:66–73 Wigner formalism, non-Poisson renewal processes, A:442–447 Williams-Landel-Ferry (WLF) equation, vitriﬁcation of liquids, B:499–502 Williams-Watts ansatz, supercooled glass formation above glass transition temperature, b-peak glass formation, A:204–209 Wright function, inverse Fourier transform calculations, B:422–424 Zeeman Hamiltonian, supercooled liquids and glasses, nuclear magnetic resonance, A:150 Zener model, fractal viscoelasticity, B:218–235 shear-stress relaxation, B:232–235 Zero-temperature ﬂuctuation-dissipation theorem, extreme quantum case, A:274–276 Zwanzig projection method generalized master equation, A:364–375 Anderson localization, A:368–374 non-Ohmic bath, A:459 non-Poisson dichotomous noise, A:415–416