Chaos, Complexity and Transport: Theory and Applications

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CHAOS, COMPLEXITY AND TRANSPORT

T-

CHAOS, COMPLEXITY AND

TRANSPORT

Theory and Applications Proceedings of the CCT '07

This page intentionally left blank

CHAOS, COMPLEXITY AND

TRANSPORT

Theory and Applications Proceedingts of the CCT '07 Marseille, France

4 - 8 June 2007

edited by

Cristel Chandre CNRS, France

Xavier Leoncini Aix-Marseille Universite, France

George Zaslavsky New York University, USA

World Scientific N E W JERSEY

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LONDON

SINGAPORE

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BEIJING

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TAIPEI

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CHENNAI

Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

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British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

CHAOS, COMPLEXITY AND TRANSPORT Theory and Applications Proceedings of the CCT'07 Copyright 0 2008 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN- 13 978-98 1-281-879-9 ISBN- 10 981 -281-879-0

Printed in Singapore by World Scientific Printers

V

PREFACE Chaos and turbulence are ubiquitous features of physical systems. Their manifestations are very diverse and not always well understood. Improving knowledge in this field is not only important for our apprehension of non-linear physics but also essential to tame or control their behaviours. Moreover the interdisciplinary character of such phenomena which were observed notably in fluid dynamics, atomic physics, plasma physics, accelerator physics, celestial mechanics, condensed matter, among others, makes research in this field quite peculiar as any advance in one direction may have strong repercussions and consequences in others (Chaos!). As a shared interest one can notably consider transport properties. These are often characterised by Ldvy-type processes, strange (fractal) kinetics, intermittency, etc. Typically one finds that portions of the trajectories are almost regular for quite a long time. This phenomenon (Ldvy flight) gives rise to strong memory effects. History comes into play, and thus rules out the traditional use of Markov processes to model transport. What makes these properties so special is that they are associated with rare events in time but are crucial for the physical behaviour of the system. One may also emphasise on the important role played by coherent structures and their impact on transport. The classical approach to study transport dynamics has been complemented by various novel approaches, based on either the development of new physical and mathematical ideas and on the implementation of sophisticated numerical codes. The concept of Ldvy processes, fractional kinetics and anomalous transport have proved to be extremely important from a conceptual point of view indicating a new direction in the non-linear dynamics. However, many questions are still open, from both a conceptual and an applied point of view. For example, the role of chaotic advection in complex situations has still to be properly addressed. The aim is to understand which (if any) of the properties commonly attributed to the processes of turbulent dispersion may be accounted for by the basic non-linear mechanisms encountered in chaotic advection. Analogously, the non-perfect nature of the tracers used in geophysical measurements and/or the possible “active” nature of some constituents turns out to be very important, determining a different behaviour of the advected particles and of true fluid particles.

vi

Further on, it is now clear that there exist regimes of anomalous transport, which may lead to a faster spreading and escape of advected quantities. Such a phenomenon is especially important in plasma dynamics, as well as in turbulent flows due to the action of coherent structures. The regimes of anomalous transport may have a truly asymptotic nature or they may be intermediate regimes encountered in proximity of significant time scales of the system; a better comprehension of these regimes is necessary for various applications. One of them being the transport in magnetised fusion plasmas, understanding transport in these systems as well as characterising the origin of anomalous behaviour is essential not only to define proper control strategies to obtain better confinement, but also to monitor what may happen near the plasma edge, where the energy is collected. All these anomalous phenomena arise once we accept the fact that uniform chaos is not often realistic. In the early days of the study of chaos, ergodic theory provided an adequate support for the kinetic approach. This is no longer the case. If we are to describe new experimental observations, data from simulations, and to develop new applications, a significantly broader notion of transport is required as well as an expanded arsenal of mathematical tools. The phase space is divided between regions where the motion is regular or irregular. Such diversity in the dynamical landscape makes transport properties more subtle than initially anticipated. In fact, many difficulties are already present in the case of few degrees of freedom Hamiltonian systems. Typically the phase space of smooth Hamiltonian systems is not ergodic in a global sense, due to the presence of islands of stability, the rate of phase space mixing in the chaotic sea is not uniform due to the phenomenon of %tickiness”, and the Gaussian nature of transport is generally lost, due to the so-called flights and trappings and the associated powerlaw tails observed in probability distribution functions. This last feature is also shared with most systems dealing with complexity. Understanding the paths from dynamics to kinetics and from kinetics to transport and complexity involves a strong interdisciplinary interaction among experts in theory, experiments and applications. The contributions are the proceedings of the conference Chaos, Complexity and Transport which was held in Marseilles (France) from June 4th to June 8th 2007. Due to the interdisciplinary character of the problem the conference made a point on balancing theoretical, numerical and experimental contributions in order to encourage the interactions between experimentalists and theoreticians in the same fields but also cross-disciplinary contributions.

vii

This book is organised into two parts. In the first part, we gather what we consider more general or theoretical contributions, while the second part is dedicated to applications. In the first part, some features of the dynamics for large N systems with long range interactions and a large number of degree of freedom giving rise to out of equilibrium phase transitions are presented in detail. One may also discover how stochastic webs in multidimensional systems can be used as a way for tiling the plane with specific symmetries. At the same time one discusses the phenomenon of chaotic transport and chaotic mixing through the course of geodesics or the construction of mixing flows using knots. Then one can learn about entropy and complexity, or about Bose-Einstein condensation of classical waves, as well as transport in deterministic ratchets. Regarding Hamiltonian systems, some new approaches to the theoretical treatment of separatrix chaos are discussed. The possibility of having a giant acceleration and about a control technique in area preserving maps are explored. The second part covers mainly applications. To facilitate reading, we have created two subdivisions. The first one deals with plasmas and fluids, while the second concerns more fields. In the plasmas and fluid subdivision, one will be able to learn in some detail, the implication of topological complexity and Hamiltonian chaos in fusion plasmas, as well as precise experimental studies of advection-reaction diffusion systems. And also nondiffusive transport observed in simulations of plasmas and the problem of solving numerically rotating Rayleigh-BQnardconvection in cylinders. Then there are the experimentally observed self-excited instabilities in plasmas containing dust particles and the clustering properties of plasma turbulence signals as well as intermittency scenario of transition to chaos in plasma. Finally, one can read about magnetic reconnection in collisionless plasmas as well as the complexity of the neutral curve of oscillatory flows. In the second subdivision, an overview of chemotaxis models using an interesting analogy with non-linear mean-field Fokker-Planck equations is presented. Then one shall learn about switchability of a flow, before moving to celestial mechanics and the formation of spiral arms and rings in barred galaxies or learning about LBvy walks for energetic electrons in space. The phenomenon of wave chaos in an underwater sound-channel is then discussed, followed by problem of Fermi acceleration in randomised driven billiards. Finally one shall learn about memory regeneration phenomenon in fractional depolarisation of dielectrics, as well as nodal pattern analysis

viii

for conductivity of quantum ring and the application of the GAL1 method to the dynamics of multidimensional symplectic maps. As already mentioned, this book reflects to some extent the presentations and the resulting discussions carried out during the conference Chaos, Complexity and Transport: Theory and Applications, which was held in the Pharo site of the Universitk de la Mkditerranke, Marseilles, France, in June 2007. In these regard, we would like to thank all participants and express our sincere gratitude to the contributing authors. We also take this opportunity to express our debt and gratitude for the support to sponsors: Centre National de la Recherche Scientifique, the GDR Phenix et GDR Dycoec, the Commissariat B 1’Energie Atomique (CEA), the Conseil Gknkral des Bouches du RhGne, the Ministkre D616guk B la Recherche, the Ville de Marseille, the GREFI-MEFI, the European Physical Society, the University de Provence and University de la Mkditerranke, The US department of Naval Research, the Delegation Generale de 1’Armement and the Centre de Physique Thkorique (UMR 6207). We also would like to thank Mrs A. Elbaz, V. Leclercq-Ortal and M-T Done1 (from the Centre de Physique Thkorique) for their help before, during and after the workshop. We would like also to thank M. Mancis and S. Foulu from Protisvalor Mkditerranke for their help. Cristel Chandre Xavier Leoncini George Zaslavsky Editors

ix

CONTENTS

Preface

V

THEORY

1

Out-of-Equilibrium Phase Transitions in Mean-Field Hamiltonian Dynamics

P.-H. Chavanis, G. De Ninno, D. Fanelli and S. Ruff0

3

Stochastic Webs in Multidimensions

G. M. Zaslavsky and M. Edelman

27

Chaotic Geodesics

J.-L. Thiffeault and K . Kamhawi

A Steady Mixing Flow with No-Slip Boundaries R. S. MacKay

40

55

Complexity and Entropy in Colliding Particle Systems

M. Courbage and S. M. Saberi Fathi

69

Wave Condensation

S. Rica

84

Transport in Deterministic Ratchets: Periodic Orbit Analysis of a Toy Model

R. Artuso, L. Cavallasca and G. Cristadoro

106

Separatrix Chaos: New Approach to the Theoretical lleatment

S. M. Soskin, R. Mannella and 0. M. Yevtushenko Giant Acceleration in Weakly-Perturbed Space-Periodic Hamiltonian Systems M . Yu. Uleysky and D. V. Malcarov

119

129

X

Local Control of Area-Preserving Maps C. Chandre, M. Vittot and G. Caraolo

136

APPLICATIONS (1) PLASMA & FLUIDS

145

Implications of Topological Complexity and Hamiltonian Chaos in the Edge Magnetic Field of Toroidal Fusion Plasmas 7'. E. Evans

147

Experimental Studies of Advection-Reaction-Diffusion Systems T. H. Solomon, M. S. Paoletti and M. E. Schwartz

177

Non-Diffusive Transport in Numerical Simulations of Magnetically-Confined Turbulent Plasmas R. Sa'nchez, B. A . Carreras, L. Garcia, J. A . Mier, B. Ph. Van Milligen and D. E. Newman

189

Rotating Rayleigh-Birnard Convection in Cylinders J . J. Sa'nchea-Alvarez, E. Serre, E. Crespo Del Arc0 and F. H. Busse

207

Self-Excited Instabilities in Plasmas Containing Dust Particles (Dusty or Complex Plasmas) M.Mikikian, M. Cavarroc, L. Couedel, Y. Tessier and L. Boufendi

218

Clustering Properties of Confined Plasma Turbulence Signals M. RajkoviC and M. jkoriC

227

Intermittency Scenario of Transition to Chaos in Plasma D. G. Dimitriu and S. A . Chiriac

237

Nonlinear Dynamics of a Hamiltonian Four-Field Model for Magnetic Reconnection in Collisionless Plasmas E. Tussi, D. Grasso and F. Pegoraro

245

On the Complexity of the Neutral Curve of Oscillatory Flows M. Wadih, S. Carrion, P. G. Chen, D. Fougbre and B. Roux

255

XI

(2) OTHERS

263

Generalized Keller-Segel Models of Chemotaxis. Analogy with Nonlinear Mean Field Fokker-Planck Equations

P.- H. Chavanis

265

On Switchability of a Flow to the Boundary in a Periodically Excited Discontinuous Dynamical System

A. C. J . Luo and B. M. Rapp

287

The Formation of Spiral Arms and Rings in Barred Galaxies

M. Romero- Gdmez, E. Athanassoula, J. J. Masdemont and C. Garcia-Gdmez

300

LBvy Walks for Energetic Electrons Detected by the Ulysses Spacecraft a t 5 AU

S. Perri and G. Zimbardo

309

Wave Chaos and Ghost Orbits in an Underwater Sound Channel

D. V. Makarov, L. E. Kon’kov, E. V. Sosedko and

M. Yu. llleysky

318

Displacement Effects on Fermi Acceleration in Randomized Driven Billiards

A . K. Karlis, P. K. Papachristou, F. K. Diakonos, V. Constantoudis and P. Schmelcher Memory Regeneration Phenomenon in Fractional Depolarization of Dielectrics V. V. Uchaikin and D. V. Uchaikin

327

337

Nodal Pattern Analysis for Conductivity of Quantum Ring in Magnetic Field

M. Tomiya, S. Sakamoto, M. Nishikawa and Y. Ohmachi Application of the Generalized Alignment Index (GALI) Method to the Dynamics of Multi-Dimensional Symplectic Maps T. Manos. Ch. Skokos and T. Bountis

346

356

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THEORY

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3

OUT-OF-EQUILIBRIUM PHASE TRANSITIONS IN MEAN-FIELD HAMILTONIAN DYNAMICS PIERREHENRI CHAVANTS

Laboratoire de Physique The‘orique, Universite‘ Paul Sabatier, 118, route de Narbonne 31062 Toulouse, h n c e E-mail: [email protected] tlse.fr GIOVANNI DE NINNO

Sincrotrone ’Prieste, S.S. 14 K m 163.5, Basovizza, 34012, l?rieste, Italy University of Nova Gon’ca, Vipavska 13, P O B 301, SI-5000, Nova Gorica, Slovenia E-mail: giovanni.deninnoOe1ettra.trieste.it DUCCIO FANELLI

Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom E-mail: Duccio.FanelliOmanchester.ac.uk STEFAN0 RUFF0

Dipartimento di Energetica “S. Stecco” and CSDC, Universitb d i Firenze, and INFN, via S. Marta, 3, 50139 Firenze, Italy E-mail: stefano.ruffoOunafi.it Systems with long-range interactions display a short-time relaxation towards Quasi-Stationary States (QSSs), whose lifetime increases with system size. With reference to the Hamiltonian Mean Field (HMF) model, we here review Lynden-Bell’s theory of “violent relaxation”. The latter results in a maximum entropy scheme for a water-bag initial profile which predicts the presence of out-of-equilibrium phase transitions separating homogeneous (zero magnetization) from inhomogeneous (non-zero magnetization) QSSs. Two different parametric representations of the initial condition are analyzed and the features of the phase diagram are discussed. In both representations we find a second order and a first order line of phase transitions that merge at a tricritical point. Particular attention is payed to the condition of existence and stability of the homogeneous phase.

Keywords: Quasi-stationary states, Hamiltonian Mean-Field model, Out-ofequilibrium phase transitions.

4

1. Introduction Haniiltonian systems arise in many branches of applied and fundamental physics and, in this respect, constitute a universal framework of extraordinary conceptual importance. Spectacular examples are undoubtedly found in the astrophysical context. The process of hierarchical clustering via gravitational instability, which gives birth to the galaxies,' can in fact be cast in a Hamiltonian setting. Surprisingly enough, the galaxies that we observe have not yet relaxed to thermodynamic equilibrium and possibly correspond to intermediate Quasi-Stationary States (QSSs). The latter are in a long-lasting dynamical regime, whose lifetime diverges with the size of the system. The emergence of such states has been reported in several different domains, ranging from charged cold plasmas2 to Free Electron Lasers (FELs),~and long-range forces have been hypothesized to be intimately connected to those peculiar phenomena. Long-range interactions are such that the two-body interaction potential decays at large distances with a power-law exponent which is smaller than the space dimension. The dynamical and thermodynamical properties of physical systems subject to long-range couplings were poorly understood until a few years ago, and their study was essentially restricted to astrophysics (e.g., self-gravitating systems). Later, it was recognized that long-range systems exhibit universal, albeit unconventional, equilibrium and out-of-equilibrium feature^.^ Besides slow relaxation to equilibrium, these include ensemble inequivalence (negative specific heat, temperature jumps), violations of ergodicity and disconnection of the energy surface, subtleties in the relation of the fluid (i.e. continuum) picture and the particle (granular) picture, new macroscopic quantum effects, etc.. While progress has been made in understanding such phenomena, an overall interpretative framework is, however, still lacking. In particular, even though the ubiquity of QSSs has been accepted as an important general concept in non-equilibrium statistical mechanics, different, contrasting, at tempts to explain their emergence have catalysed a vigorous discussion in the l i t e r a t ~ r e . ~ To shed light onto this fascinating field, one can resort to toy models which have the merit of capturing basic physical modalities, while allowing for a dramatic reduction in complexity. This is the case of the so-called Hamiltonian Mean Field (HMF) model which describes the evolution of N rotators, coupled through an equal strength, attractive or repulsive, cosine

5

interaction.' The Hamiltonian, in the attractive case, reads

where B j represents the orientation of the j-th rotator and p j stands for the conjugated momentum. To monitor the evolution of the system, it is customary to introduce the magnetization, an order parameter defined as M = IMI = ICmil/N, where mi = (cos&,sin&) is the magnetization vector. The HMF model shares many similarities with gravitational and charged sheet model^^>^ and has been extensively studiedg as a paradigmatic representative of the broad class of systems with long-range interactions. The equilibrium solution is straightforwardly worked out' and reveals the existence of a second-order phase transition at the critical energy density U, = 3/4: below this threshold value the Boltzmann-Gibbs equilibrium state is inhomogeneous (magnetized). In the following, we shall discuss the appearance of QSSs in the HMF setting and review a maximum entropy principle aimed a t explaining the behaviour of out-of-equilibrium macroscopic observables. The proposed approach is founded on the observation that in the continuum limit (for an infinite number of particles) the discrete HMF equations converge towards the Vlasov equation, which governs the evolution of the single-particle distribution function (DF). Within this scenario, the QSSs correspond to statistical equilibria of the continuous Vlasov model. As we shall see, the theory allows us to accurately predict out-of-equilibrium phase transitions separating the homogeneous (non-magnetized) and inhomogeneous (magnetized) phases.">" Special attention is here devoted to characterizing analytically the basin of existence of the homogeneous zone. Concerning the structure of the phase diagram, a bridge between the two possible formal settings, respectively'O and,12 is here established. The paper is organized as follows. In Section 2 we present the continuous Vlasov picture and discuss the maximum entropy scheme. The properties of the homogeneous solution are highlighted in Section 3, where conditions of existence are also derived. Section 4 is devoted to analyze the stability of the homogeneous phase. A detailed account of the phase diagram is provided in Sections 5 and 6, where the case of a "rectangular" and generic water-bag initial distribution are respectively considered. Finally, in Section 7 we sum up and draw our conclusions.

6

2. On the emergence of quasi-stationary states: Predictions from the Lynden-Bell theory within the Vlasov picture

As previously mentioned, long-range systems can be trapped in long-lasting Quasi-Stationary-States (QSSs),I3 before relaxing to Boltzmann thermal equilibrium. The existence of QSSs was firstly recognized with reference to galactic and cosmological applications (see7 and references therein) and then, more recently, re-discovered in other fields, e.g.two-dimensional turbulence14 and plasma-wave interactions.8 Interestingly, when performing the infinite size limit N -+ 00 before the infinite time limit, t -+ co,the system remains indefinitely confined in the &SSs.l5 For this reason, QSSs are expected to play a relevant role in systems composed by a large number of particles subject to long-range couplings, where they are likely to constitute the solely experimentally accessible dynamical regimes.2,3 QSSs are also found in the HMF model, as clearly testified in Fig. 1. Here, the magnetization is monitored as a function of time, for two different values of N . The larger the system the longer the intermediate phase where it remains confined before reaching the final equilibrium. In a recent series of p a p e r ~ , ~ i ' an ~ - approximate ~~>~~ analytical theory based on the Vlasov equation has been proposed which stems from the seminal work of LyndenBell." This is a fully predictive approach, justified from first principles, which captures most of the peculiar traits of the HMF out-of-equilibrium dynamics. The philosophy of the proposed approach, as well as the main predictions derived within this framework, are reviewed in the following. In the limit of N -+ 00, the HMF system can be formally replaced by the following Vlasov equation

a f +p-a f - (M,[f] af = 0, sin8 - Mv[f] cose) at 88 aP where f (8,p, t ) is the one-body microscopic distribution function normalized such that M [f] = f d8dp = 1,and the two components of the complex

s

magnetization are respectively given by

M , [f]=

J f cos ededp,

(3)

]

Mv[f] = f sin8dddp. The mean field energy can be expressed as

M:+M; 2

+ -.21

(4)

7

Fig. 1. Magnetization M ( t ) as function of time t . In both cases, an initial “violent” relaxation toward the QSS regime is displayed. The time series relative to N = 1000 (thick full line) converges more rapidly to the Boltzmann equilibrium solution (dashed horizontal line). When the number of simulated particles is increased, N = 10000 (thin full line), the relaxation to equilibrium gets slower (the convergence towards the Boltzmann plateau is outside the frame of the figure). Simulations are carried on for a rectangular water-bag initial distribution, see Eq. (13).

Working in this setting, it can be then hypothesized that QSSs correspond to stationary equilibria of the Vlasov equation and hence resort to the pioneering Lynden-Bell’s violent relaxation theory17 . The latter was initially devised to investigate the process of galaxy formation via gravitational instability and later on applied to the two-dimensional Euler equation. l8 The main idea goes as follows. The Vlasov dynamics induces a progressive filamentation of the initial single particle distribution profile, i.e. the continuous counterpart of the discrete N-body distribution, which proceeds at smaller and smaller scales without reaching an equilibrium. Conversely, at a coarse grained level the process comes to an end, and the distribution function f ( O , p , t ) , averaged over a finite grid, eventually converges to an asymptotic form. The time evolution of a rectangular water-bag initial distribution is shown in Fig. 2.

8

-3 -2 - 1

0

1

2

3

-3 -2

-1

0

1

2

3

-3 -2 -1

0

1

2

3

-3 -2

-1

0

?

2

3

Fig. 2. The process of phase mixing is here illustrated, showing four snapshots of the time evolution of an initial rectangular water-bag distribution. The final state (right bottom) is a QSS.

Following Lynden-Bell, one can associate a mixing entropy to this process and calculate the statistical equilibrium by maximizing the entropy, while imposing the conservation of the Vlasov dynamical invariants. More specifically, the above procedure implicitly requires that the system mixes well, in which case, assuming ergodicity (efficient mixing), the QSS predicted by Lynden-Bell, fQss(O,p,t ) ,is obtained by maximizing the mixing entropy. As a side remark, it is also worth emphasizing that the prediction of the QSS depends on the details of the initial condition,lg not only on the value of the mass M and energy U as for the Boltzmann statistical equilibrium state. This is due to the fact that the Vlasov equation admits an infinite number of invariants, the Casimirs or, equivalently, the moments M , = J f ” d 0 d p of the fine-grained distribution function. In the following, we shall consider a very simple initial condition where the distribution function takes only two values fo and 0. In that case, the invariants reduce to M and U since the moments Mn>l can all be expressed in terms of M and fo as M , = J f ” d e d p = J f;-’ x f d e d p = J f;-lTdedp = f;-lM. For the specific case at hand, the Lynden-Bell entropy is then explicitly

9

constructed from the coarse-grained DF

f and reads

We thus have to solve the optimization problema max{S[T]

T

I

U[T]= U ,M [ T ]= 1).

(6)

The maximization problem ( 6 ) is also a condition of formal nonlinear dynamical stability with respect to the Vlasov equation, according to the refined stability criterion of Ellis et aL2’ (see also Chavanis21). Therefore, the maximization of S at fixed U and M guarantees (i) that the statistical equilibrium macrostate is stable with respect to the perturbation on the microscopic scale (Lynden-Bell thermodynamical stability) and (ii) that the coarse-grained DF 7 is stable for the Vlasov equation with respect to macroscopic perturbations (refined formal nonlinear dynamical stability). We again emphasize that it is only when the initial DF takes two values fo and 0 that the Lynden-Bell entropy can be expressed in terms of the coarse-grained DF 7, as in Eq. ( 5 ) . In general, the Lynden-Bell entropy is a functional of the probability distribution of phase 1 e ~ e l s . From l~ Eq. (5), we write the first order variations as SS - PSU - abM = 0,

(7)

where the inverse temperature ,B = 1/T and cr are Lagrange multipliers associated with the conservation of energy and mass. Requiring that this entropy is stationary, one obtains the following distributionlO>ll

As a general remark, it should be emphasized that the above distribution differs from the Boltzmann-Gibbs one because of the “fermionic” denominator, which in turn arises because of the form of the entropy. Morphologically, this distribution function is similar to the Fermi-Dirac statistics so that several analogies with the quantum mechanics setting are to be expected. Notice also that the magnetization is related to the distribution function by Eq. (3) and the problem hence amounts to solving an integrodifferential system. In doing so, we have also to make sure that the critical “The momentum P = f p d e d p is also a conserved quantity but since we look for solutions where the total momentum is zero, the corresponding Lagrange multiplier y vanishes trivially” so that, for convenience, we can ignore this constraint right from the beginning.

10

point corresponds to an entropy maximum, not to a minimum or a saddle point. Let us now insert expression (8) into the energy and normalization constraints and use the definition of magnetization (3). Further, defining X = e" and m = (cos 0, sin 0) yields:

where we have defined the Fermi integrals

We have the asymptotic behaviours for t

and for t

-+

0:

+ +oo

The magnetization in the QSS, MQSS = M[fQss], and the values of the multipliers are hence obtained by numerically solving the above coupled implicit equations. It should be stressed that multiple local maxima of the entropy are in principle present when solving the variational problem, thus resulting in a rich zoology of phase transitions. This issue has been addressed inloill and more recently in,12 to which the following discussion refers to. It is important to note that, in the two-levels approximation, the Lynden-Bell equilibrium state depends only on two control parameters (U,f ~ ) This . ~ is valid for any initial condition with f ( & p , t = 0) E (0, fo}. This general case will be studied in Section 6 where we describe the phase

+

bThese parameters are related to those introduced inlo by U = €14 112, p = 27), fo = g o / N = p/(27r), k = 27r/N, x = As, y = (2/7r)Ap and the functions F inll are related to the Fermi integrals by F k ( l / y ) = 2(k+1)/2yZ(k_l),z(y).

11

diagram in the (fo,U ) plane. Now, many numerical simulations of the N body system or of the Vlasov equation have been performed starting from a family of rectangular water-bag distributions. The latter correspond to assuming a constant value fo inside the phase-space domain D :

D

=

( ( 9 , ~E) [-n,nI x

[-m,03]

I

191 < A9,

I P ~ < Ap},

(13)

where 0 5 A9 5 n and Ap 2 0. The normalization condition results in fo = 1/(4A9Ap). Notice that, for this specific choice, the initial magnetization Mo and the energy density U can be expressed as functions of A9 and Ap as

For the case under scrutiny, 0 5 MO 5 1 and U 2 UMIN(MO) = (l-M;)/2. The variables ( M o ,U )are therefore used to specify the initial configuration and hereafter assumed to define the relevant parameters space. This particular but important case will be studied specifically in Section 5 where we illustrate the phase diagram in the ( M 0 , U ) plane for the rectangular water-bag initial condition. Before that, we analytically study the stability of the Lynden-Bell homogeneous phase: two important limits, namely the non degenerate and the completely degenerate ones, are considered. We also discuss the condition for the existence of a homogeneous, non-equilibrium phase.

3. Properties of the homogeneous Lynden-Bell distribution

If we consider spatially homogeneous configurations (MQSS = 0), the Lynden-Bell distribution becomes

Using Eqs. (9), the relation between the inverse temperature energy U is given in parametric form by

p

and the

This defines the series of equilibria T ( U ) for fixed fo parametrized by X (see Fig. 3 in"). The stable part of the series of equilibria is the caloric curve. Note that the temperature T is a Lagrange multiplier associated with the conservation of energy in the variational problem (7). It also has the interpretation of a kinetic temperature in the Fermi-Dirac distribution

12

(15). If we start from a water-bag initial condition, recalling that fo = 1/(4A0Ap) and ( A P ) ~= 6[U - (1- M,3/2], we can express fo as a function of MO and U by 1 fo2 = 48[(2U - l)(A0)2 sin2 A01 ’

+

where A0 is related to MO by Eq. (14). Inserting this expression in Eqs. (16), we obtain after some algebra the caloric curve T ( U ) for fixed MO parametrized by A:

(

1 2 ,B = - -I-1/2(X)2 - 2(A0)’ sin2A0 6 Eqs. (16) can be rewritten

where G(X) is a universal function monotonically increasing with X (see Fig. 2 oflo). A solution of the above equation certainly exists provided: 1 (U - -)8n2 2 2 G(0). (20)

fi

To compute G(0) we use the asymptotic expansions (11) and (12) of the Fermi integrals. This yields G(0) = 1/12. Therefore, the homogeneous Lynden-Bell distribution with fixed fo exists only for:1° 1

1

For the rectangular water-bag initial condition, using Eqs. (17) and (21), we here find that the homogeneous Lynden-Bell distribution with fixed MO exists only for:

This result can also be obtained from Eq. (18) by taking the limit X --+ 0. Let us now describe more precisely the asymptotic limits of the FermiDirac distribution (see Fig. 3): Non degenerate limit: In the limit X -, +m, the Lynden-Bell distribution reduces to the Maxwell-Boltzmann distribution

13

P Fig. 3. Spatially homogeneous Lynden-Bell distribution function for increasing values of X (top to bottom). For X = 0, the distribution reduces to a step function (completely +m, it becomes equivalent to the Maxwell-Boltzmann distribudegenerate) and for X tion (non degenerate). In the figure, we have taken f o = 0.13 and p has been calculated from Eq. (16). ---f

Since 7 << fo, this corresponds to a dilute limit (or to a non degenerate limit if we use the terminology of quantum mechanics). The non degenerate limit corresponds, for a given value of fo, to X -+ +oo, p -+ 0 and U 4 +oo. For fo + +m, we are always in the non degenerate limit, for any p and U . In that case, the caloric curve takes the “classical” expression 1

1

a relation that can be directly obtained from Eq. (23). For the waterbag initial condition, the non degenerate limit X + +m corresponds, for a given Mo, to U -+ +oo. The non degenerate limit fo --f +m corresponds to A0 = 0 leading to MO = 1 for any U ‘. In the non degenerate limit, the Lynden-Bell statistical equilibrium state describing the QSS has the same structure as the Boltzmann distribution describing the collisional =It also corresponds to Ap = 0 leading to U = u n / i I N ( h f o ) for any Mo. However, in that case the homogeneous phase does not exist since U M I N ( M O5)Urnin(Mo) so this case will not be considered here.

14

statistical equilibrium state (but with, of course, a completely different interpretation). Completely degenerate limit: In the limit X + 0, the Lynden-Bell distribution (15) reduces to the Heaviside function

where

is a maximum velocity. The distribution (25) is similar to the Fermi distribution in quantum mechanics and p~ is similar to the Fermi velocity. Thus, the limit X + 0 corresponds to a completely degenerate limit in the quantum mechanics terminology. The completely degenerate limit corresponds to X + 0, p + +oo and U -+ Umin(fo) given by (21). This result can be directly obtained from Eq. (25). This is the minimum energy of the homogeneous Lynden-Bell distribution for a fixed fo. This is also the minimum energy of any homogeneous distribution with f E (0,fo). It corresponds to a water-bag initial condition with zero magnetization MO = 0. If we start from a water bag initial condition, the completely degenerate limit corresponds to MO = 0 for any U and to U = Umin(M0) for any Mo. A stable water-bag initial condition with MO= 0 is a maximum Lynden-Bell entropy state, so it does not mix a t all. In conclusion, we have in the general case, using the (U,fo) variables: non degenerate limit -

U fo

-

u = U m i n ( f 0 ) for any fo

+oo for any fo for any U completely degenerate limit -

0

---f

+ +m

For the water-bag initial condition, using the (U,M o ) variables, we have: 0

non degenerate limit -

0

U -++m for any MO MO = 1 for any U

completely degenerate limit - MO = 0 for any U - U = U,i,(Mo) for any MO

15

4. Stability of the Lynden-Bell homogeneous phase

We have seen that the maximization problem ( 6 ) provides a condition of thermodynamical stability (in Lynden-Bell’s sense) and a condition of nonlinear dynamical stability with respect to the Vlasov equation. We thus have to select the maximum of S at fixed U , M. Indeed, a saddle point of S is unstable and cannot be obtained as a result of a violent relaxation. Let us consider the minimization problem mjn{F[T]

=

U[?] - T S [ y ]

I M[T]= 1).

f

The criterion ( 6 ) can be viewed as a criterion of microcanonical stability and the criterion (27) as a criterion of canonical stability where F is interpreted as a free energy. Quite generally, the solutions of (27) are solutions of ( 6 ) but the reciprocal is wrong in case of ensemble inequivalence. Therefore, in the general case, criterion (27) forms a suficient (but not necessary) condition of thermodynamical and formal nonlinear dynamical stability. In the present case, it can be shown that, if we restrict ourselves to spatially homogeneous solutions , the ensembles are equivalent so that the set of solutions of (27) coincides with the set of solutions of (6). Therefore, considering homogeneous states, criterion (27) forms a necessary and sufficient condition of thermodynamical and formal nonlinear dynamical stability. We shall therefore consider in the following the minimization problem (27), which has been studied ing922for general functionals of the form S [ f ]= - J C ( f ) d e d p where C is a convex function. A simple stability criterion has been obtained in the case where the steady state is spatially homogeneous, which can be expressed in terms of the distribution function as:”

It was shown in9 that the same criterion can be expressed simply in terms of the density as

where c: = p’(p) is the velocity of sound in the corresponding barotropic gas. The equivalence between the criteria (28) and (29) is proven in.g In dConcerning spatially inhomogeneous solutions, the microcanonical and canonical ensembles are not equivalent in the region of first order phase transition. This important point will be further developed in a future contribution.

16

this Section, we apply these criteria to the Lynden-Bell distribution (15). It is shown in" that the criteria (28) and (29) can be rewritten:

If the DF satisfies (30) @ (28) @ (29), then it is (i) Lynden-Bell thermodynamically stable (ii) formally nonlinearly dynamically stable. Otherwise, it is (i) Lynden-Bell thermodynamically unstable (ii) linearly dynamically untable.^ For given fo, the relation (30) determines the range of X for which the homogeneous distribution is stable/unstable. Then, using Eqs. (16), we can determine the range of corresponding energies. Specifically, the critical curve Uc(fo) separating stable and unstable homogeneous Lynden-Bell distributions is given by the parametric equations:1°

where X goes from 0 (completely degenerate) to +GO (non degenerate). This leads to the phase diagram reported in Fig. 6. In fact, the criteria (28) and (29) only prove that f is a local entropy maximum a t fixed mass and energy. If several local entropy maxima are found, we must compare their entropies to determine the stable state (global entropy maximum) and the metastable states (secondary entropy maxima). For systems with longrange interactions, metastable states have in general very long lifetimes, scaling like e N , so that they are stable in practice and must absolutely be taken into a c c o ~ n t . ~ ~ > ~ ~ For fo + +oo, we are in the non degenerate limit X + +m and the stability criterion (30) for the homogeneous phase becomes

u2uc -- - 4.3 This returns the well-known nonlinear dynamical stability criterion (with respect to the Vlasov equation) of a homogeneous system with Maxwellian distribution function (see, e . g . , ' ~ ~ ~ This ) . also coincides with the ordinary thermodynamical stability criterion applying t o the collisional regime, for t 3 +GO, where the statistical equilibrium state is the Boltzmann distribution for f (without the bar!). On the curve U = U,i,(fo), we are in the completely degenerate limit

17

X

-+

0 and the stability criterion (30) for the homogeneous phase becomes

fo I (f0)C

=

1 7 i.e U L U --. 3, ,-12

(33)

This is the well-known nonlinear dynamical stability criterion (with respect to the Vlasov equation) of the water-bag distribution (see, e.g.,9i22). If we start from a rectangular water-bag initial condition and use the (U,M o ) variables, we must express fo in terms of U and MO using Eq. (17). Then, the critical curve U,(Mo) separating stable and unstable homogeneous Lynden-Bell distributions is given by the parametric equations

1 = 48[(2U - 1)(A8)2 sin2 A81 ’

+

Mo

=

sin(A8) ~

A8

’

(35)

where X goes from 0 (completely degenerate) to +ca (non degenerate). This leads to the phase diagram reported in Fig. 4.For MO = 1, we get X -+ +co so we are in the non degenerate limit and the critical energy is U , = 314. For MO = 0, we get X = 0 so we are in the completely degenerate limit and the critical energy is U, = 7/12.

5. The rectangular water-bag initial condition: phase diagram in the (Mo,27) plane We first comment on the structure of the phase diagram in the ( M 0 , U ) plane when we start from a water-bag initial condition. In Fig. 4 the transition line UC(Mo)divides the region of the plane where a homogeneous ( M Q S S= 0) state is predicted to occur (upper area), from that (lower area) associated to a non-homogeneous phase ( M Q S S# 0). Along the transition line two distinct segments can be isolated: the dashed line corresponds to a second order phase transition, the full line refers to a first order phase transition. First and second transition lines merge together at a tricritical point, approximately located at ( M o ,U ) = (0.17,0.61). The lateral edges of the metastability region associated to the first order transition line are also reported in the inset of Fig. 4. The correctness of the above analysis is assessed in12 where numerical simulations of the HMF model (1) are performed for different values of the system size N . The transition predicted in the realm of Lynden Bell’s theory are indeed numerically observed, thus confirming the adequacy of the

18

Homogeneous phase (stable)

_----

Uc(Mo)_

/RO 4 4

-o'61

0.25 0.6 0*/*

./

0.59 0.oq

OO

+ 111 p.1

0.2

,

0.15 I

0.4

0.6

0.8

1

Fig. 4. Theoretical phase diagram in the control parameter plane ( M o ,U ) for a rectangular water-bag initial profile. The dashed line U,(Mo) stands for the second order phase transition, while its continuation as a full line refers to the first order phase transition. The full dot is the tricritical point. The region below the thick full line U M ~ ~ J ( M O ) corresponds to the forbidden domain of the parameter space. The region of existence of the homogeneous solution is delimited from below by the thin full line Umin(Mg) (see Eq. (22)). Inset: zoom of the first order transition region. Dash-dotted lines represent the limits of the metastability region. In region (11),delimited from above by the upper dash-dotted line and from below by the full line, the homogeneous phase is fully stable and the inhomogeneous phase is metastable. In region (111), located below the full line and above the lower dash-dotted line, the homogeneous phase is metastable and the inhomogeneous phase is fully stable. These labels will appear again in Fig. 8, in connection with the discussion of the general case.

proposed interpretative scenario. Moreover, the coexistence of homogeneous and inhomogeneous phases, corresponding to different local maxima of the entropy, is also reported in12 where the probability distribution function of M is reconstructed. For all values of the energy larger than U,(Mo) (where the homogeneous phase is stable), the systems can potentially encounter a homogeneous quasi-stationary phase which slows down the approach to the thermodynamic equilibrium. Notice that above U, = 0.75, the equilibrium value of the magnetization is also zero: there is hence no macroscopic transition

19

for M ( t ) of the type displayed in Fig. 1. One has therefore to rely on other, quantitative, indicators to monitor the dynamical state of the and eventually assess the presence of a QSS. This explains why in the past QSSs regimes where believed to be localized only in specific energy windows below the critical threshold U,. Significant deviations from equilibrium are instead detected even for U > U,, as reported in Fig. 5. Single particle velocity distributions reconstructed from direct N-body simulations at U = 0.85 display a bumpy profile, analogous to the one discussed inll for the reference energy U = 0.69 < U,. Interestingly, for specific choices of the initial magnetization, the two bumps are even more pronounced than those analyzed in.” The presence of these bumps shows that relaxation is incomplete. These bumps correspond to the “vortices” (or phase space clumps) visible in Fig. 2. This state is stationary for the Vlasov equation, but Quasi-Stationary for the N-body simulation. Hence, in the long run, the two “vortices” will merge, due to finite N effects, as the HMF system proceeds towards Boltzmann-Gibbs equilibrium.

M,=O.I 0

5

~

~

0.5~

~

M, = 0.3 l ,

M, = 0.5 ~

~

-

- 0.4

0.3 -

- 0.3

0.2 -

- 0.2

0.2

-

0.1

0.4

n 0.1

a

.

-

0.1

G 5 M, = 0.7

M, = 0.9

M, = 0.95

P Fig. 5. Probability distribution function of velocities f(p), for U=0.85 and different initial magnetization, as reported in the legend of each panel.

20

6. The general case: Phase diagram in the ( t o , U ) plane

In the two-levels approximation, the Lynden-Bell equilibrium state depends only on two control parameters (U, fo). This is valid for any initial condition with f ( O , p , t = 0) E (0, fo}, whatever the number of patches and their shape. The variables ( V , M o ) used in the previous section are valid only for a rectangular water-bag configuration. Furthermore, we note that two configurations with different values of (U,M o ) in a rectangular water-bag configuration can correspond to the same values of (U,fo), hence to the same Lynden-Bell equilibrium state. To avoid any redundancy, it is preferable to use the general variables (U,fo). Therefore, in the following, we shall discuss the general phase diagram in the (U,fo) plane” and compare it with the one in the ( U ,M o ) plane for the rectangular water-bag assumption.” The stability diagram of the homogeneous Lynden-Bell distribution (15) is plotted in Fig. 6 in the ( f 0 , U ) plane. The representative curve UC(fo) marks the separation between the stable (maximum entropy states) and the unstable (saddle points of entropy) regions. We have also plotted the minimum accessible energy for the homogeneous phase Urnin(fo). Here, the term “unstable” means that the homogeneous Lynden-Bell distribution is not a maximum entropy state, i.e. (i) it is not the most mixed state (ii) it is dynamically unstable with respect to the Vlasov equation. Hence, it should not be reached as a result of violent relaxation. One possibility is that the system converges to the spatially inhomogeneous Lynden-Bell distribution (15) with MQSS # 0 which is the maximum entropy state (most mixed) in that case. Another possibility, always to consider, is that the system does not converge towards the maximum entropy state, i.e. the relaxation is incomplete.26 Let us enumerate some properties of the (fo, U ) phase diagram of Fig. 6. For U > U , = 314 (supercritical energies), the homogeneous phase is always stable (maximum entropy state) for any fo (recall that the curve Vc(fo)-+ 3/4 for fo 4 +m). On the other hand, there exists a critical point (we shall see later that it corresponds to the tricritical point of Fig. 4) located at

(fo)* = 0.10947...,

U , = 0.608...

(36)

For fo < (fo)*, the homogeneous phase is always stable (maximum entropy state) for any U 1 Umin(fo). For fo > ( f ~ )the ~ , homogeneous phase is stable for U > Vc(fo) and unstable for Umin(fo)I U < Uc(f0). This range of parameters corresponds to a second order phase transition. For (fo)* < fo < (fo),, there is an interesting regime with a “re-entrant” phase.27 The

21

0.7

I

Homogeneous phase (stable)

h>>1

'

Homogeneous phase (unstable) 0.6 (fJC

\20

C

I

Fig. 6. Stability diagram of the homogeneous phase in the (fo, U ) plane. The homogeand neous phase only exists above the line U m i n ( f o ) . It is stable above the line UC(fo) ] , is a re-entrant phase as we progresunstable below it. For fixed fo E [(fo)., ( f ~ ) ~there sively lower the energy: the homogeneous phase is stable for U > UL2'(f0), unstable for UL2'(fo) > U > VL''(f0) and stable again for U,"'(fo) > U 2 U m i n ( f o ) ,

homogeneous phase is stable for U > U , ' 2 ' ( f ~ )unstable , for U i 2 ' ( f 0 ) > U > Vi'l'(f0)and stable again for U,"l'(fo) > U 2 U m i n ( f 0 ) . This range of parameters corresponds to a first order phase transition. To make the connection between the phase diagram ( f 0 , U ) obtained inlo and the phase diagram ( M o ,U ) obtained in,12 we can plot the iso-Mo lines in the (fo,U ) phase diagram. If we fix the initial magnetization M o , or equivalently if we fix the parameter AB, the relation between the energy U and fo is

Therefore, the iso-Mo lines are of the form

22 h

A8=1.5

Homogeneous phase (stable)

0.7

,

/------

r

Homogeneous phase

0.16 Fig. 7. Iso-Mo lines in the (fo, U ) phase diagram. This graphical construction allows one t o make the connection between the (fo,U ) phase diagram of Fig. 6 and the ( M o , U ) phase diagram of Fig. 4.We can vary the energy at fixed initial magnetization by following a dashed line. The intersection between the dashed line and the curve U m i n ( f o ) determines the minimum energy Umin ( M o ) of the homogeneous phase. The intersection between the dashed line and the curve Uc(fo) determines the energy U,(Mo) below which the homogeneous phase becomes unstable.

i (w)'i,

1 with A(Ad) = goI and B(A0) = which are easily represented in the (fo,U ) phase diagram (see Fig. 7). As an immediate consequence of this geometrical construction, we can recover the minimum energy of the homogeneous phase for a fixed initial magnetization MO (or Ad). Indeed, for a given AO, the homogeneous phase exists iff U A e ( f 0 ) 2 Umin(fo) leading to

This corresponds to U > U min (M0) = U ((f0) ) leading to

u 2 Umin(M0)= -

23

Homogeneous phase Homogeneous phase (unstable) IV

0'58

-

1 I

0'&05

No homogeneous phase

v

0.11

I

0.115

I

I

12

F

Fig. 8. Zones of metastability in the (fo,V) phase diagram. In region (I), the homogeneous phase is fully stable and the inhomogeneous phase is inexistent. In region (11) the homogeneous phase is fully stable and the inhomogeneous phase is metastable. In region (111) the homogeneous phase is metastable and the inhomogeneous phase is fully stable. In region (IV) the homogeneous phase is unstable and the inhomogeneous phase is fully stable. The three curves separating these regions connect themselves at the tricritical point. In region (V) the homogeneous phase is inexistent. The corresponding caloric of) the inhomogeneous phase curves as well as the absolute minimum energy U M I N ( ~ O will be determined in a future contribution.

which is identical to (22). Note again that A Q is related to A40 by Eq. (14). Figure 7 is in good agreement with the structure of the phase diagram in the (U,Mo) plane. Indeed, along an iso-A40 line, we find that for large energies U > U,(Mo) the homogeneous phase is stable and for low energies U < UC(Mo)the homogeneous phase becomes unstable. In that case, there is no re-entrant phase. In," only the stability of the homogeneous phase has been studied, i.e. whether it is an entropy maximum at fixed mass and energy or not. The question of its metastability, i.e. whether it is a local entropy maximum with respect to the inhomogeneous phase, has not been considered. However, considering Fig. 7 and comparing with the results of,12we conclude that there must exist zones of metastability in the (fo, U ) phase diagram. They

24

have been represented in Fig. 8. In region (I) the inhomogeneous phase does not exist while the homogeneous phase is fully stable. In region (11), the inhomogeneous phase appears but is metastable while the homogeneous phase is fully stable. In region (111), the inhomogeneous phase becomes fully stable while the homogeneous phase becomes metastable. In region (IV), the homogeneous phase becomes unstable while the inhomogeneous phase is fully stable. The three curves separating these regions connect themselves a t a tricritical point. This is clearly the same as in Fig. 4. Using Eqs. (36), (37), (14) we find that it corresponds t o

(Mo)*= 0.1757... (41) with AQ, = 2.656.... Therefore, the phase diagrams in (fo,U ) and ( M o ,U ) planes are fully consistent. Note, however, that the physics is different whether we vary the energy at fixed fo or a t fixed Mo. In particular, there is no “re-entrant” phase when we vary the energy a t fixed M0l2 while a “re-entrant” phase appears when we vary the energy at fixed fO.lo

U,

= 0.608...,

7. Conclusions In this paper, we have discussed the emergence of out-of-equilibrium Quasi Stationary States (QSSs) in the Hamiltonian Mean Field (HMF) model, a paradigmatic representative of systems with long-range interactions. The analysis refers to a special class of initial conditions in which particles are uniformly occupying a finite portion of phase space and the distribution function takes only two values, respectively 0 and fo. The energy can be independently fixed to the value U . The Lynden-Bell maximum entropy principle is here reviewed and shown to result in a rich out-of-equilibrium phase diagram, which is conveniently depicted in the reference plane (fo, U).l0 When considering a rectangular water-bag distribution the concept of initial magnetization, M o , naturally arises as a control parameter and the different QSSs phases can be represented in the alternative space ( M o ,U).12In both settings first and second order phase transitions are found, which merge together in a tricritical point. These findings have been tested versus numerical simulation in,12 where the adequacy of Lynden-Bell theory was confirmed. A formal correspondence between the two above scenarios is here drawn and their equivalence discussed. It is worth mentioning that swapping from one parametric representation to the other allows us to put the focus on intriguingly different physical mechanisms, as it is the case of the “re-entrant” phases discussed in Section 6.

25

Further, in this paper, we have provided an analytical characterization of the domain of existence of the Lynden-Bell spatially homogeneous phase and investigated its stability. Homogeneous QSS are also expected to occur for U > U, = 314, a claim here supported by dedicated numerical simulations. Despite the fact that Lynden-Bell’s theory results in an accurate tool to explain the peculiar traits of QSSs in HMF dynamics, one should be aware of the limitations which are intrinsic to this approach. Most importantly, Lynden-Bell’s recipe assumes that the system mixes well so that the hypothesis of ergodicity, which motivates the statistical theory (maximization of the entropy), applies. Unfortunately, this is not true in general. Several example of incomplete violent relaxation have been identified in stellar dynamics and 2D turbulence (see some references inz6) for which the QSSs cannot be exactly described in term of a Lynden-Bell distribution. Also in this case, however, the QSSs are stable stationary solution of the Vlasov equation and novel analytical strategies are to be eventually devised which make contact with the underlying Vlasov framework. Acknowledgements D.F. and S.R. wish to thank A. Antoniazzi, F. Califano and Y. Yamaguchi for useful discussions and long-lasting collaboration. This work is funded by the PRIN05 grant Dynamics and thermodynamics of systems with longrange interactions. References 1. P.J. Peebles, The Large Scale Structure ofthe Universe, (Princeton University

Press, Princeton, NJ, 1980). 2. C. Benedetti, S. Rambaldi, G. Turchetti, Physica A 364, 197 (2006); P.H. Chavanis, Eur. Phys. J. B 52, 61 (2006). 3. J. B a d , T. Dauxois, G. de Ninno, D. Fanelli, S. Ruffo, Phys. Rev E 69, 045501(R) (2004). 4. T. Dauxois et al., Dynamics and Thermodynamics of Systems with Long Range Interactions, Lect. Not. Phys. 602, Springer (2002). 5. A. Rapisarda, A. Pluchino, Europhysics News 36,202 (2005); F. Bouchet, T. Dauxois and S. Ruffo, Europhysics News 37,9 (2006). 6. M. Antoni, S. Ruffo, Phys. Rev. E 52, 2361 (1995). 7. T. Tsuchiya, T. Konishi, N. Gouda, Phys. Rev. E 50, 2607 (1994). 8. Y . Elskens, D.F. Escande, Microscopic Dynamics of Plasmas and Chaos, IoP Publishing, Bristol (2003). 9. P.H. Chavanis, J. Vatteville, F. Bouchet, Eur. Phys. J. B 46, 61 (2005) and references therein.

26

10. P.H. Chavanis, Eur. Phys. J. B 53,487 (2006). 11. A. Antoniazzi, D. Fanelli, J. Bar&, P.H. Chavanis, T. Dauxois, S. Ruffo, Phys. Rev. E 75,011112 (2007). 12. A. Antoniazzi, D. Fanelli, S. Ruffo, Y. Y. Yamaguchi, Phys. Rev. Lett. 99 040601 (2007). 13. V. Latora, A. Rapisarda, S. Ruffo, Phys. Rev. Lett. 80, 692 (1998). 14. X.P. Huang, C.F. Driscoll, Phys. Rev. Lett. 72,2187 (1994); H. Brands, P.H. Chavanis, R. Pasmanter, J. Sommeria, Phys. Fluids 11, 3465 (1999). 15. V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64,056134 (2001). 16. A. Antoniazzi, F. Califano, D. Fanelli, S. Ruffo, Phys. Rev. Lett., 98,150602 (2007). 17. D. Lynden-Bell, Mon. Not. R. Astron. SOC.136,101 (1967). 18. P.H. Chavanis, J. Sommeria, R. Robert, ApJ 471,385 (1996); P.H. Chavanis, Ph. D Thesis, ENS Lyon (1996). 19. P.H. Chavanis, Physica A 359,177 (2006). 20. R. Ellis, K. Haven, B. Turkington, Nonlinearity 15,239 (2002). 21. P.H. Chavanis, A&A 451, 109 (2006). 22. Y.Y. Yamaguchi, J. Bar&, F. Bouchet, T. Dauxois, S. Ruffo, Physica A 337, 36 (2004). 23. M. Antoni, S. Ruffo, A. Torcini, Europhys. Lett. 66,645 (2004). 24. P.H. Chavanis, A&A 432,117 (2005). 25. A. Campa, A. Giansanti, G. Morelli, Phys. Rev. E 76,041117 (2007). 26. P.H. Chavanis, Physica A 365,102 (2006). 27. A. W. Francis, Liquid-liquid equilibrium, (Interscience, NY, 1963); C. M. Sorensen, Chem. Phys. Lett., 117,606 (1985).

27

STOCHASTIC WEBS IN MULTIDIMENSIONS G. M. ZASLAVSKY Courant Institute of Mathematical Sciences and Department of Physics, New York University, New York, NY 10012, USA City E-mail: zaslavQcims.nyu.edu

M. EDELMAN Courant Institute of Mathematical Sciences, New York University, New York, NY 10018, USA City E-mail: edelmanQcims.nyu. edu Stochastic web is a thin net that penetrates phase space of dynamical system and has dimensionality less or equal to the dimension of the phase space. Dynamics within the stochastic web is chaotic and separated from the dynamics outside the web, which could be regular. Arnold web or a web with quasicrystal symmetry are examples of the stochastic web. Particles transport can be unbounded along the webs. We review the origin of the web with symmetry in 1 1 / 2 degrees of freedom and present examples of kicked coupled oscillators with the stochastic webs for 2 1/2 and 3 1 / 2 degrees of freedom. In both cases the web has co-dimension two in the phase space. Keywords: Chaos; Stochastic web; Quasicrystal symmetry, Coupled oscillators.

1. Introduction Weak perturbation of integrable system that leads to interaction of the system’s degrees of freedom can generate in the phase space a thin net of channels inside of which trajectories are chaotic. The net is called stochastic web (SW), and its emergence is an important physical phenomenon that imposes the transport properties of systems. Arnold web is a paradigm example of the SW, and the corresponding transport is known as Arnold Arnold web is a universal characteristic of nonlinear systems. It is generated by intersection of the resonance separatrices, and it exists under the condition that the number of degrees of freedom N > 2 and the

28

system is non-degenerate:

where & ( I ) is unperturbed Hamiltonian and action I E EN.Diffusion along the Arnold web is unbounded but very slow3 and that is confirmed by simulation in.4 Another situation occurs when the condition (1) fails and the unperturbed system is degenerate

or close to the degeneracy. As an example, such situations are typical for perturbed particle dynamics in a strong constant magnetic field.5>6The case (2) for N = 1 1 / 2 (kicked harmonic oscillator) was considered in5 and it was shown that the SW exists and that thin fibers of the SW can form a net with crystal or quasicrystal type symmetry depending on some resonant conditions. Another important feature of the SW with a symmetry is that diffusion along the web is not slow and, under specific values of the parameters, could be superdiff~sive.~ The corresponding dynamical equations can be also considered as dynamical generator of symmetric patterns that appears in different areas and objects: in fluid dissipative systems,12 and condensed matter and c r y s t a l l ~ g r a p h y .Recent ~ ~ ? ~ ~developments on the use of stochastic webs of crystal and quasicrystal symmetries are related to particle dynamics in optical latticesl59l6 and photonic quas i c r y ~ t a l s . l ~Particular -~~ interest is in the atoms’ dynamics and diffusion of Bose-Einstein condensates confined by optical lattices and quasicrystal symmetry20 and the existence of band structure.21i22Different important properties of SW were studied It is easy to see that if at least one degree of freedom is degenerate in the additive Hamiltonian of the system, i.e.

and Hess Ho(I1) = 0, then the condition (2) is valid and full system is degenerate. Some systems of this kind were considered demonstrating fast diffusion process compared to the Arnold diffusion. The goal of this article is to review some results of30 and to continue study of SW for 2 1/2 and more degrees of freedom. The system under consideration consists of few coupled linear oscillators perturbed by a sequence of periodic kicks.

29

2. Kicked Two Coupled Oscillators Consider a Hamiltonian of two coupled linear oscillators perturbed nonlinearly with time-periodic kicks, 00 1 1 H = ~ ( p : t p ~ ) + ~ ( w : z : + w z 2 s ~ ) - K o T c o s ( l c l z l + k 2 z 2 ) b(t-nT),

C

n=-m

(4) where pi = Xi (i = 1 , 2 ) , KOis a perturbation parameter, and T is a period of kicks. By introducing dimensionless variables and parameters lcl =

1,

IC2

= b;

51

= -v,

XI =w~u,

~2 =

-2,

X2

= w ~ Y , (5)

we can write the equation of motion as a map between two consequent kicks, u,+1 = w, w,+1

sin a1

= v, cos a1

Y,+I = 2, sin a2

Zn+1= 2, cos a2

+ [u,+ K1 sin(v, + bZ,)] cosa1 , - [u,+ K1 sin(v, + bZ,)] sin , + [Y,+ sin(v, + bZ,)] cos - [Y, + K2 sin(v, + bZ,)] sin a1

K2

a2,

(6)

a2,

where

ai = w ~ T , Ki = kiKoT (i = 1 , 2 ) , K2 = bK1. Appearance of the SWs is related to the resonance conditions a 2

= 27r/qi,

(7)

(8)

(i = 1,2)

where qi are integers (actually qi could be rational and we consider integer qi for simplicity). Coupling of two degrees of freedom is due to the phase

9, = 21,

+ b2,

(9)

A special case is for oscillators with equal frequencies = WT= 2 ~

= a2

/q,

(10)

when an integral of motion exists. It follows from (6)

bv,) sin a + (Y, - bun) cos a , Zn+l- bw,+l = (2, - bun) cos a + (Y, - h,) sin a ,

(11)

g = (2, - bv,,Y, -bun)

(12)

Y,+1 -

bu,+l

= (2,

-

i.e. vector

rotates by a each step of the map. The invariant is its length

+

Igl2 = (2, - b ~ , ) ~ (Y, -

= inv

(13)

30

and, since q is integer, we have in addition (2, - bv,, Y, - bu,) = const,

(mod q )

(14)

i.e. points of trajectories are displayed in q invariant planes, in which the pattern (phase portrait) should be similar to the one in 2D-phase plane for N = 1 1/2. Simulations confirm the presented analysis. In Fig. 1 we present six planes of different projections of the web for q = 4. Straight lines correspond to the invariant lgI2 and Eq. (6). The web is located on four planes which intersect the (Y,u)and ( 2 , ~planes ) along 4 lines. The location of the lines depends on the initial conditions. Each of four webs, located on the corresponding four planes, can be obtained by a rotation of one of them by 27rk/q k = 0 , 1 , . . . ,q - 1).Similar SW appears for q = 5. In Fig. 2 we present only two projection planes. These figures show that the stochastic web fills 3D space but it has a tape-type structure located in a plane. More information on how the SW evolves in time can be seen from Fig. 3 (q = 8) and Fig. 4. 3. Symmetry of the Stochastic Web

The SW has invariant structure in phase space called web skeleton. For n = 1 1/2 the web skeletons were described in detail in.5 Following the same way, consider the case (10). Let us introduce the polar coordinates u =pcos$1; u = -psin$I; Y = Rcosql~; Z = -Rsin$Z;

I = w1p2/2, J = w2R2/2,

(15)

that brings the Hamiltonian (4) to the form (up to a constant)

H

= 011

+ aaJ - KoTcos(psin41 + bRsin4z) x

c 00

S(T - n)

(16)

,=-m

with dimensionless time the generating function

7.

The first two terms in H can be excluded using

F = ($1

+

- C Y T ) ~( $ 2

- CYT)J.

(17)

The new Hamiltonian is

c 00

Hq = H+dF/dr= -KoTcos[psin($l+a~)+bRsin(&+a~)] x

S(T-~),

n=-m

(18)

31

50 I

50

N

h

0

0

i U

5

-90

_1

50

0

V

0

0

50

0

50

1

V

0

50

9

0

V

Fig. 1. Six projections of the stochastic web for q = 4 present the four-fold symmetric structure in 3D space (u, v, Z ) ; K1 = 1; KZ = 0.5.

32 1000

333

> -333

-loon .__ -333

-1000

Fig. 2 .

U

1000

333

( u ,w ) and (u,Y )projections for q = 5 ; K1 = 0.44; Kz = 0.22.

where

(19) are phases in the rotational frame of reference. The invariant structure of the SW can be obtained from (18) if we present H , in the split form

H,

= V,

+ ABg(t)

(20) where V, is time independent, and perform an averaging over time. It gives 1 V, = ---K~ 2q

4

C [cos(r.ejl + U) + cos(u + b

~ejz)] .

(21)

j1jz=l

where ejl,zare unit vectors ej1,2

= (cos(27rj1,2/q),

sin(27rj1,2/q),

j1,z

= 1, * * . 14

(22)

r is a vector in (u, u ) space and R is a vector in (Y,2)space. The particular case of q = 5 was considered in.30 Expression (21) defines two coupled degrees of freedom and their interaction could, in general, lead to chaotic trajectories. Nevertheless, the conservation law (14) provides another possibility to present V, in the form

where f . = (u,V),

R = (Y,Z)

(24)

33 1000

500

333

> -333

-1000 -1000

-333

U

333

-333

-1000

1000

U

333

1000

500

167

> -167

-500

-1000

-333

,,

333

1000

333

1000

-1000

-333

U

v

333

1000

500

167

N -167

-1000

-333

Fig. 3.

v

-500

Y

500

167

Six projections of the SW for q = 8; K1 = 0.44;Kz = 0.22

with renormalized components

v = const. v ,

Z = const. z

34

,

200

%

67

> -67

-200

-200

-67

U

67

-30

200

-10

10

30

Y7

20

U

60

20

N -20

-60

-60 -20

Fig. 4.

Y 2o

-20

60

-7

Four consequent magnifications of the SW from Fig. 3.

with renormalize and the constants are defined from the initial conditions and conservation laws (4). The new form (23) for V, shows additive splitting of V, onto two independent structures in 4D phase space. By structure we consider a topological structure of the phase portrait. Projection of the 4D structures on 3D hypersurface reveals 3D structures. For example, if Y = 0, then one can present P

V, = const.

C cos(r1 .

ej)

j=O

where rl = (u,B, Z),ej for j = 0 is a unit vector along 2, and ej = ej, (jl = 1,. . . , q ) are defined in (22). The contour plots for different projections

35 -100

100

-100

I00

-inn

inn

-100

nn I O(

100

in 100

-100

Fig. 5 . SW skeleton projections: (Top row) (z, y, 0 ) , and (2,y, 1);(Bottom row) (z, 0, z ) , and ( z 1 2 , z ) . Contour lines correspond to the levels V5(z, y, z ) = - 1 , l , 3,4.

(26) are shown in Fig. 5 for q = 5. The patterns in Fig. 5 corresponds to the 2D projections of the 3D quasicrystal with 5-fold ~ y m m e t r y . ~Different ' other symmetric tilings can be obtained in a similar way by changing q.

4. More Coupled Oscillators The goal of this section is to show that system (6) can be generalized to more degrees of freedom. Here we provide an example for three coupled oscillators ( N = 3 1/2). More oscillators can be considered in a similar way. Let the coordinate and momentum for each oscillator be given as (yi,xi),

36

i = 1 , 2 , 3 . The Hamiltonian of the system is similar to (4): .

3

i=l The equation of motion can be reduced to the map similar to (6): zi,n+l

+ + Ki sin$,,] cos ai - [ X i , n + Ki sin$,] Sinai, (i = 1 , 2 , 3 ) ,

= Yi,n sin ( ~ i [xi,,

yi,n+i = Yi,n COSQi

(28)

where 3

ai = w ~ T , Ki

= IciKoT,

$n =

C ki~i,~ i=l

(29)

Consider the case of equal frequencies ai =

=2~/q,

(Vi),

(30)

where q is an integer. Symmetry of the equations (28) shows the existence of two independent integrals of motion It follows that K251,n+1 - K 1 ~ 2 , n += i (K2~1,n - Klyz,n) s i n a

+

(K2z1,n - Kiz2,n)C O S ~

K2~1,n+1- K1~2,n+i= (K2~1,n - Kiy2,n) COSQ - (K2z1,n - K 1 ~ 2 ,s ~ i n)a (31) Equations (31) preserves the length of the vector g12 =

( K 2 ~ 1- K 1 ~ 2K2zi , - Kiz2) = inv

(32)

Similarly one can obtain preservation of the vectors g13 and g23 that can be obtained from g12 by a corresponding change of subscripts. Only two of the invariants are independent. The SW in this case looks similar to what is for two coupled oscillators. It is located in 2D layers and the layers are embedded in 6D space. The invariants (32) reduce the effective dimension by 2 where chaotic diffusion is performed. That means that effectively the SW is located in the space of co-dimension 2 if the condition (30) is applied. That is similar to the case of two coupled oscillators. Examples of the SW for the cases q = 4 and q = 5 are given in Figs. 6,7 and the diffusion for q = 4 is presented in Fig. 7.

37

7

-3

1

-1

Fig. 6.

3

Xl

XI

Different projections of the SW for 3 coupled oscillators for q = 4.

60

100

20

rl

h

h“

0-

-20

-!to

-20

XI

20

60 -’YO0

x1

O

100

Fig. 7. Diffusion along the SW for 3 coupled oscillators for q = 4 and q = 5. Projections on ( $ 1 , y1) - plane.

5 . Conclusion

We have demonstrated that SW of the degenerate type can occur in N = 2 1/2 and N = 3 1/2 degrees of freedom. The same type of the SW can be assumed to exist in higher dimensions. The SW persists to have a symmetry or quasi-symmetry inherited from the N = 1 1/2 case in 2D phase space. A particle wanders in a “sandwich type” topological structure and each layer is similar to N = 1 1/2 case with rotations by 27r/q from layer to layer. The

38

main differences of t h e SW from t h e Arnold web are t h e web symmetry a n d fast particles diffusion along t h e web.

Acknowledgments This work was partly supported by t h e Office of Naval Research G r a n t No. N00014-02-0056. P a r t of t h e work was performed at CPT, Aix-Marseille University.

References 1. V. I. Arnold, Dokl. Akad. Nauk SSSR 156,9 (1964) [Sou. Math. Dokl. 5 , 581 (1964)]. 2. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (Dynamical Systems 111. Encyclopedia of Mathematical Sciences), 3rd ed. (Springer, New York, 2006). 3. N. N. Nekhoroshev, RUSS.Math. Surveys 32,1 (1977). 4. C. FroeschlC, E. Lega and M. Guzzo, Celest. Mech. Dyn. Astron. 95, 141 (2006). 5. G. M. Zaslavsky, M. Yu. Zakharov, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Sou. Phys. JETP 64,294 (1986). 6. G. M. Zaslavsky, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Weak Chaos and Quasiregular Patterns (Cambridge University Press, Cambridge, UK, 1991). 7. G. M. Zaslavsky, Phys. Report 371,461 (2002). 8. V. V. Beloshapkin, A. A. Chernikov, M. Ya. Natenzon et al., Nature 337, 133 (1989). 9. J. Gollub, Proc. Natl. Acad. Sci. U.S.A. 92,6705 (1995). 10. W. S. Edwards and S. Fauve, Phys. Rev. A bf 47, R788 (1993). 11. W. S. Edwards and S. Fauve, J. Fluid Mech. 278,123 (1994). 12. M. Field and M. Golubitsky, Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature (Oxford University Press, Oxford, UK, 1992). 13. J. S. W. Lamb, J . Phys. A 26,2921 (1993). 14. A. Baz&n, M. Torres, G. Chiappe et al., Phys. Rev. Lett. 97,124501 (2006). 15. S. Marksteiner, K. Ellinger, and P. Zoller, Phys. Rev. A 53,3409 (1996). 16. S. A. Gardiner, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 79,4790 (1997). 17. B. Freedman, G. Barta, M. Segev et al., Nature 440,1166 (2006). 18. L. Guidoni, B. DBpret, A. di Stefano, and P. Verkerk, Phys. Rev. A 60,R4233 (1999). 19. L. Guidoni, C. TrichB, P. Verkerk, and G. Grinberg, Phys. Rev. Lett. 79,3363 (1997). 20. L. Sanchez-Palencia and L. Santos, Phys. Rev. A 72,053607 (2005). 21. W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, Nature 436,993 (2005). 22. M.A. Kalitievsky, S. Brand, T.F. Krause et al., Nanotechnology 11, 274 (2000).

39

I. Dana and M. Arnit, Phys. Rev. E, 51, R2731 (1995). S. Pekarsky and V. Rom-Kedar, Phys. Lett. A 225, 274 (1997). J. H. Lowenstein, Phys. Rev. E 47,R3811 (1993). J. H.Lowenstein, Chaos 5,566 (1995). G. M. Zaslavsky, M. Yu. Zakharov, A. I. Neishtadt, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Sou. Phys. JETP 69,885 (1990). 28. A. A. Chernikov and A. V. Rogalsky, Chaos 4,35 (1994). 29. V. V. Beloshapkin, A. G. Tretyakov, and G. M. Zaslavsky, Commun. Pure A p p l . Math. 47,39 (1994). 30. G. M. Zaslavsky and M. Edelman, Chaos 17,023127.

23. 24. 25. 26. 27.

40

CHAOTIC GEODESICS J.-L. THIFFEAULT Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA E-mail: [email protected]. edu K. KAMHAWI Department of Mathematics, Imperial College London, London, S W 7 2AZ, UK When a shallow layer of inviscid fluid flows over a substrate, the fluid particle trajectories ace, to leading order in the layer thickness, geodesics on the twodimensional curved space of the substrate. Since the two-dimensional geodesic equation is a two degree-of-freedom autonomous Hamiltonian system, it can exhibit chaos, depending on the shape of the substrate. We find chaotic behaviour for a range of substrates. Keywords: shallow water flows; chaotic advection; particle transport

1. Introduction

Many well-known physical systems take the form of geodesic flow on a manifold. For instance, Euler’s equation can be though of as a geodesic flow in the space of volume-preserving diffeomorphisms, and free rigid body motion as geodesic flow in SO(3). In both cases, the metric on the space corresponds to the kinetic energy norm. It is also known that the geodesic deviation equation7 describes the stability of such flows. For instance, a space of negative curvature will lead to divergence of trajectories, and hence to chaos if the space is compact. But compact spaces of strictly negative curvature are hard to come by in the real world, to say the least. If we expect the negative curvature to lead to chaotic geodesics, we are better off looking for spaces with non-sign-definite curvature, but such that the averaging of the curvature over trajectories leads to chaos ( i e . , the negative curvature ‘wins’).

41

In this contribution we will discuss a system which is physicallymotivated and leads to chaotic geodesics. This system is the flow of a shallow layer of ideal, irrotational fluid on a curved substrate. Following Rienstra,8 we will show that, to leading order, the governing equation can be solved in terms of characteristics. Moreover, the characteristics are geodesics on the curved substrate, possibly modified by gravity if it is present. Of course, the chaotic trajectories have a nasty tendency to cross and form caustics everywhere. Hydrodynamically, caustics are usually manifested as hydraulic jumps or so-called ‘mass tube^,'^ visible as a thicker edge region of the fluid in Fig. 1 (top). Edwards et al.g have recently described

Fig. 1. Experiment in the kitchen sink, using a cut-open plastic bottle (top). The jet from the faucet impacts the inclined bottle. The pattern is qualitatively well reproduced by following fluid trajectories emanating from a point source on a cylinder (middle). However, if we pursue the trajectories further (bottom), the ideal theory presented here predicts that the flow should crawl back up the side to the same initial height. The discrepancy is clearly due to dissipation.

these mass tubes using the theory of ‘ d e l t a - s h ~ c k s . At ’ ~ ~this ~ point, we are unable to apply this theory to our problem, which means that solutions become dubious after characteristics begin to cross. Unfortunately,

42

since our characteristics are chaotic, they tend to cross a lot. Nevertheless, we believe that studying the basic properties of this geodesic flow is worthwhile as a first stab at describing the transport properties of flows on curved substrates. In addition, the geodesic flow we present is an interesting mathematical system, with rich dynamics that deserve to be studied on their own. Having gotten these disclaimers out of the way, let us proceed with the analysis. We hall do this in stages. In Section 2 we introduce a curved non-orthogonal coordinate system to describe the substrate, singling out a direction normal to the substrate. In Section 3 we use this direction to expand our fluid equations and derive a shallow-layer form. We show that the resulting equation can be solved in terms of characteristics, which are geodesics on the two-dimensional curved substrate, modified by gravity. In Section 4 we look at specific numerical solutions for particle trajectories, and in Section 5 we speculate on their chaotic nature. We offer some closing comments in Section 6. 2. Coordinate System 2.1. Separating the Shallow Direction

In our problem, fluid motion occurs over a curved substrate of arbitrary shape. The direction normal to the substrate is special in that it defines the direction in which the fluid layer is assumed 'shallow.' Hence, it is convenient to locate a point r in the fluid as

r ( x 1 , x 2y) , = x ( z 1 , z 2+ ) y63(z1,z2)

(1)

where X ( d ,x 2 ) is the location of the substrate, 63 is a unit vector normal to the substrate, and y is the perpendicular distance from r to the substrate. The coordinates x1 and x 2 are substrate coordinates used to localise points on the substrate. For example, in Section 2.2 we will use the Monge T parametrisation, X = (dx 2 f(d, x 2 ) ) , where f gives the height of the substrate. The tangent vectors to the substrate are

e , := d,X

(2)

where 8, := d/dx".The coordinate vectors associated with the coordinate system are found from (l), cZa := d,r

= e,

+ U(y) ,

dr 63 := - . dY

(3)

43

where Greek indices only take the value 1 or 2. Note that the e, are not necessarily orthogonal or normalised. We adopt the convention that quantities with a tilde are evaluated in the ‘bulk’ (away from the substrate), and thus depend on y, whilst those without the tilde are ‘substrate’ quantities and do not depend on y. Thus, &(d, x2,0) = e,(&, x2). The three-dimensional metric tensor g a b has components

Baa where

:= I?,

-

. Ep = Gap ,

-

Gap :=

-

ija3

e,

Gap := e,

:= e,

. e3 = 0 , A

= Gap

833

:= &, .&3 = 1,

+ U(y) ,

. ep .

(4)

The three-dimensional metric tensor is thus block-diagonal, and the y coordinate is unstretched compared to the Cartesian coordinate system. It measures the true perpendicular distance from the substrate to a point in the fluid. Given the substrate vectors e,, it is easy to solve for the covectors eQ, which are such that eQ . ep = &pa. Then the bulk covectors are dQ = e0l

+ (3(y),

(5)

to leading order in y. From (5), we find the inverse metrics,

Higher-order terms in y will not be needed. 2.2. Substrate Coordinates For most applications in the literature of thin films and shallow layers, orthonormal coordinates have been the coordinates of choice. This is because the main substrate shapes that have been treated are planes, cylinders, and spheres, where orthonormal coordinates are readily available. For a general substrate shape, orthonormal coordinates are difficult to construct and require numerical integration. Singularities (umbilics) also cause problems. l4 For our application-flow down a curved substrate-the Monge representation of a surface15 is the most convenient. The Monge representation is a glorified name for a parametrisation of the substrate by

44

in three-dimensional Cartesian space. Following standard notation,15 we define

The unnormalised, nonorthogonal tangents el and el = a1X = (1 O P )

T

,

e2

e2

are

= ~ Z X= (0 1 4 )

T

,

and their normalised cross product gives the normal to the substrate, 1

T

( - p -4 1)

&3 = W

,

The corresponding covectors are 1

e1 = 3 ((1+ q 2 ) -P4 P )

T 7

1 T e2 = 3 (-P4 (1+ P 2 ) 4 )

and 63 is its own covector. The metric tensor of the substrate and its inverse are

with determinant

w

=

(det G a p ) 1’2 =

d

W

.

Finally, we will need the Christoffel symbols r;,, defined by

rzp = pY ( a a ~ r+pa&,

-

and in Monge coordinates given by

The Christoffel symbols arise when taking covariant derivative^.^^^^^^^ Note that in (10) we used the usual convention that repeated indices are summed. We write the normalised gravity vector as

a

g = (sin cos 4 sin a sin 4

- cos

,

(12)

45

so that the inclination angle 0 is zero for gravity pointing downwards, and for $ E (-7r/2,7r/2) positive 6 induces flow in the positive z1 direction. Then we have the components

+ q 2 )sinecos$) /w2 , gs2 = g . e2 = - ( q cose + p q sinecos 4 - (1+ p 2 ) sinesin$) /w2 , gsl = g . el = - ( p cose + p q s i n e s i n 4 - (1

(13)

The specific parametrisation of the substrate introduced in this section will not be needed in the derivation of the equations of motion (Section 3), only in their solution. Hence, a different parametrisation could be used if called for by the geometry of the substrate. For instance, flow down a curved filament is better parametrised by cylindrical coordinates, or if the substrate has overhangs (making f multivalued) coordinates based on arc length are preferable.

3. Equations of Motion

Now that we have set up an appropriate coordinate system on our curved substrate, we need some dynamical equations of motion for the fluid. We assume an inviscid, irrotational fluid with a free surface at y = r](zl,x 2 ) , with slip boundary conditions at the substrate y = 0. The pressure on the free surface is assumed constant (zero). We also assume the flow is steady and irrotational, so that the the velocity can be written in terms of a scalar potential, u = Vp. The equations satisfied by the fluid are then

v2p= 0, 1

IVpI2 + P- - g . r P

=H,

mass conservation;

(144

Bernoulli's law;

(14b)

where H is a constant, with boundary conditions

ayp= 0 a t y = 0, V p . Vr] = ayp a t y = r]: p =0 at y = r],

no-throughflow a t substrate; (15a) kinematic condition at free surface; (15b) constant pressure at free surface.

In terms of our curvilinear coordinates, equation (14b) becomes

(15c)

46

3.1. Small-parameter Expansion Now we assume that the fluid layer is shallow, so that y is proportional to E . After replacing y by E Y , Eq. (14b) becomes

+

2P + ~ - ~ ( d ~+c p ) ~2 9 . ( X + ~ P

(GOip O ( E ) ) a,cpapcp

-

We also expand cp in powers of cp(z

1

~

6 =32H. )

(17)

E,

,32 2 Y) = V(0) + E P(1)+ 2 P(2) 3- . . . . 7

(18)

The leading-order term in (17) occurs a t order E - ~ , and gives d , ~ p ( ~=) 0. Hence, we have cp(o) = @ ( x 1 , x 2 independent ) of y, The next nontrivial terms are a t order E O , GapaOi(a dp(a

2P + (d,(p(1))2+ P

-

2 g . X = 2H.

(19)

We evaluate the whole of (19) at y = 0, and use the boundary conditions (15a) and (15c),

~@a,@ ap(a- 2 9 . x = 2

~ .

(20)

This is the equation that we need to solve to find the leading-order velocity potential @(xl,z2).We discuss the method of solution in the next section. Note that we will not need to solve the mass conservation equation (14a): at leading order, it only serves to find the fluid height once the velocity field is obtained. 3.2. Solution in Terms of Characteristics

As pointed out by Rienstra,' the trick to solving Eq. (20) is to use the method of characteristics. To do this, we differentiate (20) with respect t o x,, which gets rid of the constant H, 2GOiOaa@a,ap@

+ a,G@aa@ a p @ = 29 a,x. *

The horizontal components of velocity are k a denotes a time derivative; hence,

a&@

= ap(G&6)

= G,sdpP

From the chain rule, we have x ' = dpi?@; find, after dividing by two,

G,~z'

= G@dp@,where the

+ dpG,sP.

(21) overdot

(22)

using this and ( 2 2 ) in (21),we

+ d p ~ , s @ i + + i i ~ , ~ ~ p ~ , s ~ p , i =? kg p. ey.

(23)

mjf 1 10

47

,

0

fo=OZ -20

-10

0

10

20

30

11

-20

-10

0

10

20

21

30

-20

-10

0

10

20

30

7'

Fig. 2. A pencil of 30 trajectories starting from the origin at different angles, each with initial kinetic energy 1/2. The substrate has shape f ( z l , s2)= fo coszl coss2, and the plots are for different values of fo. Gravity is turned off g = 0. The grey background shows the periodicity of the substrate.

Now we multiply by bit of manipulation

GUY,

and obtain after an integration by parts and a

$0

+ rZ0kak0

=9.

(24)

where the rgp are defined by (11).Equation (24) describes geodesics in the curved coordinates of the substrate, under the influence of gravity. In the absence of gravity, the fluid trajectories are essentially going in straight lines in the curved substrate coordinates. (In general relativity, unlike here, the gravity determines the curvature of space.) If we define the covariant derivative of a vector V" along the trajectory,7,16-18

where T is the time (to avoid confusion with t in Eq. (8)),then the geodesic equation (24) takes the more intuitive form u -xu

= g . e' (26) Or which looks a lot like Newton's second law, but here it incorporates the constraint that fluid particles remain on the substrate. Equation (24) is a two degree-of-freedom autonomous Hamiltonian system, with the energy H defined by equation (20) as an invariant. Hence,

48

any other invariant will make the system integrable, and rule out chaos. In particular, a surface with a translational symmetry cannot exhibit chaos. 4. Fluid Particle Trajectories

To get a feel for the possible range of behaviour of fluid trajectories, we now solve the geodesic equation (24) for a range of substrates. We shall always use a set of fluid trajectories starting at the same spatial point, with the same initial kinetic energy but different direction. This models a point source, or a thin jet impacting the substrate. First, following Rienstra8 we solve the equations on a cylindrical substrate. Figure 1 (middle) shows some trajectories, all emanating from the same point. Qualitatively, the pattern captures well the observed behaviour of a jet (from a faucet) impacting the inside of a cut-out plastic bottle (Fig. 1, top). However, if we pursue the trajectories further (Fig. 1, bottom), we see that they crawl back up the side of the cylinder, with no loss

60

50 50

40

40

30

"h 30

20

20

10

10

0

0

0

10

20

30

0

20

51

40 2.1

60

60

40

40

+

N

* 20

20

0

0 -20

0

20

40

-50

0

50

Fig. 3. Same parameters as in Fig. 2, but with gravity turned on: g = 1. The trajectories exhibit chaos-like behaviour for much lower substrate height, since they begin at the top of a bump and thus have potential energy to draw upon.

49

1

35

30 25 20 15

10 5

0

10

20

30

0

10

20

10

20 10

20

0

5

10 -10

-

0

-20

0 -5 -10

-10

-15

-30

-20 -10

n

-10

0

:?

-10

20

30

0

10

20

I'

Fig. 4. Same parameters as in Fig. 3, but with initial condition (z',z2) = ( ~ / 2 , 0 ) .For larger substrate amplitudes, the system is dominated by 'rimming,' where particles skim a depression before moving t o the next one, or sometimes undergo long flights.

of energy, in contrast to the experimental picture. This comes from neglecting the hydraulic jumps that occur, as well as v i s c ~ s i t y Observe .~ that the trajectories follow a very ordered pattern, and are definitely not chaotic. This is as expected, since there is a symmetry direction, and so the motion is integrable (Section 3). Next we move on to more complex substrates. Since there is basically an infinity of choices here, we limit ourselves to periodic substrates with shape 1 2 f(z1,z2) = fo sin z sin z

(27)

for various values of fo. The other variables in the system are the strength of gravity (which can be chosen as unity if it is not zero, by rescaling the substrate height) and its orientation (as given by the angles 0 and q5 in Eq. (12)). Figure 2 shows a pencil of 30 trajectories starting from the origin a t different angles, each with initial kinetic energy 1/2, for different values of fo, in the absence of gravity. The first two cases display regular behaviour, but for substrate heights fo = 0.7 there is chaotic-like behaviour. These are, however, fairly extreme values of fo, corresponding to heavily-deformed

50

I 0 -1

0

1

2

3

4

21

Fig. 5. Same parameters as in Fig. 3, but with initial condition (x1,x2)= ( n , O ) . The particles begin at the bottom of the potential well, and they do not have the energy to escape.

substrates. Our expansion should be able to accommodate this, since the variations in the substrate height are not assumed small (only those in the fluid thickness are). For extreme heights (fo = 1.2, last case in Fig. 2), some trajectories actually backfire and come around the initial point. Figure 3 shows results for the same parameters as Fig. 2, but with gravity g = 1. The inclination is nil ( 0 = 0). It is clear that chaotic-like behaviour sets in for much smaller values of fo, even showing backscatter for fo = 0.5 in the last frame. This is because the fluid elements can now draw on the potential energy they inherit from starting at the top of the bump. This suggests that, in the presence of gravity, the results should be substantially different if we start elsewhere on the substrate. Figure 4 shows simulations with the same parameters as in Fig. 3, but starting at (xl, x2) = (n/2,0),some way down the bump. The motion is then confined to narrow channels for moderate fo. But for larger substrate amplitudes, the system is dominated by ‘rimming,’ where particles skim a depression before moving to the next one, or sometimes undergo long flights. This is a similar situation to basketball (or golf), where the ball turns around the hoop (or cup) a while before deciding to go in or out. If we take an initial condition at the bottom of the bump, (x1,x2)= (n,O),then the trajectories do not have enough energy to escape the potential well (Fig. 5). Finally, in Fig. 6 we illustrate the effect of inclining the substrate at an angle B = nl8. With q5 = -n/2 the trajectories flow ‘downhill,’ in the negative x2 direction, modified by the bumps. The system still appears to becomes chaotic for larger fo. Larger bumps induce a ‘shadow’effect, where they prevent fluid from flowing behind them (in particular for fo = 0.6).

51

-601 -20

0

-20

20

0

-20 -10

3.1

0

0

-10

-10

0

10

20

il

0 -10 -20

-20 -20

'-30

--30 -40

-40

-50

-50

-30 -40

-50

-60 -20

0

20

-20

Ll

0

20

-20

11

0

9

20

8 q5 = -7r/2. For Fig. 6. Same parameters as in Fig. 3, but with incline 0 = ~ / and larger bumps there is a 'shadow' effect, particularly for fo = 0.6.

5 . Lyapunov Exponents and Chaos

We now investigate whether the behaviour described in the previous section is chaotic or not. The first variation of the geodesic Eq. (24) gives

6X'

+ 2rzp~ ~ 6 j +. Payrgpj . a ~ f 1 6 ~=y -9.er r,",dxy

(28)

where we have used ayeu = -I?& er. This equation can be massaged into the geodesic deviation e q z ~ a t i o n , ~ ~ ~ ~

D2 -6s" Or2

+ RapyaX a X p 6 ~ ' = 0

(29)

where DID7 is defined in ( 2 5 ) , and

R~~~~:= ayrZp- a a q p+ r;&

-

rgAr&

(30)

is the Riemann curvature tensor. For two-dimensional surfaces, the curvature tensor simplifies to Rupya = 9 ( y ' S

Gpa - 6"a Goy)

(31)

52

where 9 = (rt - s2)/w4 is the Gaussian curvature and Eq. (9). Hence, a simplified form of (28) for surfaces is 0 2

-axu

0 7 2

Go, is given by

+ 9 ((i,i)ax" - (i,ax) i") = 0

where the inner product is defined by (V,W ) := G,pV"Wo. Note that the gravitational term does not enter Eq. (28) directly, though it does through (24). For the rest of this discussion we will assume g = 0, since it simplifies the discussion considerably. If that is then case, then it is easy to show that

D Or

-@,ax)

D

= (i,-ax), 07

0 2

-(k,dx) Or2

=0

(33)

which means that if we choose the initial axQ such that (x,bx) = (5,D b x / D r ) = 0, then (i, ax) remains zero for all time. With this choice initial condition, the geodesic deviation equation (32) finally takes the form

D2 -ax" 0 7 2

+ S ( i , i )62" = 0.

(34)

Now we can ask under what condition the substrate shape will be favourable to chaotic geodesics. Since Eq. (34) resembles an oscillator equation, and (5,i)1 0, we see that negative Gaussian curvature will favour divergence of trajectories. We have not yet solved Eq. (28) for bx", but a comparison of the distance between two initially very close trajectories is shown in Fig. 7, for the same parameters as in Fig. 3 (fo = 0.5). Unsurprisingly, the plot confirms exponential growth, demonstrating at least numerically that chaos is indeed present. 6 . Discussion

We have shown that the flow of a shallow layer of inviscid, irrotational fluid on a curved substrate leads to particle trajectories that follow geodesics in the curved space, subject to gravity. We have displayed the range of behaviour that these geodesics can exhibit, from regular t o chaotic. As Fig. 1 shows, the theory is not likely to be valid much beyond the point where characteristics cross, and viscosity also causes important corrections. Another effect we ignored is the possibility that centrifugal forces can cause the fluid to spin out and detach from the ~ u b s t r a t e .Experi~ ments are needed to determine to what extent the chaos observed here is

53

1oo

1o-2.

H

Lo

-

1o-6

20

40

60

80

Fig. 7. The Cartesian distance 16x1 between two trajectories, for the same parameters as in Fig. 3 with fo = 0.5. The trajectories diverge extremely rapidly, consistent with chaotic behaviour.

reproduced in reality. If chaos is indeed prevalent, then perhaps chaotic advection can be exploited in some applications to enhance mixing in shallow layers. We made the case in the introduction that chaos in the geodesic equations was a subject worthy of study on its own. The emergence of chaotic behaviour as a function of Gaussian curvature, as embodied by Eq. (32)) should be a rich subject of study, in particular because of the simple form this equation takes on a surface. We note in closing that a similar study can be made for viscous thin films.'' However, trajectories there are much less prone to chaotic behaviour. because of the diminished role of inertia.

References 1. V. I. Arnold, Ann. Inst. Fourier 16,319 (1966). 2. V. I. Arnold, Usp. Mat. Nauk. 24, 225 (1969). 3. V. I. Arnold, Mathematical Methods of Classical Mechanics, second edn. (Springer-Verlag, New York, 1989). 4. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry (Springer-Verlag,Berlin, 1994). 5. V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics (Springer-Verlag,New York, 1998). 6. Y . Watanabe, Physica D 225, 197 (2007). 7. R. M. Wald, General Relativity (Universityof Chicago Press, Chicago, 1984).

54

8. S. W. Rienstra, Z A M M 7 6 , 423 (1996). 9. C. M. Edwards, S. D. Howison, H. Ockendon and J. R. Ockendon, I M A J . Appl. Math. (2007), in press. 10. F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing, ed. B. Perthame, Advances in Mathematics for Applied Sciences, Vol. 22 (World Scientific, 1994). 11. J. Li, T. Zhang and S. Yang, Two-dimensional Riemann Problems in Gas Dynamics (Chapman & Hall/CRC Press, Boca Raton, FL, 1998). 12. H. Yang, J. Diff. Eqns. 159,447 (1999). 13. J. Li, Appl. Math. Lett. 14,519 (2001). 14. I. Kreyszig, DifSerential Geometry (University of Toronto Press, Toronto, 1959). 15. H. Flanders, Differential Forms with Applications to the Physical Sciences (Dover, New York, 1990). 16. J. L. Synge and A. Schild, Tensor Calculus (Dover, New York, 1978). 17. B. Schutz, Differential Geometry (Cambridge University Press, Cambridge, U.K., 1980). 18. J.-L. Thiffeault, J . Phys. A 34,5875 (2001). a step-by-step derivation of the geodesic deviation 19. For equation see http://io.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/ GeodesicDeviation.htm.

20. J.-L. Thiffeault and K. Kamhawi, http: //arXiv. org/abs/nlin/0607075 (2006).

55

A STEADY MIXING FLOW WITH NO-SLIP BOUNDARIES R.S. MACKAY Mathematics Institute, University of Warwick, Coventry CVq 7AL, U.K. E-mail: R.S. MacKayOwamick.ac.uk

A steady mixing (even Bernoulli) smooth volume-preserving vector field in a bounded container in W3 with smooth no-slip boundary is constructed. An interesting feature is that it is structurally stable in the class of C3 volumepreserving vector fields on the given domain of W3 with smooth no-slip boundary, thus if one could think how to drive it then it would be physically realisable. It is pointed out, however, that no flow with no-slip boundaries can mix faster than l / t 2 in time t .

1. Introduction The possibility that the motion of ideal particles in a steady or time-periodic fluid flow could be chaotic was proposed by A r n ~ l ’ dstudied ,~ by people like HBnonlg and Zel’dovich, and was part of the standard training a t Princeton Plasma Physics Lab in 1978. It was found in convectiong by 1983, but did not come to the attention of the fluid mechanics community at large until the article of Aref,’ who christened the phenomenon “chaotic advection” . The subsequent development of the subject has been reviewed in Ref. 3. The ultimate in chaotic advection would be a mixing flow, in the ergodic theorist’s sense: a flow q!~ : R x M -+ M , ( t ,x) H &(x) on a manifold M preserving finite volume p is (strongly) mizing if for all measurable A , B c M , then

(no molecular diffusion is involved). Yet as far as I am aware, no-one has made an example of a fluid flow which is proved to be mixing. Most examples in the literature have, or are suspected t o have, tiny unmixed “islands” (at fixed phase for a time-periodic 2D flow) or long thin invariant solid tori (for a steady 3D flow).

56

Furthermore, to be realistic for engineering purposes such a flow should be constructed in a container in R3 with no-slip boundary (an alternative for a physicist could be a flow in a gravitationally or surface-tension bounded ball, but let us restrict to the case of a no-slip container). So, a further 23 years on from Ref. 2, this paper constructs a steady mixing volume-preserving flow in a bounded container in R3 with no-slip boundary. Interestingly, the tools have been available in the pure mathematics literature since 1975. The paper leaves open the question of how one might drive such a flow, but makes two further significant points. Firstly, the flow can be proved structurally stable within the class of C3 volume-preserving vector fields in the interior of the given container with no-slip boundary. Thus all “nearby” flows are topologically equivalent to the given one, and any such flow is mixing. This robustness gives the hope that such an example can be realised physically. The proof will be published elsewhere. Secondly, it is proved that no C2 volume-preserving vector field with C2 no-slip boundaries mixes faster than l/t2 in time t , in a sense to be made precise. The paper concludes with a discussion of possible variants and additional results. 2. The construction

I begin from a steady vector field which I call s (Figure l),proposed by Arnol’d5 (who showed it to be irrotational Euler for a Riemannian metric to be recalled in (2)). It is the suspension vector field of the automorphism

A =

[;i]

of the 2-torus T2 = R2/Z2. This means it is the vector field

(O,O, 1) in components where

(2, y,

z ) on the quotient space M = (T2 x [O,l])/a a(x,1) = (Ax, 0)

for x = (x, y) E T2 (meaning that points (x,1) and (Ax,O)are t o be considered as the same). M is a C” manifold and s is a C” vector field on it. It preserves volume dx A dy A dz and has exponentially contracting and backwards contracting subbundles E* leading (by direct sum with the vector field) to invariant foliations .F* by the “planes” y = -yx -t c+ and y = z/y c-, respectively, where y = (1 &)/2 is golden ratio and c* denote arbitrary constants, as a result of which it is ergodic and

+

+

57

I

\

[::I

Fig. 1. The suspension flow s.

C1-structurally stable. It is not physically realisable, however, because the suspension manifold M can not be embedded in R3. The orbit of ( 0 ,0,O)is periodic and hyperbolic. Blow it up to a cylinder by the inverse of the mapping from [0,E ) x S1 x [0,1] 4 M (S' is the unit circle with angular coordinate 8) defined by ( T , 8, Z )

H

(T

cos(8

+ p),

T

sin(8

+ p),

Z)

for some E < 1/4, where p = arctan(l/y) (the inclusion of ,B is not essential but simplifies the next formulae). The identification (Y becomes a(r,8 , l ) = ( T I , 8', 0) with TI

=r

m

,

f(e) = y4C O S ~e + y4sin2 8,

(1)

tan 8' = y-4 tan 8, where 8' in the third equation is chosen from the same quadrant of the circle as 8; it defines a C" map $J of the circle (whose derivative is l / f ) . Denote the blowup manifold by N . It is a C" manifold with boundary 8 N diffeomorphic to T2. Use coordinates ( T , ~ , z near ) the boundary and (z, y, z ) elsewhere, taking into account the identifications (Y and the horizontal integer translations. If one wants to be a stickler for rigour, one can

58

make a cover of N by 10 charts, based on these two coordinate systems and the gluing map a. The vector field s on M induces one on N that I call t. It looks like s but the periodic orbit along x = 0 is blown up into an invariant torus with coordinates (0, z ) (modulo gluing by $), representing horizontal directions of approach t3 to points z of the periodic orbit. On this boundary torus the vector field has two attracting periodic orbits 8 = 0 , and ~ two repelling ones 6 = f7r/2, separating four annuli on which all orbits come from a repelling one at large negative time and go to an attracting one at large positive time (this comes out of the gluing $). The vector field t preserves the volume form d x A d y A d z in (x,y , z ) coordinates and r d r A d e A d z in ( r ,8, z ) , and inherits invariant foliations from s. Next, for a C3 function g : N -+ IR+ = {x E IR : x > 0) to be chosen later, let u = gt on N . Note that g is bounded away from 0 and +cc because N is compact. The vector field u preserves volume form i d x A d y A d z and has the same invariant foliations as t. Then, choose a C 3 function p : N -+R which is positive in fi = N \ d N (the interior of N ) and asymptotic to distance to d N near d N , measured with some C3metric. Specifically, I choose Arnol’d’s Riemannian metric

+

ds2 = ~ - ~ ” d x 2 +y4”dx5

+dz2,

(2)

where

d x - = cos p d x

+ sin p d y , + cos p d y ,

(3)

dx+ = - sinp d x and require p

-

7-47-42

Let v = p u . Then v is

sin2 e

+ 7 4 cos2 ~ e as r + 0.

C3,preserves volume form r

wg = -dr PS

1 A dB A d z = -dx PS

A d y A da

(4)

in the two coordinate systems, and is zero on a N . A remarkable fact is that N is C”-diffeomorphic to the exterior of a figure-eight knot in the 3-sphere S3 (“exterior” means the closure of the complement of a closed tubular neighbourhood). Although this is stated in many places,11~15~16*27~28~38-40 I have found it hard to locate a proof in the literature, partly because the interest in many of these references focusses on the additional fact that it can be endowed with a hyperbolic metric. Even then, most topologists are happy with existence rather than an explicit

59

diffeomorphism. In an Appendix I briefly survey the proofs of which I am aware. As the final step, I transfer the example from S3 into R3:choose the figure-eight knot to pass through the “North pole” (0, 0, 0 , l ) of S3 (considered as the unit sphere in R4)and map the rest of S3 stereographically t o the plane tangent to the “South pole”, i.e. (x,y, z , w) H The result is a C” diffeomorphism h from N to the closure of a bounded domain R of EX3 which looks like Figure 2. Think of it as an apple through the core of which a worm has eaten a tubular hole in the form of a figure-eight knot. The domain R is the remaining flesh of the apple.

w.

Fig. 2.

The domain R for the vector field w and four orbit segments.

The desired vector field is w = h*v, the image of v under h. It is C3, vanishes on the boundary of R and preserves volume h*ug.To make it preserve a pre-ordained C3 volume form vol on R (e.g. the Euclidean volume from R3),it suffices to choose the function g = where w1 is the special case of (4) with g = 1 (since all volume forms a t a point are multiples of a given one, this ratio makes sense; also it is a C3 positive function as required).

9,

60

To give some idea of what the vector field w looks like on R, Fig. 2 also indicates orbit segments approaching or departing from the four periodic orbits of the skin frzction field on the boundary ( r being distance from the boundary): they alternately attract and repel along the boundary and repel and attract from the interior. The fact that the periodic orbits go the %hart" way around the boundary is a consequence of a nice argument explained to me by Luisa Paoluzzi which I summarise in the Appendix (see also Ref. 38). Fig. 3 shows a slightly different view in which the bottom lobe of the knot has been rotated round the back to enable visualisation of the image of the cross-section z = 0 in N by h. It is based on Fig.11 of Ref. 39. Convince yourself that the cross-section is indeed diffeomorphic to a torus minus a round open disc, a space I’ll denote by To,and that it can be swept round in R, following a given co-orientation and keeping the boundary on dR, and that the action on the surface induced by sweeping once round is homotopic to A’, the blowup of the toral automorphism A. R is said to fibre over the circle, with fibre (or “Seifert spanning surface”) To and monodromy A’. More pictures of this can be found in Refs 15,27.

2

Fig. 3. The domain R and a cross-section to the vector field w, with direction of flow indicated by f.

61

A similar construction was used in Ref. 11 to make an example of a flow in R3 where the possible knot and link types of periodic orbits could be shown to be very rich (indeed Ref. 16 proved it contains all knots and links, and the same for any flow transverse to the fibration). Their vector field, however, is not volume-preserving. It was obtained from s by a DA (“derived from Anosov”) construction, perturbing the gluing map a near the fixed point (0,O) to replace it by a repelling fixed point and two saddles and then excising the repelling orbit. 3. Mixing The point of the example w is the following theorem.

Theorem 3.1. All vector fields topologically equivalent to w on R within the class of vector fields on R preserving given volume form vo1, C3 on and vanishing on the boundary are mixing. Proof. The first return map $ to the cross-section { z = 0) minus its boundary is mixing for the area form given by the flux of vol under w, by the standard Hopf argument using the existence of the invariant foliations for $J (e.g. see Ref. 12 for a nice exposition). By Anosov’s alternative1 (rediscovered in Ref. 32), the only obstacle to the flow being mixing would be if the return time function r : ‘ko -+ R+ for were a constant plus a coboundary. A “coboundary” for a map $ is a function r : $0 -+ lR of the form r(x) = a($(x)) - a(x) for some function a,so its sum along an orbit of 1c, telescopes. This is a somewhat exceptional situation. Indeed, in our case the return time goes to infinity at the boundary, so can not be a constant plus a c ~ b o u n d a r y . ~ ~ 0 $J

Actually, from mixing and a general argument of Ref. 31, it follows that the flow is Bernoulli. A nice feature of the example which makes it potentially physically realisable is that it is robust.

Theorem 3.2. w is structurally stable within the above class of vector fields. A vector field is structurally stable if all small perturbations are topologically equivalent to it. Since the proof involves many technicalities, it will be published elsewhere. How fast does the example mix? To answer this requires first a discussion about how to define rate of mixing.

62

A standard way to define the rate of mixing of a flow 4 on a manifold M preserving a volume form p is to choose a class F of functions f : M -i R and ask how fast the correlation C f g ( t = ) J f($t(z))g(z)dp(z) for f , g E F decays to the product of the means off and g, in comparison to the product of the sizes of f and g using a notion of size appropriate to the function class (or f , g can come from different function spaces). The answer depends strongly on the chosen class of functions, however. For example, if F is L2 then there is no uniform decay estimate: g could be chosen to be f o 4~ for some large T and then C f g ( T )= 11 f IILzllgIIL2.For some mixing systems, exponential decay can be proved for Holder continuous functions, but the decay rate depends in general on the Holder exponent a. An alternative is to use a metric on a space of probability measures on M and ask how fast the push-forward of an initial measure converges to p. A natural metric is the total variation metric, but for a volume-preserving flow this metric is invariant, so gives no information about mixing. A better one is the transportation metric

where Pp,qis the set of probability measures on R x R with marginals p , on the first and second factors. It is the minimum average distance that mass from one measure has to be moved to turn it into the other measure. A nice result of Ref. 21 is that

over non-constant Lipschitz functions f , where p ( f ) is the expectation of f in measure p and 11 f I I L ~ is~ the smallest Lipschitz constant for f . So the two views come close when Holder is specialised to Lipschitz ( a = 1).In particular, given an initial measure v absolutely continuous with respect to p , it can bewrittenasgpfor afunctiong E L 1 ( p ) .Then ( 4 r v ) ( f ) - p ( f )= C f g ( t )so , D(+r (v),p ) = supf c ( t ) and any upper bound on the correlation function

-

proportional to 11 f I I L ~ gives ~ a corresponding upper bound on the transportation distance. It is not clear to me, however, whether lower bounds transfer so easily, because t o obtain an accurate lower bound for the transportation distance one may have to change the choice of f as time progresses. In any case, I choose to use transportation metric.

63

Theorem 3.3. No C2 volume-preserving vector field with compact no-slip C2 boundary mixes faster than l / t 2 in time t .

Proof. Let v be a C2 volume-preserving vector field with no-slip boundary, p a C2 positive function asymptotic to distance to the boundary, and u = v/p. Then a simple calculation shows that u is tangent to the boundary. Let C = sup in a neighbourhood r 5 r1 of the boundary. Then [u,I 5 Cr for r 5 r1. Thus 1q.l 5 Cr2 for r 5 r1. It follows that fluid from outside r 5 r1 can get t o a t most distance l / ( l / r l - C t ) of the boundary in time t. Take an initial “dye” density 1 in r 5 T I and 0 outside. Then the subset r < l / ( l / r l - C t ) remains of density 1. It is of thickness of order l / t , so has volume of order l / t and the average distance that dye must be moved to achieve the average density is at least half the thickness. Thus the transportation distance to the uniformly mixed state is a t least of order 1 p . 0

%

The fact that some flows with no-slip boundaries mix like a power law was noted numerically in Ref. 18, albeit with molecular diffusion added and a different notion of mixing rate. An open question is t o determine an upper bound on the transportation distance as a function of time. This would require some study of the returntime function t o a cross-section, among other things. If one switches attention to correlation functions, there is some literature on systems with power law decay, e.g. Ref. 13 for upper and Ref. 36 for lower bounds. It seems likely to me that the correlation of many pairs of function decays like l / t for our flow. This would give rise to anomalous diffusion. Corresponding to the coordinate z of s is a quantity one can continue t o denote by z which measures how many times (plus fractional part) trajectories have crossed the cross-section of Fig. 3. Then one can examine the deviation from the mean rate of increase of z with time. If the autocorrelation function for i is integrable then the deviation would spread like normal diffusion, but if its integral is infinite then the deviation should spread anomalously. One way to obtain a handle on this would be t o use the fact that the flow has a Markov partition and compute the large deviation rate function for the increment in z (cf. Ref. 26). 4. Discussion

At the physical level, there remains the question of how to drive the flow. It suffices to compute w.Vw - YAW,where Y is the kinematic viscosity,

64

subtract off its gradient part, and apply a body force equal to the remainder. It might not be easy to implement, however. One can contrast results of Ref. 14 making an Euler flow on S3 containing all knots and links. Being an Euler flow it requires no forcing at all, but the catches are that it also requires zero viscosity, the Riemannian structure could not be specified in advance, and it is not claimed to be mixing: indeed the knots and links are supported on a proper subset. I believe it is possible to make a similar construction of a flow with stressfree boundaries, by using symplectic polar blowup instead. This ought to be C2 structurally stable. To obtain mixing, however, one would need to ensure that the speed function is nontrivial. One can ask whether the flow is a fast dynamo. The dynamics of a magnetic field in a steady conducting fluid flow may have a positive growth rate. The flow is said to be a fast dynamo if the growth rate has a positive lower bound as the magnetic diffusivity goes to zero (in principle this depends on the Riemannian metric assumed for the magnetic diffusion) (see survey in Ch.V of Ref. 6). A r n ~ l ’ dproved ~ ~ ~ that s is a fast dynamo with respect to metric (2). It would be interesting to investigate whether w is a fast dynamo. To make this problem well posed one has to specify what the magnetic field does outside R. One can ask whether there are alternative constructions of robust mixing fluid flows. I believe one would be the “pigtail stirrer”. Start from s on A4 but quotient by o ( x , y , z ) = (-x,--y,z) and blowup the orbits of both (0, 0,O) and f , 0 ) to tori. This gives a vector field in a solid torus minus a tubular neighbourhood of a knot which goes three times round the solid torus making the closure of a pigtail braid (as sketched in Ref. 25 for example). The monodromy goes back to Latt&s.22The analysis is slightly different from the example w, because the blown-up orbits are 1-prong singularities rather than regular orbits, but I think the same structural stability result should be possible. Furthermore, this example opens the possibility to make the outer boundary axisymmetric and to rotate it about its axis, so that the no-slip condition gives a non-zero field on the outer boundary. Equivalently (though different for the fluid dynamics), one could rotate the 3-braid and examine the flow in the rotating frame. Another starting point is geodesic flow on the unit tangent bundle of a surface of negative curvature, which is mixing Anosov. Birkhoff showed that blowup of 6 periodic orbits of the genus 2 case produces a suspension of a hyperbolic toral automorphism with 12 points blown up,” and I expect this can be mapped into R3.

(i,

65

What if one abandons the structural stability requirement but just asks for robust mixing, i.e. all nearby volume-preserving flows are also mixing? I believe this can be achieved by what I call a “baker’s flow” by analogy with the well known baker’s map. It is a volume-preserving flow in a container whose boundary is a surface of genus 2. The 2D stable manifold of a reattachment point on the boundary separates the volume into orbits which go round one loop from ones which go round the other loop. These two sets glue together again along the 2D unstable manifold of a separation point on the boundary. If the two manifolds are designed to intersect transversely, the eigenvalues of the skin-friction field satisfy certain inequalities at the separation and reattachment points, and the flow round the loops rotates trajectories suitably, then the return map to a transverse section in the middle is a nonlinear version of the baker’s map. The system is a volumepreserving analogue of the Lorenz system. The flows are not structurally stable, but are probably robustly mixing (just as for the Lorenz system in the good parameter regime24). Lastly, one can ask about time-periodic 2D flows. I think it might be possible to make a codimension-3 submanifold of C2 area-preserving maps of the torus, isotopic to the identity (so realisable by time-periodic flows), looking perhaps a bit like Zeldovich’s alternating sine-flow, which are mixing and topologically conjugate. The idea is to start from a pseudo-Anosov example (maybe a variant of Ref. 26), then smooth it and show topological conjugacy for all small smooth perturbations preserving the singular orbits.

Appendix

Here I survey what I have found about the diffeomorphism between the blow-up of the suspension manifold and the exterior of a figure-eight knot. The starting point is to notice that they have isomorphic fundamental groups, with isomorphism respecting the subgroup for the boundary. Then a result of Ref. 37 applies to give a homeomorphism (alternative proofs are in Ref. 30 using Ref. 29, and Cor 6.5 of Ref. 41). Stallings’ paper worries me, however, because he ends by saying that it is not clear whether fibred manifolds with isotopic monodromy are homeomorphic. All of these proofs involve various cutting and gluing operations that make it difficult to see an explicit homeomorphism and they do not address the question of smoothness (but Ref. 23 redoes it in the differentiable category). More explicit are three approaches which involve viewing the manifold as a quotient of hyperbolic 3-space W3 by a discrete group of i s o m e t r i e ~ ~ ~ ~ (see also Ref. 40).

66

Another strategy17 (also described in 10.J of Ref. 35) is to notice that the figure-eight knot has a Z2 symmetry by a half-rotation about some unknot (this was used also by Ref. 11).Quotienting by the symmetry reduces it to the closure of the pigtail braid relative to the unknot symmetry axis. Since any braid-closure is fibred, so is the figure-eight knot, and the monodromy can be seen to act like A’, the blowup of

1; ;]

to +a

L

A

With an explicit diffeomorphism it would be easy to verify the claim of Section 2 about the homotopy class of the periodic orbits on the boundary, but the following argument of Luisa Paoluzzi answers the question anyway. Choose as base point for the fundamental group n l ( N ) the point at z = 0 on b” with 0 = 0. Choose the following generators for n1(N): a translates by (1,0,0) passing over the second tube, b translates by (0,1,0) passing to the left of the second tube, c translates by (O,O, 1) up the periodic orbit at T = 0 = 0 (and glues by A ) . Then a generating set of relations is c-lac = a2b,c-lbc = ab. Also, going once round the tube anticlockwise in the plane 2 = 0 is achieved by K = b-la-lba. The preimage under the diffeomorphism h : fi -+ R of the homotopy class of a closed curve going the short way around the knot in R, cutting the Seifert surface positively, is CK? for some integer n. We want to show n = 0. The quotient of n l ( N ) by is trivial, since it is equivalent to reinserting the knot and its tubular neighbourhood into S3. The quotient of n1(N) by c is indeed trivial (the relations then imply a = a2b,b = ab, so a = b = el the identity). In contrast, one can argue that for any n # 0, the quotient of n l ( N ) by is non-trivial.

Acknowledgements

I first presented the idea at a “Mixing in fluid flows” meeting in Bristo1 in May 2004 supported by the London Mathematical Society. I thank Luisa Paoluzzi for proving the homotopy class of the periodic orbits on the boundary, Sebastian van Strien, Christian Bonatti, Yakov Pesin, Enrique Pujals, Charles Pugh and Mike Shub for believing the proof of structural stability would be possible, Jean-Christophe Yoccoz for pointing out an error in a draft, and Mark Pollicott and Lai-Sang Young for comments about mixing rates. I did the serious work on the structural stability during visits to IMPA (KO),the Fields Institute (Toronto), COSNet (Australia) and IHES (France) from July 05 to June 06. I thank all of them for their hospitality.

67

References 1. Anosov DV, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat Inst Steklov 90 (1967) 210. 2. Aref H, Stirring by chaotic advection, J Fluid Mech 143 (1984) 1-21. 3. Aref H, The development of chaotic advection, Phys Fluids 14 (2002) 131525. 4. Arnol’d VI, Sur la topologie des Bcoulements stationnaires des fluides parfaites, Comptes Rendus Acad Sci Paris 261 (1965) 17-20. 5. Arnol’d VI, Notes on the three-dimensional flow pattern of a perfect fluid in the presence of a small perturbation of the initial velocity field, J Appl Math Mech 36 (1972) 236-242. 6. Arnol’d VI, Khesin BA, Topological methods in hydrodynamics (Springer, 1998). 7. Arnol’d VI, Korkina EI, The growth of a magnetic field in a three-dimensional steady incompressible flow, Moscow Univ Math Bull 38 (1983) 50-4. 8. Anrol’d VI, Zel’dovich YaB, Ruzmaikin AA, Sokolov DD, A magnetic field in a stationary flow with stretching in Riemannian space, Sov Phys JETP 54 (1981) 1083-6. 9. Arter W , Ergodic streamlines in three-dimensional convection, Phys Lett A 97 (1983) 171-4. 10. Birkhoff GD, Dynamical systems with two degrees of freedom, Trans Am Math SOC18 (1917) 199-300. 11. Birman JS, Williams RF, Knotted periodic orbits in dynamical systems 11: knot holders for fibered knots, in Low-dimensional topology, ed Lomonaco SJ, Contemp Math 20 (Am Math SOC,1983) 1-60. 12. Burns K, Pugh C, Shub M, Wilkinson A, Recent results about stable ergodicity, in: Smooth ergodic theory and its applications, eds Katok A, Llave R de la, Pesin Ya, Weiss, Proc Symp Pure Math 69 (2001) 327-66. 13. Chernov N, Zhang H-K, Billiards with polynomial mixing rates, Nonlin 18 (2005) 1527-53. 14. Etnyre J, Ghrist R, Contact topology and hydrodynamics 111:knotted orbits, Trans Am Math SOC352 (2000) 5781-94. 15. Francis GF, Drawing Seifert surfaces that fiber the figure-8 knot complement in S3 over S’,Am Math Month 90 (1983) 589-599; and Chapter 8, A topological picture book (Springer, 1987). 16. Ghrist RW, Branched two-manifolds supporting all links, Topology 36 (1997) 423-448. 17. Goldsmith DL, Symmetric fibered links, in: Knots, groups and 3-manifolds, ed Neuwirth LP, Ann Math Studies 84 (Princeton, 1975) 3-23. 18. Gouillart E, Kuncio N, Dauchot 0, Dubrulle B, Roux S, Thiffeault J-L, Walls inhibit chaotic mixing, Phys Rev Lett, in press, 2007. 19. HBnon MR, Sur la topologie des lignes de courant dans un cas particulier, Comptes Rendus Acad Sci Paris 262 (1966) 312-314. 20. Jorgensen T, Compact 3-manifolds of constant negative curvature fibering over the circle, Ann Math 106 (1977) 61-72.

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21. Kantorovich LV, Rubenstein G Sh, On a space of totally additive functions, Vestnik Leningrad Univ 13:7 (1958) 52-59. 22. Lattits S, Sur l’iteration des substitutions rationelles et les fonctions de PoincarB, Comptes Rendus Acad Sci Paris 16 (1918) 26-8. 23. Laudenbach F, Le theoreme de fibration de JStallings, Seminaire Rosenberg, Orsay (1969) M13.369. 24. Luzzatto S, Melbourne I, Paccaut F, The Lorenz attractor is mixing, Commun Math Phys 260 (2005) 393-401. 25. MacKay RS, Postscript: Knot types for 3-D vector fields, in: Topological Fluid Mechanics, eds Moffatt HK, Tsinober A, IUTAM Conf Proc, Aug 89 (CUP, 1990), 787. 26. MacKay RS, Cerbelli and Giona’s map is pseudo-Anosov and nine consequences, J Nonlin Sci 16 (2006) 415-434. 27. Miller SM, Geodesic knots in the figure-eight knot complement, Exp Math 10 (2001) 419-436. 28. Milnor 3, Hyperbolic geometry: the first 150 years, Bull Am Math SOC6 (1982) 9-24. 29. Neuwirth L, The algebraic determination of the topological type of the complement of a knot, Proc Am Math SOC12 (1961) 904-6. 30. Neuwirth L, On Stallings fibrations, Proc Am Math SOC14 (1963) 380-1. 31. Ornstein DS, Weiss B, Statistical properties of chaotic systems, Bull Am Math SOC24 (1991) 11-116. 32. Plante JF, Anosov flows, Am J Math 94 (1972) 729-754. 33. Riley R, A quadratic parabolic group, Math Proc Camb Phil SOC77 (1975) 281-8. 34. Robbin JW, On the existence theorem for differential equations, Proc Am Math SOC19 (1968) 1005-6. 35. Rolfson D, Knots and links (Publish or Perish, 1976). 36. Sarig 0, Subexponential decay of correlations, Invent Math 150 (2002) 629653. 37. Stallings J, On fibering certain 3-manifolds, in: Topology of 3-manifolds and related topics, ed Fort MK (Prentice Hall, 1962), 95-100. 38. Thurston WP, The geometry and topology of 3-manifolds, Chs 3 and 4 (preprint, 1978). 39. Thurston WP, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull Am Math SOC6 (1982) 357-381. 40. Thurston WP, Three-dimensional geometry and topology, vol 1, ed Levy S (Princeton U Press, 1997). 41. Waldhausen F, On irreducible 3-manifolds which are sufficiently large, Ann Math 87 (1968) 56-88.

69

COMPLEXITY AND ENTROPY IN COLLIDING PARTICLE SYSTEMS M. COURBAGE and S. M. SABER1 FATHI

Laboratoire Matibre et Systbmes Complexes ( M s c ) UMR 7057 CNRS et Universitt Paris 7- Denis Diderot Case 7020, Tour 24-14.5bme ttage, 4, Place Jussieu 75251 Paris Cedex 05 / FRANCE emails : [email protected], majid.saberiQparis7.jussieu.fr We develop quantitative measures of entropy evolution for particle systems undergoing collision process in relation with various instability properties.

Keywords: Lorentz gas, hard disks, entropy, Lyapunov exponents, H-theorem

1. Introduction

There are two concepts of entropy in the theory of dynamical systems: the first one is the famous Kolmogorov-Sinai' entropy introduced by Kolmogorov in 1958. Kolmogorov, who was familiar with the Shannon entropy for random process, designed this concept and used it in order to solve the isomorphism problem of Bernoulli systems. In 1959, Sinai modified and extended the ideas and the results of Kolmogorov to any dynamical system (DS) with an invariant probability measure (also called measurable DS). It is important to note that the measure theoretical entropy is a number that characterizes the family of isomorphic dynamical systems. It is one of the main tools to classify all measurable dynamical systems. Although this theory provided considerable information about their structure, many problems are still open. On the other hand, the non-equilibrium entropy, introduced by Boltzmann in kinetic theory of gases, can be defined in the case of measurable DS. Recall that the Boltzmann H-theorem defines the entropy for the one particle probability distribution ft (z) as

70

Boltzmann showed that this quantity is monotonically increasing for all solutions of his celebrated equation. During many years until the beginnings of the twentieth century the Boltzmann H-theorem was the object of many discussions and controversies. Later on Ehrenfest proposed the urn Markov chain model for the approach to equilibrium with an H-theorem. The model consists of n = 2N balls distributed inside two halves of a box : left and right. On account of collisions between particles, Ehrenfest postulated that a t regular time interval a particle can leave the right half or to join it . So if the state space of the system is described by the number X of particles in the right hand side, the dynamics of the system would be a Markov chain where the allowed transitions are from X = m to X = m - 1, with probability m / 2 N , or from X = m to X = m 1 with the complementary probability. Mark Kac gave an exhaustive solution of this model in his book.’ Briefly speaking, it is possible to find a unique stationary probability distribution { p i } , i = 0,1,. . . ,n = 2N; such that any initial distribution { ~ ( t con)} verges t o { p i } . The non-equilibrium entropy of the distribution {vi(t)}with density ft = !!& is given by the Boltzmann like formula: Pi

+

The variable X is a “macroscopic variable” which means that a given value of X corresponds to a region in the phase space of 2N dimensions. Distinct values of X correspond t o distinct regions, {’Pi}. The set of ’Pi’s form a partition of the phase space. However, it is obvious that strictly speaking the process X ( t ) is not Markovian although some time it is claimed to be approximately Markovian. Independently, Gibbs imagined the dynamical mixing property as a mechanism of the approach to equilibrium for systems out of equilibrium. His ideas are based on the consideration of the phase space of an isolated system of N particles where the equilibrium is described by the microcanonical ensemble as an invariant measure. The system will approach the equilibrium if any initial probability distribution will converge to equilibrium under the Hamiltonian flow. According to Gibbs this will happen if the shape of any subset will change boldly under the flow, although conserving a constant volume, winding as a twisted filament filling, proportionally, any other small subset of the phase space. The famous image of this mechanism is the mixing of a drop of ink in a glass of water. Later on, Hopf found a whole class of mixing DS: the differentiable hyperbolic

71

DS where to each trajectory is attached two manifolds expanding and dilating in transversal directions. So, any domain of the phase space will be squeezed an folded filling densely any region of the phase space. For example in the baker transformation, the expanding and contracting manifolds are horizontal and vertical respectively, so that any small horizontal segments will be uniformly distributed in the phase space after few iterations of the transformation (see Fig. 1).The importance of mixing and exponential instability of trajectories for obtaining H-theorem has been discussed by Krylov.2 /.F/

/.8/ 13-

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Fig. 1.

The H-theorem for measurable dynamical systems describes the approach to equilibrium, the irreversibility and entropy increase for measurable deterministic evolutions. That is dynamical transformation T on a phase space X with some probability measure p, invariant under T , i.e. p(T-'E) = p ( E ) for all measurable subsets E of X. Suppose also that there is some mixing type mechanism of the approach to equilibrium for T , i.e. there is a sufficiently large family of non-equilibrium measures v such

72

that

vt(E) =: v ( T P t E ) s) p ( E ) , For all E.

(3) Then, the H-theorem means the existence of a negative entropy functional S(vt)which increases monotonically with t to zero, being attained only for v = p. The existence of such functional in measure-theoretical dynamical systems has been the object of several investigations during last decades see8-12,14i17i18,21). Here we study this problem for the Lorentz gas and hard disks. Starting from the non-equilibrium initial distribution v, and denoting by P a partition of the phase space formed by cells (Pl,P2,..., Pn)and by vi(t) = v o T W t ( P i )the , probability at time t for the system t o be in the cell Pi and such that .(Pi) # p(Pi) for some i, the approach to equilibrium implies that vi(t) -+ pi as t -+ oc) for any i . The entropy functional will be defined by:

which we simply denote here after S ( t ) .The H-functional (4) is maximal when the initial distribution is concentrated on only one cell and minimal if and only if vt(Pi) = p(Pi),Vi. These properties are shown straightforwardly. This formula describes the relative entropy of the non-equilibrium measure vt with respect to p for the observation associated to P. It coincides with the information theoretical concept of relative entropy of a probability vector ( p i ) with respect to another probability vector ( q i ) defined as follows: - lnpi being the information of the ithissue under the first distribution, pi In( is equal t o the average uncertainty gain of the experience ( p i ) relatively t o ( q i ) . A condition under which formula (4) shows a monotonic increase with respect t o t is that the process .,(Pi) = v o T t ( P i )verifies the ChapmanKolmogorov equation valid for Markov chains and other infinite memory chains.lOill For a dynamical system, this condition is hardly verified for given partition P . However, the very rapid mixing leads t o a monotonic increase of the above entropy, a t least during some initial stage, which can be compared with the relaxation stage in gas theory. In this paper, we will first compute the entropy increase for some remarkable non-equilibrium distributions over the phase space of the Sinai’ billiard. The dynamical and stochastic properties of the Lorentz gas in two dimensions which we consider here was investigated by Sinai and Bunimovich as an ergodic dynamical ~ y s t e m Other . ~ ~ transport ~ ~ ~ ~ properties

xi

E),

73

have been also studied numerically eel^,'^). This is a system of non interacting particles moving with constant velocity and being elastically reflected from periodically distributed scatterers. The scatterers are fixed disks. On account of the absence of interactions between particles the system is reduced to the motion of one billiard ball. We shall investigate the entropy increase under the effect of collisions of the particles with the obstacles. For this purpose, we consider the map T which associates to an ingoing state of a colliding particle the next ingoing colliding state. The particle moves on an infinite plane, periodically divided into squares of side D called "cells", on the center of which are fixed the scatterers of radius a ( Fig. 2). The ingoing colliding state is described by an ingoing unitary velocity arrow a t some point of the disk. To a colliding arrow Vl(P1) at point PI on the boundary of the disk the map associates the next colliding arrow Vz(P.2) according t o elastic reflection law. Thus, the collision map does not take into account the free evolution between successive collisions.

Fig. 2.

The motion of the particle on a toric billiard.

The billiard system is a hyperbolic system (with many singularity lines) and, in order to have a rapid mixing, we will consider initial distributions supported by the expanding fibers. Such initial measures have been used in.8i12>21For the billiard the expanding fibers are well approximated by particles with parallel arrows velocity. We call this class of initial ensemble beams of particles. We first compute the entropy increase under the collision map for these initial distributions. We will consider finite uniform partitions of the phase space as explained below. The entropy functional will be defined through (4).For this purpose, the phase space of the collision

74

map is described using two angles (p,$), where p is the angle between the , [0, f [, and 1c, E [0,7r] outer normal a t P and the incoming arrows V ( P ) ,f3 is the angle between s-axis and the outer normal a t P . Thus, the collision map induces a map: (pi,1c,i) --+ (pz,1c,2) (see Fig. A.2) and we shall first use a uniform partition of the (p,1c,) space. The computation shows that whatever is the coarsening of these partitions the entropy has the monotonic property in the initial stage. It is clear that, along mixing process, the initial distribution will spread over all cells almost reaching the equilibrium value. Physically, this process is directed by the strong instability, that is expressed by the positive Lyapunov exponent. It induces a relation between the rate of increase of the entropy functionals and Lyapunov exponents of the Lorentz gas. Our computation shows that this relation is expressed by an inequality max(S(n

+ 1)

-

~(n= ) )AS 5

C

(5)

A120

where the “max” is taken over n, which means that the K-S entropy is an upper bound of the rate of increase of this functional. Next, we shall consider the entropy increase of a system of N hard disks. Here the space in which moves a particle is a large torus divided into rectangular cells. Denoting the total number of cells by n and the number of particles initially distributed in only one region, by N , and following them until each executes t collisions with obstacles, we compute the probability that a particle is located in the ith cell as given by:

Pi(t) =

Number of particles in cell i having made t collisions

N

The equi-distribution of the cells leads to take, as equilibrium measure, pi = ,: so that this ((spaceentropy” is defined by:

The maximum of absolute value of this entropy is equal t o -Inn. So we normalize as follows:

(7) We shall also do some comparisons of the H-theorem with the sum of normalized positive Lyapunov exponents.

75

2. Entropy for collision map The entropy for the collision map is computed for a beam of N particles on a toric checkerboard with n cells. We start to calculate the entropy, just after all particles have executed the first collision. In this computation, all particles have the same initial velocity and are distributed in a small part of one cell. For each particle we determine the first obstacle and the angles (PI, $1) of the velocity incoming vector V1 ( P I )( see the figures given in the Appendix). For a uniform partition P of the space of the variables (p,$), the entropy S(t)is computed iteratively just after all particles have executed

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0

12

14

16

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6

8

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Number of collisions

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\

Y=A+B*X A

2,08466

B

-0,60266

e

0.0-

6

5

4.5:

-1.o.,

<

Fig. 3. Entropy of the collision map versus number of collisions for (a) a beam of 640 particles for a radius a =2.2, neighboring disks centers distance 1 and a partition of (B, ) space into 25 x 25 cells, (b) Logarithm of the collision map emtropy versus number of collisions for this systme.

76

the tth collision. We use the formula (4) where p ( ~ i= ) Joi” J*i+l 02

cospdpd+

(8)

+i

is the invariant measurelg of the cell Pi = [pi,&+l[x\$i,$i+l[and vt(Pi) is the probability that a particle is located after t collisions in Pi computed as Number of particles in Pi having made tcollisions N The velocity after the collision is computed from the following equation:

V(P2) = V(P1) - 2(V(Pl).n)n

(9)

where n is the normal vector at the collision point. We explain in the Appendix the main geometric formula used for this computation. This entropy increase is shown in the Fig. 3 for various partitions and various initial distributions. The absolute value of the entropy of a distribution of particles, that we call its amount of entropy, represents in fact its distance to equilibrium. It is to be noted that the amount of entropy increase under one collision is remarkably greater for the few first ones which corresponds to an exponential type increase (Fig. 3). We calculate the Lyapunov exponents by using the method of Benettin et al.4 Comparing A S = max(S(t 1) - S ( t ) )(where the “max” is taken over t ) with the positive Lyapunov exponent, A, of the collision map we verify the inequality:

+

(10)

A S < A

as shown in Fig. 4, where this exponent is 3.2. The maximal entropy increase by collision for the distribution computed in this figure is not far from this value. So it could be conjectured that in some suitable refinement limit, the entropy increase for beams tend to the positive Lyapunov exponent. The rate of the approach to equilibrium is thus related to the positive Lyapunov exponent. In order to compare the entropy increase as a function of the collisions with the entropy increase as a function of time, we compute the distribution of mean free time for the first 3 collisions. From time histogram for the first three collisions of this system ( Fig. 5), we see that a great number of particles have the same mean free time. As shown in the table 1, the mean free time vary during the first three or four collisions but after those, for the N

77

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Fig. 4. (a) Lyapunov exponent and (b) entropy of the collision map, versus of number of collisions for each particle. We see that the maximum of the entropy increase between two collisions is less than of the value of the Lyapunov exponent.

Fig. 5 .

Free time histogram for (a) first, (b) second and (c) third collision.

following collisions, rapidly the system comes near the equilibrium, where we have a constant mean free time approximately.

78 Table 1. Mean free time obtained for a beam of 640 particles for a radius a = 0.2, neighboring disks centers distance 1 and a partition of ( p ,11) space 25 x 25 cells. Collision number Mean free time Collision number Mean free time

1

1.966 8

4.162

2 26.174 9 4.208

4 3.820 11 3.863

3 5.801 10 4.212

6 4.452 13 4.051

5

3.611 12 4.272

7 4.177 14 4.397

3. Hard disks Considering a uniform space partition of a large toric space we compute the particles densities, p i ( t ) , and the normalized space entropy as a function of time by using the equation (7). Starting with a distribution of disks with localized positions in some cell and random velocities, we compute binary collisions instants and the trajectories of the hard disks. These instants are determined by checking the distance between particles, after a time interval is passed. The Lyapunov exponents of the flow are calculated by using the Benettin et al. algorithm. The result is shown in the Fig. 6. Fig. 7 is a histogram of the number of collisions, so we see that the number of collisions in a fixed time interval is reduced for large time. From Fig. 6 we see that the monotonic part of the non-equilibrium entropy is also varying exponentially with respect t o time. This shows that the collision is the main ingredient responsible of the entropy increase as described in the Boltzmann equation theory. and compute the characteristic We shall now vary the density CT = quantities. The graph of the normalized positive Lyapunov exponents spectrum per particle for the same system as in Fig. 6 is shown in Fig. 8. The

&,El

, . , .

O

mla

(a)

t

Q

,

m

. ,' . b

, Y

.

a

,

a

.

Tim.

(b)

Fig. 6. (a) Normalized space entropy and (b) its monotonic part logarithm versus time, y) for 128 hard disks with radius a=0.05 which are initially localized in the first cell of (z, space with 6 x 6 cells and a density 01 = 0.889 disks per unit area.

79

Time

Fig. 7.

Number of collisions histogram system versus time in Fig. 6.

0

50

1M)

150

200

250

300

Fig. 8. Normalized spectrum Lyapunov of exponent of system in Fig. 6.

Table 2. The data for the hard disks systems of radius, a = 0.05 and the same initial conditions, with cells 6 x 6 , in terms of the density. Density 3.555 0.889 0.222

& ZA+,(&)

n ~ s p

0.367 0.294 0.239

0.139 0.115 0.144

80

computation of the normalized sums of the positive Lyapunov exponent, shows that the inequality between maximum entropy increase and the sum of normalized of positive Lyapunov exponents is verified (Table 2),

& xAi,o(&),

ASsp=

1

c (A) .

Ai>O

4. Concluding remarks

The computations of the entropy amount of some given non-equilibrium initial distributions relatively to the equilibrium measure show an exponential type increase for all considered partitions and distributions during initial stage after which the entropy increases slowly and fluctuates near its maximal value. These computations confirm the existence of a relaxation time generally assumed in the derivation of kinetic equations3 and the origin of the rapid increase of the entropy as due to the number of collisions. The dispersive nature of the obstacles is responsible of the exponential type increase. This exponential type increase has been demonstrated for the Sinai entropy functiona121 in hyperbolic automorphisms of the torus. On the other hand, the relation of the entropy increase to Lyapunov exponents can be understood through Pesin relation and Ruelle inequality. In fact, the rate of entropy increase should be bounded by the KolmogorovSinai entropy and such bound have been found by Goldstein and Penrose for measure-theoretical dynamical systems under some assumptions. l7 An open question is to characterize the initial non stationary probability measures reaching the upper bound. Any entropy functional is not a completely monotonic function of time for any dynamical system. A completely monotonic entropy functional has been obtained when the map T on the space X is a Bernoulli system or, slightly more generally, a K-system. This means that there is an invariant measure p and some partition JO of X such that TJo becomes finer than ( 0 ( we denote it: TJo 2 50). Using the notation: T"Jo = In, we obtain a family of increasingly refined partitions, in the sense of the above order of the partitions. Moreover, J n tends, as n --+ 00,to the finest partition of X into points, and J n tends, as n --+ -00, to the most coarse partition, into one set of measure 1 and another set of measure zero. A physical prototype of a Bernoulli and a K-system is the above billiard.13i19A geometric prototype of a Bernoulli and a K-system is uniformly hyperbolic system with Sinai invariant measure." Roughly speaking, the monotonic entropy increase correspond to the process of dilation of expanding fibers.

81

Appendix A. Collision Map We shall give the formula of the collision map. We consider a particle which undergoes the first collision with the disk of center 01 with velocity V1 (pi) and the second collision with the disk of center 0 2 with velocity Vl(p2). Two cases are possible. First, we consider =crossing of the centers line as in the Fig. A . l . In this figure the angle P1P2M is a 2 - P 2 = -(a1 - Pi), where M is such that MP2 is parallel t o 0 1 0 2 . We can write

P1M = P1P2cos(p1 - cq) = d - a c o s a l - acosan. and

if we eliminate

a2

between these equations we arrive a t

Fig. A. 1. Non-crossing Collision.

i/r(PI)

n Fig. A.2. Crossing Collision.

(-4.1)

82

In crossing case which we present in Fig. A.2 we see t h a t the angle PzPlM is equal t o a2 - 0 2 = a1 - PI, and t h e length of P2M is changed to:

P2M = PlP2sin(P1 - a l ) = a s i n a l +asinaZ,

(A.4)

then, we have p2

d a

= arcsin[- sin(P1 - cq) - sinpl].

(-4.5)

To obtain p2 in the first collision between particle and obstacle Fig. A.3, we take d = OP1, p1 = 0 and a1 = 8 in t h e collision map.

Fig. A.3.

Particle obstacle Collision.

References 1. M.Kac,Probability and Related Topics in Physical Sciences, Interscience Pub,

NY, 1959. 2. N.S. Krylov, Works on the Foundation of Statistical Physics, 1950, in Russian, English trans1ation:Princeton Univ. Press, Princeton, NJ, 1979. 3. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley, New York, 1975. 4. G. Bennetin, L. Galgani, A. Giorogilli, J.M. Strelcyn, Lyapounov characteristic Exponents for smooth dynamical systems and for Hamiltonian systems; a method for all off them, Part 1 and 2, Meccanica 15 (1980) 9-30. 5. L.A. Bunimovich , Ya. G. Sin;, Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78 (1980/81), 479497. 6. L. A. Bunimovich, Ya. G. Sin;, N. I. Chernov, Markov partitions for t w e dimensional hyperbolic billiards, Russian Math. Surveys 45 (1990), 105-152. 7. N.I. Chernov, L.S. Young, Decay of correlations €or Lorentz gases and hard balls. Hard ball systems and the Lorentz gas, Encyclopaedia Math. Sci. No. 101, Springer, Berlin, 2000, pp 89-120.

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8. M. Courbage, Intrinsic Irreversibility in Kolmogorov Dynamical Systems, Physica A 122 (1983), 459. 9. M. Courbage, B. Misra, On the equivalence between Bernoulli systems and stochastic Markov processes. Physica A 104 (1980), 359-377. 10. M. Courbage, G. Nicolis, Markov evolution and H-theorem under finite coarse-graining in conservative dynamical systems, Europhysics Letters 11 (1990), 1-6. 11. M. Courbage and D. Hamdan, Chapman-Kolmogorov equation for nonMarkovian shift invariant measure, Ann. Prob. 22 (1994), 1662-1677. 12. M. Courbage, I. Prigogine, Intrinsic randomness and intrinsic irreversibility in classical dynamical systems, Proc. Natl. Acad. Sci. USA 80 (1983), 24122416. 13. G. Gallavotti, D.S. Ornstein, Billiards and Bernoulli schemes , Commun. Math.Phys. 38,(1974), 83-101. 14. P.L. Garrido, S. Goldstein, J.L. Lebowitz, Boltzmann Entropy for dense fluids not in local equilibrium, Phys. Rev. Lett. 92, (2004), 050602. 15. P. Gaspard, H. Beijeren, When do tracer particles dominate the Lyapounov spectrum? J. Stat. Phys. 314 (2002), 671-704. 16. S. Goldstein, B.Misra and M. Courbage : On Intrinsic Randomness of Dynamical Systems. J.Stat.Phys. 25, 11-126, (1981). 17. S. Goldstein, 0. Penrose, A nonequilibrium entropy for dynamical systems, J . Stat. Phys. 22 (1981), 325-343. 18. B.Misra, LPrigogine, M.Courbage, From the Deterministic Dynamics to Probabilistic Descriptions, Physica A 98 (1979), 1-26. 19. YA.G. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Russ. Math. Survey 25 (1970) 137-189. 20. YA.G. Sin;, Gibbs measures in ergodic theory, Russian Math. Surveys 27 (1972), no. 4,21-69. 21. YA.G. Sin;, Topics in Ergodic Theory Princeton University Press, Princeton 1994. 22. G.M. Zaslavsky, M.A. Edelman, Fractional kinetics: from pseudochaotic dyanamics to Maxwell’s Demon, Physica D 193 (2004), 128-147.

84

WAVE CONDENSATION SERGIO RICA

Laboratoire de Physique Statistique, ENS-CNRS, 24 rue Lhomond, F75005 Paris-Fhnce. *E-mail: rica@$s.ens. fr In this article I will review some results, in collaboration with Colm Connaughton, Christophe Josserand, Antonio Picozzi, and Yves Pomeau [Phys. Rev. Lett. 95, 263901 (2005)], on the largescale condensation of classical waves by considering the defocusing nonlinear Schrodinger equation as a represent& tive model.

Keywords: Nonlinear waves; Non linear Schrodinger equation; Wave turbulence; Kinetic equations.

1. Introduction Understanding the mechanisms responsible for self-organization processes in conservative and reversible wave systems is an arduous problem that witnessed significant interest these last years. Contrary to dissipative systems that may exhibit an irreversible evolution towards an attractor, a conservative Hamiltonian system cannot evolve towards a fully ordered state, because such evolution would imply a loss of statistical information for the system, which would violate its formal reversibility. Nevertheless, an important achievement was accomplished when Petviashvili and Yan’kov’ and Zakharov and collaborators2 reported numerical simulations performed in the framework of the focusing nonintegrable Nonlinear Schrodinger (NLS) equation. This study revealed that the Hamiltonian system would evolve, as a general rule, towards the formation of a large-scale coherent localized structure, i.e., solitary-wave, immersed in a sea of small-scale turbulent fluctuations. The solitary wave then play the role of a “statistical attractor” for the Hamiltonian system, while the small-scale fluctuations contain, in principle, all information necessary for time reversal. Importantly, the solitary-wave solution corresponds to the solution that minimizes the energy (Hamiltonian), so that the system tends to relax towards the state of

85

lowest energy, while the small-scale fluctuations compensate for the difference between the conserved energy and the energy of the coherent structure. As was initially suggested by Zakharov et a1.,2 a rigorous theoretical description of the long term evolution of the system would require a thermodynamic approach. It is only recently that statistical equilibrium models have been elaborated in the framework of statistical Remarkably it results that, whenever the Hamiltonian system is constrained by an additional integral of motion (e.g., number of particles), the increase of entropy of small-scale turbulent fluctuations requires the formation of coherent structure^,^*^ so that it is thermodynamically advantageous for the system to approach the ground state which minimizes the energy.2

Fig. 1. A time sequence of the coalescence of gas bubbles. The color scale (grayscale) represents the vapor phase by an orange (light gray) and the liquid by a violet (dark gray). The images are taken: a) at t = 36.2; b ) t = 61.1; c) t = 203.35; d) t = 634.7 units of NLS. The number of bubbles decreases in time following the law N t - l , as usually in coalescence phenomena. The final state would be a single droplet.

Similar behavior was observed in the subcritical (or quintic) nonlinear Schrodinger equation6 and in the dynamics of magnetic system in the frame of the Landau-Lifshitz e q ~ a t i o n .In ~ the former case the complex wave function satisfies

86

thus if the wave function has a low amplitude l$l < &, then a modulational instability is developed driven the system to a phase separation between two states where $ M 0 (the “gaseous” phase) arid M (the “liquid” phase). The system is finally driven by a “coarsening” to a single bubble state that minimizes the energy with a constant mass (see Fig. 1).

d m

Fig. 2. Time sequence of the coalescence of magnetic domains. The initial magnetization per surface unit is M = 0. The color scale (grayscale) represents S, M -1 by an orange (light gray) and S, i3 1 by a violet (dark gray). The images are taken: a) at t = 100; 6) t = 20000; c) t = 50000; d) t = 150000 time units .

In the later case the equation for an unitary vector differential equation:

obeys the partial

which is a Hamiltonian dynamics 8,s = 51 S [ ( V ~ S i ) ( V k S i ) + Sdx. ~ ] Moreover, the dynamics

s

and the initial magnetization M = S, dx. For an initial situation such a that S, M 1 (or S, M -1) the S, and S, components obey a focusing As in the former case non linear Schrodinger equation for $ = S, a modulational instability is developed driven the system to a phase sepa-

+

87

r a t i ~ n This . ~ behavior happens also for different initial conditions, e.g. in Fig. 2 S,= 0 and S, x 1 & S, M 0. Naturally the morphology depends on the initial magnetization. As well as in the former case a “coarsening dynamics” drives the system to a single magnetic bubble or domain. Here we shall considered the defocusing regime of the NLS equation, which is known to be also relevant for the description of weakly interacting Bose gas at finite t e m p e r a t ~ r e . As ~ >was ~ expounded independently by PomeauQ and Dyachenko et al. lo since 1992, this regime of interaction would be characterized by an irreversible evolution of the system towards a homogeneous solution (a “plane-wave” in optics terminology) with superimposed small-scale turbulent fluctuations. This scenario corroborates the general rule discussed because the homogeneous solution realizes the minimum of the energy (Hamiltonian) in the defocusing case. Accordingly, the NLS equation would exhibit a kind of “condensation” process, a salient feature that has been accurately confirmed only recently in the context of thermal Bose fields, where intensive numerical simulations of the (projected) NLS equation have been performed in 3D.7

0

2000

t

Fig. 3. Left: Evolution of the wave condensate fraction no/N in time t of a 1283 spectral simulation with periodic boundary condition. The final stage leads a 90% of wave condensation. The 3D-graphics inserted in the picture represent the iso-1Q12 surfaces for a value of 0.3 for subsequent times (from left t o right) t = 40, 120, 200, 400 and 800 time units. At the initial stage on sees that the system is dominated by linear vortices or nodes, however for time larger than 200 a vortex sate dominate the evolution. Right: Plot of the condensate fraction as a function of time for different system size. One sees that as the system size is increased the mean condensate fraction decreases.

we used a thermodynamic description, proIn Connaughton et vided by a kinetic theory of random waves: the so called ((wave turbulence” or “weak turbulence” theory, for the condensation criteria. Indeed, the classical 3D-NLS equation is shown to exhibit a genuine condensation

88

process whose thermodynamic properties are analogous t o that of of the Bose-Einstein condensation, despite the fact that condensation of bosons is inherently a quantum effect. Our analysis was based on a kinetic theory of the NLS e q ~ a t i o n , ~ >inl Owhich we introduced a frequency cut-off t o circumvent the ultraviolet catastrophe inherent t o the classical nature of a wave equation (Rayleigh-Jeans para do^).^ This allowed us to point out the essential role played by the number of modes in the dynamical formation of the condensate, so that a consistent definition of the system temperature can be determined unambiguously. Moreover, our theory revealed that in the 2D case the NLS equation does not exhibit a condensation process in the thermodynamic limit, because of an infrared divergence of the equilibrium distribution function, in complete analogy with an ideal and uniform 2D quantum Bose gas. This significant result means that the system evolves towards a state of equilibrium (maximum entropy) without generating a coherent structure that minimizes the energy (Hamiltonian), in contrast with the general rule commonly a c ~ e p t e d . ~ ~ ~ ~ ~ ~ ~ ~ ~ The present review is organized as follows. In section 2, we introduce the nonlinear Schrodinger equation as a model for “wave condensation”. In section 3, we sketch the essentials for a kinetic description for the long-time statistical behavior of random waves, that is the weak or wave turbulence theory. Next section 4 address the formation dynamics of the condensate assuming valid the assumptions for wave turbulence. We show here that the future evolution of the four wave kinetic equation realizes a finite time singularity which is a precursor of condensation. Finally, in the last section 5 we study the equilibrium resulting from the kinetic equation, for waves in the presence of a condensate in two distinct regimes the small condensate fraction and large condensate fraction. It appears that the weak turbulence theory with a condensate is in excellent agreement with the numerics of nonlinear Schrodinger equation. 2. Wave equation We shall use as an example of classical wave equation the defocusing nonlinear Schrodinger or Gross-Pitaevskii equation (NLS) in D space dimensions for the wave function Q!X

The operator A stands for the Laplace operator. This equation describes the evolution of waves interacting through the cubic collision term

89

The coefficient g being the positive for defocusing case and negative for the focusing case. We shall consider here only g = 1 for the defocusing case. The dynamics conserves the total particle number N = Jl$I2dDz, the energy H = J [ l V ~ 1 2 51$14] d D x and the linear momentum P = I m S $*V$dDx and as does its discretized version. We are questioning here the statistical equilibrium of such system of waves. The two asymptotic behaviors k + 00 (small scales) and k + 0 are of particular interest. The small scales are reached in the continuum version of the NLS equation. We are then modeling a particular case of the dramatic Rayleigh-Jeans divergences: a finite amount of energy is trying t o distribute over an infinite number of modes. It was then s u g g e ~ t e that d~~~~ the dynamics would in fact show the formation of a large coherent structure containing most of the particle number in a midst of small scale fluctuations. The system then follows a slow “effective” irreversible dynamics where the fluctuations would reach smaller and smaller scales. Briefly, one observes all the mass concentrating in the coherent structure, which is a minimizer of the Hamiltonian while the excess energy is invading the smaller scales, with a vanishing contribution to the total particle number. This particular “condensation” dynamics was observed recently for the focusing version of the NLS equation which enhance the formation of coherent structure^.^ It has been shown there that for any discretized version of the NLS equation a statistical equilibrium is reached where the energy non contained in the coherent structure is equally distributed over the fluctuating modes. On the other hand the dynamics of the large scale modes can reveal the dynamical formation of a Bose-Einstein type of condensate (BEC). To avoid the peculiar slow condensation driven by the Rayleigh-Jeans paradox, we consider further on the discretized version of NLS. The energy cannot cascade infinitely towards smaller and smaller scales, and the statistical equilibrium of3 describes an effective “thermalization” of the system, the temperature being roughly defined by the density of the excess energy between the initial energy and the energy of the minimizer. Considering the linear part of NLS only, B-E condensation should thus be observed when: D > 2 with D the spatial dimension, D = 2 describing the marginal case with logarithmic behavior.

+

3. Kinetics Theory and Bose-Einstein condensation

The longtime dynamic of random dispersive waves through a system possess a natural asymptotic closure when there is weak nonlinear interact i o n ~ . It ~ ~follows - ~ ~ that the long time dynamics is ruled by a kinetic

90

equation, similar t o the usual Boltzmann equation for dilute gases, for the distribution of spectral densities that takes account of the mode interaction through a non-vanishing collisional integral because of an “internal resonance”. Moreover, the actual kinetic equation preserves energy and momentum and an H-theorem provides an equilibrium characterized by a Rayleigh- Jeans distribution. The mathematics behind the resonant condition is formally identical to the conservation of energy and momentum in a classical or a quantum gas. Therefore, an isolated system evolves from a random initial condition to a situation of statistical equilibrium as a gas of particles does. The equation (2) could be written in Fourier space as

where

After a subtle and not easy task it is possible t o derive kinetic regime for the long-time evolution of the four wave interaction for the ensemble l k2) average of the second order cumulant ( a i l a k z )= n k I d D ) ( k -

where

A complete derivation could be found in the recent review by Newel1 et all5 As the usual Boltzmann equation, the total mass nk(t)dDk,the total momentum knk(t)dDkand the total energy k2nk(t)dDkare conserved

s

s

s

by the evolution ( 4 ) . Finally, this kinetic equation satisfies a H-Theorema: aThe entropy as defined diverges always, a better definition is S=

I

d D k l n (nk/nEq)

with n? the Rayleigh-Jeans distribution. This entropy gives, at least a zero entropy for the equilibrium distribution (5).

91

s

&S > 0 for S = d D k In n k . The equilibrium (no-flux) solution, usually named Rayleigh-Jeans distribution,

vanishes exactly Coll[niq].This is a formal solution only, because it does not yield a converging expression for the energy nor even for the total mass. Generally speaking, this kind of divergence for large frequency (or wavenumber) is removed by introducing an ultraviolet cut-off dealing with the finite number of modes present in practical numerics. In the following we shall assume that the initial condition is such a that the total initial momentum vanishes, then the equilibrium distribution is spherically symmetric and the k .w term in (5) is absent. The wave condensation arises as in the usually Bose-Einstein case for a Bose distribution if the number particle integral do not diverge in k = 0: that happens if D > 2 as can be seeing from the behavior of N / V = Co IcD-l$dk kf-' as ko + O.b For D 5 2 one has that N / V diverges for a finite energy. As the original nonlinear Schrodinger equation, in the kinetic regime the Zakharov equation preserves the total mass and energy, thus in 3 0 one hasC

s,

N/V

= 47r

ikc ikc $$ k'dk

= 4nTkc

(

1 - -arctan

kC

N

(A)) (6) J-l.

47rTk: k2dk = 3 PN

E / V = 47r

(1

-

+

+ 3 EP + 3

3

(z)3'2 arctan

.

(7)

Those equations should be understood in the following way, suppose that we know the initial mass and energy N&E then the system is driven to equilibrium by the Zakharov equation (4) to the equilibrium distribution (5) and we get equations (6,7) for the parameters there T & p , thus for a bHere C D depends on dimension D: CD = 2n for D = 2 and CD = 4n for D = 3. CIn 2D one has

N / S = 2 n l k ' k- T

dk = nTlog

(%) k2

- p

E / S = 2 7 ~k c1 7k k2 T k-2 d k = n T ( k ~ + p l o g ( ~ ) ) .

92

given pair ( N ,E ) one gets a pair (T,p ) via (6,7). As in the standard BEC for a Bose gas, one has that condensation phenomena arises if p + 0- for a finite mass (or number of particles) and energy. Explicitly, after (6) one gets

+...) for JpI<< kg, one sees that p reaches zero for a finite specific volume V / N (or T ) and one has condensation. Similar conclusion follows from the value of the energy per particle. The energy per particle is a decreasing function of p as approaches zero from the negative side. Large frequencies possesses a large (negative) chemical potential and as one decreases the energy per particle the chemical potential also decreases. There is a critical energy per particle such a that p = 0, this critical value, denoted by EBE for Bose-Einstein critical energy, is given by @= that is in three space dimensions :

y,

1

EBE = -NkZ. 3 Therefore, decreasing the energy per particle will cool the system and one reaches a finite threshold EBE below that Bose-Einstein condensation arises in the system. Besides the equilibrium solution ( 5 ) Zakharov has suggested that a Kolmogoroff-like analysis provides two other possible power law behaviors

P

pv3

nk = -

kD

where Q ( / P ) is the mass(/energy) flux in wavenumber variable per unit time. Those solutions are derived by a dimensional analysis for Q and P constant. However, it happens that the finite flux solution or KolmogorovZakharov spectra are, indeed, exact solutions of the kinetics equation.12 A remarkable property of (4). 4. Dynamics before collapse

As said, if E < E B E ,we expect condensation t o zero wavenumber, that is the spontaneous occurrence of a singularity in the solutions of (4) for k = 0 (a singularity leading t o a solution of the type n k = r ~ o & ( ~ ) ( (pk, k ) (pk

+

smooth function, see section 5 ) an interesting phenomena on its own.

93

This singularity arises also in the usual Boltzmann-Nordheim equation16 for a Bose in three spatial dimension. It is shown that equation (4) admits a selfsimilar dynamical solution which accumulates particles at zero wavenumber.17 The self-similar solution has the form (in this section we assume spherically symmetric solutions for the spectrum and we use the frequency variable w = k 2 instead of the wavenumber, k, variable):

here T is a function depending only on time. Putting (11) into (4) and imposing separation of variables depending purely on time and C = W T(t) one gets that an integro-differential equation for $(<):

(v+
(634441 + 434442 - 4142$3

- 4142$4)

(12)

>

a}

where s(

<

-

N

(t, - t ) g -+ o as t ---t t,. 5. Kinetics Theory with a condensate 5.1. Early stage At the singularity time, if our scenario of self-similar collapse holds, as seems to be confirmed by our numerical studies, the system is not yet a t equilibrium, and some exchange of mass between the condensate and the rest is necessary to reach full equilibrium, because the mass inside the singularity is still zero a t t = t,. It happens that this exchange of mass can

94

Fig. 4. Self-similar evolution17 of the distribution function nw( t )(between two neighbor curves, nw,o(t) has increased by a factor 5). Different curves in time show a clear selfsimilar evolution. One notices the build-up of a power law with an exponent -1.234 for the large frequency w-" independent on time.

be described by extending the full kinetic equation to singular distributions. As n(k = 0) diverges at t = t,, let us consider the following ansatz the distribution function for times larger than t,: nk(t) = n0(t)6(~)(k)cpk, with (Pk a smooth function, and no(t*)= 0. Introducing this ansatz into the collision integral in (4) one has after some algebra:

+

= no(t)

/

'2;3,4((Pks(Pk4 - (PkZ((Pk3

2,3,4

Here we used the notations

s,,,

+ (Pk4));

(13)

= J d 3 k 2 d 3 k 3 , and 62;3,4 = w 0 , k 2 ; k 3 , k 4=

k4)6(1)(Ici - Ic: - Ic:), and so forth. After the collapse at t , it is possible t o relate the growth of the "condensate" fraction to the nonlinear eigenvalue v before the collap~e,'~ indeed &6(D)(k2

- k3

-

95

(G)

?(t)3/2-"and q,,(t) = ?(t)-"@ with ?(t) (t - t*)*. The present scaling argument does not avoid the problem, still not achieved, to find the detailed structure of a different nonlinear eigen-problem pertinent for the post-collapse regime. Equations (13) and (14) feeds the condensate fraction until equilibrium is reached whenever all collisional integral vanish. That happens, as it could be seen directly, for the distribution function no(t)

N

Piq =

N

T

that is equilibrium distribution with zero chemical potential. The number of particles out the condensate and its energy are

ord

therefore as in standard BEC no vanishes at the critical energy EBE and no becomes the total number of particles as E + 0. This linear behavior (novs. E ) is not really compatible with the data from numerical simulation (see i Fig. 5), however should be noticed that such a equation (18) was derived for a spherically symmetric continuous distribution on waves number while in the numerics the integration in Fourier space is a cubic domain. Equation (18) should be generalized in a finite box as (19)

where

k,

=

C'

k,

are such a that -k, 5 k,, k,, k, 5 k, and exclude the origin

= k, = 0.

Those sums in (19) could be approximated by integrals only if the number of modes is very large and if they have a large occupation number. For instance, in the case of the nonlinear Schrodinger equation in three space this formula E B E = W N k : has only a sense for D

> 2.

96

Fig. 5 . Density of condensate n o / V vs. energy density E / V for a number density N / V = $. Points are numerical simulation for a 643 modes simulation with a cut-off k , = K . Plain curves are computation of integrals with same cut-off k , = K , i) is for a spherically symmetric distribution no = N - 0.3040E, ii) is estimating on the integration on a cubic domain no = N - 0.1944E as in the numerics, and iii) is the result after computing discrete sums on a cubic no = N - 0.1851E.Finally, iv) plots a numerical sum supposing that Bogoluibov spectra is the one appearing on the equilibrium distribution (see next section and Fig. 6 for details).

dimensions with a cut-off a t k, = 7r and with an infinite resolution (no inSITIT

frared cut-off) one has that for cubic domain, no = N -E*

d k z d k dk' ( k +ku+ku)

=

N - 0.19443; on the other hand, computing the exact discrete sum (19) one for a finite box of size 643one gets no = N - 0.1851E. Those behaviors together with the transition curve on a spherical domain no = N - 3 3 are plotted in Fig. 5 and compared with the given by numerics with no adjustable parameter. 5 . 2 . Late stage: The appearance of coherence and the

Bogoluibov spectra Although the theory with a condensate presented previously is in good agreement with the numerics (line iii in Fig. 5) it cannot be considered as completely satisfactory. Indeed the points from the numerics present a slight negative curvature which is incompatible with the straight line (18). Once the condensate fraction is not longer zero, the spectrum W k = k 2 changes into the Bogoluibov spectra:'O w ~ ( k = ) Jk4 2gFfLk2 an effect that could be seen in an easy way from the Hamiltonian. In the following we consider, because of simplicity (and because it makes easier a comparison

+

97

with the numerics), a discrete numeration of modes instead of a continuous one, thus +(z, t ) = C ka k ( t ) e i k ‘ = and the full Hamiltonian reads:

&

H =

c

k2aiak

k

9 +-

UilUL,Uk,Uk4b(kl

-k

k2 -

k3

-

k4) ( 2 0 )

2vk i r b , k 3 , k 4

where V is the total volume and the &function is the Kronecker discrete function, equal to zero if its argument is not zero and t o 1 otherwise. Following the principles outlined by Bogoliubov at zero temperature, the interaction part of the energy is split into pieces involving the “condensate”, i.e. the amplitudes of index zero: ao, and pieces not involving this condensate. The condensate number density appears to be po E Hence the Hamiltonian may be decomposed in five terms, depending on the way the condensate amplitude enters into those terms, however terms with three zero wavenumbers do not exist because of the &function in ( 2 0 ) , thus :

q.

where excludes the k = 0 mode. Finally, the last H4 should be regarded carefully because is not of higher order than HO nor H2, after a rapid inspection, one sees that particular terms where all four p,’s are the same or those whenever one has: p , = p,, p p = p , and p , = p , with p p = p , contributes up t o a first order. Other terms introduce correlations which are treated in a weak turbulence theory and decay rapidly to zero. The final sum in H4 reads

98

The C&laaI4 is definitively higher order and putting the other term into HO one gets for the energy up to first order 9

HO = 2~ ( N 2

+(N -no)2),

where no is by definition the total number of particles with zero momentum, i.e. no = laoI2 = poV. The kinetic equation for Hamiltonian requires to be diagonal in quadratic terms this is possible using the Bogoliubov transformation for canonical variables: a k = Ukbk

+

a; = E k b i

Vkb*k

+ 'ijkb-k

1 = IUkl2 - 1VkI2

(22)

where the third relation follows from the Poisson bracket relation { a k , a;} = i. Imposing the condition that the resulting Hamiltonian is diagonal in b i b k one has that

1

&

'uk

=

Lk

J1-ILrc(2

with

Finally, quadratic term in its diagonal form is H2 =

C ' W B ( ~with ) bibk,

~ g ( k=)

dk4+ 2gnok2.

k

Therefore, writing a nonlinear integral equation for the bk amplitudes one gets a kinetic theory similar to kinetic equations (14) but with WE(^) in the energy conservation relation instead of k 2 . The final equilibrium distribution is ( ( b i b k , ) = ( p k d ( D ) ( k- k')): rn

The mass or number of particle out condensate is directly related to ( b i b k ) via the Bogoluibov transformation, in fact

99

because of isotropy one has (bEbk) = ( b L k b - k ) and one has a t the end that

the final expression for the energy becomes

Computing numerically the sums in the following relation

+ +

+

+ +

with W B = d ( k : k$ I c , " ) ~ 2gno(k: k$ k,")in a cubic discrete box, one gets the curve iv in the figure 5, showing a great agreement with the numerics points from direct numerical simulation of the nonlinear Schrodinger equation. Moreover this agreement is also impressive for different volumes: 163, 3Z3, 643 and 12g3 units (see Fig. 6). 6. Comments and remarks

i ) Connection with Bose-Einstein condensation of weakly diluted gases. Bose-Einstein condensation of perfect quantum gases is due to the lack of mass-capacitance of the equilibrium density spectrum for low temperature. Below a critical temperature T,, a finite amount of particles accumulates in the ground state forming the so called Bose-Einstein condensate (BEC). The statistical mechanics determine the condensate fraction and the density spectrum of the excitations for given temperature and particle density. Such statistical physics describes the equilibrium state leaving open the mechanism of formation of the condensate. Starting] for instance] with a particle distribution without condensate (out of equilibrium situation)] a natural question arises: how the condensate emerges ?17 One has to consider quantum kinetic theory, that is the Boltzmann-Nordheim theory (see17 for details), t o follow the dynamics of this gas of particles. For temperature below T,, this equation exhibits a finite time singularity involving self-similar dynamics for low energies or momentum (that means wavenumbers). l7>l8 This singularity is the signature of the Bose-Einstein condensation that would begin after the singularity. In fact the Boltzmann-Nordheim theory fails before the singularity occurs when the density spectrum at low frequency is too intense. In this region an expansion of the theory is needed.

100

Fig. 6. Density of condensate no/V vs. energy density EIV for a number density N / V = The graphs a), b), c) and d) are, respectively, numerical simulations for 163, 323, 643 and l2g3 system size with a cut-off k , = T . The plain curves in a), b), c) and d) are computation of the sums (19) using the Bogoluibov spectra for 163, 323, 643 and 1283 modes in a cubic box with an ultraviolet cut-off k , = x. Note that plotted in the same graph all the four plots share essentially the same curve independently of the system size.

i.

The connection between the description of the Bose gas by the kinetic theory and the one by the NLS (or Gross-Pitaevskii) equation relies on a number of remarks. It has been emphasized several times in the literature11v12J1>22 that, by viewing the Gross-Pitaevskii equation as an equation for nonlinear waves, the kinetic wave equations for this classical field is exactly the same as the cubic part of the Boltzmann-Nordheim kinetic equation. This is not surprising because the cubic terms are dominant in the limit of the large occupation numbers, precisely the limit where the quantum fluctuations become small and where a classical field becomes a fair description of the quantum field. But this does not allow to say that the kinetic picture and the dynamics of the Gross-Pitaevskii equations are identical. The reason of this is quite obvious: for a Bose gas the condensate and the thermal particles satisfy a coupled equation very similar to the ones we have written to describe the post-blow-up dynamics (13): then the coupling term (that is the no . . term in equation (13)) plays a dominant role.

101

Without this coupling, there would be no growth of the condensed fraction. The problem represent an initial value problem with the same spectrum of fluctuations as the one given by the self-similar solution of the BoltzmannNordheim equation, one needs to take as initial spectrum the pure power w-” (w stands by energy in the context of BEC) found spectrum TI, at exactly the collapse time. But this is impossible, because this spectrum has infinite mass, because it diverges in the large energy limit. This divergence is not a problem for the Boltzmann-Nordheim kinetic equation but it makes impossible to implement an initial condition for the Gross-Pitaevskii equation with the same spectrum, since the nonlinear term in the GrossPitaevskii equation would become infinite (due to the local interaction) for an infinite mass density. Therefore the only way to get significant information on the Bose-Einstein condensate problem is by studying in detail the kinetics of a quantum gas. Considering the problem of the fluctuations in the pure Gross-Pitaevskii equation makes surely an interesting problem, but one different from the growth of a Bose-Einstein condensate. ii) Growth of Phase Coherence. A no yet satisfactorily answered question in the literature deals with the phase coherence. When the condensate does form, why it has a spatially uniform phase? It is almost obvious that no infinite range order can buildup in finite time after the occurrence of a singularity in the distribution function. This relies on the observation that, in any realistic theory, information should propagate at finite speed and after collapse the phase of the condensate is random in space. In the process of growth of a condensate the relevant information is the phase information and one expects that the correlation length of the phase increases indefinitely after collapse. It seems in the numerics (see Fig. 7) that the phase coherence appears just as is the case of pattern formation problems in situations with some symmetry (e.9. rotational invariance), the condensate would form with one phase in one region and another in another and that the two would have to come together via some kind of merging or diffusion. The initial condition for numerics considers a random wave superposition. Naturally this initial field possesses a great number of zeros or nodes of the complex wave function with a spatial distribution that probably depends on the initial spectrum. Those zeros, clearly present in a) because of the large number of lines of 27r phase jump, are, in some sense, “linear vortices” of the field and its existence do not break down the assumptions of the weak turbulence theory. However, as the condensate fraction increases many of the zeros annihilates, but some of them persist and become a

-

102

Fig. 7. u) The random phase of a typical initial condition that build-up a condensate ( E = 1 and N / S = 1) in a 642 two space dimension box, with a mesh size dx = 0.5 that is k , = 2 ~ The . time sequence b ) - f) show the complex-plane distribution, each point represents a @(I,y) at a given position (I,y), that is there are 1282 points in each graph plotted in 6) - f). One sees that the initial wave function a) and b ) is of zero average ((@) FZ 0) and the phase is uniformly distributed in (0,2n).At t = 100 units, c), the average of ([@I) is no longer zero but the phase is still almost uniformly distributed in the unitary circle, thus presumably there are a large number of zeros of @. The phase diffusion regime starts. Next sequences d) t = lo5; e ) t = 2.5 x lo5; and f) t = lo6 time units show that phase an modulus become uniform in the space.

103

-i... ,.,...--2 5 0 0 0 50000 75000 1 0 0 0 0 0 1 2 5 0 0 0 1 5 0 0 0 0 -.

,

,

~

,

,

Fig. 8. The evolution of A 4 and Ap in time. Both curves pass by with L = 64 units and D = 1.15 x

N

t

0.46e-D(2“/L)2t

“nonlinear vortex”, at this late stage the kinetic description breaks down.

A vortex dominated state has been observed in both 3D21 and 2D.22 As time goes, these vortices annihilate each other leaving a free defect zone with a more or less uniform condensate (see Fig. 3). If we define the phase and modulus standard deviations Ast, =

JS, Llp = J S with the average defined as ( f ) =

6

J , f (2)d D z , then both quantities decrease in time with a long time behavior characterized by an the exponential decaying of the slowest diffusive mode, that is A$ e-D(2TlL)2t, L the system size and D a diffusion coefficient. In fact, when the fluctuations of the phase become long ranged, their dynamics become described by the hydrodynamic limit of the perfect fluid equations. As discussed in reference,23simple scaling arguments show that, in this hydrodynamic limit, the phase becomes uniform by a diffusive like process, with an ever increasing correlation length with a power law behavior t ’ / 2 .e N

N

eIn the contex of BECs

(2)”’.

film possesses the units of a diffusion so the correlation

length

Similar scaling was previously considered by Kagan and S v i s t ~ n o v ~ ~ scales as on the basis of vortex dynamics (which follows the same scaling law than the Bernoulli equation) in a regime of “superfluid turbulence”. However in the numerics it is observed that a diffusive process is displayed in the complete absence of vortices, indicating that the mechanism proposed by Kagan and SvistunovZ4 is irrelevant.

104

iii) The end with two questions.. .

A natural question therefore arises: because the pure cubic kinetic equation displays a finite-time singularity as a precursor of a condensation, can we have a kind of singularity in the Gross-Pitaevskii or NLS equation? Signature of this singularity of the kind (11) with u = 1.234 in NLS equation is not yet accomplished satisfactorily. The major obstacle is that essentially one needs a great number of modes t o obtain a good resolution. This is feasible in the frame of equation (12) because, it is an equation for a one-dimensional field therefore one may have easily lo9 points, but we cannot expect a simulation of the NLS equation with modes anytime soon. Nevertheless, this scenario corresponds to the Boltzmann equation which derivation omits short time scales, thus the finite-time singularity should be naturally regularized in direct simulations of NLS equation. As is known, Bose-Einstein condensation does not hold in an infinite two dimensional space, thus: Does the Boltzmann equation (4) in two space

dimensions evolves to a finite time singularity? The transition rate S in Boltzmann equation (12) scales as S N w3D/2-4 in D space dimension. Therefore the Kolmogorov-Zakharov spectrum for the particle constant flux Q is n,&= Q 1 / 3 / ~ D / 2 - - 1 / 3while , for the energy flux P, one has nu = P1f3JuD/2.f Possible nonlinear eigenvalue u are such that: D/2 - 1/3 < u < D/2.19 Is known that, in an infinite two dimensional space the chemical potential p never vanishes a t equilibrium, therefore no Bose-Einstein arises formally in two space dimensions, however we do not see any objection to the existence of a solution for the nonlinear eigenvalue problem in two dimensions, indeed the previous inequality bounds u by: 2/3 < u < 1 in two space dimensions. Perhaps a singularity arises but the future evolution does not allow to feed the condensate with particles or, perhaps, simply there is no finite time singularity. This question needs more research. In conclusion, the author thanks fruitful collaboration and discussion with G. During, C. Josserand, A. Picozzi, and Y . Pomeau, he also acknowledges the Anillo de Investigacihn Act. 15 (Chile).

References 1. V. I. Petviashvili and V. V. Yankov, Rev. of Plasma Phys. 14,5 (1985). 2. V. E. Zakharov et al., Pis'ma Zh. Eksp. Teor. Fiz. 48, 79 (1988) [JETP Lett. fFor D = 2 the energy spectrum and the Rayleigh-Jeans equilibrium are the same, implying a zero energy flux. For details see.1°

105

48, 83 (1988)]; S. Dyachenko et al., Zh. Eksp. Teor. Fiz. 96, 2026 (1989) [Sov. Phys. JETP 69,1144 (1989)l. 3. R. Jordan, B. Turkington and C. L. Zirbel, Physica D 137,353 (2000); R. Jordan and C. Josserand, Phys. Rev. E 61,1527 (2000). 4. K. Rasmussen et al., Phys. Rev. Lett. 84,3740 (2000). 5. B. Rumpf and A. C. Newell, Phys. Rev. Lett. 87, 054102 (2001); Physica D 184,162 (2003). 6. C. Josserand & S. Rica, Phys. Rev. Lett. 78, 1215 (1997). 7. Yu. Kagan and B. V. Svistunov, Phys. Rev. Lett. 79, 3331 (1997); M. J. Davis, R. J. Ballagh and K. Burnett, J. Phys. B 34,4487 (2001). 8. M. J. Davis, S. A. Morgan and K. Burnett, Phys. Rev. Lett. 87, 160402 (2001); Phys. Rev. A 66,053618 (2002). 9. Y. Pomeau, Physica D 61,227 (1992). 10. S. Dyachenko, A. C. Newell, A. Pushkarev and V. E. Zakharov, Physica D 57,96 (1992). 11. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau and S. Rica, Phys. Rev. Lett. 95, 263901 (2005); see also S. Rica, “Equilibre et cinktique des systhmes d’ondes conservatifs” , Habilitation B Diriger des Recherches, Universitk de Pierre et Marie Curie, Paris VI (2007), http://tel.archivesouvertes.fr/tel-00222913/fr/. 12. V. E. Zakharov, V. S. L’vov and G. Falkovich, Kolmogorou Spectra of Turbulence I (Springer, Berlin, 1992). 13. K. Hasselmann, J. Fluid Mech. 12 481 (1962); Ibid. 15 273 (1963). 14. D.J. Benney, P.G. Saffman, Proc. Roy. SOC.London A 289,301 (1966). 15. A. Newell, S. Nazarenko and L. Biven, Physica D 152-153520 (2001). 16. L.W. Nordheim, Proc. R. SOC.London A 119,689 (1928). 17. R. Lacaze, P. Lallemand, Y. Pomeau and S. Rica, Physica D 152-153,779 (2001). 18. D. V. Semikoz and I. I. Tkachev, Phys. Rev. Lett. 74,3093 (1995); Phys. Rev. D 55,489 (1997). 19. C. Josserand, Y . Pomeau & S. Rica, J. Low Temp. Phys. 145,231 (2006). 20. N. N. Bogoliubov, Journal of Physics 11,23 (1947). 21. N. G. Berloff and B. V. Svistunov, Phys. Rev. A 66,013603 (2002). 22. V. E. Zakharov and S.V. Nazarenko, Physica 201 D,203 (2005). 23. Y. Pomeau, Phys. Scripta 67,141 (1996). 24. Yu. Kagan and B.V. Svistunov, JETP 78, 187 (1994).

106

TRANSPORT IN DETERMINISTIC RATCHETS: PERIODIC ORBIT ANALYSIS OF A TOY MODEL ROBERTO ARTUSO, LUCIA CAVALLASCA* and GIAMPAOLO CRISTADORO

Dipartimento di Fisica e Matematica and I.N.F.N., Sezione di Milano, Universitci degli Studi dell’lnsubria V i a Valleggio 11, 22100 Como, Italy *E-mail: 1ucia.cavallascaOuninsubria.it We consider a toy model of ratchet behavior, the so-called Parrondo games. We build up a deterministic version of the game and show that periodic orbit theory is able to detail its quantitative features thoroughly.

Keywords: Ratchets; Cycle expansions; Deterministic transport.

1. Introduction Deterministic transport represents one of the most remarkable features of chaotic dynamics for a number of reasons: from a conceptual point of view it shows how unstable motion is able to sustain stochastic features, while from an experimental perspective its role is crucial in several contexts: from plasma confinement (see for instance Ref. l),to collimation in particle accelerators2 , to the physics of mesoscopic devices3 . While the typical transport feature of interest in the former examples is (normal or anomalous) diffusion, recently a growing interest concentrates on “unexpected” currents, in the context of ratchet effects4, where directed transport arises in a counterintuitive direction, possibly even against an imposed bias. This field is of paramount importance in a number of context: from the conceptual foundations of thermodynamics, since the classical Feynman argument5 , to the understanding of molecular motors. We here consider a toy model of ratchet behavior, based on the (apparently) paradoxical Parrondo games (for an extensive review see Ref. 6), which will be briefly surveyed in section 2. Then the model will be reformulated as a deterministic one-dimensional chaotic systems. In section 3 we will provide analytic results on the model, by using periodic orbit expansions7 . Section 4 provides an alternative, probabilistic technique, to obtain (coincident) analytic results. Finally we comment on possible directions for further research in section 5 .

107

2. Parrondo games and their deterministic version The simplest formulation of Parrondo games is as follows: we start from two simple games A and B: A is a simple coin tossing game with winning probability p and losing probability 1 - p : each time the game is played the player toss the coin: upon winning the capital (an integer value, which we denote by X )is increased by one unit, otherwise it is decreased by the same amount. Game l3 is more involved, as it requires two “coins”, chosen according to the present value of the capital: if X = 1 m o d M winning/losing probability is p , / l - p , , while in the opposite case (X # 1 m o d M ) the winning/losing probability is p 2 / l - p2: M is a fixed integer. Now suppose that at any integer time step n (starting from zero) a game is played: X ( n ) with represent the instantaneous value of the capital. It turns out that a fine tuning of the parameters leads to a paradoxical behavior: namely take M = 3, p = 1/2 - E , p l = 1/10 - E , p2 = 3/4 - 6 , for a sufficiently small value of E : both games are in this case slightly unfair: if the player keeps on playing A or B the capital will drop down linearly, while playing A or B in random order (for instance with probability 1/2) results in a winning strategy, where on the average the capital increases linearly with time! The paradox can be explained by a Markov chain analysis8 . The deterministic version of the gamesg is obtained in terms of periodic chains of one-dimensional piecewise linear maps: for the first game A this is quite simple: we first define the map on the unit interval in the following way:

and then extend it on the whole real line by the symmetry property

fd(x + n) = fd(x) + 12

nEZ

(2)

The mapping corresponding to the game B requires a fundamental cell of size M : in the reference case of M = 3 the map is written as (see fig. (1)):

108

and extended on the real line like in eq. (2), keeping in mind that now the translation unit is of length three. In view of the technique that will

Fig. 1. The map fa on the starting cell [0,3]

be employed in the next section, together with the maps defined on the real line it is convenient to consider also the corresponding torus map: for instance if we consider (3): we may associate to it the map f ~where , the definition is taken mod 3, see fig. (2). Of course, if we follow dynamics on the torus we don’t keep track of the capital value (and no gambler would accept his capital reduced on a torus topology), but we can recover this information by assigning a label C J ~to each branch of the map: CJ = +1 for a winning branch and CJ = -1 for a losing one. Once we want to consider a random combination of the two maps

109

0

P I 1 l-pl Fig. 2.

The map

'"p,

2

3

f~ on the torus [0,3).

(games), the straightest path is to build a combined map, with the appropriate transition probabilities: if we denote by y the probability of playing game A at each step, then, once we define = YP

+ (1 - YIP1

(4)

42 = YP

+ (1 - Y)P2

(5)

41

and

we have

x E [OI4l)

x

E [4111)

+

x E [I,1 42) 2

3

-

E

[I +42,2)

x E [2,2 + 42) z E [2

+ 421 3).

with corresponding torus map fd*a shown in fig. (3). In fig. (3) we have also attached a symbol q = 1,.. . ,6 to each branch of the map: a torus orbit fully reconstructs an orbit of the lift fd*a once we assign winning or

110

Fig. 3. The torus map fd*n for the combined game on the torus [ 0 , 3 ) .

loosing indices to each branch: (TI = 0 3 = cr5 = $1 (winning intervals) and 0 2 = 0 4 = (Tg = -1 (loosing intervals). The map is of Markov type, namely if we denote by 1, the support of branch 7 each Z, is mapped onto the union of (two) other Z,l: symbolic itineraries are generated by the Markov graph of fig. (4). Due to Markov property (and the fact that the map is

Fig. 4. The Markov graph associated to the map fd*a

111

linear in each Zv), analytic results for transport properties of the map can be derived, as we show in the next section.

3. Periodic orbit theory of deterministic Parrondo games We are interested in the evolution of the capital once we play the deterministic game: if the symbolic code of the orbit (on the torus) is 771, 772, . . . , vn, the gain (or loss) is given by n.

and thus statistical properties are provided by the generating function

G'n(P)

= (, W ( n ) - X ( O ) ) ) ,

(8)

where the average is over a set of initial conditions (for instance uniformly distributed on the torus). The average on the rhs of (8) formally recalls a partition function sum: and as in statistical mechanics of lattice systems, we may introduce a transfer operator, whose leading eigenvalue will dominate the asymptotic (large n) behaviorlOill . More precisely we define a generalized transfer operator as

where a(y)

= aVj if

y E ZVj:then

where pin is the density of initial conditions, and X(p) is the leading eigenvalue of the operator (9). Thus once we know X(p) we may extracts moments (or cumulants) of the distribution, by Taylor expanding with respect to p. Periodic orbits come into play as building blocks of the dynamical zeta function c p ( z ) , whose zero closest t o the origin is exactly the inverse of X(p): we refer the reader t o Refs. 7,12,13 for a detailed proof of that, and just recall the definition of the function

where the product is over all periodic orbits p of f~*a. Each orbit (of prime period n p ) , may be uniquely labelled by its symbolic sequence

112

Vl, 7721

’

, vnp and contributes to the weights in (11) through nn

j=l

and

j=l

where Aqj is the slope of fa*a in the vj brancha. For a generic system converting (11) to a power series yields a perturbative scheme to compute the smallest zero z ( p ) = A@)-’: in the present case, due to piecewise linearity and Markov property of the map we are ableg t o compute exactly < p ( z ) , which is a polynomial (whose contributions come from non-intersecting loops of the Markov graph of fig. (4))

-

14 % %

135 264 1364 1425 2635 135264 136425 142635.

(14)

In terms of (4,s) we finally get

Cdz) = 1 - z 2 (41(1 - q 2 ) + Q Z ( 1

+

- Q1)

+ 42(1 - q 2 ) )

-z3 (e3Dqlq,2 e-3P(1 - q1)(1-

42)’)

(15)

By Taylor expanding (10) and implicit function derivation, we obtain expressions for the current and the diffusion constant in terms of the zeta function as follows:

and

D

=

21 (X”(0)

-

x’(o)2) .

(17)

where the first derivative is given by (16), while

remark that instabilities depend only on symbolic labels in the present case due t o linearity of the map in each branch.

113

So once we fix the game rules we obtain expressions for the transport indicators: for instance if we fix the original Parrondo values (19) we can express the current V as a function of both

%(f)

=

+

+

E

(small) and y as

+ +

6 ( - 8 0 ~ ~ 8(1 - y)c2 - (11(2 - y)y 4 9 ) ~ 2(1 - y)(2 - y)y) 2 4 0~ 16(1~ 7 ) ~ 11(2 - 7 ) ~169

+

(20) this proves the ratchet behavior, see figures (5). We may also devise optimal values by looking at current and diffusion as functions of both E and y,see figs (6,7). We point out that in principle we could have adopted a different strategy: starting from the transfer operators of the pair of maps f~ and f ~ and then considering the linear combination (weighted by y) of such operators: this approach has been considered in the realm of zeta function

-0.02

0

0.005

0.01

0.015

0.02

E Fig. 5 . The current V as a function of bias parameter E for small positive E (y = 1/2): for E < 0.0131 the current opposes to the bias, i . e . single games are unfair, yet their random combination results in a winning strategy.

,

114

Fig. 6.

T h e current V as a function of e and y (0 < E

< 0.0131, 0 < y < 1).

Fig. 7. The diffusion constant D as a function of e and y (0 < e

< 0.0131,0 < y < 1).

115

techniques,14 but the corresponding formalism is considerably harder to handle. 4. Periodic hopping framework

Transport coefficients may be derived with other methods too: we briefly sketch an alternative procedure15 , based on an approach to systems with space-periodic hopping, studied in Refs. 16,17. We start by introducing a lattice (sites will be denoted n: for clarity we denote (discrete) time by t ) where each site label is associated to the map fA*B on the interval [n- 1,n]. The probability of winning or loosing (notice that no outcome may result in the capital remaining constant upon playing a single game), is viewed as a hopping probability qn ( n H n 1, winning game) or pn ( n H n - 1, loosing game: so for the probability of being at site n on time t we may write a (discrete) master equation in the following way:

+

Pn(t + 1) = Qn+lPn+l(t)+Pn-lPn-l(t).

(21)

The most remarkable property of the hopping coefficients for the combined game is that they are periodic, namely Pn = 1 - qn = P n f M = 1 - 4 n - M ,

(22)

where M = 3 in the example we have discussed explicitly. The idea is t o associate an M sites model to the original one, by taking into account periodicity of the hopping (like in Floquet theory): we define

and

c 00

Sn(t) =

k=-

+

( n Mk)Pn+Mk(t),

(24)

oc)

where both gi,(t)and gn(t) again have periodicity M . Thus the idea is to write down equations for these quantities (defined on a ring of M sites), and find their long time limits, that are then employed to estimate both the current V = lim(X(t)-X(O))/t and the diffusion constant D = lim((X(t)X(0))2/(2t). By following closely Ref. 17 we get the following results

116

and

and

In particular, for M = 3 we may express our result in terms of 41, 42 defined by (4,5):

+---]1 - 42 1

42

"

I+- 1 - 4 1 42 41

r2=-

+-I

42 41 1-411-42 41 42

41

42

and 1+-1 - 4 2 42 l [ 41 u2=l + -1 - 4 2

u1=-

u3=-

42 l [ 42 l + -1 - 4 1

7

41

42

+-I

I-+

42

1-421-41 41 42 1-421-qz

+--1;

42 41 1-411-42 42 42

while transport coefficients are written as

and

-

41

117

where

d= Then we may use (31) to compute V as in fig. (5), getting exactly the same results.

5. Conclusions and perspectives In this contribution we have considered a very simple model of ratchet behavior, the so-called Parrondo games: once recast in a deterministic framework, we showed how the model can be solved by using periodic orbit theory. The advantage of this technique, with respect to other methods by which such a model can be treated, consists in providing a perturbative scheme by which extension of the model can be treated: for instance if we keep the Markov property, but relax the piecewise linearity requirement, the model cannot be solved exactly, and indeed the dynamical zeta function is a full power series. Yet, under some distorsion hypotheses, successive polynomial truncations provide exponentially converging estimates of the leading eigenvalue, so an efficient perturbative approach to nonlinear Parrondo games may be realized. By using methods devised for intermittent systemsl8?lgwe plan to study in the future “weakly-chaotic” games, where in general anomalous transport properties are expected.

Acknowledgments This work has been partially supported by MIUR-PRIN 2005 projects Transport properties of classical a n d q u a n t u m s y s t e m s and Q u a n t u m c o m p u t a t i o n with trapped particle arrays, neutral a n d charged.

References 1. Y . Elskens and D.F. Escande, Microscopic dynamics of plasmas and chaos, (Institute of Physics, Bristol, 2003). 2. B.V. Chirikov, Phys. Rep. 52, 265 (1979). 3. H.-J. Stockmann, Quantum Chaos, An Introduction, (Cambridge University Press, Cambridge, 2000). 4. P. Reimann, Phys.Rep. 361,57 (2002). 5 . R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, vol. 1, Chapter 46, (Addison-Wesley, Reading, MA, 1963). 6. G.P. Harmer and D. Abbot, Fluctuation and Noise Lett., 2 , R71 (2002).

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7. P. CvitanoviC, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quantum, ChaosBook .org (Niels Bohr Institute, Copenhagen 2005). 8. G.P. Harmer, D. Abbott, P.G. Taylor and J.M.R. Parrondo, Chaos 11,705 (2001). 9. R. Artuso, L. Cavallasca and G. Cristadoro, J.Phys. A 39, 1285 (2006). 10. R. Artuso, Phys.Lett. A 160,528 (1990). 11. P. CvitanoviC, J.-P. Eckmann and P. Gaspard, Chaos, Solitons and Fkactals 6, 113 (1995). 12. R. Artuso, E. Aurell and P. CvitanoviC, Nonlinearity 3, 326 (1990). 13. R. Artuso, in Lecture Notes in Physics ~01.618,p.145 (Springer, Berlin, 2003). 14. Yu. Dabaghian, Phys.Rev. E 63,046209 (2001). 15. L. Cavallasca, Giochi d i Parrondo e trasporto caotico, Laurea Thesis, Universit& degli Studi dell’Insubria, 2004. 16. B. Derrida and Y. Pomeau, Phys.Rev.Lett. 48, 627 (1982). 17. B. Derrida, J.Stat.Phys. 31,433 (1983). 18. R. Artuso, P. CvitanoviC and G. Tanner, Prog.Theor.Phys.Supp1. 150, 1 (2003). 19. R. Artuso and G. Cristadoro, Phys.Rev.Lett. 90, 244101 (2003).

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SEPARATRIX CHAOS: NEW APPROACH TO THE THEORETICAL TREATMENT S. M. SOSKIN Institute of Semiconductor Physics, Pr. Nauki 45, Kiev, 03028, Ukraine E-mail: smsoskinOg.com.ua Abdw Salam ICTP, Stmda Costiera 11, Weste, 34100, Italy E-mail: ssoskin9ictp.it R. MANNELLA Dipartimento di Fisica, Universitci di Pisa, Largo Pontecorvo 3, Pisa, 56127, Italy E-mail: [email protected] 0. M. YEVTUSHENKO Physics Department, Ludwig-Maximilians- Universitat Miinchen Munchen, 0-80333, Germany Abdus Salam ICTP, Stmda Costiera 11, m e s t e , 34100, Italy E-mail: bomOictp.it We develop a new approach to the theoretical treatment of the separatrix chaos, using a special analysis of the separatrix map. The approach allows us to describe boundaries of the separatrix chaotic layer in the Poincarb section and transport within the layer. We show that the maximum which the width of the layer in energy takes as the perturbation frequency varies is much larger than the perturbation amplitude, in contrast to predictions by earlier theories suggesting that the maximum width is of the order of the amplitude. The approach has also allowed us to develop the self-consistent theory of the earlier discovered (PRL 90, 174101 (2003)) drastic facilitation of the onset of global chaos between adjacent separatrices. Simulations agree with the theory. Keywords: Hamiltonian chaos, separatrix map, nonlinear resonance.

120

1. Introduction Even a weak perturbation of an integrable system possessing a separatrix results in the onset of chaotic motion inside a layer'" which we shall further call as the separatrix chaotic layer (SL). The separatrix chaos plays a fundamental role for the Hamiltonian chaos, being also relevant to various application^.^-^ The boundaries of the SL in the Poincar6 section may be easily found n ~ m e r i c a l l yHowever, .~ it is also important, both from the theoretical and practical points of view, to be able to theoretically calculate them and describe transport within the SL. One of the most powerful theoretical tools for the SL study is the separatrix map (SM), introduced in6 for the nearly integrable systems with the 3/2 degrees of freedom and called sometimes5 as the Zaslavsky separatrix map. It may also be generalized for systems with more degrees of freedom and for strongly non-integrable systems (see5 for the most recent major review). We shall further consider the case of the 3/2 degrees of freedom but the generalization of our method for other cases may be done too. One of the most interesting for physical applications relevant quantities is the width of the SL in e n e r g ~ . l - ~ > ~T-hlere l are various heuristic criterial4 based on the separatrix map and various conjectures. The width by these criteria does not depend on the angle and, as a function of a perturbation frequency w f , possesses a maximum a t wf of the order of the eigenfrequency in the stable state wo while the maximum itself is of the order of the perturbation amplitude h. However, the worklo has demonstrated in simulations for double-separatrix systems that the maximum width may be much larger as the SL absorbs one or two nonlinear resonances. The recent work" has proved this, developing a new method for the analysis of the separatrix map. The method is of a general validity, as shown in the present work. We show that the maximum width occurs a t the frequency which is typically smaller than wo by the logarithmic factor ln(l/h) while the maximum width is typically much larger than h - either by a numerical factor or by the logarithmic factor (apart from the adiabatic divergence in certain class of systems7). Besides, the method allows to describe major statistical properties of transport within the SL. Note that there were various mathematical works considering the SL in rather different contexts (see5 for the review). In particular, they analyzed the SL width in normal coordinates. However, to the best of our knowledge, these works do not specify the relation between the normal coordinates and variables conventional in physics (e.g. energy-angle or coordinatemomentum). Besides, these works just estimate the width from above and

121

below while our method allows to carry out an accurate calculation of the width in energy and, moreover, of the SL boundaries in the Poincari: section. Finally and most importantly, the methods described in5 do not make a resolution between the resonance frequency range and other frequency ranges while our method shows that the SLs in these ranges drastically differ from each other. Below, we describe the basic ideas of our method (Sec. 2), review the results of its application to the double-separatrix case (Sec. 3) and present rough estimates for the single-separatrix case (Sec. 4).

2. Basic ideas

Consider any 1D Hamiltonian system possessing at least one separatrix. Let us add a weak time-periodic perturbation,

The motion near any of the separatrices may be approximated by the separatrix map (SM).1-6J1 The map slightly differs for different types of separatrix. Our method applies to all types but, to be concrete, we consider in this section only the separatrix with a single saddle and two loops (like in a double-well potential system). Then the SM reads as" (cf. also'-6):

where E, is the separatrix energy. The quantity E is often called as the Melnikov' or Poin~arBMelnikov~ integral. The quantity S = hlcl is sometimes called separatrix split.3 For the sake of simplicity, let absolute values of all < parameters of HO and of V be 1. Then 161 1 too, if wf 1. Consider two most general ideas of the method. 1. The SM trajectory that includes any state with E = E, is chaotic since the angle of this state is not correlated with the angle of the state at the preceding step of the map, due to the divergence of w-'(E + E,).

-

-

-

122

2. The frequency of eigenoscillation as a function of energy is proportional to the reciprocal of the logarithmic factor:

w(E) =

axwo ln(AH/IE - &I)’

IE - E,I CK A H

N

E,

a=

- EjZ)

3 - sign(E - E,) 2

1

(3)

E, -EL:)

are energies of the stable states). Given that the argument of the logarithm is large in the relevant range of E , the function w ( E ) is nearly constant at a rather significant variation of the argument. Therefore, as the SM maps the state (Eo = E,,cpo) onto the state with E = El = E, aohesin(cpo), the value of w ( E ) for the given sign(aocsin(cp0)) is nearly the same for most of angles cpo (except the close vicinity of x multiples), namely

+

w ( E ) x ws*)

= w(Es f h)

for sign(aocsin(cp0)) = f l .

(4)

Moreover, even if the deviation of the trajectory of the SM from the separatrix further increases/decreases, w ( E )remains close to us*) provided the deviation is not too large/small, namely if I ln((E-E,)/h)l << ln(AH/h). If < wf then the evolution of the map (2) may be regular-like for a long time until the energy returns back to the close vicinity of the separatrix, where the correlation of angle is lost again. Such a behavior is especially pronounced if the perturbation frequency is close to us’)or us-) or to one of their multiples of a not too high order: the resonance between the perturbation and eigenoscillation gives rise to an accumulation of the energy gain for many steps of the SM, which results in the deviation of E from E , greatly exceeding the separatrix split 6 N h. As a function of w f , the largest (along the SL boundary) deviation from the separatrix takes its maximum at the frequencies slightly exceeding w$+) and ws-), for the upper and lower boundaries of the SL respectively. This corresponds to the absorption of the nonlinear resonance by the SL. The description of the regular-like parts of the chaotic trajectory in the case close to the resonance may be done within the resonance approximation. The explicit matching between the SM and the resonance approximation is carried out in.ll The resonance approximation is done in terms of slow-variables, action I = I(E) (note that d I / d E = w - l ( E ) ) and slow angle @ = @ - w f t = II, - cp, by means of the resonance Hamiltonian N

d*),

fi(1,,$).1-4,9-11

123

-ff

0

mge

ii

x

0

-x

ange ij

Fig. 1 . Schematic illustrations to the formation of the SL boundary. The separatrix energy E, corresponds to the lower boundary of the box. The GSS curve is shown by the dashed line. Solid lines show examples of resonant trajectories (RTs) overlapping the GSS curve. The SL boundary (thick solid line) is formed by: (a) the RT tangent to the GSS curve, or (b) the upper part of the self-intersecting RT (resonant separatrix).

Fig. 1 schematically illustrates the formation of the upper boundary of the SL in the Poincark section presented in the E - 1?, variables. The chaotic trajectory jumps from the separatrix onto the curve which we call the upper generalized separatrix split ( G S S ) curve, within an even T interval:

E&L($) = E,

+ SI sin(q)I,

4 E [T + 27m, 2~ + 2~7x1,

S E hiel, n

= 0, *I,

(5)

f 2 , ...

The GSS curve relates to the SM equation for E with Eo = E,. The relevance of just the even T intervals of 1?, is a consequence of a necessarily positive sign of El - Eo as far as we consider the upper boundary of the SL. Then the system follows the trajectory of the resonance Hamiltonian H ( E necessarily increases initially because $ ( O ) is within an even 7r interval), until it again reaches the GSS curve (necessarily in an odd T interval, where E decreases). After that, the system jumps onto the separatrix, where the angle correlation is lost. There is a statistical distribution of regular-like parts of the chaotic trajectory corresponding to the homogeneous distribution of initial angles. The boundary of the SL is formed by the trajectory of the resonance Hamiltonian (RT) starting from the GSS curve at such initial angle that provides for the deviation of the RT from the separatrix to be larger than that for any other initial angle. From the topological point of view, there may be two different situations: either the relevant RT is tangent to the GSS curve (Fig. l(a)) or it intersects it while being a selfintersecting trajectory i.e. a separatrix of the resonance Hamiltonian (Fig.

124

l(b)). In the latter case, the outer boundary is formed by the outer part of the separatrix (in Fig. l(b), relating to the upper boundary, this part lies above the saddle s; an example of the boundary formed by the separatrix of a different type is shown in Fig. 4(a)). The boundary strongly depends on the angle. The maximal/minimal deviation of E from E, is much larger/smaller than 6.

;::m .................

EtY

! ! L

- __

60.2

0.

...... L

.L.... ....... ~

,

........~...............~...~

0

-I

I

Cwrdinate q

2"

I

0.2

0.4

Energy E

0.S

Fig. 2. The potential U ( q ) = (0.2 - sin(q))'/2, the separatrices and the eigenfrequency w ( E ) of the unperturbed system HO = p2/2 V ( q ) ,in (a), (b) and (c), respectively.

+

7e-03 6 e - 0 3 m 5e-03 039

7e-02 5e-02

041

Fig. 3. (a). The diagram indicating (shading) the perturbation parameters range for which global chaos exists in the perturbed system, H = Ho(p, q ) - hqcos(wft). (b). The comparison of the major minimum of the diagram with the lowest-order theory (dashed lines) and the theory allowing for higher-order corrections (solid lines).

3. Application to the double-separatrix chaos

It has been found in" that, if the unperturbed Hamiltonian HO possesses more than one separatrix (cf. Fig. 2) while the perturbation is time-periodic, the onset of global chaos in between the adjacent separatrices possesses a remarkable feature: it is greatly facilitated if the perturbation frequency wf is close to certain frequencies: the perturbation amplitude h required for the chaos onset is much smaller for such frequencies than for others (Fig. 3). This is related to the characteristic shape of w ( E ) in between the separatrices: w ( E ) approaches a rectangular form in the asymptotic limit of

125

a small separation between the separatrices (cf. Fig. 2(c)). If w f is slightly smaller than the local maximum wm of w ( E ) ,then there are two nonlinear resonances that are very wide in energy: they may simultaneously overlap each other and the separatrix chaotic layers, that occurs at the value of h which is logarithmically smaller than a typical value required for the chaos onset when w f lies beyond the immediate vicinity of wm. Using the semi-heuristic approach, the lowest-order asymptotic theory based on the resonance Hamiltonian analysis was developed inlo for the minima of the global chaos boundary h,,(wf). The theory was compared to results of computer simulations for the given example (for a moderately small separation: Fig. 2). The value of w f in the minimum was well described by the lowest-order formula but the discrepancy for h,, in the minimum was nearly 50%. Besides, it was unclear how the overlap of resonances with the chaotic layers occurred and why even a small excess of h over h,, resulted in the onset of chaos in a large area of the Poincar6 section despite that chaotic layers associated with the nonlinear resonances were exponentially h,,. These problems have been resolved in our recent workll narrow for h I developing the method similar to that described in the previous section. The agreement between the theory and simulations has greatly improved (Fig. 3(b)). Our present work generalizes the above method for any separutrix (see Sec. 2 above and Sec. 4 below). The general validity of the method is brightly demonstrated by Fig. 4, that shows the direct comparison of the

12

I

I

I 08

06 04

Y

4

Fig. 4. The separatrix chaotic layers (shaded) in the plane of action I and slow angle for the system exploited in Fig. 3, for h = 0.003 and ~f = 0.401, as described by our theory. The dashed lines represent the relevant GSS curves. The solid lines represent relevant trajectories of the resonance Hamiltonian H : two solid lines with the saddles represent separatrices of the nonlinear resonances, two other solid lines are the trajectories of H which are tangent to the lower and upper GSS curves respectively. (b). Comparison of the layers obtained from computer simulations (dots) with the theoretically calculated boundaries (solid lines) shown in the box (a).

126

theoretical SL boundaries and the SLs generated by computer. Though the layers are still related to the system with two separatrices, the given perturbation amplitude is so small that the layers are well separated from each other and therefore the presence of a second separatrix does not play a significant role for any of the layers. The lower and upper layers demonstrate two characteristic situations: the lower layer has not yet absorbed the relevant nonlinear resonance (though the closeness to the resonance gives rise to a rather strong increase of the layer width) while the upper layer has absorbed the resonance so that its maximal width greatly exceeds that of the lower layer and there is a large island of stability in the layer. 4. Single-separatrix layer: estimates of the largest width

As already mentioned, the SL width in energy takes its maximum at wf

M

us+'-'. The rigorous treatment for the single-separatrix case may be done

similar to the double-separatrix case." It will be done elsewhere while, here, we present rough estimates for the width. Let us transform from p - q to action I = I ( E ) and angle $J and expand V in Eq. (1) into the double Fourier series (in t and +):I

Let us single out the term with the maximum absolute value of V ~ J (typically, it corresponds to Ic = 1 = 1) and denote it as VOeZB:

The maximum width of the SL corresponds to the perturbation frequency at which the SL has just absorbed the widest nonlinear resonance (cf.ll). From the rigorous results for the double-separatrix case,ll we may assume that the SL width is dominated by the width of the nonlinear resonance then, i.e. the width of the resonance separatrix in energy AE greatly exceeds the minimal separation in energy between the resonance separatrix and E,. Obviously, this assumption should be verified in the end. Strictly speaking, VOstrongly depends on I in the relevant range of I , end the rigorous treatment of the nonlinear resonance is complicated (cf.'-''). But for a rough estimate of the resonance width, it is sufficient to use a simple Chirikov approximation of the resonance Hamiltonian

127

H(I, which reduces to the auxiliary pendulum dynamics. The width of the corresponding resonance separatrix in energy is expressed as 4),1-4?g2'2

8hVowj Idw/dEl' The Chirkov approximation assumes that Jdw/dEJM const within the range of energies relevant to the resonance separatrix. In our case, it is not so since the quantity Idw/dEl x wy/(a7rwolE - E,I) strongly varies within the relevant range of energies. However, we may still use Eq. (8) for the rough estimate, putting in it (E- E,( A E . Then we obtain for AE the following rough asymptotic equation

-

AE

N

Vo(E = E, f A E ) h l n ( l / h ) ,

h

--t

0,

(9)

where we took into account that the relevant w j is close to w:'/-) N wo/ ln(l/h) and omitted numerical factors. The asymptotic solution of Eq. (9) essentially depends on the function Vo(x).In this context, all perturbed systems may be divided in two classes. 1. The separatrix of the unperturbed system has more than one saddle while the relevant coefficient % E G(E,$) in the Fourier expansion (in time) of the perturbation V possesses different values in adjacent saddles. An archetypal example is a pendulum subject to a dipole time-periodic pert u r b a t i o n . ' ~If~ E is close to E,, then the system stays mostly near one of the saddles, so that fi depends on $ nearly in a piece-wise manner: it oscillates between the values corresponding to the adjacent saddles. Therefore, VO (which is the absolute value of the relevant coefficient in the Fourier expansion of % in $) approaches in the asymptotic limit h + 0 some non-zero constant. As follows from Eq. (8), A E is logarithmically large:

AE

-

h ln(l/h) >> h,

h

-+

0.

(10)

This estimate agrees with the rigorous result and the result of simulations in the case considered in.ll 2. Either the separatrix has a single saddle (like for a double-well potential ~ y s t e m ~or) ~ the ) separatrix has more than one saddle while the perturbation possesses identically equal values a t different saddles. Archetypal examples are a pendulum in a wave with the same spatial period'-4 and a pendulum with the oscillating suspension point.5 Then Q(E -+ E,,$)

128

keeps nearly one and the same value for most of the period of 1c, (as it stays most of the period near the saddle/s): it significantly differs from this value only during a small part of the period, which is w(E)/wo. Hence, Vo(E = E, f A E ) N l / l n ( l / A E ) ,so that the solution of Eq. (9) is:

-

This means that the asymptotic functional dependence of the resonance width is the same as that of the SL width in frequency ranges beyond the resonance. So, the functional dependence of the SL in the resonance range is of the same type as beyond it, being 0; h. At the same time, a ratio between the resonance width and h may still be a large number. Both these conclusions are in agreement with computer simulations. Thus, for the archetypal example of the Duffing oscillator subject to the dipole timeperiodic p e r t u r b a t i ~ n the , ~ ~ratio ~ A E / h approaches in the limit h + 0 the constant value approximately equal to 20: see Fig. 3(b) in.8 For another archetypal example, namely a pendulum in the wave of the same spatial our recent simulations for the parameters exploited in1l3i4 have shown that A E l h const M 50.

hso

References 1. G.M. Zaslavsky, R.D. Sagdeev, D.A. Usikov and A.A. Chernikov, Weak Chaos and Quasi-Regular Patterns, Cambridge University Press, Cambridge, 1991. 2. A.J. Lichtenberg and M.A. Liebermann, Regular and Stochastic Motion, Springer, New York, 1992. 3. G.M. Zaslavsky, Physics of Chaos in Hamiltonian systems, Imperial College Press, London, 1998. 4. G.M. Zaslavsky, Hamiltonian Chaos and Fkactional Dynamics, Oxford University Press, Oxford, 2005. 5 . G. N. Piftankin, D.V. Treshev, Russian Math. Surveys 62,219 (2007). 6. G.M. Zaslavsky and N.N. Filonenko, Sov. Phys. JETP 27,851 (1968). 7. S.M. Soskin, O.M. Yevtushenko, R. Mannella, Phys. Rev. Lett. 95,224101 (2005). 8. S.M. Soskin, R. Mannella, M. Array& and A.N. Silchenko, Phys. Rev. E 63, 051111 (2001). 9. S.M. Soskin, R. Mannella and P.V.E. McClintock, Phys. Rep. 373, 247 (2003). 10. S.M. Soskin, O.M. Yevtushenko, and R. Mannella, Phys. Rev. Lett. 90, 174101 (2003). 11. S.M. Soskin, O.M. Yevtushenko, R. Mannella, arXiv:nlin/0612034. 12. B.V. Chirikov, Phys. Rep. 52,263 (1979).

129

GIANT ACCELERATION IN WEAKLY-PERTURBED SPACE-PERIODIC HAMILTONIAN SYSTEMS M. YU. ULEYSKY and D. V. MAKAROV* Laboratory of Nonlinear Dynamical Systems, V.I. I1 ‘ichev Pacific Oceanological Institute of the Far Eastern Branch of Russian Academy of Sciences, Vladivostok, Russia *E-mail: [email protected] http://dynalab.poi. dvo.ru Motion of a space-periodic Hamiltonian system with weak wavelike perturbation imposed is studied. We show that slow variation of the perturbation wavenumber may lead in giant explosion-like acceleration of some particles. This effect arises due to forming specific resonant channels in phase space. Giant acceleration can coexist with the ratchet phenomenon, i. e. the resulting particle current can have preferable direction. Keywords: Acceleration,Hamiltonian systems,chaos,resonance,ratchet

1. Introduction Physical systems performing large response to small external perturbations permanently attract much attention. Such effects are inherent in essentially non-equillibrium systems, usually with many degrees of freedom, when even weak influence is sufficient for qualitative changes in system’s dynamics. In the present paper we show that such phenomenon can be observed in a simple weakly-perturbed Hamiltonian system with 3/2 degrees of freedom. It should be emphasized that in our case large response coexists with the ratchet phenomenon, i.e. occurrence of directed particle current in the absence of a biased force.14 We start with considering an ensemble of unit-mass non-interacting point particles. The Hamiltonian of single particle has the form

where q!~

=

H = -P2 -COSX+ECOS~, 2 k ( p t ) z + vt, p , < ~< 1. Modulation of wavenumber k infers

130

changing of the orientation of an external perturbation. The equations of motion are the following dx -=p, d~ = -sinz+Eksinqi dt dt Assume that the parameters k and v have sufficiently large positive values, so that the phase 4 is a rapidly-varying quantity, except for resonant regions, where -d 4_- - x dk +kp+v-.0. (3) dt dt Thus the system considered has three timescales of dynamics: timescale of wavenumber variations (of order p - ’ ) , timescale of unperturbed oscillations (of order l),and timescale of perturbation oscillations (small compared with 1).Condition (3) is satisfied along the line in phase space, described by the expression v dkx p=----(4) k dt k’ Orientation of this line varies slowly with time. Further investigation of resonance (3) can be facilitated by treating z, p and k as slowly-varying parameters. In order to estimate how long a trajectory delays in resonance, we calculate the second derivative of the phase q5 and obtain a pendulumlike equation

(5)

$ - E k 2 s i n $ + f ( z , p , T) = O where

f ( z , p , T ) = -p2k,,z

- 2pk,p

+ ksinx,

(6)

and we denoted “slow” time p t as T ,k , = d k / d r , k,, = d 2 k / d T 2 . Equation (5) corresponds to the pendulum with constant torque. If the following inequality holds:

If(z,P , 7))5 & k 2 ,

(7)

equation (5) possesses oscillating solutions for the phase 4, which correspond to falling into resonance. The exact criterion of falling into resonance can be derived as f01lows.~ Equation (5) can be rewritten as a pair of the coupled first-order equations

cj = Y ,

Y = &k2sin4 - f(z,p ,

T),

(8)

referring to the Hamiltonian

1 E?(Y,4 ) = -Y2 2

+ 4 f ( z , p , 7 ) + &i2 cosd.

(9)

131

A typical phase portrait for the system (8) is demonstrated in Fig. 1. The resonant zone is bounded by the separatrix, containing one saddle fixed point. Confining values of the phase 4 by the interval (-7r : n],we obtain

w Fig. 1. Phase portrait of the pendulum with constant torque.

the desirable criterion

H 5 Hs,

(10)

where H , is the separatrix value of the Hamiltonian (9)

m z ,PI Here

7 ) = dc(5, PI T ) f ( 5 , PI 7 ) f E

m

cos4c(z, P ,

7).

(11)

4c is the coordinate of the saddle point &(z, p , T ) = 71 - arcsin

f(X1

P,

7)

&k2(r) .

Another form of this criterion was derived in.6 Both the functions, H and H , , vary along a particle trajectory. This implies that a particle, being initially outside the resonant zone, may enter it due to gradual changing of z and p . If EIC < 1, the oscillating solutions of the system (8) do exist only for some finite temporal intervals. In this case a particle, after spending some time within the resonant zone, certainly will leave it. Each visit into the resonant zone is followed by energy jump, which may be estimated using a following approximate formula6-’

where 0 < 4 < 2n,f *, p* and 4* are the values o f f , p and 4, respectively, a t the moment of hitting the resonant region. A E depends extremely on initial conditions. If the time of residence inside the resonant zone is much smaller

132

than the time between two successive visits, then scattering on resonance can be modeled as a random process.6-8 Thus multiple recurrences to the resonant area cause chaotic diffusion in phase space.5 However, if the time between two successive visits is small compared with the residence time, neighboring jumps AE may be correlated and there appears a possibility for ballistic superdiffusive behavior, corresponding to particle acceleration along the resonant line (4). The latter case deserves especial attention. The criterion (10) allows one to estimate the upper bound for the energy growth under the acceleration. Equation (4) can be rewritten as

Then we substitute (14) into (6) and (7) and leave only leading terms in the left-hand side. This yields

&k2

k,,

Ipl 5 - k-

I

- 2kT

The respective estimate for x is the following

-1 ,

The latter inequality determines the accelerating zone in the coordinate space. An example of acceleration along the resonant line was reported in.' In that work the following function k ( ~was ) considered

Ic

=

kO(l+

UCOST),

la1 < 1.

(17)

Consider temporal evolution of a particle ensemble. initially distributed with Gaussian probability density

where c o x = cop = 0.1. The parameters of the perturbation are the following: E = 0.04, k~ = 12, v = 4, a = 0.75, p = 27r/1000. Figure 2 represents the temporal dependence of mean coordinate, mean momentum and variance of momentum. It is shown that there occurs a particle flux directed toward x -+ -m. The mean momentum grows non-monotonically and abrupt accelerations are alternating with abrupt slowing-downs. Each act of acceleration is followed by step-like increasing of momentum variance. It should be emphasized that momentum variance is much larger

133

t

-1.8

'

0

zoo0

4wo

I

M)o

Boo0

loo00

M)o

8wO

loo00

t 9 ,

6

4 3

0

Zoo0

4wo

t

Fig. 2. (a) Mean coordinate, (b) mean momentum and (c) momentum variance as functions of time.

than mean momentum, that indicates the presence of particles with very high velocities. Figure 3 shows that accelerating particles form jets along the resonant line (4),which is cut according to the criterion (7). The first significant jet becomes apparent at t I I1300. Accelerating particles follow the resonant line until t 21 1400 and then leave the resonant zone. It should

134 5 ,

‘

-20 -4MM

-3000

-1000

-2000

0

1000

loo0

2000

X 10

I

a -10 O -20 -

-30

-4MM

:/

-3000

-2000

-1000 X

0

Fig. 3. Particle distribution in phase space at (a) t = 1300,(b) t = 1400.The resonant line is marked.

be noted that strongly accelerated particles never return into the resonant zone again. Forming of later jets is demonstrated in Fig. 4, where instantaneous particle distributions at t = 3200 and t = 9200 are presented. It should be noted that there also occur jets toward II: + 00, revealing themselves in the time dependence of mean momentum as slowing-downs (see Fig. 2b). Evolution of the particle cloud is also presented in the media files, which are available at.l0 In conclusion, we have shown that space-periodic Hamiltonian systems with a weak wavelike perturbation imposed can exhibit giant acceleration. The effect appears due to slow variations of the perturbation wavenumber, which lead to forming accelerating channels in phase space. Giant acceleration can coexist with the ratchet phenomenon, i. e. generation of directed particle current in the absence of a biased force. This work was supported by the projects of the President of the Russian Federation, by the Program “Mathematical Methods in Nonlinear Dynamics” of the Prezidium of the Russian Academy of Sciences, and by the Program for Basic Research of the Far Eastern Division of the Rus-

135

t

./

80

I .

&

.

*

40. 20 .

a 0 -

-80

'

-400000

I -200000

0 X

200000

400000

Fig. 4. The same as in Fig. 3 at (a) t = 3200, and (b) t = 9200.

sian Academy of Sciences. Authors a r e grateful to A.I. Neishtadt, S. Flach, A.A. Vasiliev and S.V. P r a n t s for helpful discussions during the course of this research.

References 1. S. Flach, 0. Yevtushenko, and Y. Zolotaryuk, Phys. Rev. Lett. 84,2358 (2000). 2. P. Reimann, Phys. Rep. 361,57 (2002). 3. H. Schanz, T. Dittrich, and R. Ketzmerick, Phys. Rev. E. 71,026226 (2005). 4. D.V. Makarov and M.Yu. Uleysky, JETP Lett. 83, 522 (2004). 5. D.V. Makarov and M.Yu. Uleysky, Commun. N o d Sci. Numer. Simul. 13 400 (2008). 6. A.P. Itin, A.I. Neishtadt, and A.A. Vasiliev, Physica D 141,281 (2000). 7. A.I. Neishtadt, Proc. of Steklow Inst. of Math. 2 50, 183 (2005). 8. D.L. Vainchtein, A.I. Neishtadt, and I. Mezic, Chaos 16,043123 (2006). 9. D.V. Makarov and M.Yu. Uleysky, Phys. Rev. E. 75, 065201 (2007). 10. Animation available at http://dynalab.poi.dvo.ru/bigmult.avi.

136

LOCAL CONTROL O F AREA-PRESERVING MAPS C. CHANDRE and M. VITTOT Centre de Physique ThCorique', CNRS Luminy, Case 907, F-13288 Marseille Cedex 9, France G. CIRAOLO MSNM-GPf, IMT La JetCe, Technople de Chiteau Gombert, F-13451 Marseille Cedex 20, France We present a method of control of chaos in area-preserving maps. This method gives an explicit expression of a control term which is added to a given area-preserving map. The resulting controlled map which is a small and suitable modification of the original map, is again area-preserving and has an invariant curve whose equation is explicitly known.

1. Introduction Chaotic transport arises naturally in Hamiltonian systems with mixed phase space. Achieving the control of these systems by restoring local conserved quantities is a long standing and crucial problem in many branches of physics (in particular, in plasma physics and fluid dynamics). A method for controlling continuous Hamiltonian flows has been developed based on the following idea: to find a small control term f for the perturbed Hamiltonian H = HO V (where HO is integrable), in order to have a more regular dynamics for the controlled Hamiltonian H , = HO V f. Two approaches have been developed : A global control aims at making the controlled Hamiltonian H , integrable; A local control restores a particular invariant torus (local integrability). Both approaches give a control term of order llV112.

+

+ +

*UMR 6207 of the CNRS, Aix-Marseille and Sud Toulon-Var Universities. Affiliated with the CNRS Research Federation FRUMAM (FR 2291). CEA registered research laboratory LRC DSM-06-35. +Unit6 Mixte de Recherche (UMR 6181) du CNRS, de 1'Ecole Centrale de Marseille et des Universitks de Marseille.

137

Let us stress that this method of control differs from other methods by the fact that the controlled dynamics is Hamiltonian : This makes it relevant to the control of inherently Hamiltonian systems such as beams of charged test particles in electrostatic waves, two-dimensional Euler flows or the geometry of magnetic field lines. These two control methods have been developed for continuous time flows.ly2 The global control for symplectic maps has been p r ~ p o s e d In .~ this article, we explicit the local control method for symplectic maps. In Sec. 2, we derive the expression of the control term for area-preserving maps generated by a generating function in mixed coordinates. In Sec. 3, we apply the local control of area-preserving maps to two examples : the standard map and the tokamap. 2. Derivation of the control term We consider two-dimensional symplectic maps ( A ,cp) ++ (A’,cp’) = F(A,p) on the cylinder IR x T which are &-closeto i n t e g r a b i l i t ~In . ~ this section, our aim is to find a small control term f such that the controlled symplectic map F f has an invariant curve. We consider area-preserving maps obtained from a generating function of the form

+

S(A’,cp) = A’cp

+ H(A’) + EV(A’,cp).

The map reads

+ d p V ( A ’ cp), , 9’ = cp + H’(A’)+ E ~ A V ( A9). ’, A = A’

Here a~V(A’,cp) denotes the partial derivative of V with respect to the action (first variable) and a,V(A’,cp) denotes the partial derivative of V with respect to the angle (second variable). We expand the map around a given value of the action denoted K . The generating function after the translation is :

$(A’, cp) = A’cp + H ( K + A’) + E V ( K+ A’, cp). We rewrite the generating function as :

$(A’,cp)

= A’cp

+ wA’ +

+ w(A’,cp),

EZI(~)

(1)

where

w = H’(K), 4 c p ) = V(K,cp), w(A’,p) = H ( K A’) - H ( K )- wA’

+

(2)

+ E V ( K+ A’, v) - & V ( Kp). ,

(3)

(4)

138

We notice that w\O,’p) = 0 for all ‘p E T.Without loss of generality, we assume that w(‘p)d‘p = 0. Our aim is t o modify the generating function with a control term f of order E~ such that the controlled map has an invariant curve around A’ = 0. We consider the controlled generating function

so”

sc(A’, CP)= A’P + wA’ + E V ( ( P ) -t w(A’, ‘P)

+f(~),

where we notice that the control term f we construct does only depend on the angle ‘p. The controlled map is given by

A

= A’

p’ = ‘p

+ + d,w(A’, + + w + ~ A W ( A9). ’, EV’(‘~)

’p)

(5)

f’(’p);

(6)

We perform a change of coordinates generated by

X(A0, ‘p) = Ao‘p + E X ( ’ p ) , which maps (A, ‘p) into (Ao, PO),and (A‘, ‘p‘) into (Ah, pb). The mapping becomes

A0 = Ab + E (X’(’pb) - x’(‘p0) + V’(‘p0)) + 8,w ‘ph = ‘po w ~ A (Ah W 4- EX’($‘;), Po).

+ +

We choose the function

(4+ EX’(‘pb),

Po)

+ f’(‘po), (7)

x such that x(‘p + w)- X(P) = -49).

By expanding w in Fourier series, i.e. ~ ( c p ) = ck,-z

ukeik’f’,this

reads

The control term is constructed such that the mapping in the new coordinated has A0 = 0 as an invariant curve. In order to do this, we define the function @ implicitly by

a($’>= 9 + + 8Aw ( E X ’ ( @ ( ( P ) ) ,

‘P)

a

The angle @(‘p)is obtained when Ab = 0 in Eq. (7) The expression of the control term is such that

f’(d= EX’(‘p + w)- € X ’ ( @ ( ’ p ) )

- ~ ‘ f ‘ W ( & X ‘ W ‘ p ) ) ,‘p).

(8)

From the expression of w given by Eq. (4),it is straightforward to check that if A’ is of order E then 8,w = E V ( K+ A’, ‘p) - E V ( K ‘p) , is of order E ~ Since . @(p)- (‘p + w) is of order E (again if A’ is of order E ) then EX'(@('^)) - EX’((P + w) is of order E ~ Thus, . f’ is of order E ~ .

139

The controlled mapping becomes

A0 = Ab

+ E (X’(Pb) - X’(Q,(PO))) + a,w

(Ab

+ EX’(Pb), Po)

-

%w (EX’(@(PO))lPo) I Pb = Po 4-W -I-~ A (Ab W 4- EX’(&), Po) .

(9)

(10)

It is straightforward to see from Eq. (9) that if A; = 0 then qb = @(PO) by definition of Q, and hence A0 = 0 . Since we assume that the mapping is invertible, the curve A0 = 0 is preserved by iteration of the map. Consequently] the controlled map (5)-(6) has the invariant curve with equation

A = EX’((P).

(11)

Next, we derive an approximate control term by only keeping the order expansion of f’ gives the expression of fmix,2 :

E ~ The .

n

EL

fmix,2 =

--ff”(K) 2

+

(x’(p

W))2

- E2dAV(K1 ‘p)x’(’pf W ) .

(12)

Remark 1: If we assume that w is only a function of the actions, the control term f is f(P)= X ( P + W )

-

X(@(P)) +X’(Q,(P))W’(&X’(Q,‘(‘P)))

- W(&X’(Q,(P)))I (13)

+ +

Q, is defined implicitly by Q,(cp) = ‘p w w’(~x’(Q,(p))). Remark 2: If the time step of the map is equal to 7 , i.e., if we consider controlled maps generated by

where

&(A’, cp) = A’cp

+ TWA’+

TEV((P)

+ TW(A’, + ‘p)

T~((P),

the generating function is given by

x(‘p

+TW)

-

X(V) = --74(P).

We define the operator

7-1, = and

r,

1- e-rwa, I

7

as the pseudo-inverse of 7-1, given from 7-1:I’,

= 7-1,.

The projector

R, is defined accordingly. Hence the solution for x is = (1-

r, - R , ) ~ .

(14)

The control term is f’(P) = ‘?-’& (X’(Cp

+ T W ) - X’(@T(Cp))) - a,w(Ex’(Q,T(P))l

PI1

140

+ +

where a7((p) = 'p TW T ~ A W ( E X ' ( @ ~ ( ( Pp). ) ) , We expand the expression of the control term and we neglect the order T :

f'(9) = - d ' ( ( P ) d A W ( E X ' ( ( P ) r

'P) - %w(Ex'((o),'P)

+ o(T).

Since wx' = -v, we have f('p) =

--W(--Erov',

'p)

+ o(T),

is the pseudo-inverse of wd,. This expression of the control term where corresponds to the one obtained by the local control of Hamiltonian flows.' 3. Numerical examples 3.1. Application to the standard map

The standard map S is

+ Esin'p, 'p' = 'p + A' mod 27r.

A' = A

After a translation of the action A by w , the map becomes

A' = A + Esincp, 'p' = 'p + w + A' mod 27r.

A phase portrait of this map for E = 1.5 is given in Fig. 1. There are no Kolmogorov-Arnold-Moser (KAM) tori (acting as barriers in phase space) a t this value of E (and higher). The critical value of the parameter E for which all KAM tori are broken is &std M 0.9716. The standard map is obtained from the generating function in mixed coordinates A'2 S ( A ' , v )= A ' ' ~ + W A + - + E C O S ' ~ ,

2 i.e. v(y) = cosy and w(A) = A2/2. The generating function Eq. (14) is thus

x

given by

The control term given by Eq. (13) is

1

f('p)

= x('p + w ) - x ( @ ( ' p ) + ) 2 (X'(@('p)))'

+ +

7

(15)

where @(p)= 'p w q ' ( @ ( ' p ) ) . We notice that the equation for @ is invertible for E 5 2sin(w/2).

141

Fig. 1. Phase portrait of the standard map S for

E

= 1.5.

Fig. 2. Phase portrait of the controlled standard map S m i x with the control term (15) for E = 1.5 and w = K. The bold curve is the invariant curve created by the control term.

The dominant control term is given by

and the resulting map Fig. 3.

Smix,2

generates the phase portrait displayed on

142

Fig. 3. Phase portrait of the controlled standard map S m i xwith , ~ the control term (16) for E = 1.2.

3 . 2 . Application to the tokamap

The tokamap5 has been proposed as a model map for toroidal chaotic magnetic fields. It describes the motion of field lines on the poloidal section in the toroidal geometry. This symplectic map ( A ,‘p) ++ ( A ’ ,‘p’), where A is the toroidal flux and cp is the poloidal angle, is generated by the function

A’ S(A’,‘p) = A’cp + H ( A ’ ) - E- A/ + 1 cOs’p’ It reads

A’ 1 sinvl A = A ’ + E A’ +

1 ‘p‘=(p+--

q(A’)

&

(A’

+ 1)2 cos

$0,

where q(A) = l / H ’ ( A )is called the q-profile. In our computation, we choose H’(A) = l/q(A) = 7r(2-A)(2-2A+A2)/2 and E = 9/(47r). A phase portrait of this map is shown in Fig. 4. We select a given value K of the action A for the localization. The tokamap is then generated by a function S of the form given by Eq. (1) with 4’p) =

K -m cos’p,

w ( A , p ) = H ( K + A ) - H ( K )- w A - E

cos $0.

143

cp Fig. 4. Phase portrait of the tokamap for E = 9/(47r)

cp Fig. 5 . Phase portrait of the controlled tokamap (17)-(18) for

E

= 9/(47r).

The generating function is given by Eq. (14) :

The expression of fmix,2

=

fmix,2

is given by Eq. (12) :

E2K cos(cp+w/2) 2(K 1)2 sin(w/2)

+

sin(w /2)

144

For K = 1/2 the controlled tokamap is

A=A’+E(p’=(p+

A’

1 q(A’) (1

-

2 sin(2cp

+ a) + 117r 2 9 + 2a -.z 64

sin a

&

+ A’)2 cos ‘P,

where a = ll7r/32 We notice that this control term is the same as the one obtained by performing the global control and then by expanding the control term around a given value of the action. The main advantage here is t h a t the whole series of the control term can be computed which was not the case with the global control.

References 1. G. Ciraolo, F. Briolle, C. Chandre, E. Floriani, R. Lima, M. Vittot, M. Pettini, Ch. Figarella and Ph. Ghendrih, Control of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas, Phys. Rev. E 69,056213 (2004). 2. C. Chandre, M. Vittot, G. Ciraolo, Ph. Ghendrih and R. Lima, Control of stochasticity in magnetic field lines, Nuclear Fusion 46, 33 (2006). 3. C. Chandre, M. Vittot, Y . Elskens, G. Ciraolo and M. Pettini, Controlling chaos in area-preserving maps, Physica D 208, 131 (2005). 4. J.D. Meiss, Symplectic maps, variational principles and transport, Rev. Mod. Phys. 64, 795 (1992) and references therein. 5. R. Balescu, M. Vlad and F. Spineanu, Tolcamap: A Hamiltonian twist map for magnetic field lines i n a toroidal geometry, Phys. Rev. E 5 8 , 951 (1998).

APPLICATIONS (1) PLASMA 8~ FLUIDS

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147

IMPLICATIONS OF TOPOLOGICAL COMPLEXITY AND HAMILTONIAN CHAOS IN THE EDGE MAGNETIC FIELD OF TOROIDAL FUSION PLASMAS T. E. EVANS General Atomics, P.O. Box 85608 San Diego, California 92186 USA The edge region of magnetically confined toroidal fusion plasmas, such as those found in tokamaks and stellarators, is both dynamically active and topologically complex. The topological properties of the magnetic structures observed in the active edge region of high performance poloidally diverted plasmas are qualitatively consistent with those of a time varying web of intersecting homoclinic tangles defined by invariant manifolds of the primary separatrix of the system interacting with the invariant manifolds of resonant helical magnetic islands. Here, intersections of stable and unstable manifolds produce Hamiltonian chaos in the edge magnetic field that can strongly affect the transport and stability properties of the plasma. A quantitative description of the dynamics involved in these processes requires developing a better understanding of the plasma response to such complex topologies. A review of the recent experimental observations and progress on 3D fluid, kinetic and extended magnetohydrodynamic modeling of the edge plasma in tokamaks is given in this paper. These are related to the topological structure of the invariant manifolds and the chaotic structure of the field lines based on conservative dynamical system theory.

1. Introduction 1.1. General Background

Magnetically confined toroidal plasmas, such as those found in tokamaks and stellarators, are particularly attractive for fusion energy research because of their relatively well established potential for achieving stable, steady-state, operating conditions that combine high plasma pressures with long energy confinement times and the relatively high energy densities needed to produce self-driven burning discharges. On the other hand, the edge plasma in these devices is highly susceptible to the effects of magnetic perturbations from small asymmetries in the external confinement and shaping coils. The edge plasma can also be resonantly perturbed by helical magnetic fields generated by instabilities deep inside the core plasma and by non-axisymmetric fields from external magnetic coils used to control these instabilities. Each of the perturbation fields

148

can result in helical magnetic islands that overlap to produce stochastic field line trajectories that escape from the confinement region. These open field lines connect high temperature plasma particles to material surfaces that are not intended to be exposed to such plasma conditions. Alternatively, resonant magnetic perturbation (RMP) coils can be designed to control the structure of the edge magnetic field and thus to manage the plasma properties in a thin boundary layer region of the plasma called the pedestal where steep pressure gradients couple to the plasma current causing instabilities known as edge localized modes (ELMS). Managing the power and particle exhaust from the edge of a burning fusion plasma is a formidable task in any magnetic confinement device even under ideal conditions. This is due to technological constraints imposed by the properties of high heat flux materials used in exhaust components and because of uncertainties in some aspects of the basic plasma physics operating in the boundary of these devices. In particular, the complex nonlinear nature of the interactions between the boundary layer plasma and the edge magnetic field is not well understood making this a challenging area of research. Thus, the goal of this review is to discuss recent progress made in understanding the complex relationship between structure of the edge magnetic field and the physics of plasmas that reside in the boundary region of a tokamak. Since this is an important topic, there is much more published material than can be covered in this brief review. Some of the earlier work done on stochastic boundary layer physics in circular tokamaks is covered in a 1996 review [l]. Here, the focus is on describing the implications of using a nonlinear dynamical systems approach to study magnetic field line chaos and tangled plasmas in tokamaks especially with respect to the confinement of the plasma, steady-state power exhaust issues and transient energy bursts in fusion relevant plasmas.

1.2. Steady-state Power Exhaust in Fusion Plasmas Deuterium-tritium (DT) fusion plasma discharges in ITER [2], the first net power producing experimental tokamak reactor of its kind, are expected to last up to 400 s with inductively driven plasma currents of 15 MA and fusion thermal output power levels approaching 500 MW. These plasmas will be sustained with external heating systems capable of delivering injected power levels of up to 73 MW for the duration of the discharge. Thus, in ITER the fusion power gain Q, defined as the ratio of the DT fusion output power to that of the external heating power, is expected to be 10 [2] with 50 MW of injected heating power. At this operating point, the steady-state power exhausted from the plasma along open field lines intersecting the material surfaces of the exhaust system, known as the divertor in ITER, is of order 80 MW. At least 50%

149

of this power must be radiated to the plasma facing walls before reaching the divertor surfaces in order to remain below the maximum heat flux limit of 10 MW/m2 set by the materials used on the divertor surfaces. On the other hand, an economical' fusion power generating tokamak, with DT plasmas of approximately same size as those in ITER, is projected to have thermal output power levels range between 2.5 and 3.6 GW and will require steady-state power exhaust capabilities of between 0.4 to 1.2 GW [3] or approximately 100—300 W/m2 on the divertors assuming an axisymme~ic loading distribution d radiative cooling along the open magnetic field lines. toroidal angle (o)

poloidal field • COlI **~~~ •

poloidal angle (6

+ 1m

• • .-' • •' '••' •;••..: . • •'.'_•• ' ,.•••'."••• '•' •-• •

- "'• '•;• • '... . '. ' ,-'', ' / divenur -j"

plasma

'' ••* '.". "... ..'.'.'.;.". . ' \ '/•; •' \ ,-• " • • _ . ' ->>roldal field / ,

-^ -^

«"*P'«»™

Figure 1. Cutaway 3D view of the DIII-D device, a 1.5 m class A=3.0 tokamak, showing the arrangement of the toroidal and poloidal field coils outside the vacuum vessel that contains a poloidally diverted plasma consisting of a core and divertor region.

Figure 1 shows a 3D cutaway representation of the DIII-D [4] device. DIIID is a poloidally-diverted tokamak [5] similar to ITER, with a plasma volume that is oximately a factor of 45 smaller than the plasma volume anticipated in ITER. ere, it is noted that major radius (&) of the toroidal plasma in ITER is Ro = 6.2 m and the minor (poloidal) radius (a) is a = 2.0 m. Thus, ITER ' referred to a 6 m class tokamak with an aspect ratio A E &/a = 3.1 while DIIIis a 1.5 m class tokamak with A=3.0. Assuming the power flow out of the ITER plasma and the radiation on the open field lines is axisymme~ic with respect to the toroidal angle, it should be possible to safely exhaust the steady-state heat flux produced by the plasma using open field line radiative processes. On the other hand, research in the

150

current generation of 1.5 m class tokamaks has shown that non-ideal effects such as sruall resonant and non-resonant magnetic perturbation from external field coils and internal magnetohydrodynamic (MHD) modes significantly alter the edge magnetic topology producing relatively complex 3D structures. For example, in the first poloidally diverted tokamak (ASDEX), a 1.5 m class tokamak with A 4 . 1 , the integrated power flux hitting the divertor target plates was shown to be toroidally asymmetric under all operating conditions. The magnitude of the toroidal energy deposition asymmetry in ASDEX ranged from 9 kJ/toroidal-segment at one toroidal angle to 14 kJ/toroidal-segment at a toroidal angle on the opposite side of the machine (an asymmetry factor of 1.6) during low power ohmic and ion cyclotron heated plasmas. This asymmetry factor increased to 4.5 in lower hybrid heated plasmas (30 kJ/toroidal-segment to 135 kJ/toroidal-segment) [6]. In addition, the toroidal distribution of these asymmetries was strongly affected by changes in the edge magnetic topology that were caused by edge resonant magnetic perturbations from poloidal and toroidal field coils with small non-axisymmetric displacements [7]. Toroidal asymmetries in the ITER steady-state power flow to the divertors could pose a significant challenge that must be overcome during the machine’s operational lifetime starting in 2016 and extending through 2037. 1.3. Transient Particle and Energy Bursts in Fusion Plasmas

While it should be possible to manage the steady-state heat flux in ITER, large transient energy and particle impulses associated with repetitive MHD instabilities known as ELMs [8] are of much greater concern for ITER. When scaled to ITER plasma conditions from current experiments in 1.5 m class tokamaks, ELM instabilities are expected to drive impulsive energy bursts reaching 15 MJ with rise times ranging from 300 ps to 500 ps. In ITER the ablation limit of the divertor materials is 58 MJm-*s-”*[9] and the anticipated surface area exposed to these bursts is -4.7 m2 so a 15 MJ ELM will deliver an impulsive energy flux of -67 MJm-2s-’/2 assuming a toroidally uniform distribution and a 50% radiative fraction along the open field lines. Thus, using the most optimistic assumptions for ITER, it is reasonable to expect that a relatively small number of ELMs (approximately the number occurring in less than a few hundred, full length, high confinement discharges) will rapidly erode the divertors. In addition, observations of edge plasmas in the current generation of 1.5 m class tokamaks, using high speed cameras, reveal an array of complex dynamical processes during ELMs such as the formation of rapidly growing coherent filaments that appear to rotate and interact with the divertor as well as the main chamber walls.

151

While the dynamics and topology of the ELMs in a tokamak are similar in some ways to those of coronal loops, solar flares and coronal mass ejections, they appear to be driven by rather different physical processes [lo]. This is not unreasonable to expect since the dynamics of the ELM should be governed to some degree by electromagnetic interactions with the conducting walls of the tokamak and by impurities generated when heat and particle bursts hit plasma facing surfaces. In addition to the violent energy and particle bursts due to ELMs, there is another interesting dynamical process going on in the edge plasma. This involves a relatively constant level of intermittent turbulence that ejects high-density plasma clumps between the ELMs [l l-131. These clumps propagate radially across the magnetic field and hit the main chamber walls but do not appear to cany much energy. Although the edge of high power tokamak plasmas is an active and topologically complex region, experimental observations of this region appear to be qualitatively consistent with existence of a complex web of magnetic homoclinic tangles defined by intersections of invariant manifolds of the system [ 141 and with the formation of Hamiltonian chaos [ 151 in some regions of the edge magnetic field. The following sections will review recent research results relating the properties of these magnetically tangled plasmas and their associated chaotic regions to experimental observations.

2. Hamiltonian Description of Magnetic Field Lines in Tokamak Plasmas 2.1. General Theoretical Approach A fundamental tool for understanding the structure of the vacuum magnetic fields in the edge of a tokamak is Hamilton-Jacobi theory [16] in which the Hamiltonian H(j,ij) is a constant of the motion due to the fact that the phase volume, defined by divergence-free vector fields j,;,does not change during the evolution of the system. Here, j = af/aij is the canonical momentum of the system andf(2,;) is a generating function that describes a canonical transformation of variables 2,; j,; where is an angular variable in the new canonical coordinates. The magnetic field in a tokamak is expressed as:

+

4

-

+ B4Z4= V x A

= BrZr +BOZO

where

2 is a gauge-invariant vector potential of the form:

x

is a toroidal magnetic flux coordinate and F is an arbitrary function. Here, and w is a poloidal magnetic flux coordinate while O,# are poloidal and toroidal

152

angles respectively [17-211. When expressed in this form, q is effectively a radial variable in the toroidal (q,e,$)coordinate system. Then the canonical form of the magnetic field is found by combining Eqs. (1) and ( 2 ) resulting in: B = v x ;i= s i x x v e + s i 4 x s i q

.

(3)

This automatically results in a divergence-free magnetic field:

si E = si (six xsie+si$ x siq)= o

(4)

as required by Maxwell’s equations. The rotational transform of the equilibrium magnetic field, defined as the average rate of change in the poloidal angle with toroidal angle taken over 2n: poloidal radians on a flux surface v is given by:

i.98 (G$XGq)*i%

@= );(

--= 8(0-.2n)

B*W

(sixxsie).si$

while the rate of change in the toroidal flux coordinate given by:

dq

=-

(5)

dx

x with toroidal angle is

Thus, associating q with the Hamiltonian H it is seen that the toroidal flux coordinate serves as the canonical momentum of the system and the Hamilton-Jacobi equations for the trajectories of the field lines are given by:

x

Here, the usual Hamiltonian is recognized in terms of the more familiar p,q canonical coordinates by substituting + p and 8 -+ q while associating # with time (t). In a tokamak 2 n x is the amount of toroidal magnetic flux enclosed by a surface of constant and 2 ~ = q2nH is the poloidal magnetic flux inside a surface of constant H . Figure 2 , a poloidal cross section of the DIII-D tokamak shown in Fig. 1, illustrates the nominally axisymmetric poloidal and toroidal magnetic field coils, along with various non-axisymmetric control coils, with respect to the plasma (the gray region in the center of the figure) and the axisymmetric surfaces of constant poloidal magnetic flux 2 ~ q Equations . ( 7 ) and (8) are generally

x

x

153

T~roidal

. MHD Control I-Coil ‘Surfaceof Constant Coils Figure 2 A poloidal cross section of half the DIII-D tokamak shown in Fig. 1 illustrating the locations of various (nominally) axisymmetric and non-axisymmetric magnetic coils along with the toroidal and poloidal angles and the surfaces of constant magnetic poloidal flux indicated with dashed elliptical lines in the hot plasma region (indicated by the gray region in the center of the figure).

integrable in the case of an axisymmetric plasma equilibrium but when small non-axisymmetric magnetic fields are present the Hamiltonian becomes an arbitrary function of the toroidal and poloidal angles. In this system a nonaxisymmetric symmetry breaking magnetic perturbation can be expressed in terms of a perturbed Hamiltonian &Hl(x,$,t)) where E is a small dimensionless perturbation parameter. The total Hamiltonian is then the sum of the axisymmetric part:

Hdx) = J 4 X ) d X

(9)

and the non-axisymmetric part:

H = H,(X) + EHI(X,$@) where the perturbed part of the Hamiltonian can be expressed in terms of a Fourier series as:

Here, m and n are the ooloidal and toroidal mode numbers respectively [ 2 2 ] .

154

2.2. Applications of Hamiltoniun Mapping Models to Circular Limited and Poloidully Diverted Tokamaks When small non-axisymmetric perturbations are present in the system Eq. (10) is said to be near-integrable and solutions can be found numerically. For example, in the circular limited TCABR tokamak a symplectic Hamiltonian mapping model [23] is used to calculate the perturbed magnetic field from a set of discrete saddle coils wrapped around the outside of the vacuum vessel. Here, the discretized Hamiltonian mapping function is given by:

where the sequence of 6 functions define a discretization of the action-angle variables x,8 resulting in mapping equations of the form:

=x,

~ n + l

+Ef(Xn+Ip@n?'n)

2n

A+,= @" + -

(13)

'

Nr

with

is the current in N r = 4 perturbation coils normalized to the current and E in the toroidal field coils. Equation (14) is expressed in terms of the safety factor:

which is the number of toroidal revolutions needed to complete a single poloidal revolution of the equilibrium magnetic field. Here, resonances in the helical magnetic field, due to m,n Fourier harmonics of the perturbation field occur when: 4(X,.") = m/n

(18)

is a rational number [ 161. Mapping models such as these are similar to the well-known standard map [21,24] which has been used extensively to study stochastic field line trajectories in tokamaks [21,25-281. In addition, specialized maps, such as the Wobig-

155

Mendonqa Map [29,30] and the Tokamap [31], have been formulated to more accurately account for realistic q(x) profiles in circular tokamak. A variety of specialized Hamilitonian maps have also been constructed for poloidally diverted tokamak [22,32-351 and so-called "wire" models [36,37] have been used to study magnetic footprints in poloidally-diverted tokamaks. These magnetic footprints determine the poloidal and toroidal distribution of heat and particle flux exhausted from the edge of tokamaks [23,38] and stellarators [39] on to plasma facing material surfaces. Thus, they are of substantial practical importance for designing and operating toroidal magnetic confinement fusion devices. Hamiltonian mapping codes are also an important tool for understanding both the global and fine scale structure of the edge magnetic topology in toroidal confinement systems. This structure can have a fundamental influence on kinetic and fluid transport processes [40] as well as plasma turbulence [41] and MHD stability in the edge [42,43] of the discharge. 3. Hyperbolic Fixed Points and Invariant Manifolds in Tokamak Mapping and Field Line Integration Codes

3.1. General Background Periodically perturbed continuous vector fields that define the trajectories (flows) of nonlinear dynamical systems such as magnetic field lines in toroidal confinement systems give rise to discrete planar maps referred to as diffeomorphisms [44]. More specifically, if a map is one-to-one and onto (bijective), continuous, and has a continuous, differentiable, inverse then it is referred to as being homeomorphic. Thus, a diffeomorphism is defined as a differentiable homomorphism. Given a differential equation;

c=c($)

where is a vector valued function of an independent variable $, the toroidal angle in a tokamak, then a smooth function A generates a vector field flow Q, : U -+. R" where Q,(<) = S2(<,$) is a smooth function defined for all in U and 4 over some interval I = (u,b) R" and Q, satisfies Eq. (19) since:

c

for all

< E u and t E I [45]. Then a planar map:

<

156

can be defined as a procedure that images solution curves (trajectories or orbits) of Eq. (19) on to the PoincarC plane. Thus, S2 generates a flow C2:R" + R" that can be thought of as the set of all solutions to Eq. (19). A key requirement for the study of solutions to Eq. (19) is to identify fixed points of the system for which: Q k ( C o )= 5 0

where k represents the number of periods or iterations of the map required to return to the fixed point go. Thus, a period 2 fixed point:

returns to its initial position on the Poincark plane after two iterations of the map i.e., a composition of the map with itself. Fixed points are the most basic anatomical element of a dynamical system in the sense that the properties of their local (linearized) eigenvalues Aj uniquely determine the stability properties of all the solutions. For example, a hyperbolic fixed point (sometimes referred to as a saddle point) of Hamiltonian systems has two real eigenvalues that satisfy the condition lAA < 1 <\A,\ and the asymptotic stability of the trajectories is determined by invariant sets associated with these eigenvalues. Here, the stable eigenvalue As is associated with an invariant set that contracts trajectories toward its fixed point while the invariant set associated with A,, expands the trajectories that move away from the fixed point and is said to be unstable. The stable w s and unstable w" invariant manifolds of a hyperbolic fixed point go generated by the planar map Q(C) = 5 are informally defined as:

Ws(C0)= {I;Ilirnk

+

md(5)

= fo}

(24)

and

An important question in dynamical systems theory is whether the fixed points of the system persists when perturbed. It can be shown using the implicit function theorem that the addition of a small autonomous (time independent) perturbation E does not destroy the fixed points since the system is continuous and has solutions with continuous derivatives that reside within E of the unperturbed fixed points although the fixed points may move slightly [46]. Thus, this aspect of the system is stable under small perturbations.

157

3.2. Calculations of Homoclinic Tangles and Heteroclinic intersections in Circular Limited Tokamaks

Edge RMPs were originally proposed as a possible method of controlling the heat and particle exhaust in circular tokamak by producing resonant magnetic islands on rational magnetic surfaces that overlap and create a stochastic boundary layer [47,48]. The first experimental tests of this concept were done in circular tokamaks using modular coils wound around the outside of the vacuum vessel. These coils were used to produce a so-called ergodic magnetic limiter [49,50] or an ergodic divertor configuration [I]. These modular coils were also used to carry out the first experimental tests of the resonant helical (island) divertor configuration [5 11 based on a concept originally proposed by Karger and Lackner [ 5 2 ] that was designed to more efficiently manage the heat and particle flux in circular limited tokamaks. The resonant helical (island) divertor concept, as implemented in a second experiment of this type on the JIPP T-IIU tokamak [53], is shown in Fig. 3. This concept was eventually adopted as an approach for managing the heat and particle flux in stellarators (where it is referred to as the local island divertor configuration) and has proven to be an important factor in obtaining high performance discharges in those systems [54]. ~ e ~ u r b a tcoil i ~ ncurrents

chain m,n = 3,l

Figure 3. Conceptual implementation of the resonant helical (island) divertor configuration in the JIPP T-IIU tokamak showing the modular RMP coils wrapped around the vacuum vessel and a cutaway view of the magnetic field lines with a limiter inserted into the edge of the plasmas [53]. The poloidal cross section on the right hand side of the figure illustrates the position of the limiter inside the m,n = 3,l resonant magnetic island produced by the currents in the modular coils along with the flow of heat and particles into the limiter on the low magnetic field side of the plasma.

158

It is noted that the m,n = 3,l magnetic island shown in Fig. 3 is composed of a period m=3 hyperbolic fixed point, sometimes referred to as x-points, where pairs of degenerate w s and w" manifolds intersect and m=3 elliptic fixed points are located at the center of each island lobe. The field lines inside these helical magnetic islands reside on closed flux surfaces that surround the elliptic fixed points while the field lines outside the invariant manifolds w s and w" are confined to the circular (although slightly deformed near the periodic hyperbolic fixed points) poloidal flux surfaces as illustrated in the right-hand part of Fig. 3. As shown below, the structure of the island manifolds depicted in Fig. 3, sometimes referred to as the island separatrix, is an idealization used to illustrate the basic features of the resonant helical (island) divertor concept. It is often more convenient to model the magnetic field structure in tokamaks using field line integration codes rather than mapping codes since the perturbation fields due to modular coil sets can be easily modeled using current filaments that follow the conductors of the perturbation coils such as those shown in Fig. 3. Here, the perturbation fields are calculated using a Biot-Savart algorithm [ 5 5 ] at each step along the integration path and solving a set of field line differential equations:

expressed in a cylindrical (R,@,z) coordinate system where BR,B+*BZ represent the linear superposition of the axisymmetric equilibrium magnetic field and the non-axisymmetric perturbation fields from all sources in the tokamak of interest. As shown by Bates and Lewis [ 191, Eq. (26) is Hamiltonian. When using the field line integration approach and Eq. (26), axisymmetric equilibrium field components can be constructed with N equally spaced circular toroidal field coils, represented by N toroidal magnetic dipoles, and M circular filaments represented by M poloidal magnetic dipoles distributed across the plasma to produce a toroidal current density profile [ 7 ] . Various field line integration codes have been used to model the structure of the perturbed magnetic field in circular tokamas [56-591. These codes typically take small integration steps (-2n/360) in the direction of the independent variable 4 and calculate the new R,z position of the field line after each step based on magnetic field components as specified by Eq. (26). Result from these calculations are typically displayed on the PoincarC plane by cutting circular flux surfaces along the outer equatorial plane (typically located at 0 = 0 in a toroidal coordinate system) and unfolding them into a rectangular surface where the abscissa represents the poloidal angle 0 and the ordinate represents a normalized radial variable (RN)as shown in Fig. 4. Here, the small dots represent the positions of

159

the field lines as they cross the PoincarC plane following each toroidal transit along their trajectory. Magnetic islands, identified with the m,n = 1,1, m,n = 2,l and m,n = 3,l resonant surfaces, are shown in Figure 4 along with examples of invariant manifolds (solid curves) superimposed on some of the islands [14]. The invariant manifolds are calculated with a version of the TRIPND code [59] referred to as the TRIP-MAP code [38] that uses an algorithm described by Nusse and Yorke [60] which smoothly resolves segments of the manifolds and with a more refined method described by Hobson [61].

z

U

manifolds 0

100

200

300

8 w9.1 Figure 4. A Poincare plot of magnetic field line trajectories produced by the TRIPND field line integration for a resonantly perturbed magnetic equilibrium in the Tore Supra tokamak [14,58].

The magnetic structure shown in Fig. 4 results from a statistical model that accounts for multi-millimeter random displacements of the toroidal field coils on the Tore Supra tokamak. Magnetic perturbation sources of this type are referred to as intrinsic topological noise ITN [58]. They result from the fact that tokamaks are built with discrete toroidal and poloidal magnetic field coils which are subject to practical engineering constraints such as tolerance buildups, unbalanced dynamical j x forces due to currents in the plasma and a mutual coupling of the currents in the entire tokamak coil system. In addition, design and construction irregularities, magnetic materials used as part of the heating, diagnostic or control systems and thermal forces on the magnetic components contribute to the ITN spectrum. Thus, it is reasonable to expect that each tokamak has its own unique non-axisymmetric ITN signature resulting from such contributions integrated over the entire ensemble of magnetic components that make up the device. It is also reasonable to expect that the ITN spectrum may change with time as stresses are relieved or as a result of large transient events such as disruptions of the plasma current [62] that can sometime occur in

160

tokamaks. On the other hand, so-called "field-errors'' result from displacements of individual coils that can be measured [63] and modeled using the Biot-Savart approach discussed above. Modeling of the ITN spectrum in the Tore Supra and TFTR tokamaks, in terms of a Gaussian random normal statistical process, has shown that relatively large low m,n modes can can drive resonant magnetic island chains with sizes similar to those associated with field-errors [ 5 8 ] . A closer look at the invariant manifolds associated with the m,n = 2,l and m,n = 3,l island chains shown in Fig. 4 illustrates the complexity of these structures. Figure 5 shows part of an object generically referred to in dynamical systems theory as a "homoclinic tangle"' [60,64]. The homoclinic, selfintersecting, tangle shown in Fig. 5 results from a splitting of the w s and w" invariant manifolds associated with a period 2 fixed point (right) and a period 3 fixed point (left) that defines the m,n = 2,l and 3,l magnetic island chains respectively [ 141. More generally, homoclinic tangles are formed by sets of points that converge to a hyperbolic fixed point Po (or a fixed point of period k ) under the mapping Q k ( P o ) = P o or its inverse Q-k ( C o ) = P o . Similarly, heteroclinic tangles are formed by sets of points that converge to independent fixed points Po,Zo under the mapping or its inverse.

100

140

180 8 (deg.1

220

260 f)

(deg.)

Figure 5. (left) Homoclinic tangle associated with the period 2 fixed point of the m,n = 2,l magnetic island chain and (right) a period 3 fixed point of the m,n 3,1 magnetic island chain for the case shown in Fig. 4 [14].

Intersections of the lobes making up a homoclinic or heteroclinic tangle near fixed points of the system are responsible for stochastic mixing of the field line trajectories. These intersections are permitted when stable tangle manifolds intersect unstable manifolds since this does not violate the uniqueness of

' Henri Poincark discovered these objects while trying to find a solution to a set of equations that describe the motion of a nonlinearly coupled 3-body planetaly system and wrote (in Vol3, Ch. 33, page 389, of Les MPfhods Nouvelles de la Mkcanique Ckleste, Gauthier-Villars, Paris, 1899): "One is struck by the complexity of this figure that I am not even attempting to draw".

161

solutions to the differential equations describing the system. Field lines are also exchanged between neighboring islands through intersections of stable and unstable manifolds associated with fixed points of these islands (heteroclinic intersections) as shown in Fig. 6. The dynamics of the field lines can thus be very complex and have significant implications for the transport and stability properties of the plasma as discussed in Sec. 4.

0.875 -?-.

ac"

'-..

0.750. 0.625.

0.500

0

50 100 150 200 250 300 350 0 (deg)

Figure 6 . An intersection of a stable invariant manifold associated with the tofixed point of the m,n = 2,l magnetic island chain and an unstable invariant manifold associated with the Xu fixed point of the m,n = 3,l magnetic island chain shown in Fig. 4. Magnetic flux exchanged through this type of intersection between islands results in global stochasticity and large scale, non-difhsive, transport of the field lines and the plasma confined to these field lines [14].

3.3. Calculations of Homoclinic Tangles in Poloidally Diverted Tokamaks

Homoclinic tangles are predicted to be a common feature of the primary (nominally axisymmetric) separatrix in diverted tokamak plasma equilibria [38] since relatively small non-axisymmetric magnetic perturbations such as ITN sources, field-errors, helical MHD modes in the core of the plasma, nonaxisymmetric corrections and control coils and edge instabilities such as ELMS are typically present at some level. In an ideal, perfectly axisymmetric poloidally diverted tokamak, the stable w s and unstable w" manifolds associated with each axisymmetric fixed point are degenerate resulting in an identical overlay of one manifold on the other as shown in Fig. 7(a). As first shown by Roeder, Rapoport and Evans [38], vanishing small non-axisymmetric perturbations in these systems produce transverse intersections of w s and w" that result in the formation of homoclinic tangles. The intersection points of the

162

manifolds that define a tangle are known as homoclinic intersections or points and it can be shown that an infinite number of such homoclinic points make up the structure of the homoclinic tangle [45]. Additionally, since the system is Hamiltonian and 9 = 0 the magnetic flux passing though each lobe of the tangle is conserved. Thus, as a fixed point is approached either from the stable or unstable direction the lobes get increasingly narrow and stretched out as shown in Fig. 7(b) by the structure of the W"((l;,) as it approaches the Z, (upper) fixed point along its stable direction. a.

1.5

E

N

1.5

1.o

1.o

0.5

0.5

- 0.0 h

E

0.0

N

-0.5

-0.5

-1 .a

-1 .o

-1.5

-1.5 1

1.0

1.5 R (m)

2.0

2.5

0.5

1.0

1.5

2.0

2.5

R (m)

Figure 7. (a) The primary separatrix of an ideal lower single-null poloidally diverted tokamak plasma equilibrium is composed of degenerate stable and unstable invariant manifolds associated with a single hyperbolic fixed point go.Here a section of the stable manifold has been removed to reveal the underlying unstable manifold [ 141. @) A balanced double-null poloidally diverted tokamak plasma equilibrium with two independent hyperbolic fixed points 5, and 2, that form a complex set of intersecting stable and unstable invariant manifolds when subjected to a nonaxisymmetirc perturbation due to a field-error correction coil [65].

Although Fig. 7(b) shows only a stable and unstable manifold originating from each fixed point, it should be kept in mind that there are always two stable and two unstable manifolds associated with each hyperbolic fixed point. Here, the tangles that are shown were selected because they have important implications for determining how field line trajectories from the edge of the confined plasma region interact with the material surfaces in the divertors. In other words, they define the boundaries of the magnetic footprints in which field

163

lines from the inner region hit the surfaces and thus allow these nominally closed field lines to open on surfaces in the divertor. This affects the distribution and intensity of the heat and particle flux to the divertors as well as the MHD stability of the edge plasma as discussed in Sec. 4. Figure 7(b) also demonstrates that the invariant manifolds can take on rather complex patterns in which branches of an individual homoclinic tangle shadow branches of another tangle but do not join to form a single heteroclinic tangle. This behavior is controlled by the up-down symmetry of the fixed points and shaping parameters set by the poloidal field coils such as those shown in Fig. 2 [65].

4. Implications of Topologically Tangled Tokamak Discharges on Plasma Confinement and Stability 4.1. General Background

As pointed out above, non-axisymmetric perturbations are always present to some degree in toroidally confined magnetic plasmas and can produce topologically complex structures in the vacuum magnetic fields. These structures are known to produce significant changes in the confinement and stability of the plasma that have direct effects on the power and particle exhaust as well as the overall performance of the discharge. A brief review of such effects is given in this section along with implications for controlling the steadystate heat flux and transient energy pulses as well as particle exhaust rates in ITER and in fusion prototype reactors beyond ITER. 4.2. Improved Confinement and Transport Barrier Physics in Circular Limited Tokamaks with Stochastic Magnetic Boundary Layers The first set of stochastic boundary layer experiments were done in TEXT, a circular tokamak with a ring limiter and a set of modular RMP coils known as the ergodic magnetic limiter (EML) coils [49,50]. The goal of these experiments was to test the idea (originally proposed by Feneberg [47]) that a stochastic layer could be used to cool the edge of the plasma and shield impurities from the core plasma. This process was expected to cause the edge to radiate the heat flux exhausted from the core plasma uniformly to the walls of the vacuum vessel. The physics of energy [66,67] and particle [68] transport was studied in great detail during these experiments and compared to various transport theories [40,69-711. In particular, it was noted that an anomalously large drop in the plasma density was typically observed during the RMP pulse. This was not predicted by these theories. Detailed measurements of increases in the particle flux around magnetic islands produced by the EML coil was consistent with the existence of an electric field inside magnetic islands Eisl 40V/cm that resulted

164

in Eis,x B convective cells near the plasma edge. This convective transport mechanism was found to be large enough to account for the observed drop in plasma density [72,73]. Measurements in the HYBTOK-I1 tokamak found similar results and support the conclusion that externally driven magnetic structures can significantly alter the electric field in the edge plasma [74]. Recently, this effect has been proposed as a possible mechanism to explain density drops seen during RMP experiments in high confinement (H-mode) plasmas on the DIII-D tokamak [75]. Ergodic magnetic limiter experiments in TEXT also demonstrated that the electron temperature profile could be substantially altered across an edge stochastic layer as shown in Fig. 8. Here, the temperature profile flattens over approximately 75% of the width of the stochastic layer and increases sharply over the remaining 25% of the stochastic layer. This profile has the appearance of a thermal transport barrier located at approximately -22 cm [51] and suggests the possibility that the stochastic boundary layer is creating a type of improved confinement regime. Subsequent experiments, using the modular RMP coils on JIPP T-IIU, also indicated modest improvements in the energy confinement under some conditions [53].

500

I

8

Vertical Position (cm) Figure 8. An electron temperature (TJ profile measured with a Thomson scattering system along a vertical line of sight through the TEXT plasma using the m,n = 7,3 EML coil configuration with 7 kA of current compared to the T, profile in an identical discharge with no EML coil current [51].

165

Following these initial indications of a new type of improved confinement mode related to the properties of the edge stochastic layer, a series of experiments were done with the ergodic divertor (ED) coils on Tore Supra [76] in order to verifl the existence of such a mode [77-791. In the first experiments with ohmic heating only, it was found that the particle confinement could be increased substantially by positioning the plasma against the inner bumper limiter (the high field side wall), with an edge qa = 3.0, and creating a stochastic layer with the ED coil set [77,78]. Results from the first Ohmically-heated experiment are shown in Fig. 9. When the stochastic layer forms during the ED pulse there is an increase in the line integrated density and the neutral hydrogen particle flux from the limiting surface, shown by the 2D image in Fig. 9(b), drops everywhere within the field of view of the CCD camera as seen in Fig. 9(c). Additionally, it was found that the position of the plasma with respect the high field and low field side (i.e., good and bad curvature) limiting surfaces is an important factor for determining whether the confinement increases or is reduced when the stochastic boundary is applied as shown in Fig. 10 [78].

!

34

36

38

40

Figure 9. (a) Line integrated density evolution along 3 chords across an Ohmically-heated Tore Supra discharge (#1423) with no external gas fueling after 33 s and the plasma limited on the high field side bumper limiter with a surface safety factor qa = 3.0 [77]. Here, the ergodic divertor coil was energized at 36.1 s with 36 kA of current causing the density to increase while several of the recycling diagnostics (Hat and H-3) viewing the high field side limiter showed a decrease in the neutral particle flux from the limiter. (b) A CCD camera image of the Ha emissions with a view normal to the surface of the high field side bumper limiter at 36.0 s showing regions with high levels of neutral recycling (multiple contours) without the ergodic divertor. (c) The same view of the bumper limiter as in (b) but taken at 36.1 s during application of the ergodic divertor with low neutral recycling.

166

time (s)

time (s)

time (s)

Figure 10. (a) Improved particle confinement with the plasma limited on the high field side bumper limiter during the phase of the discharge when the stochastic layer is applied, @) a relatively modest increase in the confinement is observed with a stochastic layer when the plasma is limited by both the high field side and the low field side bumper limiters and (c) the confinement is reduced during the stochastic layer phase of the discharge when the plasma is limited on the low field side bumper limiter [78].

These changes in confinement due to geometric differences in the way the plasma is limited on material surfaces suggest that topological effects such as connections between a homoclinic tangle surrounding an island near the surface of the plasma and the wall may be playing a key role in the transport physics. It is known that field lines with short connection lengths to material surfaces can change the electric potential of the plasma, which results in electric fields that affect the bulk motions of the plasma. This is supported by the fact that these improved confinement modes could only be accessed when the safety factor at the plasma surface was within a narrow range around 3.0. Since the peak in the ED perturbation spectrum in Tore Supra was located at m,n = 18,6 it is reasonable to assume that a homoclinic tangle associated with the m,n = 18,6 magnetic island in combination with a m,n = 3,l field-error island tangle was an important factor in establishing the underlying the physics of this effect. The contrast between improved and degraded confinement regimes in these plasmas was rather dramatic as seen in Fig. 11 where the current in the ED coil set was 45 kA (the maximum current possible). Here, the discharge that transitions to the improved confinement regime, Fig. ll(a), had a slow 2.7 s increase in the stored energy from 140 kJ to 210 kJ with a corresponding linear increase in the plasma pressure normalized by the poloidal magnetic field pressure p p from 0.06 to 0.10. On the other hand, the particle confinement time increased promptly by approximately 55% during the improved confinement phase of this discharge. In the other case shown in Fig. 1l(b), with the plasma

167

51

I

I

I

I

I

I

I

1 1 1.2

E

z!

0

2

4 6 time (s)

8

Figure 11. (a) With the plasma limited on the high field side bumper limiter and 45 kA in the ED coil the plasma density increases dramatically while the H, recycling drops substantially. Note that the rate of change in the density is reduced during the improved confinement phase compared to it value before the ED pulse. (b) With the plasma limited on the low field side bumper limiter the density drops dramatically and the H, recycling increases indicating a large reduction in the particle confinement time during the ED pulse with the plasma in this configuration [78].

limited on the low field side bumper limiter, the stored energy dropped from 275 kJ to 120 kJ on about the same time scale as the drop in density while P, dropped from 0.12 to 0.05. In this case the particle confinement time drops rapidly following the start of the pulse by -70% relative to that before the ED field. Similar experiments with the plasma limited on the high field side bumper limiter and qa = 3.0 were repeated using lower hybrid current drive (LHCD) with injected power levels of up to 3.3 MW [79]. In these discharges, the volume averaged particle content of the discharge increased from 0 . 8 5 ~ 1 0 'mF3 ~ during the ohmic phase to - 2 . 1 ~ 1 0 'm-3 ~ during the improved confinement phase with the LHCD and an ED coil current of 18 k.4. Although the evolution of the density and Ha recycling in these discharges resembles that observed in neutral beam-heated high confinement, H-mode, discharges in poloidally diverted tokamaks such as those required to reach Q = 10 in ITER [2], no ELMS were observed and the improvement in the confinement appeared to affect plasma particles more so than the plasma energy [79]. This suggested that an edge stochastic layer may increase the resilience of the discharge to ELM instabilities

3 68

in high confinement regimes and led to the development of an RMP concept [7] as an option for ELM suppression in H-modes [42,43,75]. These experiments also demonstrated that essentially all of the power exhausted from the core plasma could be radiated uniformly over the plasma facing surfaces due to the onset of an instability, referred to as a MARFE [SO821, which could be maintained for the duration of the combined LHCD and ED pulse. This MARFE resided in the outer 15% of the discharge (approximately the width of the stochastic layer) between the hot core plasma and the high field side bumper limiter [79]. Infrared camera measurements of the temperature distribution on the high field side bumper limiter demonstrated that surface temperature of the limiter was reduced to approximately the same level as that seen during the ohmic phase of the discharge when the ED was turned on during the LHCD pulse and the M A W E was fully formed. Improvements in the plasma confinement due to the formation of stochastic boundary layer transport barriers have recently been observed in the TEXTOR tokamak using the dynamic ergodic divertor (DED) coil set and neutral beam heating [83]. The TEXTOR results are surprisingly similar to those obtained in Tore Supra including the dependence on the positioning of the plasma with respect to the high field side limiter and the importance of rational q surfaces needed for inducing the transport barrier. Separate DED experiments have also seen the onset of MARFEs [84] that appear to be similar to those obtained in Tore Supra. Reviews of the changes in the electric fields [S5] and the transport properties [86] during DED operations in TEXTOR have provided new information on the physics of confinement changes in circular limited plasmas with edge stochastic layers. In addition, work on connecting the structure of homoclinc tangles with heat flux deposition patterns on the high field side bumper limiter has shown a correlation with the DED current [87,88]

4.3. Signatures of Homoclinic Tangles in Poloidally Diverted Tokamaks While our understanding of confinement and stability physics in circular limited plasmas with edge stochastic layers has progressed substantially due to the results reviewed above, the physics of edge stochastic layers in highly rotating, relatively collisionless, high pressure poloidally diverted H-mode plasmas is substantially unexplored and the few results that do exists are often quite surprising and theoretically challenging [43,75]. As discussed in Sec. 1, managing the steady-state power and transient energy exhaust in the next generation of tokamaks is a critical technological issue that must be solved before a prototype fusion engineering reactor can be designed and built. In

169

addition, the perturbed edge magnetic topology in poloidally diverted tokamaks is considerably more complex than in circular limited tokamaks due to heteroclinic intersections of edge magnetic islands with each other and with homclinic tangles associated with the primary hyperbolic fixed points that form single and double null diverted configuration. Finally, understanding the selfconsistent plasma-magnetic field response to the topology imposed by these complex webs of intersecting homoclinic tangles in high performance H-mode plasmas is far beyond the scope of existing theories. As discussed above, the first detailed measurements of steady-state toroidal energy deposition asymmetries were made in the ASDEX tokamak [7]. Several years later, measurements made in DIII-D using IR cameras viewing the lower divertor at two toroidally separated positions revealed the complex geometric nature of these toroidal energy exhaust asymmetries. These measurements demonstrated that the heat flux profile at one angle could have a single peak while the heat flux profile at the other toroidal position had two peaks. This resulted in approximately a factor of two asymmetry [89]. Detailed calculations of homoclinic tangles formed by non-axisymmetric field-errors and field-error correction coils produced magnetic footprints that qualitatively matched the splitting of these heat flux profiles [14,90]. It has also been shown that the toroidal phase of the single versus double peak profiles matches the calculated footprint patterns and that these patterns can be controlled with a set of nonaxisymmetric coils in DIII-D [90]. A discussion of the properties of homoclinic tangles in DIII-D and their relationship to measurement of divertor heat flux distributions during locked modes and MHD instabilities is given in Ref. 14. It is also pointed out in Ref. 90 that under some conditions the size of the tangles can be amplified by the plasma response to the externally imposed magnetic structure. The first measurements showing that ELMs are composed of nonaxisymmetric structures were also made in DIII-D [89]. Here, it was shown that large non-axisymmetric currents flow into and out of the divertor target plates during ELMs. Images of ELMs in MAST, a 1 m class spherical tokamak with A=1.3, have explicitly shown the complex 3D topological structure of these instabilities [91]. It has been suggested that the topological structure and evolution of ELM instabilities in tokamak may be related homoclinic tangles driven by ITNs, field-errors or other non-axisymmetric magnetic perturbations that are amplified by the currents flowing in the tangle [92]. It has also been shown in DIII-D that ELMs can be suppressed in ITER relevant plasmas by applying non-axisymmetric magnetic perturbations [42,43,75,93-981.

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5. 3D Plasma Modeling of the Plasma Response to Resonant Magnetic Perturbations

A fundamental element needed to understand the physics of tangled edge plasmas in toroidal systems is a theoretical framework that self-consistently couples 3D time dependent plasma equilibria, including the evolution of the plasma pressure and current density, to a set of global stability and confinement theories. This can be thought of as a unified global model of toroidally confined edge plasmas. Although substantial steps have been taken to develop individual parts of such a model an integrated picture is far from being completed. An example of a 3D equilibrium solver is the VMEC code [99] that imposes perfect magnetic surfaces (no helical islands) on perfectly conducting plasmas while guaranteeing that q*i=O.The 3D PIES code [loo] then uses VMEC equilibria to calculate the topology of resonant flux surfaces, islands and stochastic regions, with increasing plasma pressure. Although these codes were originally developed to model stellarator flux surfaces they have recently been applied to tokamaks with double-null poloidally diverted configurations that conform to stellarator symmetry [loll. Significant progress has also been made in developing 3D transport models both in the area of fluid codes [ 102-1041, that work in moderate to high plasma collisionality regimes, and in the area of gyrokinetic codes [lo51 that work in low collisionality regimes. Monte Carlo impurity transport codes [106,107] capable of simulating 3D plasmas have also been developed along with a 3D turbulence code [lo81 and a variety of analytic theories [ 109-1 141 dealing with various aspects of the transport in stochastic magnetic fields [ 1151. The final piece needed in an integrated global edge plasma model is a nonlinear 3D resistive MHD code, for example the NIMROD [ 1161 code, and a perturbed ideal MHD equilibrium code such as CAS3D [ 1171 or IPEc [ 1181. Thus, there are significant numerical resources available but the task of bringing them all together into a unified model that can be validated against experimental data is still a major step that needs to be completed. 6. Summary and Conclusions

Although magnetized toroidal confinement systems such as stellarators and tokamaks appear to be an attractive route to a steady-state burning fusion plasma regime, a significant number of scientific and technological barriers remain unresolved. For example, a critical technological issue in the next generation of tokamaks is the management of the steady-state power exhaust and transient energy bursts due to edge MHD instabilities known as ELMS. Under ideal conditions, it should be possible to manage the steady-state heat exhaust in

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ITER but transients due to ELMs must be strongly mitigated or entirely eliminated. In a prototype power producing fusion reactor new solutions will be required to reduce the steady-state heat flux exhaust to the same level expected in ITER [3] and ELMs must be completely eliminated for the duration of the discharge. Techniques based on controlling the edge magnetic topology are being developed to deal with both steady-state heat flux [3,90] and transient energy burst [42,43] issues, but in a more general scientific sense there are hndamental concerns about the number of degrees of topological freedom that are an inherent feature of toroidal confinement systems. Since the magnetic field is Hamiltonian and dynamical systems theory predicts that perturbed, divergence-free, vector field flows in toroidal systems result in complex topological structures such as intersecting webs of homoclinic tangles and Hamiltonian chaos, a diverse range of dynamical behaviors can be expected to occur in high power stellarator and tokamak plasmas. Many of these will be driven by the internal dynamics of the plasma interacting with structures produced by field perturbations from external sources that are not well understood from a basic physics point of view. In circular limited plasmas for example, experiments have shown that the perturbed magnetic field can produce either an increase or a degradation in the confinement depending on the specifics of the geometric configuration used and a presumed interaction of homoclinic tangles surrounding a magnetic island chains in the edge plasma with the limiter surface [77,78,83]. In poloidally diverted [ 14,901 and circular [87,88] tokamaks, measured heat and particle flux profiles hitting plasma-facing surfaces are found to be consistent with magnetic footprints caused by intersections of a homoclinic tangle with the surface. It has also been suggested that violent surface eruptions of the plasma, edge MHD instabilities known as ELMs, may be an expression of the intrinsic topological freedom inherent to the system. Here, it has been proposed that ELM instabilities drive currents through a preexisting separatrix tangle, caused by field-errors and intrinsic topological noise, during their linear growth phase and these currents amplify the structure of the tangle resulting in a nonlinear, explosive, growth phase during which heat and particles are transported through the lobes of the tangle to plasma facing surfaces [92]. Such a process may be indicative of the potential for the plasma to generate a wide variety of complex structures as the energy density of the system increases, and needs to be understood. It is therefore imperative to develop an interpretive unified model of the edge plasma and to validate the model with experimental data from the current generation of devices. A variety of numerical models are being developed within the fusion community as discussed in Sec. 5 but have not yet been tested critically against

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experimental data. Once the individual pieces of these physics models have been validated with experimental data, a unified model can be constructed. A primary constraint on the unified model is that it must preserve the Hamiltonian (symplectic) structure of the magnetic field, which then allows us to make contact with the global topology of the system and to quantify its stability with respect to small perturbations. When viewed from this perspective, a research program of this type can be thought of as an entirely new branch of plasma physics concerned primarily with the global topology of the plasma and may be applicable to a broad range of magnetically confined fusion, solar and astrophysical plasmas.

Acknowledgments This work was supported by the U.S. Department of Energy under DE-FC0204ER54698. I would like to thank Drs. R. Roeder and A. Wingen for their comments and suggestion on the contents of this review.

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EXPERIMENTAL STUDIES OF ADVECTION-REACTIONDIFFUSION SYSTEMS' T. H. SOLOMONt, M. S. PAOLETTI' AND M. E. SCHWARTZs Department of Physics and Astronomy, Bucknell University Lewisburg, PA 17837, U.S.A. We present the results of experiments on the effects of chaotic fluid mixing on the dynamics of reacting systems. The flow studied is a chain of alternating vortices in an annular geometry with drifting andor oscillatory time-dependence. The dynamical system is the oscillatory or excitable state of the well-known Belousov-Zhabotinsky chemical reaction. Results from two sets of experiments are as follows: (1) Fronts propagating in the oscillating vortex chain are found to mode-lock onto the frequency of the external oscillations. It is also found that the presence of a significant "wind" (drift of the vortices in the lab frame) causes fronts propagating against the wind to freeze. (2) Synchronization of oscillating reactions in an extended flow (vortex chain with large number of vortices) is found to be enhanced significantly by the presence of superdiffusive transport characterized by U v y flights that connect different parts of the flow.

1. Introduction There has been a significant amount of research during the past three decades on front propagation and pattern formation in reaction-difSuusion (RD) systems'.'. The paradigm for RD systems is the Belousov-Zhabotinsky (BZ) chemical a reaction that can oscillate almost periodically for several hours when well-mixed. When poured into a petri dish with no flow, however, target and spiral patterns form, due to an interaction between local oscillations of the chemistry and diffusive coupling between different parts of the system. The BZ system has been studied extensively since the patterns formed are similar in many respects to those found in a wide variety of RD systems, including spiral * This work is supported by US National Science Foundation grants DMR-0404961, DMR-

'

0703635 and PHY-055290. Email address: [email protected]. Current address: Department of Physics, University of Maryland, College Park, MD 20742, USA; email: [email protected]. Current address: Department of Physics, Columbia University, New York, NY 10027, USA; email: [email protected].

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waves of electrical activity in the heart5, waves of “spreading depression” in the brain that are responsible for migraine headaches6, and patterns that form in populations of slime mold in a dish7. By definition, a RD system has no fluid flows. Most fluid systems, however, are not stagnant; fluid flows dramatically enhance mixing well beyond that due to molecular diffusion alone. Despite the importance of fluid mixing on the pattern formation process, the more general advection-reaction-difision (ARD) problem has only recently begun to receive significant attention, and most of this attention has been t h e o r e t i ~ a l ~1 ~~1~2 ~~1 3’.9~1The 4~ ’ issue is of particular interest in light of ~ t u d i e s that ’ ~ indicate that fluid mixing can be chaotic even for simple, well-ordered, laminar fluid flows. The importance of chaotic mixing in ARD systems has particular relevance for the design of microfluidic devices (“factories-on-a-chip”), cellular-scale processes in biological systems and for understanding the spreading of diseases in a moving population. In this article, we present results of some experimental studies of ARD dynamics. The reaction is the Belousov-Zhabotinsky reaction, and the flow is a chain of counter-rotating vortices with both oscillatory and drifting timedependence, a flow that has been shown to produce chaotic mixing. Two experiments are discussed: (1) the effects of cellular flows on the propagation of fronts; and (2) the effects of chaotic mixing and superdiffusive transport on the collective dynamics and synchronization of oscillatory reactions.

Figure 1 . (a) Sketch of the alternating vortex chain. The entire chain of vortices can oscillate and/or drift in the lateral direction. (b) Poincar6 section for oscillating vortex chain, showing ordered regions of transport in vortex cores and chaotic region around and between vortices. (c) Poincar6 section for oscillating and drifting vortex chain.

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The alternating vortex chain flow and its mixing properties are discussed in Section 2. Details about how the flow is produced experimentally and about the chemistry are presented in Section 3. In Section 4, we describe the experiments on front propagation in this system, along with numerical simulations of the same phenomena. Experiments on the collective dynamics of oscillating reactions - and the impact of chaotic mixing on these dynamics - are described in Section 5.

2.

Chaotic mixing in the alternating vortex chain

The alternating vortex chain used in these studies is shown schematically in Figure 1. A simple model of the velocity field of this flow (assuming free-slip boundary conditions and no three-dimensional components to the flow) is as follows:

A 2x X = -U -co~(-x~)sin(-) A

2d

XY

d

2X XY y = U sin(--;-x,)cos(-) A

d

VO

x, =x+-sinuX+v,t

w

The vortex chain can oscillate and/or drift in the lateral direction with maximum oscillation speed v, and drift speed vd. If the flow is stationary (v, = vd = O), then tracers in the flow follow closed, ordered trajectories within the vortices. Long-range transport is achieved via a combination of advection of tracers around the vortices and diffusion of tracers If the vortex chain oscillates across the separatrices between vortice~’~’’~. laterally (v, # 0, v d = O), tracers near the separatrices follow chaotic trajectorie~’~”~ and move between adjacent vortices, as shown in a PoincarC section (Figure 2a). Tracers near the vortex centers follow ordered trajectories, remaining confined to a single vortex. Long-range transport in this flow has been shown experimentally to be typically enhanced diffusion, with a variance d(t)= 2D’t with D’ an enhanced diffusion coefficient. The addition of drift to the vortex chain (v, # 0, v d # 0)changes the chaotic mixing. If v d > vo, the chaotic region often divides into two separate regions, with an additional ordered, snake-like region winding around and between the vortices. Transport in this case is superdiffusive with the variance growing as d ( t ) tYwith I < y < 2. Tracers in a chaotic region in this case follow Lkvy

-

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flight trajectories, alternately sticking to ordered regions within vortex cores and undergoing long, snake-like flights between distant parts of the vortex chain. The oscillating/drifting vortex chain is notable by the fact that superdiffusion can be “turned on” or “turned off’ by adjusting the relative magnitudes of v, and vd.

Figure 2. Experimental apparatus. (a) Exploded view of magnetohydrodynamic forcing, along with the annular vortex chain. (b) Side view of apparatus.

3. Experimental methods The flow is generated with a magnetohydrodynamic technique, shown in Figure 2. Two rings of %”-diameter magnets are arranged in a circular piece of plexiglass, which is mounted coaxially onto a voltage-programmed motor. Above this magnet assembly is a shallow cylindrical container with a central electrode, an outer electrode, and two (slightly raised) plastic rings which define the region of interest. A thin layer ( 2 mm deep) of an electrolytic solution (usually the BZ chemicals) carries a radial electrical current that interacts with the magnets to produce an annular chain of vortices. Lateral movement of the vortex chain is accomplished by rotating the magnet assembly either periodically (for oscillating time dependence), with a constant angular velocity (for drifting time dependence) or a combination of the two. and Previous publications describe the chemistry used for the oscillatory22923 BZ reaction in these experiments. A key aspect of the excitable reaction is the use of Ruthenium as a catalyzer. The Ru-catalyzed BZ reaction is photosensitive; illumination with blue/green light inhibits the reaction. We use a video projector to project a red ring over most of the annular region, and blue/green everywhere else to limit the reaction to the region of interest and to control its propagation direction. Details about the techniques are described in Ref. 21.

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(a)

(b)

Figure 3. Simulations showing mode-locking with (N,M) = (1,l) for (a) and (1,2) for (b).

4.

Front propagation

4.1. Mode-locking in oscillating vortex chain

A numerical study of front propagation by Cencini et alto predicted modelocking of fronts in the oscillating vortex chain. In general, mode-locking is when an oscillating system matches its frequency to that of a periodic external driving. In the case of a front propagating in the oscillating vortex chain, mode-locking is manifested as the front moving an integer number N of wavelengths (with one wavelength equal to two vortices in the flow) in an integer number M of drive periods. Simulations of mode-locking are shown in Figure 3; experimental sequences of similar mode-locking behavior are shown in Figure. 4.

Figure 4. Experimental sequences showing mode-locking with (N,M) = (1.1) for (a) and (1,2) for (h), Each image is acquired one period of oscillation after the previous one.

182

03

P

02

01

0

oa

02

04

a5 V

as

90

12

0

02

04

v

06

08

1

Figure 5. (a) Experimental results showing non-dimensional front speed 6 as a function of nondimensional frequency V. The dotted lines show the theoretical predictions (with no fitted parameters) for mode-locked speeds. (b) Parameter-space plot showing Amol’d tongues for (1.1) and (1,2) mode-locked states. Filled diamonds denote unlocked states, whereas open squares, open circles and open triangles denote states with (l,l), (1.2) and combination (1,1)/(1,2)mode-locking, respectively.

The velocity at which a mode-locked front propagates is well defined: vf = NmMT = (N/M)@. Velocities of propagating fronts are plotted in Figure 5a, along with the theoretical predictions for mode-locked velocities. (There are no fitted parameters in these plots.) Mode-locking is apparent over a wide range of frequencies for both the (1,l) and (1,2) mode-locking regimes. There is also significant region of overlap between the two where the velocity alternately (and erratically) switches between the (1,l) and the (1,2) mode-locking values. A parameter-space diagram showing the amplitudes and frequencies for the (1,l) and (1,2) mode-locking regimes is shown in Figure 5b. Once again, a significant overlap region is found. Mode-locking behavior is quite robust in this system. It is of interest that the numerical predictions are for a two-dimensional flow with free-slip boundary conditions, whereas the experimental flow has no-slip boundary conditions and a weak, secondary, three-dimensional flow due to Ekman pumpingz4. Despite these quite significant differences in the details of the flow, the mode-locking behavior seen in the experiments is identical to that predicted numerically.

4.2. Freezing of fronts in the presence of a uniform wind The drifting vortex chain (v, = 0, vd # 0) is mathematically equivalent - if a transformation is made to a co-drifting reference frame - to a stationary vortex chain with an imposed uniform wind. For a front propagating against this wind (in the co-drifting frame), three types of behavior are possible2’: (1) if the wind (denoted non-dimensionally as & = W/V,~,where vrd is the reaction-diffusion - no

183

Figure 6 . Experimental sequences showing front propagation in the presence of an opposing, uniform wind blowing right-to-left. (a) Small wind; front propagates forward to the right. (b) Intermediate wind; front is frozen in the leading vortex. (c) Strong wind; front is blown back to the left by the wind. (d) Sequence showing front frozen in a random vortex flow with opposing wind.

flow - front propagation speed) is small, then the front propagates forward against the wind; ( 2 ) for intermediate values of E, the front remains “frozen” trapped in the leading vortex, neither propagating forward nor being blown back by the wind; (3) for large values of E, the wind blows the front backwards. These three different regimes are shown in Figures 6a, b and c, respectively. Figure 7 shows front velocities (denoted non-dimensionally by v = v f / v r d ) for three different values of the non-dimensional vortex strength p = U/vrd. The dashed diagonal line corresponds to the case with no vortex flow (U = 0); in this case, the effects of the wind are a simple Galilean transformation and v = I - E. A striking feature of this behavior is the large range of wind speeds over which

184

E

.o

Figure 7. Non-dimensional front propagation speed v as a function of non-dimensional wind E with p = 4 (ope diamonds), 12 (filled circles) and 40 (asterisks). The dashed line is the theoretical limit for p = 0.

the fronts are frozen (V = 0); for p = 40, fronts remain frozen at almost 10 times the RD front propagation velocity. The width of the frozen front regime collapses onto the p = 0 result for small vortex strength. The minimum wind for freezing of fronts is E = 1 for all values of p. This makes sense, since a front must burn across a vertical separatrix between vortices to propagate forward. Since the flow is perpendicular to the front propagation direction at the separatrix, it can’t propagate forward if the wind exceeds the RD propagation speed. Freezing of reaction fronts is not limited to ordered vortex flows. Experiments have also been done with random vortex flows in which similar freezing behavior is seen (Figure 6d). Details of the experiments with random flows are reported elsewherez5. The experiments reported in this section are all with stationary flows. Nevertheless, the results should have implications for front propagation in a wide range of time-dependent, two-dimensional flows that are dominated by vortex behavior (which is very common in 2D flows). A moving vortex in a time-dependent flow can often be viewed (temporarily) in a co-moving reference frame as a stationary vortex with an imposed wind. From this perspective, a

185

vortex passing through a reaction front traveling in the same direction should be expected to pin and drag the front. Both the freezing front behavior in this section and the mode-locking behavior from the previous section indicate the critical importance of vortex structures in the propagation of reaction fronts in a flow system. These experiments indicate the importance of considering coherent vortex structures in any general theory of front propagation in ARD systems.

5. Collective oscillatory behavior and synchronization by chaotic mixing The ARD behavior is quite different if the oscillatory version of the BZ reaction is used in the vortex chain flow. In this case, each vortex and its contents act like an oscillating node of a network, and communication between these nodes is via chaotic mixingz3. The problem is particularly interesting in light of recent studies of networks and the manner in which they are connected. As discussed in Section 1, oscillatory time-dependence of the vortex chain (or combination oscillation and drift with v d < v,) results in chaotic mixing with enhanced diffusive transport. Diffusive behavior is analogous to a network with wellordered, nearest-neighbor connections. Superdiffusive transport (for v d > v,) is associated with Levy flights that can travel long distances in a short period of time. These flights are similar in many respects to “short-cut” connections in Small- World network modelsz6. So, the oscillating/drifting vortex chain flow gives us the ability to study how the large-scale collective behavior of a fluidbased network is affected by the type of transport (diffusive or superdiffusive). If the vortices are stationary (vd = v, = O), then communication between them is via molecular diffusion across the separatrices; coupling in this case is very weak and the vortices are essentially isolated. With the oscillatory BZ reaction in this flow, the contents of each vortex oscillate almost periodically, independent of the rest. Oscillations of the BZ reaction may start off initially synchronized (due to mixing in the apparatus when the chemicals are added) but de-synchronize within a few periods of the chemical oscillations. Ultimately, the chemicals in each vortex oscillate independently of the rest. If oscillatory and/or drifting time-dependence is added to the vortex chain, three types of collective behavior are observed (Figure 8). In every situation in which the transport is diffusive (e.g, for vd < v,), aperiodic traveling waves are found, regardless of the initial conditions. (Figure 8a). Sometimes the waves travel counterclockwise around the annulus and sometimes clockwise. Often the waves emanate from a source and travel in both directions around the annulus to

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\-I

Figure 8. Sequences of images of BZ reaction in o s c i l l a t i n ~ d ~ ~ ivortex n g chain. (a) Wave behavior seen when v,i < v,. (b) Co-rotating and (c) global synchronization for v d > v,,.

a sink on the other side, with the locations of the source and sink drifting with respect to the chain. The behavior sloshes continuously and unpredictable between these different states; we have not encountered a situation where the traveling waves remain consistent in their behavior. If the transport is superdiffusive (vd > v,) then one of two types of global synchronization are observed. In most cases, “co-rotating” synchronization is observed (Figure 8b), where odd vortices are all synchronized with each and even vortices are all synchronized with each other, but where the phase difference between the two chains can be anything. The lack of synchronization between the two chains in this case is due to the isolation of the two chaotic regions that is usually (but not always) found for flows with vd > v, (see Fig. Ic). ixing in these chaotic regions skips adjacent vortices, connecting odd vortices to odd vortices and even vortices to even vortices. In some cases if v d is just

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slightly larger than vo, Levy flight trajectories and superdiffusion are possible even for a situation with only one, connected chaotic region. In this case, global synchronization is found with the BZ chemicals in all vortices oscillating in unison (Figure 8c). More details about the experimental results are presented in Ref. 23. The implication of these results is that superdiffusive transport may be a necessary (although probably not sufficient) condition for synchronization in an extended fluid system, i.e., one in which the total system size is appreciably larger than characteristic length scales of the fluid flow. These results could be important in interpreting synchronization behavior found in natural systems, such as algae blooms in the Gulf of Mexico and phytoplankton blooms in the Atlantic Ocean.

Acknowledgments This work was supported by the US National Science Foundation Grants DMR0404961,DMR-0703635,REU-0097424,and REU-0552790.

References

1. P. Grindrod, The Theory and Applications of Reaction-Diffusion Equations: Parterns and Waves (Clarendon Press, Oxford, 1996). 2. D. Ben-Avraham & S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000). 3. A. T. Winfree, Science 175,634-636(1972). 4. K.Showalter, J. Chem. Phys. 73,3735-3742(1980). 5. Z. L.Qu, J. N. Weiss & A. Garfinkel, A. J. Physiol. -Heart Circ. Physiol. 276,H269 (1999). 6. M.A. Dahlem & S. C. Muller, Annalen der Physik 13,442(2004). 7. D. A. Kessler & H. Levine, Phys. Rev. E 48,4801(1993). 8. T. Tel, A. de Moura, C . Grebogi & G. Karolyi, Phys. Rep. 413,91-196 (2005). 9. M.Abel, A. Celani, D. Vergni & A. Vulpiani, Phys. Rev. E 64, 046307 (2001). 10. M. Cencini, A. Torcini, D. Vergni & A. Vulpiani, Phys. Fluids 15,679-688 (2003). 11. G. Karolyi, A. Pentek, Z. Toroczkai, T. Tel & C. Grebogi, Phys. Rev. E 59, 5468-5481(1999). 12. G. Karolyi, A. Pentek, I. Scheuring, T. Tel & Z. Toroczkai, Proc. Nut. Acad. Sci. U.S.A. 97,13661-13665(2000). 13. Z.Neufeld, Phys. Rev. Lett. 87,108301 (2001).

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14. Z. Neufeld, I. Z. Kiss, C. Zhou & J. Kurths, Phys. Rev. Lett. 91, 084101 (2003). 15. H. Aref,J. Fluid Mech. 143, 1-21 (1984). 16. B. Shraiman, Phys. Rev.A 36,261 (1987). 17. T. H. Solomon & J. P. Gollub, Phys. Fluids 31, 1372 (1988). 18. T. H. Solomon & J. P. Gollub, Phys. Rev. A 38,6280-6286 (1998). 19. T. H. Solomon, S. Tomas & J. L. Warner, Phys. Rev. Lett. 77, 2682-2685 (1996). 20. M. S. Paoletti & T. H. Solomon, Europhys. Lett. 69, 819-825 (2005). 21. M. S. Paoletti & T. H. Solomon, Phys. Rev. E 72,046204 (2005). 22. C. R. Nugent, W. M. Quarles & T. H. Solomon, Phys. Rev. Lett. 93, 218301 (2004). 23. M. S. Paoletti, C. R. Nugent & T. H. Solomon, Phys. Rev. Lett. 96, 124101 (2006). 24. T. H. Solomon and I. Mezic, Nature 425,376 (2003). 25. M. E. Schwartz and T. H. Solomon, Phys. Rev. Lett. (in press). 26. D. J. Watts & S. H. Strogatz, Nature 393,440 (1999).

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NON-DIFFUSIVE TRANSPORT IN NUMERICAL SIMULATIONS OF MAGNETICALLY-CONFINED TURBULENT PLASMAS R. SANCHEZ1,*, B. A. CARRERAS2, L. GARCIA3, J. A. MIER3, B. PH. VAN MILLIGEN4 and D. E. NEWMAN5 Fusion Energy Division, Oak Ridge National Laboratory, Oak Ridge, T N 37831, U S A 2 B A C Solutions, Oak Ridge, T N 37830, U S A Universidad Carlos III de Madrid, 28911 Madrid, Spain Laboratorio Nacional de Fusidn, Asociaci6n E U R A T O M - C I E M A T , 28040 Madrid, Spain University of Alaska-Fairbanks, Alaska 99775, U S A In this contribution, we discuss some recent advances in the application of diagnostics derived from both continuous-time random walks (CTRWs) and fractional differential equations (FDEs) t o the characterization of non-diffusive transport features in numerical simulations of magnetically-confined turbulent plasmas in a variety of situations. An introduction to the main ideas and mathematics of CTRWs and FDEs is also provided.

1. Introduction One of the most promising methods to produce net energy from thermonuclear fusion in an economically viable way is plasma magnetic confinement in a toroidal magnetic configuration, such as a tokamak. However, a plasma confined in such a way is forced to stay very far from thermodynamical equilibrium due to the enormous density and temperature gradients that need to be sustained. Spontaneously, the plasma attempts to reduce these gradients by inducing particle and energy fluxes directed radially out of the machine via several mechanisms (namely, collisions and turbulence). These processes limit the maximum energy confinement time that may be achieved and the performance of these devices. Their understanding and control is, therefore, of extreme practical irnp0rtance.l 'Corresponding autlior: [email protected]

190

The usual theoretical approach to describe these fluxes is to relate them locally to the thermodynamic forces that induce them (pressure and temperature gradients, etc.) through diffusivities and conductivities. This is the basis of both neoclassical transport theory2 and most of the many renormalization approaches (mostly, “eddy-diffusivity”-type) to turbulent transHowever, the assumption that fluxes and forces are locally related (understanding “locally” in the sense that fluxes depend only on the local value of the forces and the state of the system), does not appear to be supported by experimental evidence. Commonplace observations such as the Bohm scaling of the energy confinement time,7 the existence of canonical profiles in the presence of on-axis and off-axis heatings or the superdiffusive propagation of cold and hot pulsesg-” are difficult to reconcile with a local (i.e., diffusive) description of transport unless mysterious, fast and inward directed radial pinches are invoked.12 These pinches are in most cases difficult to justify physically and seem to depend heavily on the experimental

condition^.'^>^^ Theoretically, an important effort has been devoted in the last few years to propose a plausible origin for these experimental findings. It was soon pointed out that the observation of power degradation and canonical profiles might be related to the existence of some local critical threshold above which a superdiffusive transport channel might become active.15 Experimental evidence of the existence of these thresholds is available in several devices.16 This ingredient was also put forward, combined with the idea of a plasma whose profiles are kept very close to their local stability limits via external drive, as a possible explanation for the observation of superdiffusive propagation. The idea was borrowed from the “self-organized-criticality” arena:17 since the profiles are locally very close to unstable, any small perturbation could trigger an avalanching event in which nearby locations are successively destabilized and the perturbation travels by almost ballistically. Following this thread of thought, it appeared to be justified that transport would thus lack any characteristic length or time scale. Indeed, the radial extent of this avalanche events will only be limited by the system radius, which may explain the observation of non-diffusive global confinement scalings in these devices. In addition, the profile erosion caused by these avalanches would also alter the future dynamics in the system, endowing it with some kind of long-term “memory” and, thus, a loss of a characteristic time scale. A certain amount of experimental evidence has been reported that seems to confirm this lack of characteristic scales in space and time, at least to a certain extent, in L-mode tokamak discharge^.^^-^^ For instance, some

191

evidence of avalanche propagation was reported in the DIII-D tokamak using temperature fluctuation measurements taken with the Electron Cyclotron Emission (ECE) system. As shown in Fig. 2 of Ref.,21 time traces of temperature fluctuations for ECE channels looking at different radial locations inside the device clearly show the presence of disturbances that propagate radially outward and inward, with the extent of their excursions only limited by the system radius. It must be said, however, that the role that these avalanche events play in energy transport and the overall power balance in the plasma remains unknown. Similar evidence is also available concerning the lack of characteristic time scales. The analysis of time-series of Eulerian edge fluctuation data reveals, from the inspection of their Hurst exponents,22 p o ~ e r - s p e c t r a ~elf-similarity~~ ,~~ and quiet-time statistic^,^^ the presence of long-term memory on the systems. The extent to which this situation impacts the global confinement properties can however only be conjectured upon at this point. Due to the lack of appropriate experimental diagnostics that could confirm these ideas, support for them has been sought in numerical simulations of turbulent plasmas of varying degree of complexity and realism. The pioneer work in this regard was that of Carreras, Newman, Lynch and Diamond using fluid interchange turbulence simulations in cylindrical geometry.26 In it, evidence was found that points to the self-consistent interaction between turbulent fluctuations and background profiles (which provides the free-energy source for the turbulence) as the cause for the disappearance of characteristic scales in that particular system. Since then, simulations have been refined and repeated in other fluid turbulent plasmas (including i n t e r ~ h a n g e ,ion-temperature-gradient28 ~~ and dissipative trapped electron mode t ~ r b u l e n c e ~in~ both ) cylindrical and toroidal geometries that seem to support this conclusion, even if the presence of other subcritical mechanisms (such as neoclassical diffusion) has caused some controversy on whether such plasma state should be termed self-organized criticality or n0t.27129 What seems to stem clearly out of this enumeration of experimental findings and theoretical conjectures is that, if it is indeed the case that the dominant transport mechanism in tokamak plasmas is scale free in space and/or time, it seems doubtful that any local (diffusive) mathematical framework could capture the relevant dynamics appropriately and reproduce the experiments observation^.^^ The main reason for anticipating this failure is that diffusivities, conductivities (and any other transport coefficient) lose their meaning if well-defined characteristic lengths and times

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at a microscopic scale (i.e., the banana orbit width or Larmor radius and the inverse collision or bounce frequencies in neoclassical transport,2 and the radial coherence length or decorrelation time for turbulence) cease to exist. Note that, by “characteristic”,we mean much shorter than the system size and/or lifespan. And clearly, avalanches with an extent only limited by the tokamak minor radius have a characteristic size (mean) that diverges with the system size: S 0: L”, v > 0. The temporal counterpart of this situation happens whenever memory effects persist in the system, in one way or another, for as long as the discharge lasts. Fueled by these ideas, a large effort has been devoted in the last few years, to look for alternative formalisms that can capture dynamics in the absence of characteristic scales and provide a useful effective description of transport in the system. Two possibilities, both well known and with widespread use in other fields of physics an s ~ i e n c e have , ~ ~ been ~ ~ ~examined in detail: continuous-time random walks (CTRWs) and fractional differential equations (FDEs). Both frameworks provide a suitable generalization of classical diffusive transport which allows the treatment of situations in which spatial and/or temporal characteristic scales are absent. Furthermore, CTRWs and FDEs are closely related. In the limit in which only long-distance, long-term dynamics are retained (the so-called “fluid limit”),most CTRWs can be recast in terms of fractional differential equat i o n ~ . Note ~ ~ >that ~ ~this fact does not imply that a one-to-one mapping exists between CTRWs and F D E s , ~ but ~ applies to the majority of CTRWs of interest for our problem. Of the several possible application of CTRWs and FDEs to fusion plasmas, we will restrict our discussion in this paper to just one of them: the characterization of non-diffusive transport features in numerical simulations of turbulence. The discussion does not intend to be comprehensive and it will intentionally show a strong bias toward our own work in this area. But before proceeding any further, it is worth mentioning that an important amount of work has also been devoted to apply CTRWs and FDEs to the construction of simple effective or paradigmatic models of (non-diffusive) radial transport in t o k a m a k ~ . ~For ~ ~more ’ information on this type of work, we refer the reader to those references and the bibliography they contain. The structure of the paper is thus as follows: first, a brief introduction to the mathematics and meaning of both concepts will be given. We will start by reviewing the fundamentals of CTRWs in Sec. 2. Then, in Sec. 3, we will discuss the basic ingredient needed to construct CTRW models: L6vy d i s t r i b ~ t i o n sIn . ~Sec. ~ 4,we will briefly introduce what frac-

193

tional differential operators are and we will show that CTRWs are indeed connected to evolution equations that contain fractional differential operators in the limit of long distances and times. Finally, in Sec. 5 , we will discuss how they can be used as a powerful diagnostic tool to characterize the onset of non-diffusive behavior and briefly discuss their application to some numerical simulations of turbulence in magnetically confined plasmas of varying complexity. Further details can be found in the references supplied. 2. Continuous-time random walks The CTRW is a generalization of the standard random walk.42 In its simplest (separable) form, it describes the motion of an arbitrary number of particles (or walkers), each of which waits at its current position r’ for a lapse of time At (a waiting-time) before taking a step of size Ar (a step-size) and moving to r = r’ Ar. After arriving at the updated location, a new waiting-time is chosen and the process is repeated over and over. Assuming that the system is invariant under time and space translations, At and Ar are drawn by each walker from two prescribed probability density functions (pdf), p(Ar) and $(At)which contain all the dynamical information of the system. Therefore, to apply the CTRW construct to any problem, we simply need to choose these two pdfs in a manner that captures the fundamental microscopic physics of the problem. That, of course, is the difficult part. We will discuss at length in Sec. 3 which pdf choices, among the infinite number available, appear to make the most physical sense. But for the time being, it is sufficient to realize that, once the pdfs are known, the time evolution of the density of walkers n(r, t)can be described by the following generalized master equation (GME):

+

that simply states that the total number of particles is conserved.43 Indeed, the first (positive) term within brackets counts how many walkers move from r’ to r by performing a jump of appropriate length. The second (negative) term in brackets, counts how many walkers leave the position r. Their sum gives the local rate of change in the number of walkers (or more precisely, their density). Two additional facts must be kept in mind regarding the GME. First, we deal with the function 4, usually called the memory function. Its

194

Laplace transform is related to the waiting-time pdf through the relationship: d ( s ) = s$(s)/(l - $(s)). Note that, were the triggerings of successive jumps uncorrelated (i.e., if there is no memory in the process), the generation of the waiting-times must then be a Poisson process, which requires that $(At)= 7F1exp(-At/TO), being 7 0 the mean waiting time. Then, use of the previous relation shows that d ( t ) = ~ G l d ( t )The . GME reduces then to a more standard Markovian master equation:

Clearly, the past history of the system no longer plays any role once the time integral has disappeared. The second choice to make refers to the pdf of step-sizes, p . The most usual choice, in the absence of motion bias, is the Gaussian law: p ( A r ) = ( 2 7 r ~ ) - e~~/ p~ ( - l A r 1 ~ / 2 u ~ This ) . choice, together with the exponential waiting-time pdf, allows to connect the CTRW with a very familiar equation: the classical diffusive equation. Indeed, rewrite Eq. 2 as:

carry out a simple Taylor expansion in around r, use the form of p ( A r ) and keep only the lowest order to obtain: dn(r t )

2= D V 2 n ( r , t ) ,

dt

D = u2 / T O .

(4)

The diffusivity, as expected, is obtained as the quotient between the average squared displacement, c2, and the average waiting-time, 70. (T gives then the magnitude of the mean step-size. These two moments provide the characteristic scale and characteristic time that must exist in order for any diffusive description to make sense. It is here that the power of the CTRW formalism is unveiled. It lies on the fact that p ( A r ) and +(At)need not be respectively Gaussian and exponential. Other pdfs can be chosen, which extends the range of applicability of CTRW models to include non-local and non-Markovian situations. In particular, if members of the family of Levy-Gnedenko distributions4’ are chosen. We briefly review these distributions in the next section. These situations often occur in practice and reveal themselves, for instance, when measuring characteristic transport exponents of passive quantities or tracers by some flow [Another beautiful example, in the context of stochastic magnetic fields, can be found in Ref.44].A well known fact about the classical diffusive equation is that it implies the following scaling for the mean

195

tracer square displacement:

-

(lArI2) Dt. (5) In many practical cases, it is however found to scale as t Z H with , H # l/2.31132When H < 1/2 one typically speaks of subdiffusion.When 1/2 < H < 1, of superdiffusion. H is known as the transport exponent, and it will be very important in what follows. 3. LQvy distributions

When implementing any CTRW model, one always has to make two choices: a waiting-time pdf and a step-size pdf. Since these choices must somehow reflect the cumulative effect of may microscopic processes (such as, for instance, molecular collisions), it seems physically well justified to use pdf forms that satisfy the central limit theorem.45 That is, limit distributions that are strictly stable with respect to the s u m of independent and identically distributed (i.2.d.) r a n d o m variables. These pdfs are known as the Lkvy (or Lhy-Gnedenko) family of pdfs. They contain as a special case the Gaussian distribution, which becomes the only stable distribution if each of the random variables is also required to have a finite variance.41 The Lkvy family is defined in terms of three parameters. Its members are denoted by L,,x,+,(y). They can be defined in closed form in terms of their Fourier transform or characteristic function as (0 < a 5 2, 1x1 5 1):41

I)?(

L,J,~(= ~ )exp [-calkla ( 1 - i X sgn(k) tan

.

(6)

The three labels [a,A, c]define the properties of each distribution. First, X measures the asymmetry of the distribution. This comes from the fact that: L+7(Y)

= La,-x,u(-y).

(7)

It can vary within -1 5 X 5 1 except when a = 1,2, for which only X = 0 is possible. Secondly, a gives the asymptotic behavior of the distribution a t large y. All Lkvy distributions exhibit heavy tails if 0 < CY < 2 . In fact, for Qf1,

La,x,u(Y)

{

c, (9) calyl-(l+a), c, (F) ,a(y(-(l+a),

where the constant is given by:

y

-+ -03

(8) y -+ +m

196

-

In the special case a = 1, the PDF decays as Ll,O,,,(y) (0/7r)IyI-~. And when a = 2, one recovers the standard Gaussian distribution. Finally is called a scale parameter because: La,x,o(ay) = pa, sgn(a)X,lalo(Y)

(10)

Extremal Livy distributions Waiting-time distributions must satisfy an additional constraint: they must be defined only for positive waiting-times! In order to satisfy this property within the LBvy family, we are limited to the subset of the extremal LBvy distributions, which are those with maximum skewness value: X = f l for a # 1,2. In this case, according to the previous equations, the power-law decay is only observed in one tail, the other decaying instead exponentially. In the case of 1 < a < 2, X = +1 implies that the exponential tail exists for y -+ -00, while X = -1 has an right exponential tail for y m. For 0 < cy < 1 the extremal distributions are ~ne-sided:~' they are defined only for y > 0 if X = 1 and for y < 0 if X = -1. In that case, the exponential tail is found in the limit y -+ O+ for X = 1, and for y -+ 0- for X = -1. An important property is that their Laplace transform is given by: --$

Moments of Livy distributions The reason why Levy distributions with a < 2 are appropriate choices for step-size pdfs if we are interested in constructing a CTRW model with non-local features is because of the following property: all moments higher than a are infinite. That is, the momenta of L , J , ~ ( Y )verify:

where the coefficient is not relevant for our discussion (it can be found in Ref.41). Thus, only the Gaussian distribution ( a = 2) has a finite variance. As a result, the characteristic transport length provided by u in the case of a Gaussian ceases to exist for a < 2. Transport is, in this sense, non-local and scale-free.

Explicit expressions of Livy distributions There are only three LBvy distributions for which an analytical expres-

197

sion exists.41 The Cauchy distribution. Its real space representation is:

the Gauss distribution,

(note that the relation of u with the usual width w of the Gaussian is thus 2u2 = w2) and the Lkvy distribution, U 112 1 J51/2,l,O(Y)= 2” - e-a’2y. y3~2 To conclude, we will give some hints on how to use the LQvy pdfs to choose the pdfs to construct a CTRW model with certain desired transport properties. As we mentioned before, CTRWs are useful to model transport in systems where subdiffusion or superdiffusion O C C U ~ S The . ~ ~ way ~ ~ to ~ do it is to remember that, if we choose the symmetric LQvypdf L , , O , ~ ( Aas~ ) step-size pdf, and the extremal Levy pdf Lo,1,7(At)as waiting-time pdf, the mean particle displacement follows the scaling:

(-)

Note that this relation implies that the transport exponent H , which we introduced at the end of the previous section, is given by H = p/a. Thus, subdiffusion ensues whenever p / a < 112 and superdiffusion when p / a > 112. The correct ratio between exponents is thus set by the observed transport exponent H . Next, appropriate considerations about locality and Markovianity can be used to determine their precise values. 4. Fluid limit of CTRWs: Fractional differential equations

To conclude our review of the fundamentals, we will show in what follows how the “fluid limit” of the CTRWs described are rewritten in terms of fractional differential operators. We start by briefly introducing what these operators are. 4.1. Crash course on fractional diffeerential operators

The Riemann-Liouville fractional derivative operators can be defined explicitly by means of the integral operator^:^^

198

In this expressions, r ( x ) is the usual Euler Gamma function, and p represents one plus the integer part of a. a [or b] is called the start [end] point of the operator. In the cases in which the start point a or the end point b extend all the way to infinity, we will use the notation:

These operators have very interesting properties. For integer a they reduce to the standard derivatives. Like them, they are linear. But it is not true that the fractional derivative of a constant is zero. Also, they must be combined appropriately with integer and non-integer derivatives and they do not satisfy the simple chain rule.46 Their non-local character comes from the fact that, to compute the value of the fractional derivative of some quantity at a given point, one has to integrate that quantity over the whole domain! So why bother with them at all if they are so complicated? The reason is that, under Fourier transformations, they satisfy that:

This property is the key to their prominence in CTRW theory, as we will show shortly. Another useful fractional operator is the so-called Riesz fractional as the symmetrization: derivative o p e r a t ~ r , ~defined '

Its usefulness comes from the fact that the Riesz operator verifies, under Fourier transform, that:

F

[c] -Ikl"f(k). 4x1" =

The last fractional operator we will introduce is the Caputo fractional derivative operator, which is defined as:46

where p is one plus the integer part of p. The Caputo fractional derivative is usually associated to derivatives in time. Its non-Markovian character is also clear: to calculate the Caputo time derivative of any quantity, one has

199

to integrate that quantity over all its past history! Its importance comes from the fact that the Laplace transform of the Caputo derivative verifies:46

[

c

P-1

L q t ) ] = so f (s) d,tP

-

sp-k-l

k=O

. -dk ( Of) , dtk

which depends only on the initial values of f ( t ) and its integer derivatives. 4.2. F i n d i n g the f l u i d l i m i t of CTRWs

We have now the tools a t our disposal to calculate the “fluid limit” of a large number of C T R W S , ~including ~ a certain family of nonlinear ones.34 But for simplicity, we will restrict the calculation to the case in which we choose the symmetric LBvy pdf L , , o , ~ ( A z )as step-size pdf, and the extremal L6vy pdf Lp,l,7(At)as waiting-time pdf. The calculation is very simple. By “fluid limit” one should understand an equation that captures the characteristic features of the CTRW transport in the limit of very long distances and very long times. Formally, we do this calculation in the limit of an infinite system. Then, the limit of long distances is equivalent to making k + 0 in Fourier space. Similarly, the limit of long times can be carried out in Laplace space by making s -, 0. Thus, we take the Fourier-Laplace transform of the GME (Eq. 2):

s n ( k ,s) - n ( k ,0) = 4(S)(P(k)- l ) n ( k ,s),

(25)

where we have applied the convolution theorem and the definition of the Laplace transform of a derivative. This equation can in fact be solved to give the Fourier-Laplace transform of the density of walkers:

where we have rewritten the memory function in terms of the Laplace transform of the waiting-time pdf. n o ( k ) is the prescribed (Fourier transform of the) initial density of walkers. Eq. 26 is known as the Montroll- Weiss equa-

t i ~ n . ~ ~ We can now take the fluid limit by taking k + 0 and s + 0 in either the Montroll-Weiss equation or in Eq. 25. To do it, we simply assume that both p and .J, are chosen from within the L6vy family, as the central limit theorem advices. Then, it is trivial t o realize using the properties we discussed before that for small k : P ( k ) = L,,o,,,(k)

2i

1- cPIIC~~.

(27)

200

Similarly, the Laplace transform of positive extremal L6vy pdfs, given by Eq. 11, behaves at small s as: $(S)

= Lp,l,,

1 - A;

1 70 0 S0 .

where we have also included the exponential law if constant: cos(q), 4 = {1,

p

(28) = 1 and defined the

p<1 p=1

Inserting now Eqs. 28 and 27 in Eq. 25, the fluid limit of the MontrollWeiss equation becomes:

n ( s ,k ) cv n o ( k ) [s

+ D[,,p]S1+IklQ] -l

.

(30)

where the coefficient D[,,pl = Apaa/rp has been defined. Eq. 30 can be rewritten as:

sn(s,k ) - n o ( k ) = -D[,,p]S1-PIklQn(s, k ) .

(31)

We can now use the properties of the fractional operators with respect to the Fourier or Laplace transforms. For instance, using Eq. 22 we can Fourier-invert Eq. 31. The result is thus an FDE in space:

Analogously, we carry out the Laplace inversion of Eq. 32 next. We do it by multiplying first both sides by sop1 and then using the properties of the Caputo fractional differential operator (Eq. 23) with respect to the Laplace transform (Eq. 24) to write the following FDE in space and time:

Of course, if a = 2 and p = 1, one recovers the standard diffusive equation. A last thing that must be stressed again before leaving this section is that the linear FDE with exponents /3 and a and fractional diffusivity D[,,pl (Eq. 33) is essentially equivalent to the CTRW with p = L,,O,~(AZ) and .II, = L p , ~ , ~ , , ( n tas ) , long as D[,,p] = Apua/.{ if we only care about long-term, long-distance behaviour. In particular, they both yield the same transport exponent H = p / a . So, in a sense, using one or the other is to a certain extent a matter of personal preference. There are however certain advantages of each of the two approaches. Linear FDEs with constant coefficients have the added advantage that their fundamental solutions or

201

propagators (i.e., the solution at any time starting with a delta function as initial condition) are known a n a l y t i ~ a l l y This . ~ ~ knowledge can be very useful as a diagnostic, for instance, as we will discuss in the next section. CTRW, on the other hand, are easier to understand intuitively. From a numerical point of view, both CTRW and FDEs can be implemented quite easily.46The main difference between them comes probably in the boundary conditions they can handle. CTRW usually require absorbing or reflecting boundary conditions. FEDs, on the other hand, can deal with Neumann or Dirichlet boundary conditions (but only after regularizing first the fractional operators close to the limits of the integrals to avoid divergence^^^). 5. Applications to numerical simulations of turbulence in

magnetically-confined plasmas Both CTRWs and FDEs can be used advantageously to develop diagnostics that can characterize the nature of transport in plasma turbulence simulations, either fluid or gyrokinetic. In the former type of codes, fluid equations are advanced in time for the plasma density, pressure and velocity fields plus some relation for the electric field (or some related quantities, specially in reduced models) in the presence of a time-independent magnetic field. In gyrokinetic codes, the kinetic equation for ions (and sometimes also electrons) is solved coupled to the Poisson equation for the electric field, also with the magnetic field prescribed. Since CTRWs and FDEs provide particle-related diagnostics, tracer particles must then be added to all fluid ~ o d e s . * Also, ~ - ~ ~tracers need to be added to those gyrokinetic codes which solve the coupled Vlasov-Poisson set of equations using continuum methods, as recently done in Ref.53 In both cases the turbulence is assumed electrostatic, and the local drift velocity is v = (E x B)/B2 (here, E and B are respectively the fluctuating electric and background magnetic field), which is used to push the tracers. The only codes for which tracers need not be added are gyro-kinetic PIC codes, which solve instead the Vlasov equation using particle (PIC) methods.54 Here one can apply the diagnostics on real gyro-kinetic particles, typically ions, as recently done with the UCAN55 PIC gyrokinetic code.56 The particle diagnostics are quite simple and have been used widely in other fields of science. In the case of CTRWs, the nature of transport requires the evaluation of the tail exponents of the waiting-time (p) and step-size ( a )pdfs. This can be easily done, but only after deciding on a reasonable definition of what a “waiting-time” and a “step-size” mean in the system of interest, which is not always easy. Then, one constructs the prob-

202

ability density function (pdf) of the waiting-time/step-sizes of the tracer particles and determine the power-law tail exponents, if present. In the case of the FDE, one can take advantage of the analytic propagator expressions for any given a and p. Since a propagator is simply the time evolution of an initial delta-function, one can numerically obtain them by concentrating all tracers around a certain location and following the spreading of the particle distribution with time. Another way to construct this propagator, particularly useful in PIC gyro-kinetic codes (for which particles are distributed uniformly initially) is to form the pdf of particle displacements with respect to their initial position as a function of time. Comparison of these numerical propagators with the analytical expressions would yield again (u and p. Yet another possibility is to estimate the transport exponent H directly. This can be done by several methods, such as by computing moments of the particle displacement pdf as a function of time, or by computing the traceraverage of the Hurst exponent or the cumulative autocorrelation function of the Lagrangian velocity time-series of each tracer.57 Then, one computes one of the other two exponents, a or p, and assume that H = P/(u holds. This was precisely the path followed by Cameras, L y n c h and Z a s l a ~ s k y . ~ ~ They added tracers to the same cylindrical fluid simulations of plasma interchange turbulence used previously to test the idea of self-organized criticality as a possible explanation of the situation experimentally observed in many tokamaks, as mentioned in the introduction.26 They then estimated H 0.84 (i.e., superdiffusion) from the second moment of the tracer displacements (see their Fig. 2), and P 0.81 from the the waiting-time pdf (Fig. 7 in their paper). It is worth to revisit their definition of waiting-time, since in turbulent systems particles rarely stay quiet. In this geometry (a cylinder which is periodic along its axis of symmetry), the most unstable radial locations are located on a discrete number of radial locations or rational surfaces (which correspond to those surfaces for which the magnetic field lines close on themselves after a finite number of transits along the cylinder axis). Whilst the locations are quiescent, tracers are trapped within them, thus exploring a very reduced radial region. At certain times, however, due to the interaction of the fluctuations with the background, avalanches may be triggered in which many of these locations become successively destabilized. Tracers are then transported very long distances, sometimes as long as the system radial size. Thus, an operative definition of the waiting-time can be given as the amount of time that the particle does not move further than a certain radial distance from its initial location. As this example shows, the

-

-

203

“waiting-time” definition must thus be user-supplied (similarly, the “stepsize” working definition). This can be problematic in cases in which particles are not clearly trapped, and the value of the exponents will depend on which precise definition is used. It is particularly in these situations that the FDE-method previously described is most useful. It does not require any user-provided definition to work [Another method also exists, similar in the sense that it does not require any user-supplied concept, to determine H and cy based instead on the statistical analysis of the tracer Lagrangian velocity time series.57]. The propagator can be computed numerically by simply book-keeping the changes in the distribution of tracers as a function of time, after having started all of them from very close radial locations. delCastillo-Negrete, Carreras and Lynch successfully applied this technique to the same cylindrical interchange simulations. They found propagators with power-law tails and determined the exponents (Y 0.75 and p N 0.5, with a related transport exponent H N 0.66 (again, superdiffusive). Variations of these two methods (and the Lagrangian one aforementioned) or combinations of them have been applied to other instances of fluid turbulence. In particular, ballooning turbulence in toroidal geometry50 and dissipative trapped electron mode (DTEM) turbulence in a periodic cylinder geometry.52 The analysis has yielded propagators with power-law tails in all cases in which fluctuations and background profiles are evolved self-consistently, with similar exponent values in all the cases. These studies have provided further numerical evidence of the central role played by the fluctuation-profile interaction loop in eliminating characteristic scales in near-critical turbulent transport, and of the relative insensitiveness of the dynamics with respect to the particular characteristics of each type of turbulence (observation that fits extremely well within the ideas of selforganized criticality). However, in the last fifteen years it has been increasingly realized that a third element must be considered if we intend to achieve a realistic understanding of transport in magnetically-confined turbulent plasmas: sheared flows and, in particular, zonal These flows, which are generated by the turbulence itself, also participate in the self-regulation of turbulent transport by the turbulence itself. They are in fact the usual suspects to explain the transport enhancement observed in H-mode in tokamaks, in which confinement improves after the appearance of an strong edge shearedGyro-kinetic codes are the ideal tool to study this interaction, since the damping of these flows is usually artificial and unrealistically strong in most fluid codes. The drawback is however that, due to the complexN

204

ity of these codes, they do not currently implement the aforementioned fluctuation-profile loop in the proper manner. So again, we can only get a partial picture of the dynamics. But it is enough to suggest that the fluctuation-flow loop is another main contributor to the establishment of scale-free transport, as recently showed by evolving flows and fluctuations self-consistently with the UCAN PIC gyrokinetic code.56 This is an area that will be of the greatest interest in the following years. To conclude, we would also like to make the reader aware that the methodology here described is also being used to study numerical simulations of turbulence which is magnetic in nature, instead of electrostatic. Such a situation would apply, for instance, to the inside of a reversed-fieldpinch c ~ n f i g u r a t i o n In . ~ ~this case, transport happens mainly across a sea of stochastic magnetic fields [Interestingly, this type of situation was previously studied with CTRWs using model fields by Balescu more than fifteen years ago44]. The different transport physics of these systems would undoubtedly make these studies extremely interesting and will open up a new area of application of the techniques described in this contribution.

Acknowledgments Work sponsored in part by the Laboratory Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for U.S. DOE under contract No. DE-AC05-000R22725. Part of this work supported by Spanish DGES Projects No. FTN2003-04587 and FTN200308337-CO4-01.

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10. K.W. Gentle, R. V. Bravenec, G. Cima, H. Gasquet, G. A. Hallock, P. E. Phillips, D. W. Ross, W. L. Rowan, A. J. Wootton, T. P. Crowley, J. Heard, A. Ouroua, P. M. Schoch and C. Watts, Phys. Plasmas 2,2292 (1995).

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12. C.C. Petty and T.C. Luce, Nucl. Fusion 34,121 (1994). 13. F. Ryter, G. Tardini, F. De Luca, H.-U. Fahrbach, F. Imbeaux, A. Jacchia, K.K. Kirov, F. Leuterer, P. Mantica, A.G. Peeters, G. Pereverzev, W. Suttrop and ASDEX Upgrade Team, Nucl. Fusion 43,1396 (2003). 14. B.Ph. van Milligen, E. De la Luna, F.L. Tabares, E. Ascasibar, T. Estrada, F. Castejon, J. Castellano, I. Garcia-Cortes, J. Herranz, C. Hidalgo, J.A. Jimenez, F. Medina, M. Ochando, I. Pastor, M.A. Pedrosa, D. Tafalla, L. Garcia, R. Sanchez, A. Petrov, K. Sarksian and N. Skvortsova, Nucl. Fusion 42 (2002) 787. 15. H. Biglari and P. Diamond, Phys. Fluids B 3,1797 (1991). 16. D.R. Baker, C.M. Greenfield, K.H. Burrell, J. C. DeBoo, E. J. Doyle, R. J. Groebner, T. C. Luce, C. C. Petty, B. W. Stallard, D. M. Thomas and M. R. Wade, Phys. Plasmas 8,4128 (2001). 17. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett 59,381 (1987). 18. P.H. Diamond and T.S. Hahm, Phys. Plasmas 2, 3640 (1995). 19. D.E. Newman, B.A. Carreras, P.H. Diamond and T.S. Hahm, Phys. Plasmas 3, 1858 (1996). 20. R. Sbnchez, D.E. Newman and B.A. Carreras, Nucl. Fusion 41,247 (2001). 21. P.A. Politzer, Phys. Rev. Lett. 84,1192 (2000). 22. B.A. Carreras, B. Ph. van Milligen, M.A. Pedrosa, R. Balbin, C. Hidalgo, D. E. Newman, E. Sanchez, M. Frances, I. Garcia-Cortes, J. Bleuel, M. Endler, S. Davies and G. F. Matthews, Phys. Rev. Lett. 80,4438 (1998). 23. M.A. Pedrosa, C. Hidalgo, B.A. Cameras, R. Balbin, I. Garcia-Cortes, D. Newman, B. van Milligen, E. Snchez, J . Bleuel, M. Endler, S. Davies and G. F. Matthews, Phys. Rev. Lett. 82,3621 (1999). 24. B.A. Carreras, B.Ph. van Milligen, C. Hidalgo, R. Balbin, E. Sanchez, I. Garcia-Cortes, M. A. Pedrosa, J. Bleuel and M. Endler, Phys. Rev. Lett. 83, 3653 (1999). 25. R. Sbnchez, B.Ph. van Milligen, D.E. Newman and B.A. Carreras, Phys. Rev. Lett. 90,185005 (2003). 26. B.A. Carreras, D. Newman, V. E. Lynch and P. H. Diamond, Phys. Plasmas 3,2903 (1996). 27. X. Garbet and R. Waltz, Phys. Plasmas 5,2836 (1998). 28. Y.Sarazin and P. Ghendrih, Phys. Plasmas 5, 4214 (1998). 29. J.A. Mier, L. Garcia and R. Sanchez, Phys. Plasmas 13,102308 (2006). 30. R. Balescu, Aspects of Anomalous Transport in Plasmas, to be published (Institute of Physics, Bristol, 2005). 31. R. Metzler and J. Klafter, Physics Reports 339,1 (2000). 32. G.M. Zaslavsky, Physics Reports 371,461 (2002). 33. A. Compte, Phys. Rev. E 53,4191 (1996). 34. R. S&nchez,B.A. Carreras and B.Ph. van Milligen, Phys. Rev. E 71,011108 (2005). 35. R. Hilfer, Physica A 329,35 (2003). 36. B.Ph. van Milligen, R. SBnchez and B.A. Carreras, Phys. Plasmas 11, 2272 (2004).

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37. R. Sinchez, B.A. Carreras and B.Ph. van Milligen, Phys. Plasmas 12,056105 (2005). 38. B.Ph. van Milligen, B.A. Carreras and R. Sinchez, Phys. Plasmas 11,3787 (2004). 39. S. Bouzat and R. Farengo, Phys. Rev. Lett. 97, 205008 (2006). 40. D. del-Castillo-Negrete, Phys. Plasmas 13,082308 (2006). 41. G. Samorodnitsky and M.S. Taqqu, Stable non-Gaussian processes (Chapman & Hall, New York, 1994). 42. E. Montroll and G. Weiss, J. Math. Phys. 6,167 (1965). 43. V.M. Krenke, E.W. Montroll and M.F. Shlesinger, J. Stat. Phys. 9 , 4 5 (1973). 44. R. Balescu, Phys. Rev. E 51,4807 (1995). 45. W. Feller, An introduction to probability theory and its applications (John Wiley & Sons, New York, 1966). 46. I. Podlubny, Fkactional differential equations (Academic Press, New York, 1998). 47. A.I. Saichev and G.M. Zaslavsky, Chaos 7, 753 (1997). 48. B.A. Carreras, V.E. Lynch and G.M. Zaslavsky, Phys. Plasmas 8, 5096 (2001). 49. D. del-Castillo-Negrete, B.A. Carreras and V.E. Lynch, Phys. Plasmas 11, 3854 (2004). 50. L. Garcia and B.A. Carreras, Phys. Plasmas 13,012508 (2006). 51. G. Spizzo, R. B. White, and S. Cappello, Phys. Plasmas 14, 102310 (2007). 52. J.A. Mier, R. Sanchez, L. Garcia and D.E. Newman, Proc. EPS Conference on Plasma Physics (Warsaw, Poland), Paper No. 4.058 (2007). 53. T. Hauff and F. Jenko, Phys. Plasmas 14,092301 (2007). 54. W.W. Lee, Phys. Fluids 26, 556 (1983). 55. R. Sydora et al, Plasma Phys. Contr. Fus. 38, A281 (1996). 56. D.E. Newman, R. Sanchez, J.N. Leboeuf, V. Decyk and B. A. Carreras, Am. Phys. SOC.Bulletin 52, 336 (2007). 57. R. SBnchez, B.A. Carreras,D.E. Newman and B.Ph. van Milligen, Phys. Rev. E 12,016305 (2006). 58. P.H. Diamond, S.-I. Itoh, K. Itoh and T.S. Hahm, Plasma Phys. Control. Fusion 47, R35-Rl61 (2005). 59. F. Wagner, G. Becker, K. Behringer, D. Campbell, A. Eberhagen, W. Engelhardt, G. Fussmann, 0. Gehre, J. Gernhardt, G. v. Gierke, G. Haas, M. Huang, F. Karger, M. Keilhacker, 0. Klber, M. Kornherr, K. Lackner, G. Lisitano, G. G. Lister, H. M. Mayer, D. Meisel, E. R. Mller, H. Murmann, H. Niedermeyer, W. Poschenrieder, H. Rapp, H. Rhr, F. Schneider, G. Siller, E. Speth, A. Stbler, K. H. Steuer, G. Venus, 0. Vollmer, and Z. Y, Phys. Rev. Lett. 49, 1408 (1982).

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ROTATING RAYLEIGH-BENARD CONVECTION IN CYLINDERS J. J.

SANCHEZ-ALVAREZ

E. T.S.I. Aeronciuticos, Uniwersidad Polite'cnica de Madrid, Madrid 28040, Spain E. SERRE* MSNM-GP UMR6181 CNRS/Uniwersite's dAix-Marseille, Marseille, 13451, Prance *E-mail: serrel Q13mgp.13m.uniw-v~r-s.fr

E. CRESPO DEL ARC0 Departamento de Fisica Fundamental, U.N. E. D. Madrid, 28080, Spain

F. H. BUSSE Institute of Physics, University of Bayreuth Bayreuth, 0-95440, Germany This article presents a numerical study of rotating Rayleigh-BBnard convection (RBC) in a fluid with Prandtl number 5.3 confined in cylindrical enclosures. Using three-dimensional numerical solutions of the basic equations in the Boussinesq-Oberbeck approximation, we have explored the transition from an initially conductive state to a nonlinear aperiodic regime. The patterns have been investigated in two cylindrical cavities with a circular and annular cross section, respectively. Different aspect ratios have been considered; in the case of the cylindrical box of height d and radius R the aspect ratio, defined as r(=R/d) = 5 has been considered while in the case of annular channels with radial extent AR = R1 - Ro (where Ro and R1 are the inner and outer radii, respectively); values L(= AR/d) 5 5 and rl(= Ri/d) = 12 : 5 have been considered. The pseudo-spectral numerical method allows the computation of three dimensional unsteady flows without any restriction on the patterns. Visualizations of the flow reproduce some experimentally observed patterns and agree with the results of linear stability analysis. The primary transition from the conductive state of no motion occurs in the form of precessing convection modes. The secondary transitions show interesting dynamical processes, which vary with different boundary conditions. When the cylindrical sidewall is thermally insulating the primary travelling wave coexists with travelling wave con-

208

vection in the bulk as the Rayleigh number is increased. In the case of a perfect thermally conducting sidewall the primary wave coexist with chaotic convection characterized by breaking rolls. In annular channels, two counter-rotating sidewall travelling waves are observed as predicted by theory. In narrower channels, these sidewall travelling waves interact and lead to a quasi-periodic time behaviour.

Keywords: Rotating Rayleigh-BBnard convection; Pattern formation.

1. Introduction

Thermal convection in a horizontal fluid layer heated from below and rotating about a vertical axis has become a prime example in theories of pattern formation and of the transition to spatio-temporal chaos. For details on recent progress in experimental and theoretical studies of the dynamics of convection patterns we refer to review articles.'>2 Rayleigh-BBnard convection in an extended horizontal layer is characterized by two parameters, the Rayleigh number R a = gcyd3AT/vrc and the Prandtl number cr = u/rc, where d is the height of the fluid layer, AT is the temperature difference applied between upper and lower boundaries, g is the acceleration of gravity, LY is the coefficient of thermal expansion of the fluid, and u and IE are its kinematic viscosity and thermal diffusivity, respectively. In the presence of rotation about a vertical axis the Coriolis number R = 2nfd2/v, where f is the rotation frequency, enters as an additional parameter, and several new features such as oscillatory onset of convection and subcritical finite amplitude convection are found. From the theoretical point of view, the infinitely extended layer is a convenient idealization since periodic boundary conditions can be applied for the mathematical formulation of the problem. Fortunately, experimental realizations of Rayleigh-Bknard convection show good agreement in many respects with the theoretical predictions as long as the aspect ratio of the layer is not too small. Of particular importance for the pattern formation problem is the Kiippers-Lortz instability3 which causes the instability of all steady convection flows for Coriolis numbers R above a critical value RKL = O(10). It was first observed experimentally by Heikes & B ~ s s e who ~>~ studied a parameter regime different from that of the earlier investigation of Rossby.' In the past two decades the Kiippers-Lortz instability has become a subject of intense experimental7-'' and theoretica111i12 investigations. In contrast to the non-rotating case the sidewalls of the rotating layer play a prominent role. While the sidewalls of the Rayleigh-BBnard layer exert a stabilizing influence on the onset of convection, a destabilizing effect has been found in

209

the case of a rotating layer. Rossby' already noticed that convection sets in at Rayleigh numbers below those predicted for an infinitely extended layer. The correct theory for the onset of sidewall supported convection was given by Goldstein et al.13914(referred to by GOL93 in the following). More general boundary layer analyses have been presented by Herrmann & Busse15 and Kuo & Cross.16 The direction of propagation of the sidewall supported convection waves is in the direction of 6 x n' where n' is the normal vector of the sidewall pointing into the fluid. The travelling sidewall convection has been studied experimentally by Ning L! Eckeg and by Liu & E ~ k e . ' ~ - 'Good ~ agreement with the prediction of linear theory was found. Nonlinear properties including the stability with respect t o the Benjamin-Feir instability have been investigated by Liu & Ecke17 through appropriate experimental controls of the wavenumber. This procedure allowed the determination of the coefficients of the complex Ginzburg-Landau equation which turned out to be capable of describing the experimental results quantitatively for a wide range of parameters for slightly supercritical Rayleigh numbers. For the numerical simulations of the flow, we have used a spectral method of high accuracy. This approach is well suited to study threedimensional time dependent rotating flows including the effect of confinement exerted by sidewalls20i21and particularly to simulate convection in the circular cells used in experimental studies. The goal of this work is to explore the effects of the sidewalls and their thermal boundary conditions on rotating Rayleigh-BBnard convection. In rotating systems, direct numerical simulations have shown their ability to reproduce and predict many interesting nonlinear phenomenal ,12922123

2. Numerical integration of the Boussinesq equations Rayleigh-BBnard convection is described by the well-known Boussinesq equations. Using d, d2/n, n/d, as scales for length, time and velocity, respectively, we write the Boussinesq equations relative to the rotating frame of reference as follows :

v.? = 0

dT at

-

-.-.. = V 2 T + V.VT

210

where R is the Coriolis number, CJ is the Prandtl number, and Ra is the Rayleigh number defined in the introduction. 6 is the unit vector in the axial direction (opposite to the direction of gravity). It is convenient to write these equations using a cylindrical polar coordinate system ( T , 0, z ) relative to the rotating frame of reference, The velocity components are = (u, w, w), T is the temperature and p is the pressure. The centrifugal force is not included because the limit of ( 2 ~ f Rlg ) ~ << 1 will be assumed. This limit is usually also approached in e ~ p e r i r n e n t s Centrifugal .~~ effects in rotating convection have been recently i n ~ e s t i g a t e .The ~ ~ temperature is made dimensionless using the temperature difference applied between the boundaries: T = (T*- T,)/AT with AT = That - Tcold and T, = (That Tcold)/2,where T* represents the dimensional temperature. No-slip boundary conditions are considered a t all walls, (u= 'u = w = 0) apply since these are fixed in the rotating frame. Conducting and insulating thermal boundary conditions a t the vertical sidewall have been considered. Insulating boundary conditions are more relevant t o experiments using Plexiglas. The numerical solutions of Eqs. (1-3) are obtained with a numerical method based on a pseudo-spectral collocation-Chebyshev expansion in both the radial and the axial directions ( T , z ) and a Fourier-Galerkin method is used in the azimuthal direction. This choice takes into account the orthogonality properties of Chebyshev polynomials and, in particular, provides exponential convergence, referred to as spectral accuracy. 26 The time scheme is semi-implicit second-order accurate. It corresponds to a combination of the second-order Euler backward differentiation formula and the Adams-Bashforth scheme for the non-linear terms.

+

3. Results for circular boxes

The computations presented here were performed for confined layers of fluid with Prandtl number CJ = 5.3. The spatio-temporal behaviour of the convective flow has been studied in a rotating cylinder. The values of the rotation rates considered in the present computations are moderate, fl = 30, R = 60. In every case presented here, the stable solution a t low Ra is a purely conductive state and for increasing Ra, the first transitions are directly induced by the presence of the sidewall. Two kinds of oscillatory solutions are found representing the onset of convection in the form of either wall-attached travelling waves or slowly precessing bulk modes. The onset of convection in the form of a precessing pattern in the interior is not yet well understood. While the results of the linear stability analysis for

21 1

small containers have demonstrated the onset of this precessing mode of convection (GOL93), there is no clear evidence for the interior bulk pattern from experiments. Present numerical results using thermally insulating sidewalls in a cavity of aspect ratio I' = 3 and rotating at R = 30, show that the first unsteady mode is a slow bulk mode. Even though the results of the linear stability analyses are only available for I' I 2.5 and I' = 00,they clearly show that at low rotation rate the slow bulk modes are preferred at the onset; thus for a cavity with insulating sidewall and aspect ratio = 2.5, the slow bulk modes appear first when R 38.5 (GOL93). Experimental results by Liu & E ~ k with e ~ water ~ in a cavity of r = 5, show that for large R the onset occurs in the form of sidewall travelling wave and with increasing Ra there is a transition to bulk convection. As R decreases the interval of Ra by which sidewall travelling waves precede the onset of bulk convection decreases as well and for 52 x 50 the Rayleigh numbers for the onset of sidewall and bulk convection coincide in the case I' = 00. As already mentioned, for r = 3 and R = 30 the flow pattern at onset of convection is a bulk mode which rotates rigidly in the retrograde direction relative to the rotating system. In figure 1 it is shown the pattern for R a = 3175, the frequency of rotation is w = -0.012. This low frequency indicates that this pattern is related to the slow rotating modes predicted by linear stability analysis of GOL93. The maximum amplitude of the pattern is located on the interior of the cavity and it has lower amplitude near the sidewall (see Fig. 1).

Fig. 1. Iso-lines of temperature at mid-height for r = 3, u = 5.3, R = 30 and Ra = 3175 (thermally insulating sidewall). In this and all following figures the rotation of the convection layer is in the counter-clockwise direction. Dark regions are warm upflows and bright regions are cold downflows. The pattern is shown at three subsequent time steps. The time between two figures (from i to iii) is 37.74(d2/n)

212

In the comparison with the linear stability theory of GOL93 we notice that the found pattern is consistent with theory. We have performed computations using different initial conditions and we have not found any other final stable solution at Ra = 3175. We thus conclude that the higher values of R used in experiments are the reason that the first transition to a slow body mode has apparently not yet been seen by experimentalists. Increasing the rotation rate of the cavity to R = 60 the initial convective mode is the so called wall-mode, which consists in a wall-attached traveling wave. In Fig.2 is shown the convective pattern at Ra = 5600, the frequency of this pattern is w = -10.11 where the sign of the frequency is defined by the dependence of the dominating Fourier component expi(rn8 w t ) (in this case rn = 11).

+

Fig. 2. Iso-lines of temperature at mid-height for I? = 3, o = 5.3, R = 60 and Ru = 5600 (thermally insulating sidewall).

!

esults in an annular cavity In this Section, we consider the nonlinear flow in an annular cavity, with insulating thermal boundary conditions at both sidewalls. Using a wide channel, L, = 5, (I'o = 7.5, I'l = 12.5) and R = 60 the two sidewall travelling waves were observed for Ra = 5600. These waves (Fig. 3a) at I'o and I'l have nearly the same absolute values of their frequencies and wavenumbers, wo = -9.87, w1 = +9.99, and ko = 4.13, kl = 4.08, respectively, but travel in the opposite direction while in the interior between the walls the fluid is nearly at rest. The independence of the two sidewall travelling modes in the case L = 5 gives rise to the question of their interaction in a narrower channel. To study

213

Fig. 3. Isotherms at mid-height in an annular cavity at R = 60, Ra = 5600 (insulating thermal boundary condition). (a) Sidewall travelling waves ( L = 5, r o = 7.5,rl = 12.5). (b) Interaction of two counter-rotating wave in a narrow annular cavity (L= 1.5,ro = 7.5,rl = 9.0).

this problem the outer radius has been decreased to L = 1.5 and L = 0.75 while the inner radius remained unchanged (I'o = 7.5 and l?l = 9.0,8.25). For L = 1.5 the sidewall modes interact in the bulk region (Fig. 3b). The pictures of the flow show mo = 28 and ml = 33 rolls near the sidewalls. A pairing between the convection structures on opposite sides of the annular layer can be observed leading to transverse rolls. Owing to the opposite drifts of the sidewall modes the rolls become stretched in the direction parallel to the sidewalls until they break and new transverse rolls are formed. A similar situation for somewhat different parameters is shown in figure l a of Scheel et al. (2003).28In the configuration considered here the wavenumbers of these travelling waves are slightly different, ICo = 3.73 and Icl = 3.66 and the angular frequencies observed in the power spectra are wo = -9.41, w1 = +9.60 and the period of 31.5 corresponding to their sum, W b = w1 wo = +0.19 is clearly evidenced in the bulk region where the effects of the two sidewall travelling waves are almost linearly superimposed. When the width of the channel is of the same order of magnitude as the radial extent of the sidewall travelling modes the latter can no longer be realized as separate waves. Instead they combine to form a nearly steady pattern of convection in the form of rolls oriented nearly perpendicular to the sidewalls as shown in figure 4 for the case of L = 0.75. In the limit of an infinite radius corresponding to a straight channel both sidewalls are equivalent and a steady convection pattern must be expected. But as has been point out recently by K. Zhang et al.29 since the spatial structures of both waves are different a standing-wave solution is not feasible. In the present configuration the difference in the curvature of the sidewalls gives rise to a

+

214

-=-I. 1p

Ra=17000

I

/Ha 8000

0 1

-0 I

a ?L

ir

L Ra-16400

Ra=12000

Fig. 4. Convection patterns in the narrow channel (L= 0.75, ro = 7.5, rl = 8.25) for R = 60, Pr = 5.3 for Ra = 5600, 6000, 8000, 12000, 16400, and 17600 (from bottom t o top first right then left). The patterns drift with the angular frequencies w = 1.41, 1.36, 1.09, 0.90, 0.91, and 0.91 (from top t o bottom first left then right) in the direction opposite t o rotation. The wavenumber is m = 38 for Ra = 5600, 6000 and m = 30 for the other cases.

pattern drifting steadily in the retrograde direction with a frequency somewhat larger than the sum of the two side wall mode frequencies (see values given in the caption of Fig. 4). The orientation of the rolls is also not strictly radial but exhibits a spiral like inclination turning inward in the prograde direction] at least for lower values of Ra. At a value slightly below 12000 the direction of spiralling changes sign and a t high Rayleigh numbers the angle reflects a spiral turning outward with the sense of rotation (see Fig. 4). This change can be attributed to an increasing mean shear in the channel. The steady convection is associated with a mean axisymmetric azimuthal flow. In the limit of infinite radius of the channel the flow will be antisymmetric with respect to the vertical midplane of the channel. The slight reversals of this shear flow near the upper and lower boundaries are typical and can be seen also, for instance] in the related problem treated by Plaut (2003).30 The average over the height of channel of this flow is shown in Fig. 5 and it is evident that its amplitude increases strongly with the Rayleigh number.

215

At Ra = 12000 its amplitude is high enough to reverse the inclination of the rolls relative to the radial direction.

d

1

0

7

h > V

o

-1

-2

-3 -4 r Fig. 5. Vertically averaged profiles of the axisymmetric component of the azimuthal velocity 'u for the cases of Fig. 4.

5 . Concluding remarks

Boundary and finite size effects on rotating Rayleigh-Bbnard convection have been numerically investigated in cavities with small and moderate values of the aspect ratio. Using numerical solutions of the three-dimensional Navier-Stokes equations in the Boussinesq-Oberbeck approximation, we have explored the transition from an initially conductive state to a nonlinear regime at two rotation rates s2 = 30 and s2 = 60. The first transitions in both, the circular layer and the annular channel, are to precessing solutions as it was expected from the results of the linear stability analysis obtained for a circular layer of small aspect ratio. - In the cylindrical box at low rotation rate, s2 = 30, the first transition from the conductive state is to a slow precessing mode within the bulk. This bulk pattern has not been reported in experiments but its drift frequency seems to indicate that it is related to the slow bulk mode predicted by the linear stability results of GOL93. - In the annular channel, the first transition is to fast sidewall travelling waves which travel in opposite directions at the two walls in very good

216

agreement with t h e behaviour predicted by Herrmann & Busse (1993) and Kuo & Cross (1993). T h e two counter-travelling sidewall waves begin t o interact as t h e width of t h e channel decreases leading first to a quasi-periodic time dependence a n d then to a nearly steady pattern of convection.

References 1. E. Bodenschatz, W. Pesch and G. Ahlers, Annu. Rev. Fluid Mech. 32,709 (2000). 2. E. Knobloch, Int. J. Eng. Sci. 36,1421 (1998). 3. G. Kuppers and D. Lortz, J . Fluid Mech. 35,609 (1969). 4. K. E. Heikes and F. H. Busse, Ann. (N. Y.) Acad. Sci. 357,28 (1980). 5. F. H. Busse and K. E. Heikes, Science 208,173 (1980). 6. H. Rossby, J . Fluid Mech 36,309 (1969). 7. F. Zhong, R. Ecke and V. Steinberg, Chaos 2,163 (1992). 8. F. Zhong, R. Ecke and V. Steinberg, J . Fluid Mech. 249,135 (1993). 9. L. Ning and R. E. Ecke, Phys. Rev. E 47,3326 (1997). 10. Y . Hu, R. E. Ecke and G. Ahlers, Phys. Rev. Lett. 74 (1995). 11. Y.Tu and M. C. Cross, Phys. Rev. Lett. 69,2215 (1992). 12. J. MillAn-Rodriguez, M. Bestehorn, C. Perez-Garcia, R. Friedrich and M. Neufeld, Phys. Rev. Letters 74,530 (1995). 13. H. F. Goldstein, E. Knobloch, I. Mercader and M. Net, J . Fluid Mech. 248, 583 (1993). 14. H. F. Goldstein, E. Knobloch, I. Mercader and M. Net, J . Fluid Mech. 262, 293 (1993). 15. J. Herrmann and F. H. Busse, J. Fluid Mech. 255,183 (1993). 16. E. Y . Kuo and M. C. Cross, Phys. Rev. E 47,R2245 (1993). 17. Y . Liu and R. E. Ecke, Phys. Rev. Lett. 78,4391 (1997). 18. Y . Liu and R. E. Ecke, Phys. Rev. E 59,p. 4091 (1999). 19. R. E. Ecke and Y . Liu, Int. J. Eng. Sci. 36,1471 (1998). 20. E. Serre, E. C. del Arco and P. Bontoux, J . Fluid Mech. 434,65 (2001). 21. E. Serre and P. Bontoux, J . Fluid Mech. 459,347 (2002). 22. Y . Hu, W. Pesch, G. Ahlers and R. E. Ecke, Physical Review E 58, 5821 (1998). 23. J. J. SBnchez-Alvarez, E. Serre, E. Crespo del Arco and F. H. Busse, Phys. Rev. E 72,p. 036307 (2005). 24. F. Zhong, R. Ecke and V. Steinberg, Physical Review Letters 67,2473 (1991). 25. F. MarquBs, I. Mercader, 0 . Batiste and J. M. Massaguer, J . Fluid Mech. 580 (2007). 26. R. Peyret, Spectral methods for incompressible viscous pow, Applied Mathematical Sciences, Vol. 148, 2002). 27. Y . Liu and R. E. Ecke, Physical Review E 59 (1998). 28. J. Scheel, M. Paul, M. Cross and P. Fischer, Phys. Rev. E 68,1 (2003). 29. K. Zhang, X. Liao, X. Zhan and R. Zhu, Phys. Rev. E 75,p. R55302 (2007). 30. E. Plaut, Physical Review E 67 (2003).

217

SELF-EXCITED INSTABILITIES IN PLASMAS CONTAINING DUST PARTICLES (DUSTY OR COMPLEX PLASMAS) M. MIKIKIANI, M . CAVARROC, L. COUEDEL, Y . TESSIER, L. BOUFENDI

G R E M I , Groupe de Recherches sur 1 'Energe'tique des Milieux Ionise's, UMR6606, CNRS/CJniversite' d'orle'ans, 14 rue d'lssoudun, BP6744, 45067 OrlCans Cedex 2, France * E-mail: maxime.mikikian@univ-Orleans. fr Dusty plasmas are complex systems where new phenomena arise from the presence of solid dust particles inside a plasma. Dust particles acquire a negative charge by attaching free electrons and are thus trapped in the plasma. This charge loss can be drastic for the plasma equilibrium and instabilities can appear due t o the strong interdependence between the dust particle cloud and the plasma. In this paper, various types of self-excited instabilities consisting in regular or chaotic oscillations, are presented. They are obtained in low pressure radio-frequency discharges where dust particles are grown by using a reactive gas or by sputtering a surface exposed t o the plasma. The complex evolution scheme of these instabilities is brought t o the fore thanks to various diagnostics.

Keywords: Dust; Plasma; Instabilities; Nanoparticle; Fourier analysis.

1. Introduction Dusty plasmas1i2 are partly ionized gases (mixture of neutral atoms and/or molecules, positive and/or negative ions, free electrons) containing solid dust particles with sizes ranging from few nanometers to centimeters. These media are observed in astrophysics (comet tails, planetary atmospheres) and also in industry where dust particles are usually fatal for the processes in microelectronics but useful to build small objects in nanotechnologies or to design nanostructured materials with interesting properties. Thermonuclear fusion is also concerned by dust production due to wall erosion. To study these media, dust particles are either artificially injected in the plasma or formed using reactive gases or a target sputtered by the plasma ion bombardment. These methods lead to the formation of a dense cloud of submicron dust particles filling the whole plasma volume.

218

Reactive gases are often used due to their ability to easily form dust particles by following a succession of different growth phases. One of the most studied and well-known reactive gas is silane (SiH4).3-7 The interest for silane based plasmas is related to their implication in microelectronics and nanotechnology. Indeed, in the late 1980s, dust particle formation in the gas phase has been evidenced in reactors used for silicon-based device f a b r i ~ a t i o n .Cleanliness ~~~ is often a major requirement for microelectronics processes and a lot of studies began for understanding dust particle formation and growth in order to avoid their deposition. More recently, silicon nanocrystal formationg became of high interest for their incorporation in thin films in order to improve film properties. For example, the use of silicon nanocrystals in solar cell technology enhances the optoelectronic properties of the deposited films.lo In nanotechnology, single electron devices (SED) like transistors or memories can also be built, thanks to these crystalline nanoparticles." Dust particle formation in hydrocarbon-based gases like methane (CH4)12J3 or acetylene (C2H2)12-15 has also a high industrial implication for deposition of diamond-like carbon (DLC) films16 or nanocrystalline diamond17 with unique properties like extreme hardness. Hydrocarbon gases are also of great interest for the astrophysical community dealing with planet atmospheres like Titan, where dust particles are created due to the presence of methane.18 Dust particle production can also be achieved by material sputtering.19-21 This phenomenon occurs in industrial reactors or in fusion devices22like Tore Supra23as well where graphite walls can be severely eroded by ion bombardment. Produced dust particles can strongly limit performances of the fusion plasma and raise radio-toxicity issues due to tritium retention. This aspect is of great importance for the future ITER reactor. Once dust particles reach a few nanometers in size, they are trapped in the plasma due to the negative charge they acquire by attaching free electron^.^^-^^ This charge loss can be drastic for the plasma equilibrium and instabilities can appear due to the strong interdependence between the dust particle cloud and the plasma. Various types of low-frequency instabilities (from few Hz to few kHz) are observed (with various frequencies and shapes) either during dust particle growth process20~28-30 or once the dust cloud is completely f ~ r m e dIn. this ~ ~ last ~ ~case, ~ instabilities are related to the presence of a dust-free region in the plasma center. This region, named "void", exhibits successive contractions and expansions phases. This instability is usually named "heartbeat" instability due to this characteristic behavior.

219

2. Instabilities during dust particle growth In this section, instabilities occurring during the dust particle growth process are analyzed. These dust particles are produced in two different experimental setups and by using material sputtering or silane diluted in Ar.

2.1. Dust particles f o r m e d by sputtering Dust particle growth i n ~ t a b i l i t i e s(DPGI) ~~ typically appear a few tens of seconds after plasma ignition. At this stage, dust particles have grown to a typical size of few tens of nanometers. DPGI appearance is well observed on different diagnostics like electrical and optical measurements. These instabilities follow a well-defined succession of phases as observed in electrical measurements shown in figure l ( a ) . In this figure, the beginning of DPGI is detected around 40 s and successive phases are numbered from 1 to 7. These different phases are better evidenced by performing a Fourier analysis of the electrical signals. A typical spectrogram is given in figure 1(b). In order to emphasize small ordered domains, the spectrogram intensity has been normalized inside each 100 s range (from 0 to 100 s intensity has been normalized to its maximum value inside this time domain and so on). We can identify seven different regimes:

0 0 0

0

Three ordered phases PI, P2, P 3 (from N 40 s to N 80 s). Chaotic phase P4 (from N 80 s to N 405 s). High frequency phase P5 (from v 405 s to N 435 s). Chaotic phase becoming more and more regular P6 (from N 435 s to II 600 s). Regular oscillation phase P 7 (from N 600 s to I I680 s)

Instabilities begin with regular oscillations (Pl, P2, P3) followed by a long chaotic regime (P4). This chaotic phase suddenly ends with a high frequency phase (P5) and starts again (P6) becoming more and more ordered. Finally, the system reaches a regular phase (P7) that can be sustained for a long time (in the example given here the plasma has been switched off at 680 s). The P1 and P2 phases are short and are not detected in all experiments while the P3 phase lasts longer and is regularly observed. The three phases are well separated and evolve as a function of time. In figure 2(a), the P1 phase is characterized by wide separated peaks with a mean frequency of about 40 Hz. The transition from P1 to P2 corresponds to the growth of two small peaks between these higher amplitude patterns. The small peaks continue to grow (P2 phase) and the higher amplitude ones

220

Time (s) Fig. 1. Dust Particle Growth Instabilities: the different phases observed on (a) electrical measurements, (b) FFT of (a).

decrease. Finally, all peaks reach the same amplitude characterizing the P 3 phase. The frequency of the P 3 phase is then approximately three times the P2 frequency (around 94 Hz). The chaotic regime P4 (after 80 s) is characterized by less ordered patterns and a strong increase in DPGI amplitude. Then, the amplitude slowly decreases during the whole phase. P4 is also characterized by structured oscillations appearing in a transient manner as shown in figure 2(b). These structured regions are identified by some bright spots on electrical and optical spectrograms. On time series they appear as bursts of order. During the chaotic regime, structured oscillations in electrical signals always appear following a three peak structure (for example between 94 s and 94.04 s in figure 2(b)) that could be related to the three peaks observed in the P2 phase. Indeed, it could be a reemergence of this phase during the chaotic regime. In some experiments, the chaotic regime is suddenly interrupted by a strong frequency change (at 405 s in figure 1). Indeed, DPGI turn into low-amplitude and high-frequency (around 500 Hz) oscillations. After the high-frequency phase, a second chaotic regime is usually observed. It corresponds to a phase similar to P4 but with more and more ordered regions. The final step of DPGI often corresponds to a regular oscillation phase characterized by a spectrogram with a typical frequency around 18

221

Hz. This regular oscillation is similar to what is obtained during the heartbeat instability] which corresponds to regular contractions and expansions of the void size.21i32 0.021

. .

'1'

? J 0.01

d

-a 3

. I

0

5-0.01

43.25 43.3 43.35 43.4 43.45 43.5 43.55 43.6 43.65 43.1

.o1 0 .01

93.1

93.8

93.9

94

Time (s)

94.1

94.2

94.3

Fig. 2. (a) TYansitions between the first three organized phases, (b) burst of order during the chaotic regime.

2.2. Dust particles formed b y reactive gases In silane based plasmas, the dust particle growth follows specific steps. When dust particle size reaches about 2 nm, these clusters tend to agglomerate to form bigger size dust particles. When this aggregation occurs, the amount of plasma electrons lost on dust particle surface is suddenly changed. This drastic change leads to plasma in~tabilities~>~' as observed on the electrical measurements in figure 3(a)-(b). The corresponding time evolution of the instability frequency is given in figure 3(c). We can first observe an increase of the frequency that could correspond to the onset of the instability. Once the instability is established, the frequency decreases until the end of the phase where it becomes difficult to define. In some very specific conditions a particular case of the instability was observed. Figure 4 shows the shape of the instability in this case. In the first part, it behaves in the same way as the "common" instability, then it finishes before briefly restarting. The second part seems to be a sort of replica of the end of the first part, with exactly the same frequencies. This phenomenon is

222 -65 ,-66

m

-69 0.2

0.21

0.22

0.23

0.2

0.21

0.21

0.212

0.214

0.22

0.23

Time (s)

Time (s)

0.216

0.218

0.22

0.222

0.224

0.226

0.228

Time (s) Fig. 3. Aggregation instability observed on electrical diagnostics (a) V d c , (b) 3H, (c) F F T of (b).

highly sensitive to the operating conditions, appearing only for a very tight range of rf power and silane flow rate. This particular case of the instability seems to correspond to dust critical formation conditions, as the slightest modification of one of the parameters leads to the disappearance of the phenomenon. It underlines the threshold dependence of this instability.

-341

I

Fig. 4. Particular case of the aggregation instability: the instability stops and briefly restarts.

As long as silane is provided in the discharge, the dust particle growth continues as a cyclic phenomenon. Other kinds of instabilities are thus ob-

223

served. As the mean size of dust particles trapped in the discharge is higher, characteristic frequencies are smaller than for the aggregation instability (< 100 Hz instead of few kHz). Observed instabilities are related to the formation of new generations of particles, growing in the discharge center and pushing out the older generation of bigger size dust particles. These instabilities are characterized by two phases: a less-ordered and a highordered. The transition between these two regimes is sudden as observed in figure 5. The less-ordered regime corresponds to low amplitude noisy oscillations while the high-ordered one has higher amplitude and better defined oscillation frequency.

W-0.02‘

-0.25

-0.2-0.15

-0.1

-0.05

I

,

0

0.05

Time (s)

l , l , ’ , 0.1

0.15

0.2

I 0.25

Fig. 5 . Transition between the two characteristic phases (less and high ordered phases) of successive generation instabilities.

3. Instabilities of the grown dust cloud In the sputtering discharge, successive generations of dust particles are not systematically obtained like in silane discharges. Thus, situations with a stable cloud of trapped dust particles can be obtained. Nevertheless, a dustfree region is often observed in the dust cloud center and is caused by the different forces acting on dust particles. This egg-shape region called “void” can exhibit self-excited oscillation^^^^^^ of its size with typical frequencies about few tens of Hz. Near the instability threshold or just before it stops, a transition regime characterized by “failed peaks” on both electrical and optical signals can be observed. Indeed, the oscillation pattern shows the occurrence of lower amplitude oscillations intercalated between two main oscillations. This phenomenon is repeated with more and more failed peaks and finishes by the instability stop. Figure 6 shows a transition between 1 and 2 failed peaks in the optical signal. The number of failed peaks increases (> 10) and then the instability stops (figure 7(a)).

224

I

I

I

0.05

0

0.1

0.2

0.15

0.25

0.3

0.35

0.4

0.45

Time (s) Fig. 6 . Last beats of the instability: more and more failed peaks appear before the instability stops. The transition between one and two failed peaks is shown.

,

il h7A,

0.666' 0

, 2

4

6

8

10

12

14

16

18

20

Time (s)

Fig. 7. Slow evolution of the heartbeat instability toward its end: (a) more and more failed peaks until the stop around 18 s, (b) 3D and (c) 2D phase spaces.

As observed for other instabilities (figure 4), this behavior underlines the threshold dependence of this instability and the way it evolves toward its end. This alternation of failed and main peaks seems to correspond to mixed-mode oscillations often encountered in chemical systems or neuronal dynamics. The 3D and 2D phase spaces corresponding to the time series presented in figure 7(a) are plotted in figure 7(b)-(c).These attractors have been obtained by using an appropriate time delay calculated using the mutual information method.

225

Acknowledgments T h e PKE-Nefedov chamber has been made available by t h e Max-PlanckInstitute for Extraterrestrial Physics, Germany, under the funding of DLR/BMBF under grants No.50WM9852. We would like t o thank S. Dozias for electronic support and J. Mathias for optical support. This work was partly supported by CNES under contract 02/CNES/4800000059.

References 1. A. Bouchoule, Dusty Plasmas :Physics, Chemistry and Technological Impacts in Plasma Processing (Wiley, Chichester, 1999). 2. P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics (Institute of Physics, Bristol, 2002). 3. R. M. Roth, K. G. Spears, G. D. Stein and G. Wong, Appl. Phys. Lett. 46, p. 253 (1985). 4. Y . Watanabe, M. Shiratani, Y. Kubo, I. Ogawa and S. Ogi, Appl. Phys. Lett. 53,p. 1263 (1988). 5. A. Bouchoule, A. Plain, L. Boufendi, J.-P. Blondeau and C. Laure, J . Appl. Phys. 70,p. 1991 (1991). 6. A. Howling, C. Hollenstein and P. J. Paris, Appl. Phys. Lett. 59,p. 1409 (1991). 7. A. Bouchoule, L. Boufendi, J. Hermann, A. Plain, T. Hbid, G. Kroesen and W. W. Stoffels, Pure Appl. Chem. 68, p. 1121 (1996). 8. G. S. Selwyn, J. Singh and R. S. Bennett, J. Vac. Sci. Technol. A 7,p. 2758 (1989). 9. M. Cavarroc, M. Mikikian, G. Perrier and L. Boufendi, Appl. Phys. Lett. 89, p. 013107 (2006). 10. P. Roca i Cabarrocas, P. Gay and A. Hadjadj, J . Vac. Sci. Technol. A 14, p. 655 (1996). 11. A. Dutta, S. P. Lee, Y . Hayafune, S. Hatatani and S. Oda, Jpn. J. Appl. Phys. 39,p. 264 (2000). 12. C. Deschenaux, A. Affolter, D. Magni, C. Hollenstein and P. Fayet, J . Phys. D: Appl. Phys. 32,p. 1876 (1999). 13. S. Hong, J. Berndt and J. Winter, Plasma Sources Sci. Technol. 12, p. 46 (2003). 14. I. StefanoviC, E. KovaEevic, J. Berndt and J. Winter, New. J. Physics 5,p. 39 (2003). 15. K. De Bleecker, A. Bogaerts and W. Goedheer, Phys. Rev. E 73,p. 026405 (2006). 16. J. Robertson, Mater. Sci. Eng. R 37,p. 129 (2002). 17. D. Zhou, T. G. McCauley, L. C. &in, A. R. Krauss and D. M. Gruen, J. Appl. Phys. 83,p. 540 (1998). 18. C. Szopa, G. Cernogora, L. Boufendi, J. J. Correia and P. Coll, Planet. Space Sci. 54,p. 394 (2006).

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19. B. Ganguly, A. Garscadden, J. Williams and P. Haaland, J. Vac. Sci. Technol. A 11,p. 1119 (1993). 20. G. Praburam and J. Goree, Phys. Plasmas 3,p. 1212 (1996). 21. M. Mikikian, L. Boufendi, A. Bouchoule, H. M. Thomas, G. E. Morfill, A. P. Nefedov, V. E. Fortov and the PKENefedov Team, New J . Phys. 5, p. 19 (2003). 22. J. Winter, Plasma Phys. Control. Fusion 40, p. 1201 (1998). 23. C. Amas, C. Dominique, P. Roubin, C. Martin, C. Brosset and B. PBgouriB, J . Nucl. Mater. 353,p. 80 (2006). 24. B. Walch, M. Horanyi and S. Robertson, IEEE Trans. Plasma Sci. 22, p. 97 (1994). 25. C. Arnas, M. Mikikian and F. Doveil, Phys. Rev. E 60, p. 7420 (1999). 26. A. A. Samarian and S. V. Vladimirov, Phys. Rev. E 67,p. 066404 (2003). 27. L. Couedel, M. Mikikian, L. Boufendi and A. A. Samarian, Phys. Rev. E 74, p. 026403 (2006). 28. D. Samsonov and J. Goree, Phys. Rev. E 59, p. 1047 (1999). 29. M. Mikikian, M. Cavarroc, L. Couedel and L. Boufendi, Phys. Plasmas 13, p. 092103 (2006). 30. M. Cavarroc, M. C. Jouanny, K. Radouane, M. Mikikian and L. Boufendi, J. Appl. Phys. 99, p. 064301 (2006). 31. M. Mikikian and L. Boufendi, Phys. Plasmas 11,p. 3733 (2004). 32. M. Mikikian, L. Couedel, M. Cavarroc, Y . Tessier and L. Boufendi, New J . Phys. 9, p. 268 (2007).

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CLUSTERING PROPERTIES OF CONFINED PLASMA TURBULENCE SIGNALS MILAN RAJKOVIC Institute of Nuclear Sciences VinEa, P. 0. Box 522, 110001 Belgrade, Serbia E-mail: milanrovin. bg.ac.yu MILOS SUKORIC National Institute for Fusion Science, 322-6 Oroshi-cho, Toki, Giju, Japan E-mail: [email protected] Intermittency in turbulent boundary layers of fusion devices is studied by considering only the zero-crossing information of signals for various confinement regimes. Certain common features with neutral fluid turbulence are recognized in the low confinement regime (L-mode) of fusion plasma while completely different characteristics are noticed for the dithering H mode (L/H-mode) and the high confinement regime (H-mode). Spectral characteristics of approximate signals (containing only the zero-crossing information) in each regime are compared with each other and with the spectral scalings of the velocity signals of neutral fluid turbulence. A scaling exponent characterizing the tendency of small scales to cluster is introduced and its relationship with large scale clustering is investigated.

1. Introduction

Study of turbulence in magnetic confinement devices represents one of the most important issues in the pursuit of fusion energy production since turbulence hinders confinement, suppresses reactions as it causes particle and energy losses. Control of turbulence requires a thorough knowledge of its dynamics in the core of magnetic confinement devices as well as on the edges beyond the last closed flux surface, in the region known as the scrape-off layer (SOL). A major breakthrough in the confinement improvement occurred by the end of the eighties, when the high confinement (H-mode) was discovered in contrast to the well known L- or low confinement mode.' The H- or high confinement mode manifests itself by self-organization of a region just inside the poloidal field separatrix where the transport coefficients are reduced by up to an order of magnitude compared with the L-mode forming a pedestal in the plasma pressure. As a consequence a n improvement in

228

the global confinement is usually increased by a factor of two in the case of toroidal devices while it is in the order of -20% in the case of large helical devices (LHD). The thickness of this barrier-type region is about equal to the ion poloidal gyroradius or the width of an ion banana orbit. Properties of intermittency both in the case of neutral fluids and plasmas are usually deduced from the analysis of temporal and/or spatial fluctuations of one or several relevant quantities. In the case of neutral incompressible fluids one or all three components of the fluid velocity represent the basic quantity from which all other relevant quantities, such as dissipation, may be derived. In the case of confined plasmas these quantities are usually the ion saturation current, recorded at one or more spatial locations, from which plasma density fluctuations may be inferred and floating potential recorded at different poloidal positions from which radial velocity fluctuations may be determined. In spite of many universal features these two types of turbulence have important differences. Nonlinearites in plasma turbulence are more numerous having different spectral cascade directions in addition to the most important E x B nonlinearity, leading to more complex fluctuating characteristics. Also, time and space measurements in plasmas lead to different information on the structure of turbulence. Turbulence in confined plasmas is created and damped at the same spatial location where the measurements are taken so that spatial and temporal informations are interwoven and Taylor’s frozen flow hypothesis cannot be applied, a common practice in neutral fluid turbulence studies. For the same reason the inertial range14may exist only locally in space or in time, and the extent of this range changes along the temporal scale as well as along space, for example along poloidal direction. Two features of intermittency are its clustering property and the variability in amplitude. Namely, different events cluster together creating uneven density in space and time, and events reflected in the highly variable amplitude are dispersed in space and time disproportionately. Much insight into the nature of intermittency may be gained from the study of approximations of turbulent signals which neglect amplitude aspect , 2 3 In this approach amplitude variations and local frequency of oscillations are separated by retaining only the zero-axis crossings (frequency) information. In a related approximation conveying somewhat different information about the original signal, turbulent signal is approximated by a zero-mean process constructed from the original signal x ( t ) as

229

[ ( t ) is a binary approximation (BA) of the original signal and hence may assume values of 1 and 0 only. In Section I1 we examine spectral characteristics of approximated neutral fluid turbulence signal and approximated intermittent signals recorded in different plasma confinement regimes. In Section 111 the clustering properties of these signals are studied based on variance properties of the zero-crossings number in a given time period. 2. Power spectra of approximate signals In Fig. 1 we present power spectra of local velocity fluctuations measured in the turbulent region of a round free jet5 and its binary approximation. Binary approximations of velocity fluctuations, which have been studied previously and as explained recently in,2 are characterized by a spectral slope exhibiting “-4/3” scaling in contrast to the Kolmogorov “-5/3” scaling in the inertial range of the original velocity fluctuations signal. In2 a general relation relating the slope uwof the complete turbulent signal and the slope CJBA of its BA is proposed and heuristically proved, namely

Plasma intermittency in this presentation consists of the ion saturation current ( I S A T )fluctuations recorded by the movable Langmuir probe located at the outboard midplane on MAST device.6 Sampling rate was 1 MHz and during the discharge the distance from the plasma edge to the probe changed slowly. Three signals, each recorded during different confinement regimes are used: 6861 L-mode, 9031 dithering H-mode (denoted as L/H mode and representing an unstable state during which plasma switches intermittently from low to high confinement) and 5738 H-mode. In Fig.2 the spectra of 6861 L-mode and of its BA approximation are presented and inspection of these spectra supports the claim about similarity of neutral fluid and low confinement plasma t ~ r b u l e n c e . ~ In Figs. 3 and 4 the same type of spectra are presented for the L/Hmode and the H-mode and the slope values for each spectrum are indicated. Evidently, relationship (1) is not valid for these confinement regimes. The most striking feature of these spectra is increasing difference between slopes of the signal and its BA approximation as confinement increases. Also the slope of BA approximation approaches the slope of Gaussian white noise spectrum in the H-mode, suggesting perhaps similarity in certain aspects of H-mode with white Gaussian noise. However, rn will be seen in the next Section this is far from being the case.

230

102

. . . . 10-

k (m-1)

Fig. 1. Spectra for incompressible fluid turbulence.

:

104100

101

I 0'

f

Fig. 2.

Fig. 3.

Spectra for the 6861 L-mode.

Spectra for the 9031 L/H mode.

231

Fig. 4. Spectra for the 5738 H-mode.

3. Clustering information in the approximate signal

Zero-crossing (or for that matter crossing of any particular level of interest) may offer important insight into the underlying process whose temporal variations are studied. The average number of zero-crossings of stationary Gaussian process in a specific time interval may be analytically determined and is given by the celebrated Rice formula,' which we here present in the following form: PT

N ( T )= lim

J, G(rc(t)) (a(rc(t)/at(d t ,

?%-+a

where limn--ta S(z) is the Dirac delta function. Important information on the clustering properties of the signal is however contained in the expression for the variance of the number of zero crossings. The expression for variance again may be derived analyti~ally,~ and is directly proportional to the time interval T,i.e.(N2(T)) T . Based on this expression for Gaussian process out goal is to contrast clustering properties of turbulent signals with the white Gaussian noise. For this purpose a running average within a time interval T of the number of spikes of BA approximation is constructed; this quantity is actually equal to N ( T ) ,the average number of zero-crossings in T . Then fluctuations of the running average are ~ N ( T= ) N ( T )- ( N ( T ) ) , where the brackets denote long-time average, possibly the time of the whole signal. We are interested in the scaling of the variance

-

232

Fig. 5 . Standard deviation of the running density fluctuations vs. fluid turbulence.

T,

for incompressible

For a white Gaussian noise p = 1/2. We call p the clustering exponent and since white noise has no clustering, the value of 1/2 indicates lack of clustering. In Fig. 5 we show the standard deviation of the running density fluctuations for a neutral fluid. Two scaling intervals of the type (3) appear dividing the scales of interest into two groups which we interpret using the Taylor's frozen flow hypothesis. The scaling interval with exponent 0.5 suggests that there are no clustering effects for scales larger than the integral scale of the flow. This is an indication that large scales behave as white noise. The scaling on the left, in the range corresponding to the dissipative and inertial range scales with an exponent value less than 1/2 showing tendency of small scales to cluster. Analogous quantities are presented in Fig. 6 for the L-mode signal. In comparison with the turbulent neutral fluid, it is evident that clustering in plasma turbulence takes place on a lesser number of small scales, i.e. the extent of scales on which clustering takes place is smaller. Also, the clustering exponent is somewhat larger (-0.335 in comparison with -0.36). The extent of large scales is, on the other hand, greater and this is due to the large structures of confined plasma turbulence known as blobs or avaloids. These structures do not exhibit clustering since they behave as white noise. Note that the attribute of scales being large or small should be taken in restricted sense, since Taylor's frozen flow hypothesis may not be applicable in the case of confined plasma turbulence. In the next figure, Fig. 7, standard deviation of the running density fluctuations is presented for the L/H mode which again shows white noise scaling for large scales ( p = 0.5), however temporal extent (and possibly N

233

Fig. 6.

Standard deviation of the running density fluctuations vs.

7,for

6861 L-mode.

100

9331 UH

Fig. 7. Standard deviation of the running density fluctuations vs. T , for 9031 L/H mode.

spatial) of this region is smaller than for the L-mode. The implication is that the presence of blobs is diminished in this regime The clustering exponent (exponent corresponding to smaller scales) increases i.e. the slope decreases indicating increasing tendency to cluster. Increased confinement, resulting in the H-mode, may generate at certain points in time huge coherent structures known as edge localized modes (ELMs) whose temporal evolution is presented in Fig. 8. On the bottom the signal with large coherent structures, known as edge localized modes (ELMs) is presented while a part of the signal without ELMs is presented on the top. Corresponding standard deviations of the running density fluctuations are presented in Fig. 9 (H-mode without ELMs) and Fig. 10 (with ELMs). In the absence of ELMs the clustering

234 10 5-

:

- L o - - -

5d -5

-5-10-

-15r -20

!

2

3

4

5

time (m)

6-

3

d

op -5-

-2

'

'

'5738H-i&d.e

10

1

1

I

-10-15

6

-

-20

Fig. 8. Ion saturation current (plasma density) of 5738 H-mode as a function of time. In the top figure (first 10ms) the ELMs are absent, while in the bottom figure ELMS are dominant events.

Fig. 9. Standard deviation of the running density fluctuations vs. (ELMs absent).

T,

for 5738 H-mode

is evident on all scales with an exponent 0.36 and there are no structures without clustering effects. Introduction of ELMs causes intense clustering (small slope, exponent -0.2), which involves large scales. In comparison with the L-mode which shows no clustering related to large structures as blobs (or avaloids), the large scale structures of H-mode (ELMs) are conN

235 100 .

.

,

. . . . . .,

,

,

,

. , ., , ,

. . . , . . , .,

,

. . . . . .. I

5738bH-made

-

s

10-3

'

'

"""'

'

' " " " I

'

' " " " I

'

Fig. 10. Standard deviation of the running density fluctuations vs. (ELMs present).

'

" " "

T,

for 5738 H-mode

centrated sets formed by particles clustering and possibly by accumulation of vorticity. The physics behind clustering in plasma turbulence is rather difficult to express in a form amenable to rigorous analysis but present analysis offers some interesting conclusions and opens up new areas for understanding plasma turbulence. First, blobs (large scale structures of L-mode) have no clustering properties and are very much different from edge localized modes which are produced by clustering effects. Even small ELMs have different temporal (and most likely) spatial characteristics from blob filaments. Moreover, the overall extent of scales corresponding to blobs surpass the scales corresponding to large scale structures of incompressible fluid turbulence. In the H-mode clustering effects are present on all scales relating this effect to the formation of transport barrier and zonal flows. Since transport is to a large extent suppressed in the H-mode, the value of the clustering exponent can be related to the transport coefficient.1° Finally, clustering effects may offer new insight about the hierarchy of length scales and their role in the creation of coherent structures

Acknowledgment The authors are grateful to Ben Dudson for providing the MAST data and to Ruchard Dendy for stimulating discussions.

References 1. The ASDEX team, Nucl. Fusion. 29 (1989) 1959. 2. K. R. Sreenivasan, A. Bershadskii, J. Stat. Physics, 125 (2006) 1145-1157.

236

3. A. Bershadskii, J. J. Niemela, A. Praskovsky, K. R. Sreenivasan, Phys. Rev. E 69 (2004) 056314. 4. U. Frisch, Turbulence, the Legacy of A . N. Kolmogorov, Cambridge University Press, Cambridge, 1995. 5. C. Renner, J. Peinke and R. Friedrich, J. Fluid Mech., 433 (2001) 383-409. 6. B. D. Dudson, R. 0. Dendy, A. Kirk, H. Meyer and G. F. Councel, Plasma Phys. Control Fusion 47 (2005) 885-901. 7. M. RajkoviC, M. SkoriC, K. S ~ l n aand G. Antar, Nucl. Fusion (2008). 8. S. 0. Rice, Bell Syst. Techn. J. 23 (1944) 282-332, 24 (1945) 46-156; these papers are also in Selected Papers on Noise and Stochastic Processes, ed. Nelson Wax, Dover, New York, 1954. 9. M. R. Leadbetter and J. D. Gryer, Bull. Am. Math. SOC.71 (1965) 561. 10. M. Rajkovid, M. SkoriC (in preparation).

237

INTERMITTENCY SCENARIO OF TRANSITION TO CHAOS IN PLASMA D. G. DIMITRIU,S . A. CHIRIAC Faculty of Physics, Alexandru loan Cuza University, I 1 Carol I Blvd. Iasi, RO-700506, Romania Experimental results are presented that clearly line out a scenario of transition to chaos in plasma by type I intermittency in connection with the nonlinear dynamics of a double layer structure. The intermittencies were recorded in the time series of the current through the plasma conductor as random bursts that interrupt regular oscillations. The oscillations are triggered by the nonlinear dynamics of the double layer structure.

1. Introduction

Chaotic evolution is a frequent phenomenon in filament-type discharge plasma, occurring in relation to sheath in~tabilitiesl-~. Plasma is a nonlinear system where a wide variety of transition from ordered to low and high dimensional chaotic states were identified through different types of scenarios: period d ~ u b l i n g ’ ~ ~ , intennitten~ies~’~, q~asiperiodicity~ and torus breakdown6. In plasma devices, the chaotic states were observed by time series analysis of the ac components of the discharge c ~ r r e n t ” floating ~, potential of a probe3, or the current collected by a positively biased electrode immersed in the p l a ~ m a ~ ~ ~ . ~ . Type I intennittency is associated with a saddle-node bifurcation (tangent bifurcation in one-dimensional maps). Since Pomeau and Maneville did pioneering work on the analysis of low dimensional systems’ transition to chaos7, different types of intennittencies were classified. For type I intermittency, the theory is developed on a quadratic map, based on which we can numerically or theoretically derive other characteristic features such as the probability distribution of the laminar length and the llfpower spectrum. The duration of the periodic state (so-called laminar length) seems to be at random due to the stochastic occurrence of bursts, which lead to intermittent states. Here we report on type I intermittency in plasma, related to the nonlinear dynamics of a double layer structure. Double layers are localized nonlinear potential structures’ consisting of two adjacent positive and negative space

238

charge sheaths, sustaining a potential difference equal to or higher than the ionization potential of the background gas, depending on the gas pressure and plasma density. One common way to obtain a double layer structure is to positively bias an electrode immersed into a plasma being in equilibrium. In this case, a complex space charge structure in form of a quasi-spherical intense luminous body attached to the electrode is obtained. Experimental investigations revealed that such a complex space charge structure consists of a positive nucleus (an ion-enriched plasma) surrounded by a nearly spherical double layer9*". The stability of the structure is ensured by the balance between the charges lost by recombination and diffusion and the charges created by ionizations and accumulated in the adjacent regions. By increasing the potential applied on the electrode, the rate of the ionization processes increases and the balance needed for the double layer existence is perturbed. Consequent on this, the structure disrupts, passing into a dynamic state. This dynamic state is a periodic one and determines the appearance of strong oscillations of the plasma parameters, such as the plasma density or the current collected by the electrode. Our results indicate that this dynamics evolves chaotically under certain experimental conditions. For gradually increasing the voltage on the electrode, we recorded the time series of the ac components of the electrode current. By statistical analysis of these time series, we identified a scenario of transition to chaos by type I intermittency.

2.

Experimental results and discussion

The experiments were performed in a hot-filament discharge plasma diode, schematically shown in Fig. 1. The plasma is created by volume ionization processes between energetic electrons from the hot filament (marked by F in Fig. 1) and gas atoms. The chamber wall acts as anode (marked by A in Fig. 1) and is made from non-magnetic stainless steel, being grounded. The discharge current is Zd = 40 mA. The plasma parameters, measured by mean of emissive and cold probes, were plasma density npl E 5 ~ 1 ~0 m ~ -electron ~, temperature T, G 2-3 eV, for an argon pressure p = 5x10" mbar. The plasma diffuses into the chamber, were an additional electrode of 3 cm in diameter (marked by E in Fig. 1) is introduced and positively biased in respect to the plasma potential (and also to ground). Figure 2 shows the static current-voltage characteristic of the electrode, obtained by gradually increasing and subsequently decreasing the potential on the electrode E, V,. The sudden jumps of the current collected by E, marked by ZE, are related to the generation and dynamics of the double layer structureg and

239

u1 uz

A

EP

I

1

X.

E

+Y

-

/A9

-

Figure 1. Experimental setup (F - filament, E - additional electrode, A - anode, U1 power supply for heating the filament, U2 - power supply for discharge, PS - power supply for the electrode bias, R, R2 - load resistors, EP - emissive probe, PP - plane probe, X, Y - to the oscilloscope).

Figure 2. Static current-voltage characteristic of the additional electrode (the small letters mark the positions on the characteristic where the behavior of the plasma system changes).

show hysteresis". After the first sudden jump, marked c-d in the static currentvoltage characteristic of the electrode in Fig. 2, a quasispherical luminous structure appears in front of E (photo in Fig. 3). Its appearance implies a process by which thermal energy of the electrons extracted from the surrounding plasma

240

Figure 3. Photo of the complex space charge structure obtained in front of the positively biased additional electrode.

is converted into the electric field energy of the double layer at the border of the structure”. The second jump of the current through the electrode E, marked by e-f in the Fig. 2, is related to the transition of the structure in the dynamic state. The periodical disruptions and re-aggregations of the double layers cause a modulation of the current collected by the electrode, as shown in Fig. 4a. The fast Fourier transform (FFT) amplitude graph of these oscillations and the reconstructed 3D state space (by time delay m e t h ~ d ’ ~of) the plasma system dynamics are shown in Figs. 5a and 6a, respectively. y increasing the voltage applied on the additional electrode from 55 V to 64 V, we recorded the time series of the ac component of the current collected by the electrode E (Fig. 4), with a sampling rate of 500 kHz.From these time series we calculated the FFT amplitude graphs (Fig. 5) and we reconstructed the 3D state spaces of the plasma system dynamics (Fig. 6). From the figure 4 we observe a transition to a chaotic state, due to intermittencies.The FFT amplitude graphs indicate the evolution to a chaotic state by embedding the fundamental frequency in broadband noise, associated with the onset of the intermittencies. The reconstructed 3D state spaces indicate the loss of stability of a periodic attractor (limit cycle) through a succession of bursts. The mechanism of reinsertion of trajectories in the closed loop of the attractor is relevant for proving the intermittency route to chaos’.

54-

4

3-

3

-

-= 2

2-

2

1-

-w

1

0

0-

1

1-

I 0

2

I 1

2

4

3

5

.

0

, 2

3

4

5

54

i

E

2

-w

1

1

Figure 4. Oscillations of the current collected by the additional electrode, for different values of the voltage applied on it: (a) 55 V, (b) 58 V, (c) 62 V, (d) 64 V.

Frequency (kHz)

0.3

7

Frequency (kHz)

1

(d) 0.2

0.2

-?

IJ.0.1

t

-+:

c t

0.0

0.1

0.0

0

Frequency (Wz)

Frequency ( W z )

Figure 5. FFT amplitude spectra of the corresponding signals from Fig. 3.

242

Figure 6 . 3D reconstructed state spaces of the plasma system dynamics from the corresponding signals from Fig. 3.

A typical fingerprint of type I intennittency is the presence of a tangent bifurcation (saddle-node point of bifurcation), represented in the return map shown in Fig. 7. We reconstructed this map by plotting the maxima and minima of the time series.

3. Conclusion The evolution to a chaotic state of a complex space charge structure dynamics in plasma through intennittency is experimentally investigated. The results from the time series analysis, spectral analysis, 3D reconstructed state space of the current oscillations and the return map confirm the existence of the type I intennittency.

243

a4-

a3 a2-

-: ?

-2 aiI

I

0.0

0.1

0.2

0.3

0.4

In(a.u.)

Figure 7. Return map, reconstructed by plotting the maxima and minima of the time series.

Acknowledgments This work was financially supported by the National Authority for Scientific Research - Romanian Ministry for Education, Research and Youth, under the Excellence Grant No. 1499/2006, cod ET 69.

References 1. P. Y. Cheung and A. Y. Wong, Phys. Rev, Lett. 59,551 (1987). 2. P. Y. Cheung, S. Donovan and A. Y. Wong, Phys. Rev. Lett. 61, 1360 (1988). 3. J. Qin, L. Wang, D. P. Yuan, P. Gao and B. Z. Zhang, Phys. Rev. Lett. 63, 163 (1989). 4. S. Chiriac, D. G. Dimitriu and M. Sanduloviciu, Phys. Plasmas 14, 072309 (2007). 5. W. Ding, W. Huang, X. Wang and C. X. Yu, Phys. Rev. Lett. 70, 170 (1993). 6. S. Chiriac, M. Aflori and D. G. Dimitriu, J. Optoelectron. Adv. Mater. 8, 135 (2006). 7. Y. Pomeau and P. Maneville, Commun. Math. Phys. 74, 189 (1980). 8. C. Charles, Plasma Source. Sci. Technol. 16, R1 (2007). 9. M. Sanduloviciu and E. Lozneanu, Plasma Phys. Control. Fusion 28, 585 (1986).

244

10. B. Song, N. D’Angelo and R. L. Merlino, J. Phys. D: Appl. Phys. 24, 1789 (1991). 11. S. Chiriac, E. Lozneanu and M. Sanduloviciu, J. Optoelectron. Adv. Muter. 8, 132 (2006). 12. E. Lozneanu and M. Sanduloviciu, Chaos, Solitons and Fractals 30, 125 (2006). 13. D. Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989.

245

NONLINEAR DYNAMICS OF A HAMILTONIAN FOUR-FIELD MODEL FOR MAGNETIC RECONNECTION IN COLLISIONLESS PLASMAS E. TASSI' and D. GRASS0 Dipartimento di Energetica, Politecnico di Torino, Torino, 10189, Italy * E-mail: [email protected]

F. PEGORARO Dipartimento di Fisica, Universitd di Pisa, Pisa, 56127, Italy In this contribution we present some aspects of the nonlinear dynamics of a model which describes the phenomenon of magnetic reconnection (MR) occurring in plasmas where particle collisions can be neglected. The concept of MR is introduced in the framework of the singlefluid description of a plasma. The model under consideration is then reviewed with focus on its non-canonical Hamiltonian structure. Numerical solutions of the model equations show that in the nonlinear phase a secondary instability of Kelvin-Helmholtz type occurs at moderate values of the parameter p, which indicates the ratio between thermal and magnetic pressure, and also in the presence of finite electron compressibility. This represents a novel feature, in the nonlinear dynamics of reconnection, with respect to previously investigated models valid only for very low values of p. Keywords: Magnetic reconnection, non-canonical Hamiltonian systems, KelvinHelmholtz instability

1. Introduction Astrophysical and laboratory plasmas often exhibit nonlinear behaviors which manifest themselves, for instance, through processes such as selforganization, chaos and turbulence.' Plasmas therefore represent a fertile ground for the investigation of nonlinear phenomena. Among these, the process of MR plays an important role, also due to its implications for space and fusion plasma^.^^^ In the former context MR has indeed been suggested to play a key role in triggering solar flares and magnetic substorms

246

whereas in the latter case it is believed to be one of the processes causing the so-called sawtooth oscillations observed in tokamak fusion devices. In the framework of a single-fluid description of a plasma, MR can be thought of as result of the violation of the so-called frozen-in condition, according to which, in a perfectly conducting plasma, if two infinitesimal volumes of fluid are threaded a t some time by the same magnetic field line, then they will always be joined by the same magnetic field line a t any later time. The mathematical justification for this condition is a consequence (see e.g.’l4) of the equation

aB at

--

Vx

(V x

B ) = 0,

which is valid for an infinitely conducting plasma. In (1) B is the magnetic field and v is the plasma fluid velocity. In many laboratory and astrophysical contexts (1) is satisfied to a good extent. However, there can be localized spatial regions where effects such as collisional resistivity, electron inertia or turbulence can prevent the plasma from behaving as a perfect conductor and break the constraint of the frozen-in condition. Infinitesimal plasma volumes initially joined by a single magnetic field line can then disconnect from that line and connect to two distinct field lines a t a later time (see Fig. 1). This phenomenon is accompanied by a conversion of magnetic energy into plasma kinetic and thermal energy, thus explaining, for instance, why MR is believed to be responsible for the huge releases of energy occurring during solar f l a r e ~ . Among ~t~ the various physical effects determining the violation of the frozen-in condition and the consequent possibility for MR to take place, electron inertia has been suggested t o be a relevant one in particular in the context of fusion plasmas, for it provides characteristic reconnection time scales comparable with those observed during sawtooth

-

magncticfisld lins

. . . . . . 3

pbrnavclofityfisld line

Fig. 1. Figure sketching the mechanism of violation of the frozen-in condition: pairs of small plasma volumes (the black and the white squares) are initially joined by the same magnetic field line. At a later time, due t o reconnection, the black and the white squares, initially disjoined, are connected by the same magnetic field line.

247

oscillations in t o k a m a k ~ Models .~ in which MR is caused by electron inertia and where dissipative effects are negligible (which are for instance applicable namely t o the high temperature, weakly collisional tokamak plasmas), possess the remarkable property of admitting a Hamiltonian formulation, which, in addition to its formal elegance, makes it possible to take advantage, in the analysis, of the huge amount of results available for Hamiltonian systems. In this contribution we present results related to the nonlinear dynamics of a model for MR driven by electron inertia. In Sec. 2 we briefly introduce the model and in Sec. 3 we present its non-canonical Hamiltonian formulation. In Sec. 4 we present the results from the numerical simulations and we focus on the onset of secondary fluid-like instabilities taking place after the initial instability that led to reconnection. The final section is devoted to conclusions. 2. The model The dynamical model for MR that we consider here is the four-field model derived in Ref.6v7 Such model describes MR processes caused by electron inertia in a plasma where particle collisions can be neglected. To describe the geometry of the problem a Cartesian coordinate system (z, y, z ) is adopted, with z as ignorable coordinate. The dynamics is then two-dimensional with reconnection of magnetic field lines taking place in a slab lying in the zy plane. The model equations, written in a standard dimensionless form, read

dU

+ [%VI

(5)

-

In the above equations the flux function +, the stream function p and the fields 2 and w are related to the magnetic field B and to the plasma average velocity field v by the following relations: B(z,y, t ) = V+(z, y, t) x i (B(O) c p Z ( z , y , t ) ) i and v ( z , y , t ) = - V p ( z , y , t ) x z v ( z , y , t ) i . The constant B(O) is the so called guide field which mimics the toroidal

+

+

+

248

field present in tokamak devices. The parameter cp is defined as cp = where j3 indicates the ratio between the plasma background pressure and the magnetic pressure based on the guide field B(O). The other parameters present in the system are the electron skin depth d , and d p = dicp, with di indicating the ion skin depth. Finally the Poisson bracket is defined by [ f , g ]= ( O f x V g ). z. The above model can be derived starting from the standard two-fluid description of a plasma. In particular (2) comes from the electron momentum equation, (3) from the electron vorticity equation, (4) from the average plasma vorticity equation and (5) from the average plasma momentum equation. Notice that the term d2aV2+/at, in Eq. (2), is proportional to the electron inertia and it is namely the presence of this term that breaks the frozen-in condition allowing MR to take place. If electron mass were neglected the magnetic flux would be a scalar quantity advected by the stream function (p dpZ and its contour lines (corresponding to the projections of magnetic field lines in the xy plane) could not reconnect.

d

m

+

+

3. Hamiltonian structure

As many dissipation-free models, the system (2)-(5) admits a non-canonical Hamiltonian formulation.8 In short this means that the above model can be written in the form

xi

-at ={&,H}, i = l , . . ., 4 (6) are field variables, H is a Hamiltonian functional and the bracket

where ci {, } is an antisymmetric bilinear form satisfying the Jacobi identity. For the model under consideration it has been showng that the dynamical equations for the field variables = - dZV2@,(p, Z and u can be obtained from the Hamiltonian

+, +

's

H =2

d 2 z ( d Z J 2+ IV(pI2+ u2 + lV+I2+ Z 2 )

(7)

and from the bracket defined by

{ F ,GI =

1

d2x ( ~ [ F uGu] ,

+ +e(

[F$,7 Gu]

+

+ P U , G$J - dP([FZ,G$,I+ [F?be, GZI) + cp([FvrGzl [ F z ,GI)) + ~ ( [ F Z , G+ U [Fu,Gz] ] -dpde2[F~,,G+,,] +cpde2([Fv,G$,] [ F QG ~ v,] )

+

+

Gvl - a [ F z , Gzl - C P Y P v , GI)+ V([Fv,GU] [Fu,

+ [F+,,Gzl)- c p ~ ( [ F v , G z+] [ F z , G ~ ] ) ) ) , +~pde~([Fz,G$,I

(8)

249

+

where F and G are two functionals of the dynamic variables, a = d p cpde2/di,y = d e 2 / d i and subscripts indicate functional differentiation. It has also been shown that the bracket (8) possesses four infinite families of so-called Casimir invariants, i.e. functionals C of the field variables such that {C, G} = 0,

for every functional G.

(9)

Casimir functionals are constants of motion and can provide important information about the nonlinear dynamics of the system. For the four-field model the four families of Casimirs are given by

C1 = C,

where D

=

/

d2zwF(D),

/d2zg+ ( D

-

“pw

ff

-

C2 =

/

d2z’H(D),

E)/ -Z

d2zg+(T+),

=

(11)

w = V 2 p + Z / a ,T&= &(l/2a)&&4D-(a/cp)u~ whereas F , N , g+ and g- are arbitrary functions. When written in terms of the variables D , w , T* the model equations (2)-(5) take the form

d*Z),

= $e+div,

aD at

- = -[P,DI,

aw

- = -[p,w] at

+ de2 +1 di

2

P7

$13

This representation shows that the fields D , T* turn out to be Lagrangian invariants advected by corresponding velocity fields. In particular the field D gets advected by the actual average plasma velocity whereas T* get advected by “virtual” velocity fields associated to stream functions composed by a fluid (associated to cp) and a magnetic (associated to $) component.

250

4. Numerical simulations

The model equations (2)-(5) have been solved numerically, adopting a finite volume scheme, over the rectangular domain {(x, y) : -2n 5 5 5 2n, -7r 5 y 5 n} discretized over 1024x512 grid-points. The initial condition corresponds to the choice: $eq = 1/ cosh2(x), peg = 0, Z q = 0,V e g = 0 and double periodic boundary conditions have been imposed.

@

-0

t=O

5

10

-7

-1

0 x/n

1

2

0

-1

1

x/.

Fig. 2. Contour plot of the flux function at two different times showing how initially disconnected magnetic field lines get joined at a later time due t o MR.

Figure 2 shows contour plots of the 1c, function obtained from the simulations at two different times. The comparison of the plots shows how initially disconnected magnetic field lines get connected a t later time due t o the on-going MR process. In addition t o the investigation of the MR process itself the numerical simulations make it also possible to observe and study further nonlinear processes which occur as a by-product of MR. In the analysis presented in this contribution we focus on the observation of plasma vorticity structures and on their dependence on the parameters cp and d p , while keeping d, fixed to the value 0.24. Looking a t the vorticity field is significant in order t o detect the presence of fluid-like instabilities and allows the comparison with previous work based on low-p models. Indeed, numerical simulations of the four-field model run in the low-p regime, whose results are not presented here, correctly tend to reproduce the vorticity and current density behavior already observed in the analysis of low-/3 models. In particular for values of cp, dp lop3, in which limit the four-field model reduces to the two-field model investigated in one observes the formation of two vorticity and current density jets directed along the vertical direction. Such jets N

251

subsequently propagate toward each other until they collide and undergo a Kelvin-Helmholtz-type instability. On the other hand if cp l o v 3 and dp 10-1 the simulations of the four-field model reproduce the behavior observed in the low-p two-field model investigated in.l3?l4In this limit vorticity and current density tend to concentrate along the separatrix lines of the magnetic island formed after reconnection, and to create filamented structures. This behavior was explained in terms of the constraints imposed to the dynamics by the presence of two Lagrangian invariants in the model equations. One of the novel aspects introduced by the four-field model is that it makes it possible to investigate nonlinear dynamics also in higher-/? regimes. In Fig. 3 it is possible to see the evolution of the vorticity field V2cp through four different times (time here is normalized with respect to the Alfvkn time L,/ij/Bo where L,Bo and p are characteristic length scales and intensity of the magnetic field, and plasma mass density, respectively) for cp = 0.2 and dp = 0.48. The contour plot a t t = 50 shows the presence of a localized vertical structure a t the center of which a quadrupolar structure is formed. Contour plots a t earlier times show that such structures form due to the headon collision between two vertical jets moving in opposite directions. The vertical structure is enclosed by a region where vorticity is filamented, and whose boundary turns out to coincide with the contour of the magnetic island formed due t o reconnection,. At subsequent times pairs of vortices of the quadrupolar structure start to drift in opposite directions along the y = 0 axis. At t = 65 the vortices reach the filamented region and one can also observe that the vertical structure starts to break up indicating the onset of a Kelvin-Helmholtz type instability. In the higher-P regime, where w and 2 are not decoupled from the system, the vorticity is given by N

N

I

U = L&,/----(T+ 1 2d,

dz+d:

- T-)

+w ,

The vorticity can then be interpreted as the superposition of two contributions. One contribution corresponds to the terms with T+ and T-. These fields behave similarly to two Lagrangian invariants already investigated in the very low-/? limit. Indeed such fields get advected by the fields cp+ = c p f and are responsible for the appearance of filamented

2 J$- / ,

vorticity structures observed in the contour plots. As explained in Ref.13 such filamented structures are the result of the stretching of the fields T*

252

0.5

$

0.0

---0.5 -1.0 2

1

0

1

Z , h

05 I=

00

\

3

-0 5 10

a

1

0

1

X/7T

0.5 0.0

s ---

0.5 1 .li

z

1

0

1

.c/n

Fig. 3. Contour plots of the vorticity field at different Alfvkn times. The values of the parameters are: cp = 0.2 (corresponding to /3 = 0.04), do = 0.48, d, = 0.24.

253

operated by the “virtual” velocity fields 96 which rotate in opposite directions. The localized vertical structure is then due t o the presence of the remaining contribution to U , i.e. the one coming from w , which had no corresponding term in the expression for U in the low-p limit. Contour plots of U at subsequent times show that such vorticity structures undergo a KelvinHelmholtz type instability as it is the case in the very low-p regime when the value of the electron compressibility parameter ps = a / e B is much smaller than dele . Recalling that for the low+ regime under consideration cp f l ,so that dp = p s , our results show that the Kelvin-Helmholtz instability, which is inhibited in the presence of finite ps a t very low-p, can take place even if ps # 0 , if the value of p is increased. The effect of increasing p introduces a shift between -Z/a and U , which yields a finite generalized vorticity w . Indeed, as above explained, it is the generalized vorticity that is related to the appearance of the vorticity jets undergoing Kelvin-Helmholtz instability. Moreover, at higher ,f3 values, an effective coupling with Eq. (5) occurs, whereas w decouples from the system for very small values of p.

-

5. Conclusions In this contribution we presented new results concerning the nonlinear dynamics of a collisionless reconnection process described by a four-field Hamiltonian model. Numerical simulations reproduce already known results in the regime of very low-p. At higher p values a new regime is found, where both filamentation and formation of vorticity jets are observed. It is shown that in this new regime a secondary Kelvin-Helmholtz instability can take place even if ps is finite.

References 1. W. Horton and Y.-H. Ichikawa, Chaos and Structures in Nonlinear Plasmas (World Scientific Publishing Co., 1996). 2. E. R. Priest and T. G. Forbes, Magnetic Reconnection (Cambridge University Press, 2000). 3. D. Biskamp, Magnetic Reconnection in Plasmas (Cambridge University Press, 2000). 4. G. Hornig and K. Schindler, Phys. Plasmas 3,781 (1996). 5. J. Wesson, Nucl. Fusion 30,p. 2545 (1990). 6. R. Fitzpatrick and F. Porcelli, Phys. Plasmas 11,4713 (2004). 7. R. Fitzpatrick and F. Porcelli, Phys. Plasmas 14,p. 049902 (2007). 8. P. J. Morrison, Phys. Plasmas 12,058102 (2005). 9. E. Tassi, P. J. Morrison and D. Grasso, Hamiltonian structure of a collisionless reconnection model valid for high and low @ plasmas, in PTOC.Workshop

254

10.

11. 12. 13. 14.

Collective Phenomena in Macroscopic Systems, eds. M. R. G. Bertin, R. Pozzoli and K. Sreenivasan (World Scientific, Como, Italy, August 2007). D. Del Sarto, F. Califano and F. Pegoraro, Phys. Plasmas 12, p. 012317 (2005). D. Del Sarto, F. Califano and F. Pegoraro, Mod. Phys.Lett. B 2 0 , 931 (2006). D. Grasso, D. Borgogno and F. Pegoraro, Phys. Plasmas 14,p. 055703 (2007). E. Cafaro, D. Grasso, F. Pegoraro, F. Porcelli and A. Saluzzi, Phys. Rev. Lett. 80, 4430 (1998). D. Grasso, F. Califano, F. Pegoraro and F. Porcelli, Phys. Rev. Lett. 86, 5051 (1994).

255

ON THE COMPLEXITY OF THE NEUTRAL CURVE OF OSCILLATORY FLOWS M.WADIH, S.CARRION, P.G.CHEN, D.FOUGeRE, B.ROUX MSNM-GP, UMR 6181 CNRSKJniversitksAix-Marseille, France In this paper, the linear stability property of oscillatory flow in a circular pipe under a periodic pressure gradient is examined. Our results of neutral curve show the existence of a much more remarkable finger-like protrusions structure. In this case, we may speak about two neutral curves: one inside which corresponds to a rather late critical threshold and the other one which makes this threshold weaker but closer to experiment observations. The underlying mechanism is explained both from mathematical point of view and from physical point of view.

1. Introduction The linear stability property of oscillatory flows such as flat Stokes layers usually exhibits regular and smooth enough neutral curves, suggesting a light influence of unsteadiness of the basic flow; the unsteady effect when compared to classic steady flows is often limited itself to advance or enhance transition. Many studies, whether theoretical or numerical in nature even failed to predict transition thresholds onto less orderly regimes as observed in some experimental works. From large body of literature dealing with the linear stability of various types of oscillatory flows one can distinguish a lot of works, started from the pioneer work in 1930’s by Schlichting [l],followed by Stuart [2] and Riley [3], who analyzed the steady streaming induced by an oscillatory flow. Venezian [4] considered BCnard problem with periodic temperature gradient and obtained solutions by assuming the amplitude temperature gradient be small and expanding all quantities in powers of this parameter. The same problem was studied by Rosenblat & Herbert [5] with a WKB method. Analytical studies of purely oscillatory flows begun with the paper of von Kerzeck & Davis[6] for a high frequency oscillatory, the study was followed by Hall [7] in the case of the classical flat Stokes layer.

256

The recent works of Blennerhassett & Bassom [8, 91 on the linear stability of Stokes layers provide a new insight into the investigation of oscillatory flows for a better prediction of various transition modes. The most intriguing feature of their results is the fine structure of the neutral curve, which has thin fingerlike protrusions from an essentially smooth curve in some range of wavenumber. The present work concerns the linear theory of neutral stability curve for the oscillatory flow in a circular pipe under a periodic pressure gradient. The calculations of stability based on Floquet theory lead to a relationship between Reynolds number and wavenumber of the first instability mode. The finger-like protrusions appear on the neutral curve in such a way that we can speak of two neutral curves. The formulation of the problem is described in Section 2. Our results are presented and discussed in Section 3. Conclusions are given in Section 4.

2. Formulation of the problem An oscillatory flow in a circular pipe with infinite length, radius R is considered with periodic flow with frequency w and amplitude Q,, . Only the axisymmetric case is considered herein with non-dimensional coordinates (z, r) where the zaxis is in direction of the pipe axis. Lengths, velocities are respectively scaled with the radius R and $ / ( n R 2 ) ,the time being z = ot .

Figure 1. Schematic of flow configuration.

The dimensionless parameters usually introduced are the Reynolds number R e = Q o / (nRv) and the frequency 52 = R2co/v where v is the kinematic viscosity. The basic motion is described by the equation of continuity and the Navier-Stokes equations:

v.v=o 51a,v+R,v.Vv=-R,Vp+V2V completed with a non-slip condition at the wall.

(1)

257

As the basic flow V is generated in the direction of longitudinal axis, the solution of ( 1 ) takes the form:

-

, v = w = o , (2) where J o and J 2 respectively denote Bessel functions of order zero and two. The linear stability analysis is made by introducing an axisymmetric perturbation u ' e , + v ' e , . Then, by the mean of the introduction of stream function disturbance of the form:

with a the real wavenumber of the disturbance, the use of Galerlun method as described by Siouffi et al. [lo] leads to ordinary differential equations for fm

(4 f DKf, = p42 r

,

(4)

where D , K are operators defined by (see also Siouffi et al. [ l o ] ) : d2

1d

1 d2

I d

) ' K = -r ( d-----a2 r2 r d r

-+d

+---

D=-dr2 r dr (r12 with the boundary conditions:

Then following the usual Galerkin technique, the first-order linearized differential system is obtained:

c-dG

=AG+B(~)G,

dz

where C, A and B are matrices representing, for N basic functions of the stream function, linear operators which depend only on Ll and a :

258

where operator L is defined by L =

3. Results

3.1. Numerical method Matrices A, B and C are first calculated for fixed parameters R and a . Then the numerical solution of the system ( 5 ) was obtained by using a fourth-order Runge-Kutta method. For a fixed parameter R e , the number of equations N is increased until to find a minimal number N to ensure the convergence of the solution.

3.2. Strong stability The strong stability corresponds to the resolution of (5) for one initial condition as it was studied by Siouffi et al. [lo] for R=100. They found a critical Reynolds number of 3500 for a = 2.5 which is in agreement with experimental results [ l l ]and [12]. This study gives a sufficient condition of stability.

3.3. Weak stability The use of Floquet theory implies that the stability is ensured in the sense where some disturbances can grow inside a cycle but be globally attenuated in the next cycles as it is showed in figure 2. This stability is insured if the real part of the Floquet exponents is zero. Starting from N initial conditions such as G(z = 0) = I (I is the identity matrix), we found that for SZ =loo, N = 50 is necessary to ensure the convergence of the Floquet exponents.

259

lcll 70 60

-

-

^

-

_

0,OO

_

-

Re=9000

3,14

~

6,28

Re=9745

9,42

-

12,56

Re=9780

15,70

z

18,84

Figure 2. Temporal evolution of the magnitude of the fundamental matrix of system (5) for c;2 =lo0 and a=2. Re = 9000: stable, R, = 9745: critical value, Re = 9780: unstable.

The neutral curve ( a ,Re ) in figure 3 which corresponds to the case of zero value of the real part of the greatest exponent Floquet shows protrusions. The inner points (on the right) correspond to complex conjugate Floquet exponents while the other points correspond to real Floquet exponents. We may define two curves: an outer curve along the extremities of the fingers and another one along inner points. We obtain the critical values aC= 1.997 and Re, = 9631.34,

ff

2,20

2,oo

1,80

1,70 9600

9700

9800

9900

Re

10000

Figure 3. Neutral curve for 51 =loo.

It is seen in figure 4 that the two greatest Floquet exponents a, and O, become alternatively complex conjugate or real as R, is increased.

260

0,060

0,000

-0,060

-0,120 9400

9500

9600

9700

9800

R

Figure 4. Real part ofthe two greatest Floquet exponents as a function of R e for S2 =lo0 and a=2.

When a is increased, fluctuations appear on the Floquet exponent (figure 5) so that it becomes numerically impossible to obtain the neutral curve for value of a greater than 2.7. 0,005 3

qr 0,000

-0,005

-

-

d

,v

4

I

I,

---- 9 r

A

I

-0,010

4

-0,020

!

13768

t

I I

13770

I\

I I

13772

U

13774

Re

13776

Figure 5 . Variation of the real part of the greatest Floquet exponents with Re for S2 =lo0 and ~~=2.62.

4.

Conclusions

Our results of neutral curve show the existence of a much more remarkable finger-like protrusions structure. Therefore, we may speak of two neutral curves: one inside which corresponds to a rather late critical threshold and the other which makes this threshold weaker but closer to experiment observations.

261

The mechanism of the generation of fingers on the neutral curve can be explained from a mathematical point of view by an alternation of zones for real or complex Floquet exponent, and from a physical point of view by the coalescence of two propagating waves in opposite directions. In addition, this mechanism is generic and when the transition takes place in a zone with real value, a finger appears on the neutral curve which decreases the critical threshold and introduces a second curve closer to a smaller Reynolds number. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

H. Schlichting, Phys. Z. 33, 327 (1932). J. T. Stuart, J. FluidMech. 24, 673 (1966). N. Riley, J. Inst. Math. Appl. 3,419 (1967). G. Venezian, J. Fluid Mech. 35,243 (1969). S. Rosenblat and D. M. Herbert, J. Fluid Mech. 43,385 (1970). C. von Kerzeck and S. Davis, J. FluidMech. 62,753 (1974). P. Hall, Proc. R. SOC.Lond. A 359, 151 (1978). P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 464,393 (2002). P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 556, 1 (2006). M. Siouffi, S. Carrion and M. Wadih, C.R. Me'canique 330,641 (2002). M. Hino, M. Savamoto and S. Takasu, J. Fluid Mech. 75, 193 (1976). D. M. Eckmann, J. B. Grotberg, J. Fluid Mech. 222, 329 (1991).

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(2) OTHERS

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265

GENERALIZED KELLER-SEGEL MODELS OF CHEMOTAXIS. ANALOGY WITH NONLINEAR MEAN FIELD FOKKER-PLANCK EQUATIONS PIERREHENRI CHAVANISt Laboratoire de Physique The'orique, Universite' Paul Sabatier, 118 route de Narbonne 31 062 Toulouse, France $E-mail: chavanis0irsamc. ups-tlse.fr We consider a generalized class of Keller-Segel models describing the chemotaxis of biological populations (bacteria, amoebae, endothelial cells, social insects,...). We show the analogy with nonlinear mean field Fokker-Planck equations and generalized thermodynamics. As an illustration, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). We also discuss the analogy between biological populations described by the Keller-Segel model and self-gravitating Brownian particles described by the Smoluchowski-Poisson system.

1. Introduction

The name chemotaxis refers to the motion of organisms induced by chemical signals.' In some cases, the biological organisms (bacteria, amoebae, endothelial cells, ants...) secrete a substance (pheromone, smell, food, ...) that has an attractive effect on the organisms themselves. Therefore, in addition to their diffusive motion, they move preferentially along the gradient of concentration of the chemical they secrete (chemotactic flux). When attraction prevails over diffusion, the chemotaxis can trigger a self-accelerating process until a point at which aggregation takes place. This is the case for the slime mold Dictyostelium discoideum and for the bacteria Escherichia coli. This is referred to as chemotactic collapse. A model of slime mold aggregation has been introduced by Patlak2 and Keller & Sege13 in the form of two coupled differential equations. The first equation is a drift-diffusion equation describing the evolution of the concentration of bacteria and the second equation is a diffusion equation with terms of source and degradation describing the evolution of the concentration of the chemical. In the simplest model, the diffusion coefficient D and the mobility x of the bac-

266

teria are constant. This forms the standard Keller-Segel model. However, the original Keller-Segel model allows these coefficients t o depend on the concentration of the bacteria and of the chemical. If we assume that they only depend on the concentration of the bacteria, the general Keller-Segel model becomes similar to a nonlinear mean field Fokker-Planck equation. Nonlinear Fokker-Planck (NFP) equations have been introduced in a very different context, in relation with a notion of generalized thermodynami c ~ As . ~ far as we know, the connection between the general Keller-Segel model and nonlinear mean field Fokker-Planck equations has been first mentioned in Chavanis5 and developed in subsequent papers (see6 and references therein). This analogy makes possible t o interpret results obtained in chemotaxis in terms of a generalized thermodynamics. At the same time, chemotaxis becomes an example of great physical importance for which a notion of (effective) generalized thermodynamics is justified. The standard Keller-Segel (KS) model has been extensively studied in the mathematical literature (see the review by Horstmann7). It was found early that, above a critical mass, the distribution of bacteria becomes unstable and collapses. This chemotactic collapse leads ultimately to the formation of Dirac peaks.8-20 Recently, it was shown by Chavanis, Rosier & Sire21 that, when the equation for the evolution of the concentration is approximated by a Poisson e q ~ a t i o n the , ~ standard ~ ~ ~ ~ Keller-Segel ~ ~ ~ (KS) model is isomorphic to the Smoluchowski-Poisson (SP) system describing self-gravitating Brownian particles. The chemotactic collapse of biological populations above a critical mass is equivalent to the gravitational collapse of self-gravitating Brownian particles below a critical temperature.22Assuming that the evolution is spherically symmetric, Chavanis & Sire21-28were able to describe all the phases of the collapse (pre-collapse and post-collapse) analytically in d dimensions, including the critical dimension d = 2. Recently, some authors have considered generalizations of the standard Keller-Segel (KS) model. Two main classes of generalized Keller-Segel (GKS) models of chemotaxis have been introduced: (a) Models with filling factor: Hillen & Painter29>30considered a model with a normal diffusion and a density-dependent mobility x ( p ) = x(1 p/oo) vanishing above a maximum density 00. The same model was introduced independently by C h a ~ a n i s in ~ ?relation ~~ with an “exclusion principle” connected to the Fermi-Dirac entropy in physical space. In these models, the density of bacteria remains always bounded by the maximum density: p(r, t ) I no. This takes into account finite size effects and filling fac-

267

tors. Indeed, since the cells have a finite size, they cannot be compressed indefinitely. In this generalized Keller-Segel model, chemotactic collapse leads ultimately to the formation of a smooth aggregate instead of a Dirac peak in the standard Keller-Segel model. This regularized model prevents finite time-blow up and the formation of (unphysical) singularities like infinite density profiles and Dirac peaks. Therefore, the Dirac peaks (singularities) are replaced by smooth density profiles (aggregates). (ii) Models with anomalous diffusion: Chavanis & Sire3' studied a model with a constant mobility and a power law diffusion coefficient D ( p ) = DpY-' (with y = 1 l / n ) . This lead to a process of anomalous diffusion connected to the Tsallis entropy.33 For 0 < n < n3 = d / ( d - 2), the system reaches a self-confined distribution similar to a stable polytrope (e.g. a classical white dwarf star) in astrophysics. For n > n3, the system undergoes chemotactic collapse above a critical mass (the classical chemotactic collapse related to the standard Keller-Segel model is recovered for n -+ + w ) . ~ ' In the pre-collapse regime, the evolution is self-similar and leads to a finite time singularity. A Dirac peak is formed in the post-collapse regime. For n = 723, the dynamics is peculiar and involves a critical mass similar to the Chandrasekhar limiting mass of relativistic white dwarf stars in a s t r o p h y ~ i c s .The ~ ~ case of negative index n < 0 is treated in35 with particular emphasis on the index n = -1 leading to logotropes. In the present paper, we discuss a larger class of generalized Keller-Segel models and interpret these equations in relation with nonlinear mean field Fokker-Planck equations and generalized thermodynamics. For illustration, we present for the first time a model incorporating both a filling factor and some effects of anomalous diffusion.

+

2. The generalized Keller-Segel m o d e l 2.1. The dynamical equations

The general Keller-Segel model3 describing the chemotaxis of bacterial populations consists in two coupled differential equations

dP

at = v . (DzVp) - v . (DlVC) ,

that govern the evolution of the density of bacteria p ( r ,t ) and the evolution of the secreted chemical c ( r , t ) . The bacteria diffuse with a diffusion

268

coefficient D2 and they also move in a direction of a positive gradient of the chemical (chemotactic drift). The coefficient D1 is a measure of the strength of the influence of the chemical gradient on the flow of bacteria. On the other hand, the chemical is produced by the bacteria with a rate f(c) and is degraded with a rate k(c). It also diffuses with a diffusion coefficient D,.In the general Keller-Segel model, D1 = D ~ ( p , cand ) 0 2 = D2(p1c) can both depend on the concentration of the bacteria and of the chemical. This takes into account microscopic constraints, like close-packing effects, that can hinder the movement of bacteria. If we assume that the quantities only depend on the concentration of bacteria and write Dz = D h ( p ) , D1 = x g ( p ) , k(c) = k 2 , f ( c ) = X and D, = 1, we obtain

dP

d t = V . (Dh(P)VP- X S ( P ) V C ) dC

E-

at

= AC- k2c

1

+ Xp.

(3)

(4)

For c = 0, Eq. (4) becomes the screened Poisson equation

AC- k2c = -Xp.

(5)

Therefore, we can identify k-' as the screening length. If we assume furthermore that k = 0, we get the Poisson equation

AC= -Xp.

(6)

The generalized Keller-Segel (GKS) model (3) can be viewed as a nonlinear mean-field Fokker-Planck (NFP) e q ~ a t i o n .Written ~ in the form dtp = V . ( V ( D ( p ) p )- x ( p ) p V c ) , it is associated with a stochastic ItoLangevin equation

with

where R(t) is a white noise satisfying ( R ( t ) )= 0 and (Ri(t)Rj(t'))= & S ( t - t') where i = 1,...,d label the coordinates of space. The standard Keller-Segel model is obtained when the mobility x and the diffusion coefficient D are constant. This corresponds to h(p) = 1 and g ( p ) = p. In that case, the stochastic process (7) and the Fokker-Planck equation (3) are similar to the ordinary Langevin and Smoluchowski equations describing the diffusion of a system of particles in a potential @ ( r , t )= - c ( r , t )

269

that they produce themselves through Eq. (4). For example, when Eq. (4) is approximated by Eq. ( 6 ) , the system becomes isomorphic to the Smoluchowski-Poisson system describing self-gravitating Brownian particles.21122The steady state of the standard Keller-Segel equation is p e 5 ‘ . This is similar to the Boltzmann distribution p e-’@IT of statistical equilibrium provided that we introduce an effective temperature T through the Einstein relation T = D / x . In the present study, we shall consider more general situations and allow the mobility x ( p ) and the diffusion coefficient D ( p ) to depend on the local concentration of particles p ( r , t ) . This is an heuristic approach to take into account microscopic constraints that affect the dynamics of the particles at small scales and lead to non-Boltzmannian distributions at equilibrium. Indeed, it is not surprising that the mobility or the diffusive properties of a particle depend on its environment. For example, in a dense medium its motion can be hampered by the presence of the other particles so that its mobility is reduced.

-

-

2 . 2 . Generalized free energy and H-theorem

We define the energy by

E=-1 / 2x

For

E

+

[ ( V C ) k2c2] ~ dr -

(9)

= 0, this expression reduces to

E=

‘s

pcdr.

--

2

On the other hand, we define the temperature by

D T=X’ Therefore, the Einstein relation is preserved in the generalized thermodynamical framework. We also set p = 1/T. We introduce the generalized entropic functional

S=-

J’

C(p)dr,

(12)

where C ( p ) is a convex function (C” 2 0) defined by

s

This defines the entropy up to a term of the form A M + B where M = p d r is the mass (which is a conserved quantity). We can adapt the values of the

270

constants A and B in order to obtain convenient expressions of the entropy. Finally, we introduce the generalized free energy

F=E-TS. (14) The definition of the free energy (Legendre transform) is preserved in the generalized thermodynamical framework. The free energy is the correct thermodynamical potential since the system is dissipative. Thus, it must be treated within the canonical e n ~ e m b l e . ~ > ~ ~ A straightforward calculation shows that

F

‘J

= --

( - A c + k2c - Xp)2dr -

- Xg(p)Vc)2dr.

XE

For

E

= 0,

this equation reduces to

Therefore, F 5 0 (in all the paper, we assume that E , X , X , D , h~ ,are positive quantities). This forms an H theorem in the canonical e n ~ e m b l e ~ ~ ~ ~ for the nonlinear mean field Fokker-Planck equation (3). This also shows that the free energy F [ p ,c] is the Lyapunov functional of the generalized Keller-Segel model (3)-(4). It is sometimes useful to introduce the Massieu function

J=S-pE, (17) which is related to the free energy by J = -pF. Clearly, we have J 2 0. We can now consider particular cases: if D = 0 (leading to T = 0), we get F = E so that E 5 0. If x = 0 (leading to p = 0), we have J = S so that s 2 0. 2.3. Stationary solution

The steady state of Eq. (3) satisfies F leads to

AC - k2c = -Xp,

=

0. According to Eq. (15), this

Dh(P)VP

- XS(P)VC = 0.

(18)

Using Eqs. (11) and (13), the second equation can be rewritten

C”(p)Vp

- p v c = 0,

(19)

which can be integrated into C ’ ( p ) = pc - a,

(20)

271

where a is a constant of integration. Since C is convex, this equation can be reversed to give p ( r ) = F(-Dc(r)

+a),

(21)

where F ( z ) = (C’)-l(-z) is a monotonically decreasing function. Thus, in the steady state, the density is a monotonically increasing function p = p(c) of the concentration. We have the identity n

Substituting Eq. (21) in Eq. (5), valid for a stationary state, we obtain a mean-field equation of the form

+ k2c = XF(-Dc + a ) .

-Ac

(23)

The constant of integration a is determined by the total mass M (which is a conserved quantity). Finally, we note that the generalized entropy (12) is related to the distribution (21) by: C(p)= -

/

P

F-l(z)dz.

(24)

Equation (21) determines the distribution p(r) from the entropy S and Eq. (24) determines the entropy from the density. 2.4. M i n i m u m of free energy

The critical points of free energy at fixed mass are determined by the variational problem

SF

+ TaSM = 0,

(25)

where a is a Lagrange multiplier. We can easily establish that

SE

‘s

= --

x

(Ac - k2c + Xp)Scdr -

65’= -

J’

(26)

C’(p)Spdr.

The variational problem (25) then leads to

A c - k2c = -Xp,

C ’ ( p ) = DC - a.

(28)

Comparing with Eq. (20), we find that a stationary solution of Eq. (3) is a critical point of F at fixed mass. On the other hand, we have established that

FIO,

F=O*&p=O.

(29)

272

According to Lyapunov’s direct m e t h ~ d this , ~ implies that p(r) is linearly dynamically stable with respect to the NFP equation (3)-(4) iff it is a (local) minimum of F a t fixed mass. Maxima or saddle points of F are dynamically unstable. In conclusion, a steady solution of the GKS model/NFP equation (3)-(4) is linearly dynamically stable iff it satisfies (at least locally) the minimization problem:

I

{F[p,cl

min P>C

M[Pl = MI.

(30)

In this sense, dynamical and generalized thermodynamical stability in the canonical ensemble coincide. Furthermore, if F is bounded from below a, we can conclude from Lyapunov’s theory that the system will converge to a stable steady state of the GKS model for t + +co. Finally, we note that the GKS model can be written ap =

at

v.

[

El

xg(p)V-

,

where b/Sp is the functional derivative. This shows that the diffusion current J = -Xg(p)V(bF/bp) is proportional to the gradient of a quantity SF/Sp that is uniform a t equilibrium ( ( ~ 5 F l b p=) ~-Ta ~ according to Eq. ( 2 5 ) ) . This corresponds to the linear thermodynamics of Onsager. The same result can also be obtained from a generalized Maximum Free Energy Dissipation (MFED) principle which is the variational formulation of Onsager’s linear thermodynamics. 2.5. Particular cases If we take h(p) = 1 and g ( p ) = l/C”(p), the NFP equation (3) becomes

In that case, we have a constant diffusion D ( p ) = D and a density dependent mobility x ( p ) = x/(pC”(p)).If we take g ( p ) = p and h(p) = pC”(p), the NFP equation (3) becomes

’_ a - V . (DpC”(p)Vp- x p V c ) . -

at

(33)

note that for the standard Keller-Segel model, or for the Smoluchowski-Poisson system, the free energy is not bounded from below. In that case, the system can either relax toward a local minimum of F at fixed mass (when it exists) or collapse t o a Dirac peak,24 leading to a divergence of the free energy F ( t ) + -co. The selection depends on a complicated basin of attraction. The same situation (basin of attraction) happens when there exists several minima of free energy at fixed mass.

273

In that case, we have a constant mobility x ( p ) = x and a density dependent diffusion D ( p ) = D p [ C ( p ) / p ] ’ . Note that the condition D ( p ) 2 0 requires that [ C ( p ) / p ] ’ 2 0. This gives a constraint on the possible forms of C ( p ) . Finally, if we multiply the diffusion term and the drift term in the NFP equation ( 3 ) by the same positive function X(r,t ) (which can be for example a function of p(r, t ) ) ,we obtain a NFP equation having the same free energy (i.e. satisfying an H-theorem P 5 0) and the same equilibrium states as the original one. Therefore, for a given entropy C ( p ) , we can form an infinite class of NFP equations possessing the same general properties. 2.6. Generalized Smoluchowski equation

The NFP equation (33) can be written in the form of a generalized Smoluchowski (GS) equation

2 =v at

’

[ x ( V p-

(34)

with a barotropic equation of state p ( p ) given by

P’(P) = TPC”(P).

(35)

Since C is convex, we have p’(p) 2 0. Integrating Eq. (35) twice, we get

Therefore, the free energy (14) can be rewritten

F = _I 2x

/

[ ( V C+) ~ k2c2] d r - / p c d r

+ /p/”#dp’dr.

(37)

With these notations, the H-theorem becomes

‘s

+

J ip

( A c - k2c Xp)’dr - - ( V p - p V c ) 2 d r 5 0. (38) A€ The stationary solutions of the GS equation (34) satisfy the relation

F = --

Vp-pVc=Q,

(39)

which is similar to a condition of hydrostatic equilibrium. Since p = p ( p ) , this relation can be integrated to give p = p ( c ) through

This is equivalent to

274

This relation can also be obtained from Eqs. (35) and (22). Therefore, we recover the fact that, in the steady state, p = p ( c ) is a monotonically increasing function of c. We also note the identity

Finally, we note that the relation (40) can also be obtained by extremizing the free energy (37) at fixed mass writing 6F-adM = 0. More precisely, we have the important result: a steady solution of the generalized Smoluchowski equation (34)-(4) is linearly dynamically stable iff it is a (local) minimum of the free energy F [ p ,c] at fixed mass M [ p ]= M . This corresponds to the minimization problem (30). The generalized Smoluchowski equation (34) can also be obtained formally from the damped Euler equation^:^ dP

-

dt

+ v .( p u ) = 0 ,

dU

1

- + (u . V ) u = - - v p at P

<

(43)

+ v c -
(44)

For = 0, we recover the usual barotropic Euler equations of hydrodynamics. Alternatively, if we consider the strong friction limit -+ +m, we can formally neglect the inertial term in Eq. (44) and we get < u = - LVp Vc O(<-‘). Substituting this relation in the continuity equation P (43), we obtain the generalized Smoluchowski equation (34) with x = 1/<. These hydrodynamic equations (hyperbolic model) have been proposed in the context of chemotaxis to describe the organization of endothelial ~ e l l s . This ~ ~ ~inertial ~ - ~ model ~ takes into account the fact that the cells do not respond immediately to the chemotactic drift but that they have the tendency to continue in a given direction on their own. Therefore, the inertial term models cells directional persistence while the general density dependent pressure term -Vp(p) takes into account anomalous diffusion or the fact that the cells do not interpenetrate. Finally, the friction force - ~ ~ - ~ These filaments, or networks patterns, are not obtained in the Keller-Segel model (parabolic model), corresponding to + +oo, which leads to pointwise blow up or round aggregate^.^>'^ Note finally, that the GS equation

<

+ +

<

<

275

(34) can be derived rigorously from kinetic models in a strong friction limit .+ t o o , using a Chapman-Enskog expansion4’ or a method of moments.6

2.7. Kinetic derivation of the generalized Keller-Segel model

As discussed previously, the generalized Keller-Segel model (3) can be viewed as a nonlinear Fokker-Planck equation where the diffusion coefficient and the mobility explicitly depend on the local concentration of particles. Such generalized Fokker-Planck equations can be derived from a kinetic theory, starting from the master equation, and assuming that the probabilities of transition explicitly depend on the occupation number (concentration) of the initial and arrival states. Below, we briefly summarize and adapt to the present situation the approach developed by Kaniadakis41 in a more general context. We introduce a stochastic dynamics by defining the probability of transition of a particle from position r to position r’. Following K a n i a d a k i ~ , ~ ~ we assume the following form 7r(r

4

r’) = w(r, r

-

r’)a[p(r,t)]b[p(r’,t ) ] .

(45)

Usual stochastic processes correspond to a ( p ) = p and b(p) = I: the probability of transition is proportional to the density of the initial state and independent on the density of the final state. They lead to the ordinary Fokker-Planck equation (64) as will be shown below. Here, we assume a more general dependence on the occupancy in the initial and arrival states. This can account for microscopic constraints like close-packing effects that can inhibitate the transition. Quite generally, the evolution of the density satisfies the master equation

2 =/ at

[7r(r’4

r) - 7r(r + r’)] dr’.

Assuming that the evolution is sufficiently slow, and local, such that the dynamics only permits values of r’ close to r, one can develop the term in brackets in Eq. (46) in powers of r - r’. Proceeding along the lines we obtain a Fokker-Planck-like equation

with

276

and

(50) The moments

ci

and

cij

are fixed by the Langevin equation

dr (51) dt = Ji, c i j = DSij, the kinetic equation (47) becomes

- = ~ V CJ2DR(t). +

Assuming isotropy

ci

at

(52)

Now, according to the Langevin equation (51), D is independent on r and J = - x V c . Thus, we get

the foregoing equation can be rewritten ap =

at

v . (Dh(P)VP

-

XdP)VC)>

(55)

and it coincides with the GKS model (3). We note that In ~ ( p=) C ’ ( p ) .

(56)

We also have the relations =

Inversely,

V%GG=

mec‘(p)’2,

(57)

277

It seems natural to assume that the transition probability is proportional to the density of the initial state so that a(p) = p. In that case, we obtain an equation of the form

Note that the coefficients of diffusion and mobility are not independent since they are both expressed in terms of b ( p ) . Choosing b(p) = 1, i.e. a probability of transition which does not depend on the population of the arrival state, leads to the standard Fokker-Planck equation, or standard Keller-Segel model (64). If, now, we assume that the transition probability is blocked (inhibited) if the concentration of the arrival state is equal to an upper bound DO, then it seems natural to take b(p) = 1 - p/ao. In that case, we obtain

a p = v * ( D V p - x p ( 1 - p/ao)Vc) , at which will be considered in Sec. 3.5. Inversely, we can wonder what the general form of the mobility will be if we assume a normal diffusion h(p) = 1. This leads to b(p) - pb’(p) = 1 which is integrated in b(p) = 1 K p where K is a constant. Interestingly, we find that this condition selects the class of fermions (X = -1) and bosons ( K = +1) and intermediate statistics (arbitrary K ) . The corresponding NFP equation is

+

ap =

at

v . ( D V p- xp(1+ K p ) V c ) .

3. Examples of generalized Keller-Segel models

In this section, we consider generalized Keller-Segel models of chemotaxis and show their relation with a formalism of generalized thermodynamics. 3.1. The standard Keller-Segel model: B o l t z m a n n entropy

If we take h(p)

=

1 and g(p) = p, we get the standard Keller-Segel model

- _ - V . ( D V p - xpVc) . at

It corresponds to an ordinary diffusion D ( p ) = D and a constant mobility x ( p ) = x. The associated stochastic process is the ordinary Langevin equation

dr - = X V C+ m R ( t ) . dt

The entropy is the Boltzmann entropy

S=

-

s

plnpdr,

(66)

and the stationary solution of Eq. (64) is the Boltzmann distribution = &-a--l.

(67)

The standard Keller-Segel model is isomorphic to the Smoluchowski equation with an isothermal equation of state

P(P) = PT.

(68)

3 . 2 . Generalized Keller-Segel model with power law

digusion: Tsallis entropy If we take h(p) = qp4-l and g(p) = p , we obtain the GKS model - = V . ( D V p 4 - xpVc)

at

.

(69)

It corresponds to a power law diffusion D ( p ) = Dp4-l and a constant mobility x(p) = x. The associated stochastic process is dr

+d%pqR(t).

d t = XVC

This model can take into account effects of non-ergodicity and nonextensivity. It leads to a situation of anomalous diffusion related to the Tsallis statistic^.^^ For q = 1, we recover the standard Keller-Segel model with a constant diffusion coefficient, corresponding to a pure random walk (Brownian model). In that case, the sizes of the random kicks are uniform and do not depend on where the particle happens to be. For q # 1, the size of the random kicks changes, depending on the distribution of the particles around the “test” particle. A particle which is in a region that is highly populated [large p ( r , t ) ]will tend to have larger kicks if q > 1 and smaller kicks if q < 1. Since the microscopics depends on the actual density in phase space, this creates a bias in the ergodic behavior of the system. Then, the dynamics has a fractal or multi-fractal phase space structure. The generalized entropy associated to Eq. (69) is the Tsallis entropy

s=--q - 1

/(Pq

- p)dr,

and the stationary solution is the Tsallis distribution

(71)

279

The generalized Keller-Segel model (69) is isomorphic to the generalized Smoluchowski equation (34) with an equation of state P ( P ) = TP4. (73) This is similar to a polytropic gas with an equation of state p = K p Y (with y = 1 l / n ) where K = T plays the role of a polytropic temperature and q = y is the polytropic index. For q = 1, we recover the standard KellerSegel model (64).For q = 2, we have some simplifications. In that case, the GKS model (69) becomes

+

The entropy is the quadratic functional

S= and the stationary solution is

J

1 p = --(-pc 2

p2dr,

+a),

(75)

(76)

corresponding to a linear relation between the density and the concentration. In that case, the differential equation (23) determining the steady state reduces to the Helmholtz equation. Finally, the pressure is P ( P ) = TP2.

(77) In the context of generalized thermodynamics, the NFP equation (69) was introduced by Plastino & P l a ~ t i n oand ~ ~the generalized stochastic process (70) was introduced by B ~ r l a n dWhen . ~ ~ the NFP equation (69) is coupled to the Poisson equation ( 6 ) , we obtain the polytropic Smoluchowski Poisson system describing self-gravitating Langevin particles. When the NFP equation (69) is coupled to the field Eq. ( 4 ) ,we obtain a generalized KellerSegel model of chemotaxis taking into account anomalous diffusion. These models have been introduced and studied by Chavanis & Sire.6y32For the particular index n 3 = d / ( d - 2 ) or q 4 / 3 = y4/3 = 2(d- l ) / d , the GKS model presents a critical dynamics.34

3.3. Generalized Keller-Segel model with logarithmic dinusion: logotropes If we take h ( p ) = l / p and g ( p ) logarithmic diffusion

=

p , we obtain a GKS model with a

- = V ( D V l n p- xpVc).

at

280

The generalized entropy associated to Eq. (78) is the log-entropy (79) and the stationary solution is p=

-. 1 a

-

pc

The pressure law is

(81)

P(P> = T l n p .

This is similar to a logotropic equation of state.44 This is also connected to a polytropic equation of state (or Tsallis distribution) with y = q = 0. Indeed, the logotropic model (78) can be deduced from Eq. (69) by writing DVpq = DqpQ-lVp, taking q = 0 and re-defining Dq -+ D . When the NFP equation (78) is coupled to the Poisson equation (6), we obtain the logotropic Smoluchowski-Poisson system. When the NFP equation (78) is coupled to the field Eq. (4), we obtain a generalized Keller-Segel model of chemotaxis. These models have been introduced and studied by Chavanis & Sire.35

3.4. Generalized Keller-Segel models with power law diffusion and power law drift: Tsallis entropy We introduce here an extension of the GKS model (69). If we take h(p) = qpq+fi"-l and g ( p ) = pfi+l, we obtain

dP = V . (Dqpq+P-lVp - xpfi+'Vc dt

1.

(82)

This corresponds to a power law diffusion D ( p ) = *pq+fi-l and a power q+P law mobility x ( p )= x p p . The associated stochastic process is

(83) Since pfi can be put in factor of the diffusion current in Eq. (82), this model has the same equilibrium states (72) and the same entropy (71) as Eq. (69). For p = 0, we recover Eq. (69) with a constant mobility and a power law diffusion. For ( p , q ) = (O,O), we recover the logotropic Smoluchowski equation (78) provided that we make the transformation Dq + D . For p = 1 - q, we have a normal diffusion and a power law mobility -=V

at

. (DqVp - ~ p ' - ~ V c )

(84)

281

For q = 2, we get -=V

at

. ( 2 D V p - XVC),

(85)

which has the same entropy and the same equilibrium states as Eq. (74). Finally, for q = 0 (making the transformation qD 4 D ) , we obtain -=V .

at

(DVp-

which has the same entropy and the same equilibrium states as Eq. (78). When the NFP equation (82) is coupled to the field equation (4), we obtain a generalized Keller-Segel model of chemotaxis taking into account anomalous diffusion and anomalous mobility.

3.5. Generalized Keller-Segel models with a filling factor: Fermi-Dirac entropy If we take h(p) = 1 and g ( p ) = p(1- p / a o ) , we obtain a GKS model of the form

This corresponds to a normal diffusion D ( p ) = D and a mobility x(p) = x(1- p / a o ) vanishing linearly when the density reaches the maximum value pmaz = 00. The associated stochastic process is

dr dt The generalized entropy associated with Eq. (87) is a Fermi-Dirac-like entropy in physical space - = x(1-

and the stationary solution is a Fermi-Dirac-like distribution in physical space

From Eq. (go), we see that, in the stationary state, p < no. This bound is similar to the Pauli exclusion principle in quantum mechanics. In fact, we can show that p ( r , t ) remains bounded by g o during the whole evolution. For 00 4 +m, we recover the standard KS model (64).

282

An alternative GKS model, with the same entropy and the same equilibrium states, is obtained by taking h ( p ) = 1/(1- p/ao) and g ( p ) = p. This leads to -=

at

v . ( - ~ u O vln(1-

p/co) - x p ~ c ) .

(91)

This corresponds to a nonlinear diffusion with D ( p ) = -ao(D/p)ln(l p / a o ) and a constant mobility x ( p ) = x. Equation (91) can be put in the form of a generalized Smoluchowski equation (34) with a pressure law

p ( p ) = -Too In(1 - p/ao).

(92)

For p << 00, we recover the “isothermal” equation of state p = pT leading to the standard Keller-Segel model (64). However, for higher densities, the equation of state is modified and the pressure diverges when p -+ 60. This prevents the density from exceeding the maximum value no. The NFP equation (87) has been introduced by Kaniadakis & Q ~ a r a t i ~ ~ to describe fermionic systems and by Robert & S ~ m m e r i in a ~the ~ statistical mechanics of two-dimensional turbulence (see also47). In the context of chemotaxis, the model (87) has been introduced by Hillen & Painter2’ and, independently, by C h a ~ a n i s . ~It>provides ~l a regularization of the standard Keller-Segel model preventing overcrowding, blow-up and unphysical singularities. The filling factor (1- p / a o ) takes into account the fact that the particles cannot interpenetrate because of their finite size a. Therefore, the maximum allowable density is 00 l / a d . It is achieved when all the cells are packed together. In the model (87), it is assumed that the mobility vanishes when the density reaches the close packing value ( p + 00) while the diffusion is not affected. The alternative model (91) has been introduced by C h a ~ a n i s . ~In> ~ that l case, the mobility is assumed to be constant and the regularization preventing overcrowding is taken into account in the pressure law (92). We can also multiply the diffusion term and the mobility term in the NFP equation (3) by the same positive function X(r, t ) in order to obtain a more general model with the same entropy and the same equilibrium states in which both diffusion and mobility are affected by overcrowding. N

3.6. Generalized Keller-Segel models incorporating

anomalous diffusion and filling factor The previous models focus individually on two important effects: anomalous diffusion (see Secs. 3.2-3.4) and exclusion constraints when the density becomes too large (see Sec. 3.5). Here we introduce a mixed model which

283

combines these two effects in a single equation. If we take h(p) = qpq+P"-I and g(p) = p P + l ( l - p / a o ) , we obtain dP =

v . (Dqpq+P"-lvp-

xpfi+l

at

(1- P / a o ) V c )

.

(93)

This corresponds to a power law diffusion such that D ( p ) = [Dq/(q p)]pq+P-' and a mobility x ( p ) = x p p ( 1 - p / a o ) . The associated stochastic process is

+

The generalized entropy corresponding to Eq. (93) is obtained by integrating twice the relation

A first integration gives C'(p) = q a p D q--2

);(

,

where

Therefore, the generalized entropy can be expressed as

s,

flu0

C(P) = 9 4

@¶-2(t)dt.

(98)

On the other hand, the equilibrium density is given by p = ao@;?,[(pc cy)/qn:-l]. Note that these results not depend on p since the term pP can

be put in factor of the diffusion current in Eq. (93). Let us consider some particular cases. (i) For q = 1, Eq. (93) has the same entropy and the same equilibrium states as Eq. (87). (ii) For (TO + +co, we recover Eq. (82). (iii) For p = 0 and q = 2, we have ap -=

at

v . (DVp-2

-

x p ( 1 -p/ao)Vc) .

The generalized entropy is

s = -20:

J (1 $) In (I - 5) dr, -

(99)

284

and the stationary solution is

[

= uo 1 - ,(-Pc+a)/Zmo

I+

.

(101)

For 00 -+ +cm,we recover Eq. (76). Dividing the diffusion and the drift term by 1- p / a o , we can also consider the alternative model

which has the same entropy and the same equilibrium states as Eq. (99). The pressure law is P ( P ) = -2Ta: [ln(l- P/OO)- p/aol.

(iv) For ( p , q ) = (0,O) and performing the transformation qD directly taking h(p) = l / p and g ( p ) = p(1 - p / a o ) , we obtain -=

at

v . (DVIn p - x p ( 1 -

p/ao)vc).

(103) +

D , or

(104)

This corresponds to a logarithmic diffusion and a modified mobility taking into account an exclusion principle through the filling factor. The generalized entropy is obtained from the relation

leading to

and finally to the explicit expression

We can consider the alternative model

with the same entropy and the same equilibrium states. The associated pressure law is

285

4. Conclusion

In this paper, we have discussed a generalized class of Keller-Segel models describing the chemotaxis of biological populations. We have shown their analogy with nonlinear mean field Fokker-Planck equations and generalized thermodynamics. We have given explicit examples corresponding to different entropy functionals. In particular, we have considered the case where the particles (cells) experience anomalous diffusion and the case where they experience an exclusion constraint (volume filling). We have introduced a mixed model taking into account these two effects in a single equation (93). Of course, we can construct other types of Keller-Segel models which may also be of interest. The general study of these models, which combine both long-range interactions and generalized thermodynamics, is very rich and can lead to a wide diversity of phase transitions and blow up phenomena. These nonlinear meanfield Fokker-Planck equations are therefore of considerable theoretical i n t e r e ~ t . ~ References 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20.

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P.H. Chavanis, C. Rosier and C. Sire, Phys. Rev. E 66,036105 (2002). P.H. Chavanis, Physica A 384,392 (2007). C. Sire and P.H. Chavanis, Phys. Rev. E 66,046133 (2002). C. Sire and P.H. Chavanis, Phys. Rev. E 69,066109 (2004). P.H. Chavanis and C. Sire, Phys. Rev. E 70,026115 (2004). J. Sopik, C. Sire and P.H. Chavanis, Phys. Rev. E 72,026105 (2005). P.H. Chavanis and C. Sire, Phys. Rev. E 73,066103 (2006). P.H. Chavanis and C. Sire, Phys. Rev. E 73,066104 (2006). T. Hillen and K. Painter, Adv. Appl. Math. 26,280 (2001). K. Painter and T. Hillen, Can. Appl. Math. Q. 10,501 (2002). P.H. Chavanis, Eur. Phys. J. B 54,525 (2006). P.H. Chavanis and C. Sire, Phys. Rev. E 69,016116 (2004). C. Tsallis, J. Stat. Phys. 52,479 (1988). P.H. Chavanis, C. Sire, [arXiv:0705.4366] P.H. Chavanis, C. Sire, Physica A 375,140 (2007). P.H. Chavanis, Physica A 361,55 (2006). A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. di Talia, E. Giraudo, G. Serini, L. Preziosi, F. Bussolino, Phys. Rev. Lett. 90,118101 (2003). F. Filbet, P. LaurenCot, B. Perthame, J. Math. Biol. 50,189 (2005). P.H. Chavanis, Eur. Phys. J. B 52, 433 (2006); P.H. Chavanis, C. Sire [arXiv:0708.3163] P.H. Chavanis, P. LaurenCot, M. Lemou, Physica A 341,145 (2004). G. Kaniadakis, Physica A 296,405 (2001). A.R. Plastino, A. Plastino, Physica A 222,347 (1995). L. Borland, Phys. Rev. E 57,6634 (1998). D.E. McLaughlin, R.E. Pudritz, Astrophys. J. 476,750 (1997). G. Kaniadakis, P. Quarati Phys. Rev. E 49,5103 (1994). R. Robert, J. Sommeria, Phys. Rev. Lett. 69,2776 (1992). P.H. Chavanis, J. Sommeria, R. Robert, Astrophys. J. 471,385 (1996).

287

ON SWITCHABILITY OF A FLOW TO THE BOUNDARY IN A PERIODICALLY EXCITED DISCONTINUOUS DYNAMICAL SYSTEM ALBERT C.J. LUO AND BRANDON M. RAPP Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805, USA [email protected]

Abstract This paper presents the switchability of a flow from one domain into another one through the boundary in the periodically driven, discontinuous dynamical system. The normal vector-field product for a flow switching on the separation boundary is introduced, and the passability condition of a flow to the discontinuous boundary is developed analytically, and the sliding and grazing conditions to the separation boundary are presented as well. Numerical illustrations of periodic motions with and without sliding on the boundary are given, and the normal vector fields are illustrated to show the flow switchability on the discontinuous boundary.

Keywords: Disocntinuous system, Flow switchability, Flow grazing, Sliding motion.

1. Introduction In 1964, Filippov [ l ] investigated the motion in the Coulomb friction oscillator from a mathematical point of view, and he presented a differential equation theory for dynamical systems with discontinuous right-hand sides. To determine the sliding motion along the discontinuous boundary, the differential inclusion was introduced via the set-valued analysis. The detailed discussion of such discontinuous differential equations can be referred to Filippov [Z]. One applied the Filippov’s concept to control dynamical systems. In 1974, Aizerman and Pyatniskii [3,4] extended the Filippov’s concepts to develop a generalized theory for discontinuous systems. In 1976, Utkin [5] used such a generalized theory to develop methods for controlling dynamic systems based upon the discontinuity (i.e., sliding mode control). Further, this sliding mode control was further developed. In 1988, DeCarlo et aZ.[6] gave a review on the development of the sliding mode control. In 2005, Renzi and Angelis [7] used the sliding mode control method to study the dynamics of variable stiffness structures. The Filippov’s theory mainly focused on the existence and uniqueness of the solutions for non-smooth dynamical systems, and the Filippov’s differential inclusion provides a set of possible candidates for motion switching or sliding. However, the sliding mode control theory has a difficulty to achieve an efficient

288

control because the local singularity caused by the separation boundary was discussed incompletely. Therefore, the further investigation of the local singularity in the vicinity of the separation boundary should be completed. In 2005, Luo [8] developed a general theory for the local singularity of a flow to the boundary in non-smooth dynamical system with connectable domains. The local singularity of non-smooth dynamical systems near the separation boundary was discussed. The imaginary, source and sink flows were introduced in Luo [9] to determine the sliding and source motions in non-smooth dynamical systems. The comprehensive discussion of the singularity and dynamics of discontinuous dynamical systems can be referred to Luo [lo]. Based on the local singularity theory, the grazing motion to the separation boundary and the sliding motion on the separation boundary in discontinuous dynamical systems were discussed through the piecewise linear systems (e.g., [ 113-[131) and friction-induced oscillator (e.g., [14]-[17]). The inclined line control law in phase space will be considered in this paper to demonstrate the flow switchability on the discontinuous boundary. Through the straight line control law, the phase domain is separated into two domains, and the vector fields of the discontinuous dynamical system will be switched, and the two vector fields in the two domains are different. Owing to this discontinuity, the discontinuous dynamical systems will have more complicated dynamical behavior. To investigate the switchability of a flow from one domain into another one in the periodically driven, discontinuous dynamical system, the normal vectorfield product for flow switching on the separation boundary will be introduced, and the passability condition of flow to the discontinuous boundary will be presented. In addition, the sliding and grazing conditions to the separation boundary will be presented. Numerical illustrations of periodic motions with grazing to the boundary andor sliding on the boundary will be presented, and the normal vector fields will be presented to show the analytical criteria.

ra

-

Fig. 1 . Mechanical model with an inclined line control law.

289

2. Equations of Motion Consider a mass-spring-damper model with a mass rn which is connected with a switchable spring of stiffness k, and a switchable damper of coefficient r, in the a-region (a=1,2), as shown in Fig. 1. A periodical force exerts on the mass of the oscillator, i.e.,

where Q and L2 are excitation amplitude and Erequency, and constant force is represented by U, . The coordinate is defined by (x,t). Both t and x are time and mass displacement, respectively. The control law for this discontinuous dynamical system is given by ax+bi=c

(2)

where a , b and c are constants. For a mass-spring-damper discontinuous system, the equation of motion is:

x + 2 d a i + c,x = A,, cos(Slt + 4) + b,

(3)

where

3. Analytical conditions

In phase plane, the vectors are introduced by x A ( x , ~ )~~ ( x , y and ) ~ FA(Y,F)~.

(5)

From Eq. (2), this control logic generates a discontinuous boundary in the system. To analyze dynamics of the system, the three states are given by Q1 = {(x,y)l ax

+ by > c } and LIZ= { ( x , y)l ax + by < c } .

(6)

asZs ( a,p = 1,2 ) is defined as

The separation boundary aL2, = fi, n

a q , = asz,,

=a,nil, = {(x,y) I pl,(x, y ) = OX+ by --c

=

o}.

(7)

290

I

ax+by=c

Fig. 2 . Sub-domains and boundary.

The domains and boundary are sketched in Fig. 2. The boundary is depicted by a dotted straight line, governed by Eq. (2). The two domains are shaded. The arrows crossing the boundary indicate the flow possible directions to the boundary. If a flow of the motion in phase space is in domain Q a ( a = 1,2 ), the vector fields in such a domain is continuous. However, if a flow of motion from a domain Q a ( a € (1,2}) switches into another domain 52, ( PE(1,2) , P # a ) through the boundary &Zap, the vector field in domain Q a ( a €{1,2)) will be changed into the one in domain Qp ( P = {1,2)) accordingly. Because of the discontinuity, the flow may not pass over the boundary under a certain condition. The flow motion may be along the boundary, which is called the sliding flow. As discussed in Luo [lo], the sliding motion on the boundary has an equilibrium point (E,O) where E = c/u . Based on the equilibrium point, the parabolicity and hyperbolicity in vicinity of such an equilibrium was discussed as in Figs. 3 and 4.

Fig. 3. Phase portraits for flow parabolicity near equilibrium ( E . 0 ) :(a) n x b > 0 and (b) a x b c 0 .

291

Fig. 4. Flow hyperbolicity in the vicinity of equilibrium ( E , O ) :(a) n x b > 0 and (b) a x b < 0

From Eq. (2), with initial condition ( x ~ ' ) , X ~ ' ) on ) the boundary, the displacement and velocity for sliding are given by JO)

1

(qw- c>exp[-% ( t -ti >I,

=L a +1 a

y(O) = - - bJ - ( ~ A +-~c)' exp[-

( t - ti >I.

The equations of motion can be further described as

x = F@)(x,t ) for R E (0,a}

(8)

(9)

where

F@)(x,tk( ~ , F , ( x , t ) ) in ~ 52, ( a ~ { 1 , 2 ) ) , F(')(x,t)= ( ~ , - f y ) ~for sliding on aPaP(a,.BE{1,2}), F(')(x,t)= [F(m)(x,t),F(m(x,t)] for non-sliding on an,, Fa (x,t ) = -2d, y - C

+ 4 COS(Pt + @)+ b, .

~ X

(1 1)

From the theory of discontinuous dynamical systems in Luo [8, lo], for a sliding motion on aP, with the corresponding normal vector nJQafl pointing to domain 52, (i.e., nand -+ P, ), the necessary and sufficient conditions of the sliding motion on the switching boundary are G(03n(xm,t,-) = nanM.F(n)(xm,tm-) T
G ( o . B ) ( ~ m , t ,=- )n~nM.F(B)(xm,tm-) > 0.

292

where

~ , P (1,2} E and a # P

with

n&&= VP,,

aV&

3%

T

=(T,T$(xm,ym)'

(13)

where V = (a/ax,a/ay) is the Hamilton operator. The necessary and sufficient conditions of a motion switchable to the boundary aQ, with nand + SZ, are from Luo [8,9],

Note that t, is switching time for the motion to the switching boundary and t,, = t,, f0 reflects the responses in domain rather than boundary. The grazing motion to the separation boundary aQ, is from Luo [8,9], i.e.,

where

DF'"' (x,t ) = (Fa(x,t ) ,VF, (x, t)*F'"'(x,t ) + -)T.

(16)

The conditions presented in this paper are valid only for straight line boundary or k"-order contact between the motion flow and separation boundary. Substitution of Eq. (7) into Eq. (13) gives nJQlz

- nJQz, = ( a 9 b ) T '

(17)

From the forgoing equation, the normal vector always points to the domain Q, Therefore,

.

293

G(03")(~,,t,) =nkuB -F(")(x,,t,) = uy, +bF,(xm,t,), T G ( 1 3 a ) ( ~ m=, naQ,, tm ) .DF(")(x,,~,)= ~F,(x,,t,)

(X,

(18)

.

+b[VF,(x,t)*F'")(~,t)+-a;-] aFn(X,t) -1, )

From Eqs. (12) and (18), the conditions for sliding motion on the switching boundary are:

G(o~')(xm,t,,-) < 0 and G(0s2)(~,,t,-) >0

(19)

From Eqs. (14) and (18), the switchability conditions for motion on the switching boundary are: G(02')(xm,tm-) < 0 and G ( 0 , 2 ) ( ~ , , t , ,<, +0,) for 52'

+Q2;

G(031)(x,,,,t,,,+) > 0 and G ( 0 * 2 ) ( ~ , , t>m0,- ) for Q2 -+ 52,.

}

(20)

From the theory for non-smooth dynamical systems in Luo [8,9], the vanishing conditions for sliding motion on the separation boundary are: G(031)(xm,tm-) < 0 and G ( 0 , 2 ) ( ~ , , t ,=- )0 for aQ,, -+52,;

G(03')(xm,t,,,) = 0 and G ( 0 , 2 ) ( ~ , , t > , - 0) for aQI2+ 52'.

1

(21)

From Eq. (16), the onset condition of the sliding motion on the switching boundary is given by

]

G(o*')(xm,tm-) < 0 and G ( 0 2 Z ) ( ~ , , t=, +0) for Q,

+ aQ,,;

G(O.l)(xm,t,+) = 0 and G ' o ~ 2 ' ( x , , t m>- 0 ) for 52,

+aQ12.

(22)

4. Illustrations The phase plane and the displacement response will be presented for illustration. The normal vector field is very important to determine the grazing and sliding bifurcations, and the normal vector €ields versus displacement will be presented, and the normal vector field time-history will be given to observe the switchability of the switching dynamical systems. The closed-form solutions in Appendix will be used for numerical computation of motions for such a switching dynamical system. Consider a periodic motion with mapping structure pZlozl for 52 = 2.0, as shown in Fig. 5. In Fig. 5(a), the trajectory in phase plane shows the switching of the motion on the boundary. The switching boundary is depicted by the dotted

294

line. The switching points are labeled by the circular symbols, and the starting point is marked by a large filled circle. The mapping structure is pZlozl = Pzo 4 o p, o Pz04.The sliding motion along the switching boundary is shaded for further discussion. To show the switchability of motion on the discontinuous boundary, the normal vector field plays an important role. The normal vector field distribution along the displacement is illustrated in Fig. 5(b). The motion switchable on the separation boundary requires the normal vector fields in vicinity of switching point on the boundary to have the same sign, and such a switchable condition is given in Eq. (20). For the sliding motion on the boundary, the two normal vector fields in vicinity of switching points have opposite sign, as in Eq. (19). Once a motion flow arrives to the boundary, only if Eq. (19) is satisfied, the sliding motion along the switching boundary in phase plane can be observed. However, when one of the two sets of inequality in Eq. (21) is satisfied, the sliding motion will disappear and the flow will get into the corresponding domains. In Fig. 5(b), the solid curves give the real normal vector field and the dashed curves show the imaginary normal vector field, which can be referred to Luo [8,9]. After the motion is switched, the previous vector field is still used to control the flow as an imaginary flow. From the initial condition, both the normal vector fields in the two domains are positive ( G‘o”’ > 0 and G‘032’ > 0 ), the motion flow should be in domain 9,, which is labeled. Once the motion flow comes back to the switching boundary, both of the two normal vector fields for the switching point are negative (i.e., G‘oxl’< 0 and G‘032’ < 0 ), so it implies that the motion flow will be switched into domain Q Z . When the motion flow in domain 51, returns back to the switching boundary, both of the two normal vector fields possess the opposite sign, i.e., G‘O”’ < 0 and G‘O”’ > 0 . When G‘”” = 0 and G‘o*2’ > 0 , the sliding motion on the boundary disappear, and the motion flow enters into domain 51,. When the motion flow returns again to the switching boundary, the normal vector fields ( G‘O”’ < 0 and G‘”’ < 0 ) are observed. Thus, the motion flow switches into the domain QZ. Finally the motion flow returns to the initial point from domain 51, . The periodic motion ~ i t h , P 2 ~ 0is2formed. ~ The displacement and normal vector field time-histories are also presented in Fig. 5(c) and (d). The responses in each domain are labeled by the mapping and the normal vector fields in each domain are labeled by G‘os”and G(0.2’. The sliding responses are shaded. The time-history of the normal vector fields also shows the analytical conditions for the switchability of the motion flow on the switching boundary, which will not be discussed. The response between the two dashed lines is for a period.

295 15

-c

200

v

%

0

2 n

A

-g 5

o

y.

::

?8

-15

-200

2 -30

-400

-2.0

0.0

(4

2.0

4.0

Displacement x

0.0

-2.0

2.0

4.0

Displacement x

(b) 400

-

3 -c -

200

m

y.

5

f 2 00

20

40

60

80

0

-200

-400 00

20

40

60

80

Fig. 5. Periodic motion of P,,,,, : (a) phase plane, (b) normal vector field versus displacement, (c) displacement and (d) normal vector field time-history. Initial condition is to = 0.6478 , x,, = -1.4848 and yo =-1.1517. ( r , = O S , k, =50, r, =1.0, k, =150, gl=gz= 1, =150, sz=2.0, @ = O , a = b = l , c = - 3 ) .

Consider another periodic motion relative to mapping Pz121with 2excitation periods. The same parameters are used again except for SZ = 17.5. Based such parameters, the initial condition (i.e., to = 0.0282, xo = -1.0243 and yo = -1.9098) are used in Fig.6. The trajectory in phase plane and normal vector field distribution along displacement are illustrated in Fig.6(a) and (b). For this periodic motion in Fig.6(a), it is observed that no sliding motion on the switching boundary exists. Therefore, both the normal vector fields in two domains for all switching points on the switching boundary are with the same sign. In other words, the motion flow switching from SZ, to SZ2 or from SZ2 to requires the normal vector fields are negative or positive, which are

296

observed in Fig.6(b). In a similar fashion, the displacement and normal vector field time-histories are illustrated in Fig.6 (c) and (d). For each two-excitation periods, the complete periodic cycle is observed.

-480 -1.6

-30 -1.6

0.0

-0.8

08

1.6

24

Displacement x

(a>

"

on

'

'

'

"

' I' '

' '

05

'

'

10

I

0.0

,

08

I

1.6

2.4

Displacement x

(b) s t

- 2 0 1I " ' " ' '

I

-0.8

200

3P

O

2 2

~200

Bf

-400

"

15

00

05

10

15

q2,),

Fig. 6 . Periodic motion of : (a) phase plane, (b) normal vector field versus displacement, (c) displacement and (d) normal vector field time-hstory. Initial condition is to = 0.0282 ,xo = -1.0243 and yo=-1.9098. ( r ; = O . 5 , k , = 5 0 , r,=I.O, k2=150, g , = l , gz=I, Qo=lSO, Q=17.5, @ = O , a = b = l , ~ = - 3 ) .

5. Conclusions

The switchability of a flow from one domain into another one in the periodically driven, discontinuous dynamical system is investigated through the boundary. The normal components of the vector fields for a flow switching on the separation boundary is introduced. The switchability conditions of a motion flow on the discontinuous boundary are developed, and the sliding and grazing conditions to the separation boundary are presented as well. The normal vector fields are illustrated to demonstrate the analytical criteria. This investigation will help one better understand the sliding mode control.

297

Appendix

With the initial condition(xi,ii,ti), solution for Eq. (3) in two regions 51,

298

ulp'

=

JX

c,Ca) ( x i ,ii,t i )= xi - A(")cos Qti - B(")sin Qti - da), 1 Cp) (xi,X i , t i )= -[Xi - (d,A(") + B(")Q)cos Qt, Ulp'

-( d for Case 111(i.e.,

a B(")

- A(")Q)sin Qti + d, [ x i - C ( a ) ) ] .

4 = ca >,

References

1. Filippov, A.F., 1964, "Differential equations with discontinuous right-hand side", American Mathematical Society Translations, Series 2 , 42, pp. 199231. 2. Filippov, A.F., 1988, Differential Equations with Discontinuous Righthand Sides, Dordrecht: Kluwer Academic Publishers. 3. Aizerman, M. A., Pyatnitskii, E.S., 1974, "Foundation of a theory of discontinuous systems. 1," Automatic and Remote Control, 35, pp. 10661079. 4. Aizerman, M. A., Pyatnitskii, E.S., 1974, "Foundation of a theory of Discontinuous Systems. 2," Automatic and Remote Control, 35, pp. 12411262. 5. Utkin, V. I., 1976 ,"Variable structure systems with sliding modes," ZEEE Transactions on Automatic Control, AC-22.pp. 212-222. 6. DeCarlo, R.A., Zak, S.H., Matthews, G.P., 1988, "Variable Structure

299

7. 8.

9. 10. 11.

12. 13. 14.

15. 16. 17.

control of nonlinear multivariable systems: A tutorial,” Proceedings of the IEEE, 76, pp. 212-232. Renzi, E., Angelis, M. D., 2005, “Optimal semi-active control and nonlinear dynamics response of variable stiffness structures,” Journal of Vibration and Control, 11(10), pp.1253-1289. Luo, A.C.J., 2005, “A theory for non-smooth dynamical systems on connectable domains,” Communication in Nonlinear Science and Numerical Simulation, 10, pp.1-55. Luo, A.C.J., 2005, “Imaginary, sink and source flows in the vicinity of the separatrix of non-smooth dynamic system,” Journal of Sound and Vibration, 285, pp.443-456. Luo, A.C.J., 2006, Singularity and Dynamics on Discontinuous Vector Fields, Elsevier: Amsterdam. Menon, S . and Luo, A.C.J. 2005, “A global period-1 motion of a periodically forced, piecewise linear system”, International Journal of Bifurcation and Chaos, 15, pp. 1945-1957. Luo, A.C.J., 2005, “The mapping dynamics of periodic motions for a threepiecewise linear system under a periodic excitation”, Journal of Sound and Vibration, 283,723-748. Luo, A.C.J. and Chen, L.D., 2005, “Periodic motion and grazing in a harmonically forced, piecewise, linear oscillator with impacts”, Chaos, Solitons and Fractals, 24, pp. 567-578. Luo, A.C.J. and Gegg, B.C. 2005 “On the mechanism of stick and non-stick periodic motion in a forced oscillator with dry-friction,’’ ASME Journal of Vibration and Acoustics, 128, pp.97- 105. Luo, A.C.J. and Gegg, B.C., 2006, “Stick and non-stick periodic motions in a periodically forced oscillator with dry-friction,’’ Journal of Sound and Vibration, 291, pp.132-168. Luo, A.C.J. and Gegg, B.C., 2006, “Periodic motions in a periodically forced oscillator moving on an oscillating belt with dry friction,” ASME Journal of Computational and Nonlinear Dynamics, 1, pp.212-220. Luo, A.C.J. and Gegg, B.C., 2006, “Dynamics of a periodically forced oscillator with dry friction on a sinusoidally time-varying traveling surface,” International Journal of Bifurcation and Chaos, 16, pp.35393566.

300

THE FORMATION OF SPIRAL ARMS AND RINGS IN BARRED GALAXIES M. ROMERO-GOMEZ' and E. ATHANASSOULA Laboratoire d'Astrophysique de Marseille, Obseruatoire Astronomique de Marseille Provence, 2 Place Le Verrier 13248 Marseille, France *E-mail: merce.romerogometQoamp.f r

J.J. MASDEMONT

I. E.E. C €d Dep. Mat. Aplicada I, Universitat Politkcnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Spain

c. GARC~A-GOMEZ D. E.I. M., Universitat Rovira i Virgili, Av. Paasos Catalans 26, 43007 Tarragona, Spain We propose a new theory to explain the formation of spiral arms and of all types of outer rings in barred galaxies. We have extended and applied the technique used in celestial mechanics to compute transfer orbits. Thus, our theory is based on the chaotic orbital motion driven by the invariant manifolds associated to the periodic orbits around the hyperbolic equilibrium points. In particular, spiral arms and outer rings are related t o the presence of heteroclinic or homoclinic orbits. Thus, R1 rings are associated to the presence of heteroclinic orbits, while RlRz rings are associated to the presence of homoclinic orbits. Spiral arms and Rz rings, however, appear when there exist neither heteroclinic nor homoclinic orbits. We examine the parameter space of three realistic, yet simple, barred galaxy models and discuss the formation of the different morphologies according to the properties of the galaxy model. The different morphologies arise from differences in the dynamical parameters of the galaxy. Keywords: galactic dynamics - invariant manifolds - spiral structure - ring structure

1. Introduction Bars are very common features in disk galaxies. According to Eskridge et al. [l]in the near infrared 56% of the galaxies are strongly barred and

301

6% are weakly barred. A large fraction of barred galaxies show either spiral arms emanating from the ends of the bar or spirals that end up forming outer rings (Elmegreen & Elmegreen [2]; Sandage & Bedke [3]). Spiral arms are believed to be density waves (Lindblad [4]). Toomre [5], finds that the spiral arms are density waves that propagate outwards towards the principal Lindblad resonances, where they damp. So other mechanisms for replenishment are needed (see for example Lindblad [6]; Toomre [5,9];Toomre & Toomre [7]; Sanders & Huntley [8]; Athanassoula [lo] for more details). Rings have been studied by Schwarz [ll-131. The author studies the response of a gaseous disk galaxy to a bar-like perturbation. He relates the rings with the position of the principal Lindblad resonances. There are different types of outer rings and they can be classified according to the relative orientation of the principal axes of the inner and outer rings (Buta [14]). If the two axes are perpendicular, the outer ring has an eight-shape and it is called R1 ring. If they are parallel, it is called R2 ring. There are galaxies where both types of rings are present, in which case the outer ring is simply called R1 R2 ring. Our approach is from the dynamical systems point of view. We first note that both spiral arms and (inner and outer) rings emanate from, or are linked to, the ends of the bar, where the unstable equilibrium points of a rotating system are located. We also note that, so far, no common theory for the formation of both features has been presented. We therefore study in detail the neighbourhood of the unstable points and we find that spiral arms and rings are flux tubes driven by the invariant manifolds associated to the periodic orbits around the unstable equilibrium points. This paper is organised as follows. In Sec. 2, we give the characteristics of each component of the model and the potential used to describe it. In Sec. 3, we give the equations of motion and we study the neighbourhood of the equilibrium points. In particular, we give definitions of the Lyapunov periodic orbits, the invariant manifolds associated to them, and of the homoclinic and heteroclinic orbits. In Sec. 4, we present our results and in Sec. 5, we briefly summarise.

2. Description of the model

We use a model introduced in Athanassoula [15] that consists of the superposition of an axisymmetric and a bar-like component. The axisymmetric component is the superposition of a disc and a spheroid. The disc is modelled as a Kuzmin-Toomre disc (Kuzmin [16]; Toomre [17]) of surface

302

density C ( T ) (see also left panel of Fig. 1): -3/2

C ( r ) = =2TTd (l+;)

,

where the parameters v d and rd set the scales of the velocities and radii of the disc, respectively. The spheroid is modelled using a spherical density distribution, p ( r ) (Eq. 2), characteristic for spheroids. In the middle panel of Fig.1, we plot the isodensity curves for this density function: p(r) = P b

(

1f

3-3/2,

(2)

where Pb and rb determine the central density and scale-length of the spheroid. Bars are non-axisymmetric features with high ellipticities. We will use three different bar models. In the first one the bar potential is described by a Ferrers ellipsoid (Ferrers [ I S ] )whose density distribution is:

+

where m2 = x 2 / a 2 y2/b2. The values of a and b determine the shape of the bar, a being the length of the semi-major axis, which is placed along the x coordinate axis, and b being the length of the semi-minor axis. The parameter n measures the degree of concentration of the bar and po represents the bar central density. In the right panel of Fig. 1, we plot the density function along the semi-major and semi-minor axes of the Ferrers ellipsoid with index n = 2 , and principal axes a = 6 and b = 1.5. We also use two ad-hoc potentials, namely a Dehnen’s bar type (Dehnen [19]) and a Barbanis-Woltjer (BW) bar type (Barbanis & Woltjer [ 2 0 ] )to compare to the results obtained with the Ferrers ellipsoid. The Dehnen’s bar potential has the following expression:

where the parameter (Y is a characteristic length scale and wo is a characteristic circular velocity. The parameter E is related to the bar strength. The BW potential has the expression: %(T,

e) = Z f i ( r 1 - T ) cOs(2e),

(5)

303

where the parameter the bar strength.

r1

is a characteristic scale length and 2 is related to

Fig. 1. Characteristics of the components. L e f t panel: Density function of the KuzminToomre disc (red solid line) with r d = 0.75 and Vd = 1.5.M i d d l e panel: Isodensity curves for the spherical distribution representing the spheroid with parameters r b = 0.3326and pb = 23552.37. R i g h t panel: Density along the semi-major axis (black solid line) and the semi-minor axis (red dashed line) of a Ferrers bar with n = 2, a = 6, b = 1.5 and po = 0.0193.

a,

The bar-like component rotates anti-clockwise with angular velocity = OPz,where 0, is a constant pattern speed a .

3. Equations of motion and dynamics around

L1

and Lz

The equations of motion in a frame rotating with angular speed 0, in vector form are

i: = -V!D-2(aP x i ' ) -a, x

(a, x r),

(6)

where the terms -2aPx i- and -Clp x (52, x r) represent the Coriolis and the centrifugal forces, respectively, CP is the potential and r is the position vector. We define an effective potential C P e ~= CP - :Og (x2 y2), then Eq. (6) becomes i: = -V!D,,ff - 2(OP x i'), and the Jacobi constant is

+

which, being constant in time, can be considered as the energy in the rotating frame. The surface C P e ~= EJ ( E J defined as in Eq. (7)) is called the zero velocity surface, and its intersection with the z = 0 plane gives the zero velocity curve. All regions in which @,,a > EJ are forbidden to a star with this energy, and are thus called forbidden regions. For our calculations we place ourselves in a frame of reference corotating with the bar, and the bar semi-major axis is located along the 2 axis. In this aBold letters denote vector notation. The vector z is a unit vector.

304

rotating frame we have five equilibrium points, which, due to the similarity with the Restricted Three Body Problem, are also called Lagrangian points (see left panel of Fig. 2). The points located symmetrically along the 5 axis, namely L1 and Lz, are linearly unstable. The ones located on the origin of coordinates, namely L B ,and along the y axis, namely L4 and Lg, are linearly stable. The zero velocity curve defines two different regions, namely, an exterior region and an interior one that contains the bar. The interior and exterior regions are connected via the equilibrium points (see middle panel of Fig. 2). Around the equilibrium points there exist families of periodic orbits, e.g. around the central equilibrium point the well-known 5 1 family of periodic orbits that is responsible for the bar structure. The dynamics around the unstable equilibrium points is described in detail in Romero-G6mez et al. [21]; here we give only a brief summary. Around each unstable equilibrium point there exists a family of periodic orbits, known as the family of Lyapunov orbits (Lyapunov [22]). For a given energy level, two stable and two unstable sets of asymptotic orbits emanate from the corresponding periodic orbit, and they are known as the stable and the unstable invariant manifolds, respectively. The stable invariant manifold is the set of orbits that tends to the periodic orbit asymptotically. In the same way, the unstable invariant manifold is the set of orbits that departs asymptotically from the periodic orbit (i.e. orbits that tend to the Lyapunov orbits when the time tends to minus infinity), as seen in the right panel of Fig. 2. Since the invariant manifolds extend well beyond the neighbourhood of the equilibrium points, they can be responsible for global structures. In Romero-G6mez e t al. [23], we give a detailed description of the role invariant manifolds play in global structures and, in particular, in the transfer of matter. Simply speaking, the transfer of matter is characterised by the presence of homoclinic, heteroclinic, and transit orbits. Homoclinic orbits correspond to asymptotic trajectories that depart from the unstable Lyapunov periodic orbit y around Li and return asymptotically to it (see Fig. 3a). Heteroclinic orbits are asymptotic trajectories that depart from the periodic orbit y around Li and asymptotically approach the corresponding Lyapunov periodic orbit with the same energy around the Lagrangian point a t the opposite end of the bar L j , i # j (see Fig. 3b). There also exist trajectories that spiral out from the region of the unstable periodic orbit, and we refer to them as transit orbits (see Fig. 3c). These three types of orbits are chaotic orbits since they fill part of the chaotic sea when we plot the Poincar6 surface of section (e.g. the section (x,k)near L1).

305

LD

12

< z > L1

A 0

ro I L5

-5

5

0

Fig. 2. Dynamics around the L1 and Lz equilibrium points. Left panel: Position of the equilibrium points and outline of the bar. Middle panel: Zero velocity curves and Lyapunov periodic orbits around L1 and L z . Right panel: Unstable (in red) and stable (in green) invariant manifolds associated to the periodic orbit around L1. In grey, we plot the forbidden region. From Romero-Gbmez et al. 2006, Astronomy & Astrophysics, 453, 39, EDP Sciences.

-10

-5

5

0

X

10

10

-5

0

5

X

10

LO

5

0

5

LO

X

Fig. 3. Homoclinic (a),heteroclinic (b) and transit ( c ) orbits (black thick lines) in the configuration space. In red lines, we plot the unstable invariant manifolds associated to the periodic orbits, while in green we plot the corresponding stable invariant manifolds. In dashed lines, we give the outline of the bar and, in (b) and ( c ) , we plot the zero velocity curves in dot-dashed lines. From Rornero-Gcjmez et al. 2007, Astronomy and Astrophysics, 472, 63, EDP Sciences.

4. Results

Here we describe the main results obtained when we vary the parameters of the models introduced in Sec. 2. One of our goals is to check separately the influence of each of the main free parameters. In order to do so, we make families of models in which only one of the free parameters is varied, while the others are kept fixed. Our results show that only the bar pattern speed and the bar strength have an influence on the shape of the invari-

306

ant manifolds, and thus, on the morphology of the galaxy (Romero-G6mez et al. [23]). Our results also show that the morphologies obtained do not depend on the type of bar potential we use, but on the presence of homoclinic or heteroclinic orbits. If heteroclinic orbits exist, then the ring of the galaxy is classified as rR1 (see Fig. 4a). The inner branches of the invariant manifolds associated to y1 and 7 2 outline an inner ring that encircles the bar and is elongated along it. The outer branches of the same invariant manifolds form an outer ring whose principal axis is perpendicular to the bar major axis. If

0

0 -1

i

3

in

in-

h 0

"

' '

I

"

"

I

"

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I

~

"

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1-

h 0 -

m

in

I

I

0

0

-

3

3

-

I

1

c

-

rR1RZ ring s t r u c t u r e

-

' ' ' " 1 , ' -

d

-

Barred spiral s t r u c t u r e

-7

-7 -80

-10

0

10

80

-80

-10

0

10

a0

Fig. 4. Rings and spiral arms structures. We plot the invariant manifolds for different models. (a) rR1 ring structure. (b) rR2 ring structure. ( c ) RlRz ring structure. (d) Barred spiral galaxy. From Romero-G6mez et al. 2007, Astronomy and Astrophysics, 472, 63, EDP Sciences.

307

the model does not have either heteroclinic or homoclinic orbits and only transit orbits are present, the barred galaxy will present two spiral arms emanating from the ends of the bar. The outer branches of the unstable invariant manifolds will spiral out from the ends of the bar and they will not return to its vicinity (see Fig. 4d). If the outer branches of the unstable invariant manifolds intersect in configuration space with each other, then they form the characteristic shape of R2 rings (see Fig. 4b). That is, the trajectories outline an outer ring whose principal axis is parallel to the bar major axis. The last possibility is if only homoclinic orbits exist. In this case, the inner branches of the invariant manifolds for an inner ring, while the outer branches outline both types of outer rings, thus the barred galaxy presents an R1 R2 ring morphology (see Fig. 4c). 5. Summary

To summarise, our results show that invariant manifolds describe well the loci of the different types of rings and spiral arms. They are formed by a bundle of trajectories linked to the unstable regions around the L1/L2 equilibrium points. The study of the influence of one model parameter on the shape of the invariant manifolds in the outer parts of the galaxy reveals that only the pattern speed and the bar strength affect the galaxy morphology. The study also shows that all the different ring types and spirals can be obtained when we vary the model parameters. We have compared our results with some observational data. Regarding the photometry, the density profiles across radial cuts in rings and spiral arms agree with the ones obtained from observations. The velocities along the ring also show that these are only a small perturbation of the circular velocity. Acknowledgements

MRG acknowledges a “Becario MAE-AECI” . References 1. P.B. Eskridge, J.A. F’rogel, R.W. Podge, A.C. Quillen, R.L. Davies, D.L. DePoy, M.L. Houdashelt, L.E. Kuchinski, S.V. Ramirez, K. Sellgren, D.M. Terndrup, G.P. Tiede, AJ, 119, 536 (2000). 2. D.M. Elmegreen, B.G. Elmegreen, MNRAS, 201,1021 (1982). 3. A. Sandage, J. Bedke, “The Carnegie Atlas of Galaxies”, Carnegie Inst, Washington (1994).

308

4. 5. 6. 7. 8. 9.

10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

B. Lindblad, Stockholms Observatorium Ann., Vol. 22,No. 5 (1963). A. Toomre, ApJ, 158,899 (1969). P.O. Lindblad, Stockholms Observatorium Ann., Vol. 21,No. 4 (1960). A. Toomre, J. Toomre, ApJ, 178,623 (1972). R.H. Sanders, J.M. Huntley, ApJ, 209,53 (1976). A. Toomre, “The structure and evolution of normal galaxies”, eds. S.M. Fall and D. Lynden-Ball, Proc. of the Advanced Study Institute, Cambridge, pp. 111-136 (1981). E. Athanassoula, Phys. Rep., 114,319 (1984). M.P. Schwarz, ApJ, 247,77 (1981). M.P. Schwarz, MNRAS, 209,93 (1984). M.P. Schwarz, MNRAS, 212,677 (1985). R. Buta, ApJS, 96,39 (1995). E. Athanassoula, MNRAS, 259,328 (1992). G. Kuzmin, Astron. Zh., 33,27 (1956). A. Toomre, ApJS, 138,385 (1963). N.M. Ferrers, Q.J. Pure Appl. Math., 14,1 (1877). W. Dehnen, AJ, 119,800 (2000). B. Barbanis, L. Woltjer, ApJ, 150,461 (1967). M. Romero-Gbmez, J.J. Masdemont, E. Athanassoula, C. Garcia-Gbmez, A & A , 453,39 (2006). A. Lyapunov, Ann. Math. Studies, 17 (1949). M. Romero-G6mez, E. Athanassoula, J.J. Masdemont, C. Garcia-G6mez, A & A , 472,63 (2007).

309

LEVY WALKS FOR ENERGETIC ELECTRONS DETECTED BY THE ULYSSES SPACECRAFT AT 5 AU S. PERRI' and G. ZIMBARDO Department of Physics, University of Calabria, Rende, I-87'036, Italy *E-mail: [email protected]

We consider the propagation of energetic particles, accelerated by interplanetary shock waves, upstream of the shock. In connection with superdiffusive transport and L&y random walks, we consider the relevant non-gaussian p r o p agator, which has power law tails with slope 2 < p < 3. We show that in the case of superdiffusive transport, the time profile of particles accelerated at a traveling planar shock is a power law with slope y = p - 2 , rather than an exponential decay as expected in the case of normal diffusion. We analyze a dataset of interplanetary shocks in the solar wind detected by the Ulysses spacecraft between July 1992 and November 1993.We find that the time profiles of energetic electrons correspond to power laws, with slopes y N 0.60-0.98, implying superdiffusive transport. On the other hand, the propagation of protons seems to be diffusive even when a non Gaussian statistics is involved.

Keywords: anomalous transport; energetic particles; interplanetary shocks.

1. Introduction One of the most far reaching consequences of chaos is that anomalous, non diffusive transport regimes are possible for systems which exhibit non Gaussian statistics and long range correlations. This is the case of particle motion in periodic or quasiperiodic potentials, where chaotic trajectories may alternate long flights, or jumps, to dynamic trapping,lI2 as well as of turbulent fluids, where long range correlations may be found in the velocity of particle^.^ Very similar phenomena are found in plasma physics, and here we show how anomalous transport is useful in describing the propagation of energetic particles in space plasmas. Energetic particles observed in the heliosphere are accelerated at solar impulsive events, as solar flares and coronal mass ejections (CMEs), and at interplanetary shocks waves, as those associated to corotating in-

310

teraction regions (CIRs), which are compressive zones in the solar wind environment formed when the fast wind (KwN 800 km/s) encounters the slow one (Vsw N 400 km/s); this gives rise to a pair of collisionless shocks, one moving forward and the other one in the reverse d i r e ~ t i o n Ener.~~~ getic particles propagate in the interplanetary space interacting with magnetic turbulence, which causes pitch angle ~ c a t t e r i n g The . ~ ~ ~propagation and acceleration properties of ions seem to be well described by the diffusive shock acceleration (DSA) mechanism,'-1° which takes into account the scattering of particles by magnetic irregularities, located upstream and downstream of the shock this mechanism causes a particle Brownian-like motion during the acceleration process. However, there are experimental evidences of different transport r e g i m e ~ for ,~~ instance ~~ the transport of solar energetic particles (SEPs) varies from diffusive to scatter-free. 1 2 3 1 3 In addition, anomalous transport regimes, characterized by a mean square displacement which grows as (Ax2(t)) 0: t a , both slower ( a < 1) and faster ( a > 1) with respect to normal diffusion ( a = l),have been observed in various system^.'>'^)^^ The superdiffusive regime, i.e., a > 1, can be characterized by L6vy random walks whose jump lengths distribution exhibits At this point an important question is whether, under power law appropriate conditions, anomalous transport of energetic particles can be observed in the heliosphere. For this aim, it is possible to look at the particle spatial distributions, which can be sampled by a spacecraft moving in the solar wind. If normal diffusion works, we expect an exponential decay for the time profiles of energetic particles upstream of an interplanetary shock.1°>17However spacecraft data show that there is a large variety of different behaviours.ls In this work we derive the expressions of particle fluxes by using the propagator formalism both under the hypothesis of normal diffusion and in the case of superdiffusive transport; finally we show that many fluxes of electrons accelerated a t CIR shocks exhibit power law tail corresponding t o a superdiffusive transport, while proton time profiles are often well fitted by an exponential decay. 2. Deriving energetic particle profiles by the propagator formalism

A spacecraft in the solar wind measures particles accelerated at various times and positions, and the actual time profile reflects the propagation properties from the source to the observer, as well as the source evolution. We consider the propagation of particles, accelerated by interplanetary

311

shock waves, away from the shock. For the present analysis, we consider a large scale planar shock: this assumption is reasonably well satisfied by interplanetary shocks (e.g. Ref. 17), which may be due either to fast CMEs or to CIRs, thanks to their large size (compared to the relevant transport scales). Accordingly, we consider a steady state, one dimensional shock model. We suppose that the energetic particle fluxes measured by a spacecraft at (x,t ) are the superposition of the energetic particles accelerated at the shock moving according to x’ = Kht‘ (in other words, at t’ = 0 the shock will be at the origin of the coordinate system), with Kh assumed to be constant. To fix the ideas, we consider that the observer is at x = 0, upstream of the shock, which is coming from x = -m; then, t < 0 for the relevant time interval. The particle omnidirectional distribution function f(x,E , t ) at the observer will be expressed in terms of the distribution function f & ( dE, , t’) of particles accelerated at the shock as f(X, E , t ) =

S

P ( Z - Z’,

t

- t’)fsh(d, E , t’)dddt’

(1)

with f s h ( d , E , t ’ ) = fo(E)6(x’ - Kht’), where fo(E) represents the distribution function of particles of energy E emitted by the shock. Here, P ( x - x’,t - t’) is the probability of finding a particle at position x at time t , if it was injected at x’ and t’. Also, x - x‘ is the distance upstream of the shock (the source of energetic particles), t - t’ > 0 , and D is the Gaussian diffusion coefficient. This form, P ( x - x’,t - t’),emphasizes the space-time translational invariance of the propagator. In the case of normal diffusion, particles accelerated at the shock are spread in space according to the Gaussian propagator (e.g., Ref. 14,19)

P ( x - x’,t - t’) =

1

(x - x’)2

& G q c - q e x p [-4D(t - t’,1 .

(2)

Using this expression of the propagator we obtain

(3) Exploiting the 6 function, considering that the observer is at the origin of the coordinate system x = 0 and by introducing the variable T = t - t‘ we obtain

312

It is easy to show that the integral I ( t ) is finite, since the integrand goes to zero for r 0, and decays exponentially for r -+ 00. Application of the Laplace transform given by Eq. (2.12) of Ref. 19,

taking into account that t are left with

< 0, gives I ( t ) = Vs;’ exp(-Kiltl/2D).

Then we

which coincides with the exponential decay obtained by Ref. 10,17 for the energetic particle distribution function upstream of the shock, starting from a diffusion equation including the convective term I/sha f /ax. Conversely, in the case of superdiffusive propagation, transport can be described in the framework of continuous time random walks. For Lkvy random walks, a jump probability $(r, t ) = Alrl-Pb(t - T / U ) of making a jump of length r in a time t can be adopted.l4?l6The power law behaviour of $ ( r , t ) reflects the fact that very long jumps of length T have small but non-negligible probability, contrary to the case of Gaussian random walk. For p < 3, the probability $ ( r , t ) has diverging second order moment, which corresponds to an infinite value of the mean free path; however, this does not imply an infinitely fast transport because long jumps require long times, as implied by the space-time coupling expressed by b(t - T / u ) . In general, the propagator can be obtained in the Fourier-Laplace space and its explicit inversion is only possible in limiting cases: close to the source, i.e., for Ix - 2’1 << kE/2(t- t ’ ) ’ / ( p - ’ ) , the propagator behaves as

where a0 and p are constantd6 and k , is an anomalous diffusion constant (different from the anomalous diffusion coefficient D, introduced below), whose physical dimensions are [ Z 2 / t 2 / ( p - ’ ) ] ; far off the source, i.e., for (x>> kh’2(t-t’)’/(,-1) the propagator has a power law behaviour described

x’I

by

P ( x - X I , t - t’) = b

t - t’ (x - X ’ ) P

where b and p are constants with respect to position and time, but they may depend on particle velocity and on the relevant transport process;16 the propagator in Eq. (8) goes to zero for x - x’ > v(t - t’), with u the

313

particle velocity. For 2 < p < 3 superdiffusion with ( x 2 ( t ) )= 2 D 3 is obtained for large t , with a = 4 - p,16,20while for 3 < p < 4 transport is diffusive, but the propagator has non Gaussian, power law tails as above. l6 We assume to be a t large distance from the shock, that is in the tails of the probability distribution, and we determine the energetic particle profile by using the expression in Eq. (1) and the non Gaussian propagator in Eq. (8). A straightforward calculation, reported in Ref. 21,22, yields

where we have assumed that the shock starts at t o = --oo and that the observer is at x = 0; then far off the shock the time profile of the accelerated particles is a power law decay with slope y = p-2. Accordingly, an energetic particle profile with 0 < y < 1 implies superdifisive transport with a = 4 - p = 2 - y,while 1 < y < 2 implies a non Gaussian propagator like that in Eq. (8) and a long-tailed distribution for jump lengths, but a diffusive transport with a mean square deviation growing linearly in time. We leave the derivation of the power-law particle distribution function by means of a transport equation including fractional derivatives, which describe anomalous diffusion, for future work.

3. Data analysis We analyze a dataset consisting of repeated shock crossings observed by the Ulysses spacecraft in 1992-1993, a period of low influence of the solar impulsive events due t o the decline in solar activity. From July 1992, Ulysses detected a long series of forward-reverse shock pairs associated with CIRs23124and its heliocentric distance was more than 5 AU; this implyes that the shock can be considered planar with a good approximation. We study fluxes of electrons and protons accelerated both a t the forward and at the reverse shock of the CIRS obtained from the CDAWeb service of the National Space Science Data Center (cdaweb.gsfc.nasa.gov). The data analysis is performed by considering particle time profiles at some distance from the shock front because close to it the propagator can be Gaussian-like even for the case of LBvy walks, (see Eq. (7)). In our analysis we study various shock events, during which Ulysses was a t a heliocentric distance N 4.5-5.0 AU and at a latitude E 25"-30" S. As an example, we show in Figure 1 the event of January 22, 1993. From top to bottom, panels show one hour averages for the plasma radial velocity and the plasma temperature from SWOOPS (PI D. McComas), the

314

120ow 14 Jon 1993

1s:ww 18 Jan 1993

zowoo 22 Jan 1993

0ow:w 26 Jan 1993

MWW 30 Jan 1993

Fig. 1. CIR shock event of January 22, 1993, as observed by Ulysses spacecraft. Fkom top to bottom: solar wind radial velocity, solar wind temperature, proton fluxes and electron fluxes. The lower two panels are in semi-log scale. Energies as indicated (from Ref. 21).

semi-log plots of proton fluxes measured by LEFS 60 of HI-SCALE (PI L. Lanzerotti) and those of electrons measured by LEFS 60. We consider the particle time profiles upstream of the reverse shock. In this event, and also in all considered cases, particle fluxes vary slightly more than one order of magnitude over about 200 hours. We notice that several flux fluctuations, with time scale of 20-30 hours, can be distinguished in the time profiles, as well as in the following events. These irregularities are due to the low frequency magnetic t ~ r b u l e n c e which , ~ ~ has a correlation length X 2 35 x lo6 km a t 1 AU (3-4 hours in time scales).26 In addition, turbulence affects the magnetic connection between the spacecraft and shock, causing temporal changes in the energetic particle profiles with the corresponding time scales.27 In Figure 2 we show a comparison between the electron fluxes and the

315 100:

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'

'

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'

'

' '

""'

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........548-781 keV

m

0.11 1

........

. . . . . . . . 10 100 A t (hours) I

I

.

1 1

10 A t (hours)

100

Fig. 2. Comparison between electron fluxes (left panel) and proton fluxes (right panel) of the event indicated in Figure 1. In order to make clear power law decays plots have been displayed in log-log scale. Energies as indicated (from Ref. 21,22).

proton ones upstream of the reverse shock of January 22, 1993; solid lines indicate the best fits. These plots are in log-log scale to better appreciate the power law decay; the time difference At has been calculated as It - tshl, where t is the upstream observation time and t& is the shock crossing time. Since the energetic particle flux J is related to f(x,E , t), we assume a power law decay J = A(At)-y. Calculating the reduced chi-square values for the fits, we obtain for both electron and proton time profiles that power law fits the data better than an exponential decay, J = Kexp(-At/.r), expected for a Gaussian transport (see Figure 2). Note that the power law decay is obtained over more than one decade in electron flux, and over almost 200 hours in time, so that the variations due to the turbulence do not affect the fit. In this event for electrons fluxes we find values of y = 0.81-0.98, implying p = y 2 = 2.81-2.98, i.e., superdiffusion with (Ax2) N t4-p = t1.02-t1.1g,while for protons y = 2.0-2.33 and p = 4.0-4.3. As we argued in Section 2, values of p greater than 3 lead to a diffusive behaviour, even if, as in this case, the statistics is non Gaussian. Other two events, i.e., May 10, 1993 and January 10, 1992, exhibit power law decays in electron time profiles with y E 0.60-0.85, which leads to a mean square displacement (Ax') 2 t1.2-t1.4.Looking a t the event of September 12,1992, associated to a CIR forward shock, in Ref. 22 it has been highlighted that the proton time profile, far from the shock front, displays an exponential decay, typical of a diffusive transport. Finally, we stress that in this kind of analysis we have neglected some effects as solar wind convection and adiabatic deceleration.10>28

+

316

4. Conclusions

In this work, by analyzing electron Ulysses data at 5 AU, we have highlighted that the energetic electron fluxes are well fitted by a power law decay with a slope y E 0.60-0.98 over a period of 100-200 hours, i.e., superdiffusion, while proton fluxes exhibit either an exponential decay and a power law profile but, in the latter case, the exponent y is greater than 1, so the transport can be considered normal. It is interesting to notice that superdiffusive regimes have been observed both in fluids3 and in pla~rnas.~’ In those cases, superdiffusion is related to the existence of series of vortices or magnetic islands, which may exhibit a gerarchy of scale lengths, typical of turbulence, which induce a power law distribution of jump lengths. On the other hand, in the case under consideration the superdiffusive transport is mostly parallel to the background magnetic field, so that the magnetic islands, commonly found in the presence of magnetic turbulence in the plane perpendicular to B0,30 are not so relevant to the creation of a power law distribution of jump lengths. We argue that in this case superdiffusion is due to the weak wave electron interaction, which is caused by small Larmor radii, so that pitch angle diffusion, which drives particles to reverse their direction of propagation, is not very effective. The weak interaction leads to long correlations in the electron parallel velocity, which results in a superdiffusive transport. Further investigations are needed in order to understand the statistical details of such a process. However, we argue that, at least for electrons, shock acceleration mechanisms should be reformulated either in terms of non Gaussian probability distributions or in terms of fractional Fokker-Planck equation^.^?^^ These results promise to have application to models of cosmic ray propagation in space, as well as to the spreading of energetic particles throughout the heliosphere.

Acknowledgments We acknowledge D. J. McComas of Southwest Research Institute for the use of the SWOOPS Ion Measurements data of Ulysses, L. J. Lanzerotti of Bell Laboratories for the use of HI-SCALE LEFS 60 data and M. Lancaster and C. Tranquille of the Ulysses Data System.

References 1. G. M. Zaslavsky, R.Z. Sagdeev, D.K. Chaikovsky, A.A. Chernikov, JETP, 68, 995 (1989). 2. G. M. Zaslavsky, Phys. Reports, 371,461 (2002).

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3. T. H. Solomon, E. R. Weeks and H. L. Swinney, Phys. Rev. Lett., 71,3975 (1993). 4. A. J. Hundhausen, Introduction to Space Physics (eds. M.G. Kivelson and C.T. Russell, Cambridge, 1995). 5. D. V. Reames, Space Sci. Rev., 90,413 (1999). 6. J. R. Jokipii, Astrophys. J., 146,480 (1966). 7. J. Giacalone and J. R Jokipii, Astrophys. J., 520,204 (1999). 8. A. R. Bell, MNRAS, 182,147 (1978). 9. R. D. Blandford and J. P. Ostriker, Astrophys. J. Lett., 221,L29 (1978). 10. L. A. Fisk and M. A. Lee, Astrophys. J., 237,620 (1980). 11. J. E. Mazur, G. M. Mason, J. R. Dwyer, J. Giacalone, J. R. Jokipii and E. C. Stone, Astrophys. J. Lett., 532,L79 (2000). 12. W. Droge, Astrophys. J., 589,1027 (2003). 13. R. P. Lin, Adv. Space Res., 35,1857 (2005). 14. R. Metzler and J. Klafter, Phys. reports, 339,1 (2000). 15. S. Perri, F. Lepreti, V. Carbone and A. Vulpiani, Europhys. Lett., 78,40003 (2007). 16. G. Zumofen and J. Klafter, Phys. Rev. E, 47,851 (t993). 17. M. A. Lee, J . Geophys. Res., 88, 6109 (1983). 18. D. Lario et al., in Proc. Solar Wind 11-Soho 16, (Whistler, Canada, 2005). 19. G. M. Webb, G. P. Zank, E. Kh. Kaghashvili and J. A. le Roux, Astrophys. J., 651,211 (2006). 20. T. J. Geisel, J. Nierwetberg and A. Zacherl, Phys. Rev. Lett., 54,616 (1985). 21. S. Perri and G. Zimbardo, Astrophys. J. Lett., in press (2007). 22. S. Perri and G. Zimbardo, J. Geophys. Res., submitted (2007). 23. S. J. Bame, B. E. Goldstein, J. T. Gosling, J. W. Harvey, D. J. McComas, M. Neugebauer and J. L. Phillips, Geophys. Res. Lett., 20,2323 (1993). 24. A. Balogh, J. A. Gonzalez-Esparza, R. J. Forsyth, M. E. Burton, B. E. Goldstein, E. J. Smith and s. J. Bame, Space Sci. Rev., 72,171 (1995). 25. M. Neugebauer, J. Giacalone, E. Chollet and D. Lario, J . Geophys. Res., 111,A12107 (2006). 26. W. H. Matthaeus and M. L. Goldstein, Phys. Rev. Lett., 57,495 (1986). 27. G. Zimbardo, P. Pommois and P. Veltri, J. Geophys. Res., 109, A02113 (2004). 28. D. Ruffolo, Astrophys. J., 442,861 (1995). 29. S. Ratynskaia et al., Phys. Rev. Lett., 96,105010 (2006). 30. G. Zimbardo, Plasma Phys. Control. Fusion, 47,B755 (2005).

318

WAVE CHAOS AND GHOST ORBITS IN AN UNDERWATER SOUND CHANNEL D. V. MAKAROV’, L. E. KON’KOV, E. V. SOSEDKO and M. YU. ULEYSKY Laboratory of Nonlinear Dynamical Systems, V.I.I1 ‘ichev Pacific Oceanological Institute of the Far Eastern Branch of Russian Academy of Sciences, Vladivostok, R w s i a *E-mail: [email protected] http://dynalab.poi. d v 0 . r ~ Sound wave propagation in a weakly-inhomogeneous acoustic waveguide in the deep ocean is studied. Vertical oscillations of the inhomogeneity result in strong chaos of rays propagating with small angles with respect to the channel axis. Increasing of amplitude of the inhomogeneity results in multiplication of short periodic orbits, and their phase space distribution becomes irregular. However, the Floquet modes calculated with signal frequency of 100 Hz reveal well-ordered peaks within the chaotic area. We link occurrence of these peaks with periodic orbits of resonance 1:8, which survive as ghost orbits.

Keywords: wave chaos, ray chaos, underwater sound channel, scattering on resonance, ghost orbits, scar.

1. Introduction The semiclassical approximation is often used as the first step for describing wave or quantum dynamics. In this way one first calculates classical trajectories and then imposes wave-based corrections, associated with diffraction and interference. The well-known advantages of semiclassical technique are simplicity (as compared with full-wave calculations) and the possibility to give an intuitive “physical” interpretation of the processes observed. On another front, one must always keep in mind that the semiclassical approximation is nothing but the short-wavelength asymptotics for original wave equations and its applicability has severe restrictions. One necessary restriction is that wavelength must be much smaller than the lengthscale of refractive index (or potential) variability. Another one is concerned with caustics, where Maslov method must be appealed.’ Additional problems in ray-wave (or quantum-classical) correspondence occur when classical ray dynamics

319 0 0.5 1

4

1.5

bi

2

2.5 3

0

20

40

60

80

100

r, km Fig. 1. Rays in a range-independent underwater sound channel.

possesses chaotic properties. Wavefield manifestations of ray chaos are referred to as wave chaos. In this case one introduces the so-called mixing (or log) time T A1 Inko, where A1 is the maximal Lyapunov exponent and ko is the wavenumber, playing the role of the inverse Planck constant. Beyond the mixing time classical paths intersect each other in an irregular manner, and interference cannot be neglected. Nevertheless, semiclassical calculations accounting interference between distinct paths can give accurate predictions a t times beyond the mixing timee2l3Furthermore, it is well realized that significant impact into a structure of wave eigenfunctions is given from classical periodic orbits. In particular, one of the most intriguing effects concerned with periodic orbits is the so-called scarring, i.e. forming of intensity peaks at unstable short periodic orbit^.^?^ The present paper is devoted to long-range sound propagation in the ocean. Increasing of hydrostatic pressure with depth, combined with warming of the upper oceanic layer, results in non-monotonic dependence of the sound speed on depth. According to Snell’s law, there occurs a waveguide confining sound waves within a restricted water volume and preventing their contact with lossy bottom. Since low-frequency sound waves are insignificantly attenuating inside the water column, they are capable to propagate over distances of thousands kilometers. Typical ray trajectories of sound rays in the horizontally homogeneous ocean are depicted in Fig. 1. In recent decades long-range sound propagation has attracted increasing attention in the context of hydroacoustical tomography - monitoring of large-scale variability of the ocean using sound signals. The traditional N

320

scheme of hydro-acoustical tomography relies upon analysis of signal travel times along different eigenrays, connecting source and receiver. However, it is realized that even weak longitudinal sound speed variations, mainly associated with ocean internal waves, are sufficient for Lyapunov instability and chaos of sound rays. Ray chaos leads to exponential proliferation of eigenrays with increasing distance between source and receiver, therefore the solution of the respective inverse problem is strongly degenerate under chaotic conditions. Thus ray chaos poses severe limitations on the possibility of extracting information about environment from the hydro-acoustial data.7 Predictability horizon for ray motion, determined by the reciprocal Lyapunov exponent, depends on the topology of a ray. In realistic models of environment this quantity is of approximately 100 km for near-axial rays and of approximately 300 km for steep ones.’ We have shown that strong instability of near-axial rays arise due to resonant scattering on small-scale vertical oscillations of a sound-speed p e r t u r b a t i ~ n . ~These J ~ oscillations are associated with the so-called internal wave structure. On the other hand, vertical wavelength of almost horizontal low-frequency sound waves is not small, that implies weak sensitivity to small-scale sound-speed structures.l1 Thus it is still not enough clear what mechanism accounts for fuzzy wavefield pattern corresponding to near-axial rays. In this way it seems to be worth to examine how chaotic ray dynamics reveals itself in sound propagation with relatively low frequencies, when one should not expect one-to-one correspondence between wave and ray dynamics. In the present paper we report about a specific conflict of ray and wave descriptions, when a wave pattern reveals features having “classical” nature but not allowable on the classical level. These features relate to the scarring phenomenon. The distinctive property of the case we consider is that scars are supported by the so-called ghost periodic orbits,12 which correspond to complex rays.13 The paper is organized as follows. In the next section we briefly describe properties of ray dynamics. Section 3 contains analysis of the Floquet modes. In conclusion we summarize and discuss the results obtained.

2. Classical ray dynamics Consider a two-dimensional waveguide with with the sound speed c presented in the form

321

where co is a reference sound speed, A c ( z ) represents the range-independent depth change of the sound speed due to the waveguide, and dc(z, r ) is a small term varying with range r. We start with considering ray dynamics. Ray trajectories obey the Hamiltonian equations

dz dr dp dr

-=--

dH dz

-

dH

-= P , aP 1 dAc co dz

1 ddc co dz ’

(3)

where p is tangent of ray grazing angle. In the small-angle approximation the Hamiltonian looks as follows

H

=

p2 Ac(z) -1+-+-+-. 2 co

dc(z, r ) CO

(4)

Ray equations ( 2 ) , (3) and the Hamiltonian (4)admit simple interpretation by means of the principle of optical-mechanical analogy: they describe motion of unit-mass point particle in a potential well A c ( z ) with small time-dependent perturbation 6c(z, r ) . In this fashion T plays the role of time, and p is often called as ray momentum. In the present paper we shall consider a range-periodic waveguide with A c ( z ) and 6c(z, r ) given by the following expressions:

21rz 21rr 6c(z, r ) = ~ c o F ( zsin ) -sin -, A, A, where co = c ( z = h) = 1535 m/s, y = exp(-ah), h = 4.0 km is depth of the ocean bottom, 1.1 = 1.078, a = 0.5 km-’,b = 0.557, A, = 0.2 km, A, = 5 km. F ( z ) is an envelope function, given by expression

where B = 1 km. The channel axis, the depth where function A c ( z ) takes on the smallest value, is given by the formula 1

z, = -In

a

2

-N 1 km. P+Y

The respective unperturbed sound-speed profile is depicted in Fig. 2.

322

0

1

4 2 N"

3

4 1480 1500 1520 1540

c, m / s Fig. 2.

The unperturbed sound-speed profile.

We use the model of range-periodic sound-speed perturbation. Although horizontal periodicity is a strong idealization, our model is relevant for qualitative analysis of the sound propagation in realistic environments. Amplitude of the perturbation term E is a small constant. In this paper we shall consider two cases: E = 0.0005 and E = 0.005. The sound-speed perturbation (6) can represented as a superposition of two waves, propagating upwards and downwards, respectively. This yields 1 ddc

--

co d z

~ e - ____ -

2B

[ g) ~

(1

~

-

/

~

(cosa- - COS@+) 1

1

- k,z (sin@- - sin@+)

(9)

,

where we denoted @* = k,z f k,r, k , = 27~/X,, k, = 27r/X,. Since k , is a large parameter, phases vary rapidly along unperturbed ray paths, except for resonant regions, where

or

d@dr

-= k,p - k,

N

0.

323 0

1

E

A 2

6 3

I

4

-0 2

a)

-0 I

0

0.1

$3

0.2

b)

P Fig. 3.

._-

-0.2

Poincarh map. (a)

E

-0.1

0

0.1

0.2

I

0.3

P

= 0.0005, (b) E = 0.005.

It is easy to show that these resonances mainly affect near-axial rays, making them strongly chaotic (seeg~l0 for details). This is well demonstrated in Poincar6 sections (see Fig. 3). There occurs a chaotic layer in the part of phase space, corresponding to near-axial rays. The layer is bounded by invariant curves, and steeper rays, not being influenced by resonances, perform stable dynamics.

3. Wave dynamics In the present section we study how chaotic layer, corresponding to nearaxial rays, reveals itself in structure of the Floquet modes, which are calculated at the signal frequency of 100 Hz. The Floquet modes u, are the eigenfunctions of the shift operator F , defined as F + ( z ,r ) = +(z,?-

+ Ar).

(12)

Phase space representation of the Floquet modes can be obtained by use of the Husimi distribution function

Here Az is the smoothing scale, which we took of 100 m. Mixing ray dynamics inside the chaotic layer presupposes existence of strongly extended Floquet modes with irregular distribution of Husimi zer o ~Nevertheless . ~ ~ the Floquet modes calculated with E = 0.005 reveal almost regular pattern. Although few modes cover wide area in phase space,

324

1

-0.3 -0.2 -0.1

0

0.1

0.2

0.3

P Fig. 4.

4 -0.3 -0.2 -0.1

An extended Floquet mode.

7--

0

P

011

-0.2 -0.1

0

0:1

0:2

0.3

P

Fig. 5 . Floquet modes with chainlike topology.

their Husimi zeros are located along certain curves. Since that these modes could be classified rather as weakly irregular than strongly chaotic. One of such modes is illustrated in Fig. 4. We want to focus attention on those Floquet modes which have the specific chainlike topology of peaks, as it is demonstrated in Fig. 5. All the peaks belong to the chaotic sea of the classical system. On another front, a comparison with Fig. 3(b) yields that the peaks in Fig. 5a are located at the elliptic orbits of KAM resonance 1:8 and the peaks in Fig. 5b are placed at the hyperbolic ones. This implies that the respective Floquet modes are localized at the unstable periodic orbits and relate to cars.^^^'^

0.3

* : 0.2 -

0

'

s"

0

0:.

0

0 0

*.

0

.

0.0

0

0.1

. .

0.

..tt8

0

0

. 0 :

0

.

O 0 . .

). 0

*.

0

0 0

: 0

0

0.0

0

..*@.

dbo 0

O..O

@-

0

Fig. 6. Phase space locations of periodic orbits in the normalized action-anglevari-

ables. Is is the most accessible value of the action for guided rays propagating without reflections from the lossy bottom. Perturbation strength e is of 0.005. KAM resonance 1:8 corresponds to the line I/Is = 0.2.

However the situation we met is more complicated because increasing perturbation amplitude strongly alters phase space distribution of periodic orbits, especially in the range of low values of action, as it is illustrated in Fig. 6. According to this figure, peaks on Husimi plots don't supported by certain periodic orbits. This infers that the ordered peaks are localized at the so-called ghost orbits,12 which correspond to complex ray paths. Occurrence of prominent complex paths may be considered as some manifestation of tunneling through narrow classical barriers, which appear due to rapid depth oscillations of the sound-speed perturbation.

4. Conclusion

In the present paper we have examined interplay between ray and wave behavior in the model of a range-periodic underwater sound channel. We have shown that small-scale vertical oscillations of the perturbation result in strong chaos of rays propagating with small angles with respect to the channel axis. Increasing the perturbation amplitude results in multiplication of periodic orbits, and their phase space distribution becomes strongly irregular. On the other hand, Husimi plots for the Floquet modes reveal a chain of well-ordered peaks. Phase space locations and number of the peaks indicates on unambiguous link to KAM resonance 1:8. We suppose that periodic orbits of resonance 1:8 survive as the ghost orbits due to tunneling through small-scale sound-speed oscillatghions.

326

This work was supported by t h e projects of the President of t h e Russian Federation, by t h e Program “Mathematical Methods in Nonlinear Dynamics” of t h e Prezidium of t h e Russian Academy of Sciences, a n d by t h e Program for Basic Research of t h e Far Eastern Division of t h e Russian Academy of Sciences. Authors a r e grateful to S.V. Prants, A.I. Neishtadt, A.L. Virovlyansky a n d A.I. Gudimenko for helpful discussions during t h e course of this research.

References 1. V.P. Maslov and M.V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics (Reidel, Boston, 1981). 2. S. Tomsovic and E.J. Heller, Phys. Rev. Lett. 67,664 (1991). 3. S. Tomsovic and E.J. Heller, Phys. Rev. E. 47,282 (1993). 4. E.J. Heller, Phys. Rev. Lett. 53,1515 (1984). 5. E.B. Bogomolny, Physica D, 31 169 (1988). 6. W. Munk and C. Wunsch, Deep-sea Res. 26A, 123 (1979). 7. F.D. Tappert and Xin Tang, J. Acoust. SOC.Am. 99, 185 (1996). 8. F.J. Beron-Vera, M.G. Brown, J.A. Colosi, S. Tomsovic, A.L. Virovlyansky, M.A. Wolfson, and G.M. Zaslavsky, J . Acoust. SOC.Am. 114,1226 (2003). 9. D.V. Makarov and M.Yu. Uleysky, Acoust. Phys. 53,495 (2007). 10. L.E. Kon’kov, D.V. Makarov, E.V. Sosedko, and M.Yu. Uleysky, Phys. Rev. E. 76,056212 (2007). 11. K.C. Hegewisch, N.R. Cerruti, and S. Tomsovic J . Acoust. SOC.Am. 117, 1582 (2005). 12. M. Kus, F. Haake, and D. Delande Phys. Rev. Lett. 71,2167 (1993). 13. Yu.A. Kravtsov and Yu.1. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1999). 14. P. Leboeuf and A. Voros, J . Phys. A 23,1765 (1990). 15. 1.P Smirnov, A.L. Virovlyansky and G.M. Zaslavsky, Chaos 14,317 (2004). 16. I.P. Smirnov, A.L. Virovlyansky, M. Edelman and G.M. Zaslavsky, Phys. Rev. E. 72,026206 (2005).

327

DISPLACEMENT EFFECTS ON FERMI ACCELERATION IN RANDOMIZED DRIVEN BILLIARDS A.K. KARLIS*, P.K. PAPACHRISTOU, F.K. DIAKONOSt Department of Physics, University of Athens, GR-15771 Athens, Greece E-mail: * [email protected] t fdiakono Qphys.uoa. gr

V. CONSTANTOUDIS Institute of Microelectronics, NCSR Demokritos, P. 0. Box 60228, Attiki, Greece

P. SCHMELCHER Physikalisches Institut, Universitat Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany Theoretische Chemie, Im Neuenheimer Feld 229, Universitat Heidelberg, 69120 Heidelberg, Germany Fermi acceleration of an ensemble of non-interacting particles evolving in a stochastic Fermi-Ulam model (FUM) and the phase randomized harmonically driven periodic Lorentz gas is investigated. As shown in [A. K. Karlis, P. K. P a pachristou, F. K. Diakonos, V. Constantoudis and P. Schmelcher, Phys. Rev. Lett. 97, 194102 (ZOOS)], the static wall approximation, which ignores scatterer displacement upon collision, leads to a substantial underestimation of the mean energy gain per collision. Herein we clarify the mechanism leading to the increased acceleration, through the investigation of a randomized FUM, comprising one fixed and one moving wall oscillating according to a sawtooth time-law. Moreover, it is shown that the presence of a particular asymmetry of the driving function leads to a different particle energy gain compared to the one observed when a symmetric driving law is considered. Furthermore, the impact of scatterer displacement upon collision on the acceleration of particles in the case of a harmonically driven phase randomized Lorentz gas is investigated in terms of a Markov process in momentum space. Generalizing the recently introduced hopping wall approximation (HWA) for application in the Lorentz gas, the time-evolving distribution of particle speeds is obtained. The analysis reveals that the underestimation made by the static approximation in both the Lorentz gas and the FUM, assuming a harmonic driving, is the same. Keywords: Fermi acceleration; Lorentz gas; Diffusion.

328

1. Introduction

Fermi acceleration was originally proposed in 1949l for the explanation of the high-energy cosmic ray particles. The basic idea was that cosmic ray particles would, on the average, gain energy by scattering off timedependent magnetic inhomogeneities -for a review see Ref. 2. Ever since, Fermi acceleration has been investigated in the context of a variety of systems, pertinent to different areas of physics, including a s t r o p h y ~ i c splasma ,~ physic^,^ atom optics5 and has even been used for the interpretation of experimental results in atomic physics.6 Conceptually, the simplest dynamical system which allows for the investigation of Fermi-acceleration is the Fermi-Ulam model (FUM).7 FUM comprises two infinitely heavy walls, one fixed and one oscillating, and an ensemble of non-interacting particles bouncing between them. FUM and its variants have been extensively studied both theoretically (see Ref. 8 and references therein) and experiment all^.^-^^ Despite the simplicity of the model, the equations defining the dynamics are, in general, of implicit form with respect to the collision time, which hinders the analytical treatment and greatly complicates numerical simulations. For these reasons, a simplification originally introduced by Lieberman et a1.,12 known as the static wall approximation, has become over the time the standard approach for studying the FUM.13-17 The SWA simplifies the process in that the displacement of the moving wall is ignored. However, the time-dependence in the momentum exchange between particle and wall on the instant of collisions is retained, as if the wall were oscillating. More recently, similar simplifying approximations were employed for the investigation of higher-dimensional, spatially extended billiards, such as a periodic Lorentz gas,18 consisting of circular scatterers oscillating harmonically, with their equilibrium positions placed a t the nodes of a square 1 a t t i ~ e . I ~ However, as has been recently shown by Karlis et a1.,20 the SWA leads to a considerable underestimation of the energy gain of the particles evolving in a stochastic FUM. Moreover, in Ref. 20 with the introduction of the so-called hopping wall approximation (HWA) it was made clear that the movement of the oscillating wall in the configuration space affects deeply the diffusion in momentum space. In spite of the externally imposed stochasticity, small additional fluctuations in the time of free flight, caused by the displacement of the wall, act systematically leading to the increased acceleration compared to that predicted by the SWA. In Sec. 2, the physical mechanism a t work, leading t o the increased acceleration due to the displacement of the moving wall, is exemplified through

329

the investigation of a stochastic FUM with one fixed and one moving wall oscillating according to a sawtooth time-law. In addition, it is shown, as indicated in Ref. 20, that in the presence of a particular asymmetry of the sawtooth driving function the increase in acceleration is different from that found considering a symmetric time-law, for all finite number of collisions. However, when the same asymmetric driving is applied on the two-moving wall variant of the FUM, the increase in acceleration due to the displacement of the wall upon collision is once more constant and coincides with that obtained on the assumption of a symmetric force function. These findings open up the prospect of designing specific devices combining driving laws with different symmetries in order to achieve a desired acceleration behavior. In order to determine the effect of scatterer displacement upon collision on particle acceleration in the case of higher-dimensional billiards, the driven phase-randomized periodic Lorentz gas is investigated in Sec. 3. Generalizing the recently introduced hopping wall approximation (HWA) for application in the Lorentz gas, the time-evolving distribution of particle speeds is obtained. The analysis reveals that the increase in acceleration due to scatterer displacement on collision takes place also in higher-dimensional driven billiards and moreover that the increase coincides with that observed in the stochastic FUM assuming harmonic driving. Thus, it can be inferred that the increase of particle energy gain upon collision with an oscillating scatterer is common to many driven dynamical systems and features in any billiard allowing for Fermi acceleration to develop.

2. Fermi-Ulam model with a sawtooth wall driving function In order to clarify the physical mechanism which accounts for the effect of the wall displacement on the acceleration of the particles, a FUM comprising one fixed and one oscillating wall driven according to a sawtooth time-law is investigated. The specific choice for the driving function is justified due to its simplicity, which helps elucidate the important physical aspects of the mechanism in a more intuitive and straightforward manner. The phase of the oscillating wall is shifted randomly (according to a uniform distribution) after each collision with the fixed wall. The stochastic component in the oscillation law of the wall simulates the influence of a thermal environment on wall motion and leads to Fermi a c ~ e l e r a t i o n . ' ~ -The ~~>~~

330

following class of time-periodic laws for the moving wall are considered:

f i^

^e[o,a)

Xi(t) = xo,i+ { -&%%&

^ € [a,b]

I jh^ 1A 2A

u(t)= k

1-6 T

(1)

^e(6,i] QpE[O,a) @yE[u,b] T

te

(2)

Vw' A i

where T is the period of the oscillation, A the amplitude, XO,(L,R) the equilibrium position of the left or right moving wall and r\ a random number uniformly distributed in [0, 2TT). Let us assume that the oscillating wall is on the left. According to the SWA, which treats the oscillating wall as fixed, the time of collision t is given by the intersection of a particle trajectory with the line corresponding to the equilibrium position of the wall, as shown in Fig. l(a). On the other hand, the true collision time t', as predicted by the exact map,22 is the intersection of the line representing the particle trajectory with the piecewise linear function defined by Eq. (1). Furthermore, in Fig. l(a) we observe that if the particle trajectory lies in zone 2 —zones are marked with double arrows in Fig. l(a) and further are delimited by dashed lines— then due to the displacement of the wall toward the right, the actual collision time —obtained using the exact map— compared to that estimated by the SWA (lying in zone 2) is smaller (shifted in zone 1), on which particle and wall have opposite velocities (head-on collision). Moreover, in zone 4, again due to the displacement of the wall, the collision time is shifted toward greater values (in zone 5) in comparison to the that predicted by the SWA (lying in zone 4), when again, particle and wall velocities have opposite directions. Thus, in both occasions, the SWA would render a head-on (energy increasing) collision to a head-tail (energy decreasing) collision. On the other hand, the estimation of the change of particle energy upon collision for trajectories confined between the borderlines of the zones 1, 3 and 5 is the same using either the exact model or the SWA, as the collision times obtained using either of the models correspond to the same wall velocity. Consequently, the SWA leads to the underestimation of the acceleration rate of particles. Our attention now shifts to the dependence of the increase of particle energy gain caused by wall displacement on the specific characteristics (symmetries) of the wall driving function. The symmetry u (m^ + j — tj = —u (m-j + j + t) , (TO = 0,1,2,...), characterizing the harmonic driving

331

studied in Ref. 20, in the context of a stochastic FUM, revealed that wall displacement doubles the mean particle energy gain. Let us now investigate the effect on the increase of particle acceleration of the breaking of this particular symmetry. In order to quantify the relative efficiency of the mechanism leading to the increased acceleration between setups with different driving force laws, the ratio Rh ( n ) =

( (V,") - (V,'))

~

((V,")-(V,"))s,,

-

5 (6Viz)eroct

i=l

2(6V:)swa is introduced. The

i=l

specific definition of Rh(n),compared to that given in Ref. 20, is more convenient for numerical calculations as it converges much more rapidly in terms of ensemble averaging. For the harmonically driven FUM (with randomized phase of oscillation) the correction factor Rh(n) can be proved" (within the leading order of (+)) that it is independent of n and equals 2. As already mentioned the ensemble of particle trajectories can be divided into two sets: The first set consists of trajectories for which the acceleration process is identical to that of the exact model. The second set consists of trajectories for which the acceleration in the SWA is underestimated. Since the phase is uniformly distributed the size of the zones determines also the statistical weight of each zone. Thus, denoting the statistical weight of the zones with pi, (i = 1 . ..5) and the particle velocity with V the average energy gain over the phase upon collision is ((6V:)) = C;=lpibV:,i. The ensemble average can be extracted by integrating ((6V:)) over the P D F of the particle velocities (see Ref. 22) to obtain for n >> 1:

where the plus (minus) sign corresponds to a FUM with an oscillating wall on the left (right). Thus, a FUM with a moving wall on the right leads to a slower acceleration rate compared to its counterpart. This is illustrated in Fig. l ( b ) , where numerical results for R h , L ( n )obtained for a = 5 x and b = 0.67 are shown along with the analytical result of Eq. (25) of Ref. 22. As demonstrated in this figure, for the particular choice of the parameter values, the function R h , ~ ( ndeviates ) from the value 2 even for relatively large n. Obviously, for the case of a two-moving wall FUM the factor Rh is given by &(n) = R h z ~ ( n )2+ R h s ~ ( n )= 2. consequently, for a two-moving wall FUM the correction factor for the increased acceleration due to the wall displacement on collision, assuming that both walls follow the dynamics of Eq. (l),is independent of n and equal to that obtained for the case

332

of harmonic driving. In conclusion, Rh(n) for the FUM setups with one moving wall depends on the number of collisions n and tends to the value Rh (00) = 2 , in a manner specified by the particular oscillation law, which is characterized here by the parameters a , b. On the other hand, when both walls of the FUM are allowed to move, then unless a specific choice for the dynamics of each of the walls is made, Rh(n) is rendered independent of n and equals that obtained when a harmonic driving is considered.

Particle trajectory

a

1

b

tlT

t

l.g 1.8'

0.2

0.4

0.6

0.8

1

1.2

1.4

Number of collisions n

1.6

1.8

I

2

lo5

Fig. 1. (a) The oscillation law for the FUM (with one oscillating wall). The borderlines of the zones discussed in the text are delimited by the dashed lines of slope X = The zones are also marked by double arrows. (b) Numerical results ( x ) as well as an& lytical results (see Eq. (25), in Ref. (22)), for R h , ~ ( nin) the original FUM, employing and b = 0.67. sawtooth driving for uo = 0.01, a = 5 x

w.

3. Time-dependent Lorentz gas In the theory of dynamical systems Lorentz gas18 acts as a paradigm allowing us to address fundamental issues of statistical mechanics, for instance, ergodicity and and transport processes, such as diffusion in

333

configuration pace.^^^^^ A generalization of the original periodic Lorentz gas model has recently been introducedlg allowing the velocity of the scatterers to oscillate radially on a square lattice, i.e. static approximation. Due to the inherent strong chaotic dynamics of the static Lorentz gas, owing to the convex geometry of the scatterers, one intuitively expects13 that the introduction of time-dependence induces Fermi acceleration, resulting in unbounded growth of the velocity of the tracer particle. This is in contrast to the FUM, where, on the supposition of smooth periodic force functions, the particle energy remains bounded, due to the presence of a set of spanning KAM curves in the phase space8?l2and only in the presence of external stochasticity does Fermi acceleration become feasible. The acceleration problem can be treated as a Markov process in momentum space. Therefore, the evolution of the probability distribution function (PDF) of the magnitude of particle velocities p(lV1,n) can be determined by the Fokker-Planck equation. In Ref. 19 the study of a square periodic Lorentz gas consisting of “breathing” circular scatterers, i.e. with oscillating boundary of the scatterer in the radial direction, conducted by means of the static approximation, concluded that p(lVI , n ) is a sum of spreading Gaussians. Furthermore, general arguments presented in Ref. 28, where a random time-dependent Lorentz gas has been investigated, also suggest that p(lVl , n) is a Gaussian. However, the numerical results shown in Fig. 2, which correspond to two snapshots for n = 5 x lo4 and n = 5 x lo5, obtained through the iteration of the exact map,22 point to the direction of p(lVl , n) being a Maxwell-Boltzmann like distribution. As mentioned above, p(JV1,n) can be analytically obtained through the solution of the Fokker-Planck equation (FPE)

In Eq. (3), P is the probability of a particle possessing the velocity IVI if it had the velocity IVI - AlVl, A n collisions earlier. Assuming that An = 1, B(V) = (S(V1)and D(V) = ((SlV1)2),where (SlVl) is the mean increment of the magnitude of the particle velocity during one mapping period, i.e. in the course of one collision. To do so the transport coefficients must be calculated taking into account the displacement of the scatterers. However, the implicit form of the exact map describing the dynamics of the system prohibit an analytical calculation. For this reason, the hopping wall approximation (HWA) originally introduced for the stochastic FUM2’ is generalized for application in the Lorentz gas. The key simplifying assumption of the hopping approximation is that

334

.. -

0.6

> 0.4

v

Q

0.2

n 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

IVI 0.4

I

t

n

c..

- 0.3 0.2 -

>

W

Q 0.1

0 0

2

4

6

8

10

12

14

Fig. 2. Numerically computed P D F for the magnitude of the particle velocity using an ensemble of lo4 trajectories and following the exact dynamicszz for: (a) n = 5 x lo4 and In each case the analytical result provided (b) n = 5 x lo5 with E = 2 215 and llVoll = by Eq. ( 3 ) is also shown (solid line). Velocities are measured in units of ww (w=2.15, angular frequency w = 1)

g.

a scatterer on the instant of the n t h collision occupies the position it held on the (n - 1)th collision, provided that the time of free flight between two successive collisions is small compared to the period of oscillation, i.e. high particle velocities. Thus, within the framework of this approximation, the deterministic component of the phase of the scatterer's position a t the nth collision is taken to be equal to the deterministic phase component of the scatterer's velocity at the (n - I ) t h collision. This leads to an explicit mapping" with regard to the instant of collision t,, which permits the analytical estimation of the transport coefficients of the FPE. The specific values of the coefficients" together with reflective boundary conditions at V = 0 lead to the following solution of the Fokker-Planck equation:

335

where c is the dimensionless ratio of the amplitude of oscillation over the spacing between the scatters centers a t equilibrium w. In Fig. 2 the analytical result for the PDF, Eq. (4),is also plotted for the sake of comparison. Clearly, the analytics provides an accurate description for the PDF p( IVI, n) for n >> 1. From Eq. (4) if follows that for n >> 1 the ensemble mean square velocity is V," = 2e2n &. Therefore, the mean change of particle energy is (6V,"),,,,t = (V,") - (V,"-l) = 2c2. If the static approximation is applied,22 one finds (6V,")static = c2. Therefore, Rh ( n ) =

+

2(6y2)ezact/(6y2),t,ti, 2 , independent of the number of collisions n n

=

i=l

i=l

and equal to that obtained in the case of a stochastic FUM with symmetric wall driving.

4. Concluding remarks

The basic findings of the present work are: a

Rh(n), which quantifies the increase in acceleration due to the displacement of the moving scatterer, was shown that on the assumption of a driving time law featuring the symmetry -u (m; $ -t) = --u (m; $ t ) , (m = 0 , 1 , 2 , . . .), is independent of the number of collisions n, regardless of the specific billiard, being a FUM or a Lorentz gas. On the other hand, if the driving time-law acting on the moving wall of the FUM -with one fixed and one moving wall- is asymmetric, then Rh(n) is different from 2 for any finite number of collisions n and becomes equal to 2 only asymptotically. The PDF of particle velocities in the driven Lorentz gas resembles a Maxwell-Boltzmann distribution. This is in contrast to the result obtained by the application of the standard SWA, i.e. by not taking into account the displacement of the scatterer, where the corresponding PDFs are found spreading Gaussians.

+ +

+

As a final remark, we note that the understanding of the dependence of particle acceleration behavior on the symmetries of the driving law, helps open up the prospect of designing specific devices combining driving laws with different symmetries in order to achieve a desired acceleration behavior.

336

Acknowledgment T h e project is co-funded by the European Social Fund a n d National Resource (EPEAEK 11) PYTHAGORAS.

References 1. 2. 3. 4. 5.

E. Fermi, Phys. Rev. 75,1169 (1949). R. Blandford, D. Eichler, Phys. Rep. 154,1 (1987). A. Veltri and V. Carbone, Phys. Rev. Lett. 92 143901 (2004). A. V. Milovanov and L. M. Zelenyi, Phys. Rev. E 64, 052101 (2001). F. Saif, I. Bialynicki-Birula, M. Fortunato, W. P. Schleich, Phys. Rev. A 5 8 ,

4779 (1998). 6. G. Lanzano et al, Phys. Rev. Lett. 83,4518 (1999). 7. S. Ulam, in Proceedings of the Fourth Berkley Symposium on Mathematics, Statistics, and Probability, California U.P., Berkeley, 1961, Vol. 3,p. 315. 8. A. J . Lichtenberg, M. A. Lieberman, Regular and Chaotic Dynamics, Applied Mathematical Sciences 38, Springer Verlag, New York, 1992. 9. Z. J. Kowalik, M. Franaszek and P. Pieranski, Phys. Rev. A 37,4016 (1988). 10. S. Warr et al, Physica A 231,551 (1996). 11. S. Celaschi and R. L. Zimmerman, Phys. Lett. A 120,447 (1987). 12. M. A. Lieberman and A. J. Lichtenberg, Phys. Rev. A 5,1852 (1972). 13. E. D. Leonel and P. V. E. McClintock, 3. Phys. A: Math. Gen. 38, 823 (2005). 14. A. J. Lichtenberg, M. A. Lieberman and R. H. Cohen, Physica D 1, 291 (1980). 15. E. D. Leonel, P. V. E. McClintock and J. K. d a Silva, Phys. Rev. Lett. 93, 014101 (2004). 16. E. D. Leonel and P. V. E. McClintock, Phys. Rev. E 73,066223 (2006). 17. D. G. Ladeira and J. J. Leal d a Silva, J. Phys. A: Math. Theor. 40, 11467 (2007). 18. H. A. Lorentz, Proc. Roy. Acad., Amsterdam, 438, 585, 684, (1905). 19. A. Yu. Loskutov, A. B. Ryabov, and L. G. Akinshin, J. Exp. Theor. Phys. 89,966 (1999); J. Phys. A: Math. Gen. 33 7973 (2000). 20. A. K. Karlis, P. K. Papachristou, F. K. Diakonos, V. Constantoudis and P. Schmelcher, Phys. Rev. Lett. 97,194102 (2006). 21. G. M. Zaslavskii and B. V. Chirikov, Sov. Phys. Dokl. 9,989 (1965). 22. A. K. Karlis, P. K. Papachristou, F. K. Diakonos, V. Constantoudis and P. Schmelcher, Phys. Rev. E, 016214 (2007). 23. Ya. G . Sinai, Russ. Math. Surveys 27,21 (1972); 25 137 (1970). 24. L. A. Bunimovich, Comm. Math. Phys. 65,295 (1979). 25. L. A. Bunimovich and Ya. G. Sinai, Commun. Math. Phys. 78, 247 (1980); 78,479 (1980). 26. H. v. Beijeren, Rev. Mod. Phys. 54,195 (1982). 27. C. P. Dettmann, in Encyclopaedia of Mathematical Sciences, edited by D. Szasz (Springer, Berlin, ZOOO), Vol. 101, Chap. The Lorentz gas: A paradigm for nonequilibrium stationary states, p.315. 28. F. Bouchet, F. Cecconi and A. Vulpiani, Phys. Rev. Lett. 92,040601 (2004).

337

MEMORY REGENERATION PHENOMENON IN FRACTIONAL DEPOLARIZATION OF DIELECTRICS V. V. UCHAIKIN* and D. V. UCHAIKIN Department of Theoretical Physics, Ulyanovsk State University, Ulyanovsk, 432 970, Russia * E-mail: uchaikinQsv.uven.ru The depolarization of dielectrics with polar molecules is considered as diffusion of points over a spherical surface. The normal (Gaussian) diffusion leads to the Debye relaxation, while the subdiffusion regime expressed in terms of fractional differential equation leads to a non-Debye relaxation law of the inverse power type. When the order a becomes 1, the relaxation takes the exponential form, but when a is close to 1 (but is not strictly equal to it) the relaxation follows firstly exponential law and then inverse power law. Because of the property of fractional derivatives, the end part of the process depends on its prehistory. This phenomenon is interpreted as regeneration of memory.

Keywords: Depolarization; Memory; Relaxation; equation.

Fractional

differential

1. Introduction

There exist numerous evidences that the dielectric response of a wide variety of dielectrically active materials manifests deviation from the Debye

relaxation law

f ( t ) = foe-+,

t 2 0,

7

> 0.

(1) Rich experimental data collections had been gathered and analyzed in Jonscher's books,1,2 proceeding^,^-^ reviews6 and other editions. It is no need t o think, that these observations are quite new: the first of them was made more than hundred years ago7lg and is known now as the Curie-von Schwei-

dler law:

f ( t )0: t-", t

00,

ck

> 0,

(2) Such kind of relaxation is observed in many other natural processes: luminescence decay, chemical reaction processes, mechanical creep and viscoelasticity, trapping charges in p - n junctions and so on. It is so widespread --f

338

way of approaching to an equilibrium state that Jonscher named it the “Universal Relaxation Law” .2 The non-Debye systems manifest a new, very important quality - the “memory”. This means that the real law of relaxation depends not only on its value at the switch-off moment t = 0 as Eq (1) predicts but on the whole previous history of the process f ( t ) , t < 0. The rise of the memory has been noted by a few authors of theoretical and experimental works (one of them gave his articleg an expressive title: “Dead matter has memory!”). We present here a new effect connected t o the memory phenomenon: when a being close to 1, the memory about a previous history arises not immediately after time t=O but after a some interval (O,t,) during which the relaxation approximately follows the exponential law independently of the prehistory t < 0. 2. Diffusion model of the relaxation Let us recall the classical image of a polar dielectric as a set of polar molecules freely moving in a boiled neutral liquid representing the influence of heat motion. Each molecule possess its own randomly oriented dipole moment plus a non-random component P caused by an external electrical field. This component is much smaller than the basic one, but namely it produces the polarization, because the mean value of the basic component equals 0. After switching off the external field, the P becomes accessible t o action of the heat bath, changes randomly its direction and the mean value of its projection on the initial direction ( P Z ( t )decreases ) with time. As a result, the polarization of the volume V containing N molecules decreases as f ( t ) = N(Pz(t )) . Two assumptions lie in the base of the model: 1) independent behavior of different molecules and 2) the Markovian character of motion of each of them. They allow us to reduce the relaxation process to a random motion of a single molecule in the direction space or, in other words, t o Brownian motion of a point on the sphere of the unit radius. The probability initially concentrated a t the upper pole of the sphere ( z = 1) gradually spreads over it and eventually fills it uniformly. The relaxation function being proportional to the mean z-coordinate of the distribution goes to 0 as t 4 00, its behavior determines a relaxation law. In terms of spherical coordinates 29, cp, the angular probability density of the point p(29, cp, t ) obeys the diffusion equation

339

where K is a positive diffusivity in the direction space, and Do,, denotes the angular part of Laplacian. On multiplying both sides of the equation by fo cos 6 sin 6d6dy and integrating over the sphere we arrive at the following equation

dfo = dt

-T-lf(t)

for the seeking function

f ( t )= fo(cos 6 ( t ) )= fo

//

+ foS(t)

(4)

p ( 6 , (p, t )cos 29 sin 6d6dy.

The solution of the equation is exactly the Debye relaxation function Eq (1).What can be improved in this model t o get a solution with the property

(2)? Discussing why relaxation in solids is differ from relaxations in gases, Jonscher fairly notes that the dipoles can not be considered as independent anymore, molecules may unite in clusters. It is naturally to suppose that the clumsy clusters hinder to each other in motion, that they have to wait until they have a space for rotation decreasing their potential energy. A dipole stays almost motionless as if were in a trap (in the direction space). After some random (perhaps very long) time TI it makes a jump to some other direction being close to the previous one. Following the procedure described in" (see alsoll) one can prove that the relaxation process will be of the Debye type in long-time region for any distribution p(t) with a finite mean time ( T ) .However, in the case when 00

( T )= / t p ( t ) d t = cq 0

one can get another kind of asymptotic behaviour of the relaxation. Namely, assuming that

Prob(T

> t ) cx tKa-', t -+ co, a E (0,1),

we obtain a solution having the property (2). The correspondent diffusion (more exactly subdiffusion) process is described by the time-fractional subdiffusion equation generalizing Eq (3)

where

340

and -mDF is the Riemann-Liouville fractional differential operator: t

The corresponding generalization of relaxation equation (4) takes the form of an ordinary fractional equation:

-oD,af(t) = -7-

--a

f(t) + for(l

tT“ - a )7

t>0,

O
(5)

The last term of the equation arises from the initial condition f(0)= fo at time t = 0 and does not include any information about a previous history of the process. Such statement of the problem, being very natural for the integer-order process (4), looks to be incomplete for the fractional process (5) because of its memory. It seems to be reasonable the following modification of the equation (5):

-mDt”f(t)= - T - ” f ( t )

+F(t),

0 < Q < 1.

(6)

Remember, that integrating over a period preceding an observation moment reflects a special quality of a process called “limited after-effect” property or, for a shortness, the memory. These processes were considered by outstanding Italian mathematician Vito Volterra. l2 Following his ideas, we extended the domain of integration to the whole preceding semiaxis --co < t‘ < t , but the “short-memory” principle which means taking into account the behaviour of f ( t ) only in the “recent past”, allows us to restrict ourselves by finite interval (-a,O), a > 0, where a is an effective “memory length”.13 For the sake of convenience we will suppose that f ( t ) = 0 and F ( t ) = 0 for t 5 -a and the system undergoes on an external force during period (-a, 0):

- a D ? f ( t ) = -r-”f(t) Note that in case a time.

#

+F(t),

> 0, -a < t < 00, O < < I.

(7) 1 the scale factor r does not mean a mean relaxation (Y

3. Numerical calculations

On the assumptions formulated above the solution of Eq (7) is represented in the form

/ t

f(t)=

-a

G(t - t’)F(t’)dt’,

341

where Green's function is expressed through two-parametric Mittag-Leffler functions

via relation13 G(t) = ta-lEa,a(-(t/r)a). Remember some simple cases of this function:

El,2(.) =

E2,2(Z) =

e" - 1 -1

Z

sinh(&)

&

*

For further details, see.l3?l4A more complicated Green function was used in.15 We compute the solution f ( t ) = U(t) of (7) for the special case F(t)=

{

0, t I -8, E, - e < t I o , 0, t > 0 ,

where 0 < 8 < a. A little bit conditionally, this case can be interpreted as a process of charging of a dielectric capacity by constant electrical field E during time 8 till t = 0, variable V(t) represents a voltage on the capacity as a function of time during charging (t < 0) and following discharging a real dielectric. Using Eqs (9)-(11) in (8) we get

Interchanging summation and integration and using the gamma function property yields

342

According to (9)

and finally we have for V ( t ;a , 6')

= U ( t ) / U ( O ) ,t > 0:

6')/71") v(t;a , e) = (t + 6')"-&,a+l(-[(t 6'" +E","+ 1 (-

-

l+(t)taE","+1(-(t/7)")

(6'/7)")

(12) Each value of parameter a E ( 0 , l ) produces a family of curves respective to different durations of charging 6' and only one value of a produces the only curve independently of 8, this is Q = 1. Really, using special formula (11) we get from (12):

In the last case the relaxation follows the Debye law and does not depend on 6': the memory is absent (Fig.1, left panel). However, computing (12) for Q = 0.8 and different charging durations 6' we obtain different relaxation curves. They are represented on the right panel of Fig.1 for charging durations 6' = 1.0 (the upper line), 0.6, and 0.3 (the lower line). It is clearly seen that the process at t > 0 depends on its prehistory at t < 0: we observe the memory effect. This results are in agreement with experimental data discussed in the a r t i ~ l e . ~ An amazing behavior of the solution is observed when Q is very close to 1 but not equal it: in this case the polarization falls according to Debye exponential law for a long time but after some crossover moment we observe splitting with respect to different values of 6' and transition to non-Debye power laws (Fig.2). It seems as if the memory returns to the system after some interval of time. Such behavior may be called the regeneration memory

effect. 4. Where come long waiting times from?

No doubt the key-assumption of this subdiffusional approach is an inverse power type distribution of random waiting (trapping) time T

p T ( t ) 0: t--l

, t --t

00, Q

E

(0,l).

For most physicist, it would be more easy to agree with an exponential distribution, and quantum mechanics confirms this: quasi-classical calculations really show the exponential distribution of trapping time for a particle

343

2

Fig. 1. Charging-discharging of a dielectric capacity. The left panel: a = 1; the right panel: a = 0.8 (0 = 1(1), 0.6(2), 0.3(3))

!2

Fig. 2. Transition Debye relaxation to non-Debye one at large times ( a = 0.99, 2, e = qi), ioo(z), iooo(3))

which can leave a potential hole through an energy barrier of hight tunneling: pT(tlE) = X(€)e-X(')t.

T

E

=

by

(13)

Here the jump rate X has the Arrhenius form

A(€) = vexp(-PE)

(14)

with P being an inverse proportional to the absolute temperature. The presence of hard acting environment in dense matters may shake this point of view, but where come the power law from? What is its physical cause? One could hope that a certain answer will be found from a strict theory taking into account all essential physical details of the process, but a direct

344

many-body calculation is extremely difficult to perform. Now, there exists a few approximation models demonstrating origin of inverse type distribution on some assumptions replacing strict proofs. We refer here to one of them called the random activation energy model (RAEM).16*19This model is based on relations (13)-(14) under additional assumption that the activation energy E is an exponentially distributed random variable: +(E)

= a!Pe-"PE.

In this case

A survey of reactions in condensed media with power type reaction rates connected with thermally Assisted Tunneling can be found in. l7 5. Summary

The following statements are worth to be stressed in the conclusion. 1. The idea of trapping of dipole moments carriers in solid dielectrics relaxation dates back to l?ro1ich18 2. The inverse power type relaxation is observed in experiments.2 3. The inverse power type asymptotical behavior of a waiting-hopping process on a sphere is possible only under condition p ~ ( t cx) t-"-' , t - + 00, a! E (0,1).10 4. The inverse power distribution of trapping time can be considered as a main cause for asymptotical behavior of relaxation to be described by fractional equation. This seems to be a more natural explanation then assumptions about existing fractal structures or even about of fractal time itself.

345

5. Finally, t h e regeneration memory effect can be considered as a n indication that one and t h e same material can reveal Debye and non-Debye relaxation in different time domains. Acknowledgment T h e authors thank t h e Russian Foundation for Basic Research (grant No. 07-01-00517).

References 1. A. K. Jonscher, Dielectric Relaxation in Solids, (Chelsea Dielectric Press,

London, 1983). 2. A. K. Jonscher, Universal Relaxation Law, (Chelsea Dielectric Press, London, 1996). 3. Non-Debye Relaxation in Condensed Matter, T. V. Ramakrishnan, M. Ray Lakshmi (Eds), (World Scientific, Singapore, 1987). 4. Relaxation in Complex System and Related Topics, I. A. Campbell, C. Giovannella (Eds), (Plenum Press, New York,1990). 5. Complex Behaviour of Glassy Systems, M. Rubi, C Perez-Vicente (Eds), (Springer Verlag, Heidelberg, 1997). 6. W. T. Coffey, J . Molecular Liquids 114,5 (2004). 7. M. J. Curie , Ann. d e chimie et de physique ser. 6, 18,203 (1889). 8. E. R. von Schweidler, Ann. der Physdk 24,711, (1907). 9. S. Westerlund, Physica Scripta 43,174 (1991). 10. V. V. Uchaikin, J . Exper. Theor. Physics 88,1155 (1999). 11. V. V. Uchaikin, V. M. Zolotarev, Chance and Stability. Stable Distributions and their Applications (VSP, Utrecht, the Netherlands, 1999). 12. V. Volterra, Theory of functionals and of integral and integro-differential equations (Dover Publications Inc., 1959). 13. I. Podlubny, Fractional Differential Equations (Academic Press, New York London, 1999). 14. R. Gorenflo, F. Mainardi, in: Fractals and Fractional Calculus in Continuum Mechanics, Carpinteri A, Mainardi F (eds) (Springer Verlag, Vienna - New York. 1997) 223. 15. V. V. Uchaikin, Intern. J . Theor. Phys. 42 121 (2003). 16. B. K. P. Scaife, Principles of Dielectrics (Oxford Univ. Press, London, 1998). 17. A. Plonka, J Phys Chem B 104,3804 (2000). 18. H. Frohlich. Theory of Dielectrics, (Clarendon Press, Oxford, 1958). 19. M. 0. Vlad, Physica A184,303(1992).

346

NODAL PATTERN ANALYSIS FOR CONDUCTIVITY OF QUANTUM RING IN MAGNETIC FIELD MITSUYOSHI TOMIYA', SHOICHI SAKAMOTO, MASAKI NISHIKAWA and YOSHIFUMI OHMACHI Department of Materials and Life Science, Seikei University, 3-3-1 Misashino-shi, Tokyo 180-8633, Japan 'E-mail: tomiyaOst.seikei.ac.jp The electron transport inside two-dimensional nanostructure is numerically studied, by the nodal pattern analysis of the wave functions. Especially the doubly connected quantum ring(QR) structure provides us interesting features under an external electro-magnetic field. Recent experimental techniques in nanostructures is now sophisticated enough, for example, to control the electron number in a two-dimensional system. Then one of the main interests on electron transports in nanodevices is to know characteristics of a magnetoresistance. El-om the nodal pattern analysis, it is found that every state cannot contribute to the electron transport, when the overall shape of the device is not near integrable. We also find the wavefunctions that stick to the inner wall of the QR under a weak magnetic field. Classically such strength of magnetic field makes the cyclotron radius near the radius of the inner hole of the device. It implies that there must be the noble relation between the ring size of the inner hole and the cyclotron radius of an electron motion and the electron transport is dependent on the relation. Consequently, the resistance is affected by the drive of an ac electro-magnetic field which is capable of exciting the electron to the higher eigenstate.

1. Introduction

It is known that production of the heterojunctions is getting increasingly promising and great flexibility. Lots of these efforts lead us from submicron t o more smaller sized structures. Then theoretical study for systems of single electron or a few electrons in two dimensions1>2has now urgent needs. On the other hand, the two-dimensional electron gas(2DEG) is still be considered as many electrons in the pm sized semiconductor devices. The pm sized devices already has shown many exotic features, especially under the electro-magnetic field, such as the Aharonov-Bohm(AB) effect,

347

Fig. 1. Schematic illustration of the device shapes. The symbols p and q represent the labels of the lead parts. The size of their rectangle frame is 47.9nm x 14.3nm. (a)nanowire model, (b)strait model, (c)quantum ring(QR) model. At dark parts, the potential is put very large value. The shape of a circle model is the same as the QR model without the inner hole in the middle.

the Altshuler-Aronov-Spivak (AAS) effect the quantum Hall effect, the Shubnikov-de Haas(SdH) oscillation and so on. All these properties are purely quantum mechanical phenomena under external magnetic field.3>4 Moreover, recently the experimental results under the microwave(MW) drive also have attracted considerable attenti~n.~-lO Recently the noble feature of the magnetoresistance of the 2DEG in the pm sized devices has been found by Prof. K. Fhjii's gr0up.l' Their specimen has a hole in the middle of the structure and we shall call it a quantum ring(QR)(fig.l). Thus the structure is doubly connected and makes the physical properties more sensitive especially against the magnetic field. The strong static magnetic field also induces the SdH oscillation in the specimen, however, they uniquely have the dip of the magnetoresistance in the intermediate strength range of the static magnetic field which is perpendicular to a two-dimensional structure, under a radiation of MW. The similar result has been reported in the case of simply connected and pm sized dot ~ a s e . ~ The - ~magnetoresistance has the oscillatory property in the smaller strength of a magnetic field than the range where the SdH oscillation is observed. It is also the oscillatory property of the magnetoresitance, if the condition

is satisfied. Here j corresponds to node of the oscillatory property, w is the frequency of the maicrowave and w c = e H / m c is the cyclotron frequency. Prof. Fujii's group has found the dip instead of the oscillation with respect to the strength of the magnetic field." It can be very easily expected that quantum mechanical study must be inevitable for much smaller systems, which is our main interest. Thus, to investigate the more precise properties of nano-sized devices, we should see the behavior of a single electron more closely inside the devices. Fortunately the progress of computers has been astonishing and very

348

intensive calculation that was beyond the reality a decade ago is now available. Quantum mechanical calculation consumes the computational resource tremendously more than classical calculation in general, however, even the direct study of the nodal pattern of the wavefunctions has been made possible lately.12 2. Simulation method

For the computational simulation, we have made the model of the device structure in the 900~270(=243,000)mesh points(fig.1). The shape of a nanowire is just a rectangle of this size. We put the lattice constant a=la.u.(=O.O529nm). Then, our model size of the wire turns out to be 47.9nm x 14.3nm. In this work, a.u. represents the Hartree atomic unit, i.e. me = e2 = fi = 1. We also take a strait, a QR and a circle shaped model in consideration(fig.1). The circle model has the same size as the QR model and no hole inside. They are all molded in the original rectangle area of the size of 900x270 meshes. Outside the area its potential V is given very large value numerically instead of the infinity. The conductance of the models g is evaluated by the Landauer formula13

where Tqpis the transmission function from the lead p to the lead q. The transmission function is derived from the matrix formed Green's function. Due to the discretization of the structure of the devices. By virtue of Data's m e t h ~ d the , ~ retarded Green's function can involve the contribution of the leads up to some degree as

G R = [ E I - 'H - CR]-',

(3)

where 'H is the Hamiltonian, and P

is the contribution from the connection between the leads and the device model. Here pi represents the point in the lead which is adjacent to point i in the device. Then we have

S,R(Pi,Pj)= 4

- l

c

xm(Pi)e~~(ik,a)X,(pj),

(5)

m

and t = fi2/2ma2, Xm is the m-th eigenfunction of the transverse mode in the lead, k, = J2m(E - &)/ti and + I , ,is the energy of xm. Finally the

349

Fisher-Lee relation makes the evaluation of the transmission function pos~ i b l e . ~we > lget ~ the transmission function from the matrix Green function:

Tqp= t$"GRI',GA], where the formula

and the relation urn = hkm/m is used. We can also calculate the wavefunctions inside the device. Thus the nodal patterns of the wavefunctions can be examined in detail, solving the Schrodinger equation . a@

2-

at

= 7-N,

and the Hamiltonian

'H= 1 2 (-iv+q)2+V.

(9)

We choose the vector potential A in the Landau gauge A = (0, xH,0) to have the static magnetic field H = (O,O, H ) . Note that the Hartree atomic unit is applied in this work. Outside the device models, we put V = lO3Oo that is almost as large as possible for the double precision variable. If it is necessary to simulate the situation of actual experimental condition when the longitudinal resistance is detected, then the scalar potential for the longitudinal electric field is added in the potential V. 3. Nodal pattern analysis

By the simulation on the mesh, the conductance can be enumerated from eq.(2) and eq.(6). We can check that the conductance or the transmission function is clearly quantized in idealistic cases such as the wire and the straight cases. The shape of the nanowire model and, at least, the narrow strait part of the strait model(fig.1) are just rectangular, and integrable in the classical mechanical point of view. On the contrary, the conductance of the circle dot model oscillates violently. The vibration pattern looks partly similar between the circle and the QR models. The sharp staircase disappears even without a magnetic field and represents just the actual upper limit of the conductance(fig.2). Thus it is the effect of the shape of the dots which have the wider part in the middle of the strait part. It

350

0

0.2

0.4

0.6

energy [eV] Fig. 2. Plots of transmission function(eq.(6)) of the nano-device: the strait, the circle and the QR model.

implies that the electron is refracted by the dot in the middle of the device, and it depends on the shape of the dot. In the case of the nanowire and the strait, they have clear nodal patterns that is typical property in integrable systems. They can be divided into the series that are defined by the number of the nodes in the transverse mode. It originates the staircase shape of the conductance(fig.3a,b). It steps up just at the energy where the base state of the new series appears. The base state has one more nodes in the transverse mode than the previous series and only one node in the longitudinal direction. On the contrary, the QR can not define such clear series of the wave functions. In QR, sometimes the wave functions have the regular-look nodal patterns in strait parts that bridge the leads and the dot in the middle. They also often have the irregular and even occasionally localized patterns(fig.3~). Of course, in general, they would not contribute to the conductivity much. Especially the localized wavefunctions, e.g., the lower left of f i g . 3 are ~ robust

351

Fig. 3. Nodal patterns of the wavefunction inside the nano-device. The square of the wavefunctions of (a)nanowire, (b)strait and (c)QR models are shown.

even after connecting the leads to the device12 and they would not be able to contribute to the conductivity. The circle is itself known integrable, though, the circle model has similar non-regular patterns, due to the chaotic property of whole shape of the device including the leads. Therefore the nodal pattern should have the key role in the physical properties of the devices. A static electro-magnetic field drastically changes the nodal patterns of the wavefunctions. Especially the pattern of the circle and QR models are closely investigated here(fig.4). We often can find the

Fig. 4. Nodal patterns of the wavefunctions inside the nano-devices with and without static electro-magnetic field. (a-1) Circle without EM field, (a-2) Circle with static EM field(E=O.OOla.u., H/c=O.OOJa.u.), (b-1) Circle without EM field, (b-2) Circle with static EM field(E=O.OOla.u., H/c=O.OOJa.u.). All sequences of the wavefunctions are from the 19th t o the 24th state.

352

Fig. 4.

(Continued)

regular circular shaped wavefunctions both in the circle and QR under B certain strength bf the magnetic field. Most of them seem to be attached to the outside wall. We also can discover the wavefunctions that stick to the inner waII of the QR. Moreover, we check that their appearance is very robust against the shape of the inner walls. They are similar to the edge states in the context of the Hall effect. Therefore we call the former states as the inner edge states and the latter states as the outer edge states.

353

4. Discussions

The dip in the magnetoresistance is a recent discovery of Prof. Fujii’s group. It seems to be mainly determined by they size of the radius of the inner hole. The strength of the magnetic field at the dip makes the cyclotron radius closed to the radius of the inner wall. On the other hand the averaged size of circular electron motion measured by the experiments of the AB effect tell us that the radius should be a bit larger. It means that the method detects only the electrons which really contribute to the conductivity and the measured radius of the circle that is just in the middle of the QR. Under the MW drive, some another quantum mechanical effect detects the size of the inner hole. It is not dependent on the frequency of MW, as far as it occurs. The existence of the inner edge states can explain this property of the dip under the MW drive. They can only detect the size of the inner hole, because the probability density of electrons concentrates in the close neighborhood of the inner wall. Thus this kind of wavefunctions do not contribute to the conductivity in general. Under the MW drive, the interaction between these wavefunction and the usual conducting wavefunction must be activated. The interaction would usually be scattering, and then the conductance would become smaller under the MW. The activation of the interaction with the inner edge state makes the longitudinal conductance, then the reistance also becomes proportionally smaller. The relation between two-dimensional longitudinal conductance oxxand resistance pij (2, j = z, y) leads rxx =

P I X

PZX

+ P&

‘

Then the change of the longitudinal resistance would be proportional to the change of the longitudinal conductance

under the condition pxx << ply, which is actually satisfied in Prof. Fujii’s specimen. Therefore the negative change of the longitudinal conductance makes the resistance also smaller from eq.(ll). Independently it has been reported that, under the perpendicular static magnetic field, small longitudinal electric field would lead the zero or small resistance.8 Correspondingly, it has been published that the 2DEG under the MW with Landau levels would have the zero resistance by the impurity scattering mechanism at faint longitudinal ~ u r r e n tRoughly .~ speaking, electrons can travel against the very small static longitudinal electric field

354

that is added to make the longitudinal current flow and to detect the longitudinal reistance, if the scattering occurs under some circumstance. It has been also reported that the grid of the asymmetric shaped dots(semi-circles) can lead the current under the MW drive and no longitudinal dc-electric field drive.1° Our nano-structure system is about a hundred times smaller than the Fujii’s specimen, however, the above discussion can be essentially considered as the single electron approximation. Therefore, the naive scaling property of quantum physical quantities is expected to be sufficiently available and, the scale of energy has t o be ten thousands times larger in our system. Then, this kind of resonant behavior should occur under an infrared radiation, instead of the MW range.

5. Conclusion It is shown that the nodal patterns of the wavefunctions for the twodimensional nanostructure show the detailed information of the physical properties. From the recent magnetoresistance experiments of pm sized twodimensional devices under the MW drive, it is predicted that the infrared radiation can induce the dip in the magnetoresistance of the nano-sized device under the weak static magnetic field by the naive scaling property.

References 1. Y . Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha, Phys. Rev. Lett.

96,047202 (2006). 2. H. Sano, A. Endo, S. Katsumoto and Y. Iye, J. Phys. SOC.Jpn. 76,094707 (2007). 3. S. Data, Electron Runsport in Mesoscopic System (Cambridge University Press, Cambridge, 1995). 4. D. K. Ferry and S. M. Goodnick 1997 Runsport in Nunostructures (Cambridge University Press, Cambridge, 1997). 5. M. A. Zudov, R. R. Du, J. A. Simmons and J. L. Reno, Phys. Rev. 64, 201311(R) (2001). 6. R. G. Mani, J. H. Smet, K. von Klintzing, V. Narayanamurti, W. B. Johnson and V. Umansky, NATURE 420, 646 (2002). 7. A. C. Durst, S. Sachdev, N. Read and S. M. Girvin, Phys. Rev. Lett. 91, 086803 (2003). 8. A. V. Andreev, I. L. Aleiner and A. J. Millis, Phys. Rev. Lett. 91,056803 (2003). 9. M. G. Vavilov and I. L. Aleiner, Phys. Rev. B69,0353030 (2004). 10. A. D. Chepelianskii, D. L. Shepelyansky, Phys. Rev. B71,052508 (2005). 11. K. Fujii, private communications, (2006); A. Yagara, master thesis, Osaka university (2006).

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12. J. P. Bird, R. Akis, D. K. Ferry, D. Vasuleska, J. Cooper, Y. Aoyagi and T. Sugano, Phys. Rev. Lett. 82,4691 (1999); R. Akis, J. P. Bird and D. K. Ferry, Appl. Phys. Lett. 81, 129(2002); J. Bird, Rep. Prog. Phys., 66,583 (2003). 13. M. Buttiker, Y. Imry, R. Landauer and S. Pinhas, Phys. Rev. B31 6207 (1985). 14. D. S. Fisher and P. A. Lee, Phys. Rev. B23 6851 (1981).

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APPLICATION OF THE GENERALIZED ALIGNMENT INDEX (GALI) METHOD TO THE DYNAMICS OF MULTI-DIMENSIONAL SYMPLECTIC MAPS T. MANOSa9bi*, Ch. SKOKOS‘ and T. BOUNTISa a Center for

Research and Applications of Nonlinear Systems ( C R A N S ) , Department of Mathematics, University of Patms, GR-26500, Pat% Greece. Observatoire Astronomique de Marseille-Provence ( O A M P ) , 2 Place Le Verrier, 13248, Marseille, France. Astronomie et SystBmes Dynamiques, IMCCE, Observatoire de Paris, 77 Av. Denfert-Rochereau, F-75014, Paris, France. * E-mail: thanosmOmaster.math.upatras.gr (T. Manos) We study the phase space dynamics of multi-dimensional symplectic maps, using the method of the Generalized Alignment Index (GALI). In particular, we investigate the behavior of the GALI for a system of N = 3 coupled standard maps and show that it provides an efficient criterion for rapidly distinguishing between regular and chaotic motion. Keywords: Symplectic maps, Chaotic motion, Regular motion, GALI method.

1. Introduction The distinction between regular and chaotic motion in conservative dynamical systems is fundamental in many areas of applied sciences. This distinction is particularly difficult in systems with many degrees of freedom, basically because it is not feasible to visualize their phase space. Thus, we need fast and accurate tools to obtain information about the chaotic vs. regular nature of the orbits of such systems and characterize efficiently large domains in their phase space as ordered, chaotic, or “sticky” (which lie between order and chaos). In this paper we focus our attention on the method of the Generalized ALignment Index (GALI), which was recently introduced and applied successfully for the distinction between regular and chaotic motion in Hamiltonian systems.l The GALI method is a generalization of the Smaller Alignment Index (SALI) technique of chaos detection.24 We present some preliminary results of the application of GALIs on the dynamical study

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of symplectic maps, considering in particular the case of a 6-dimensional (6D) system of three coupled standard maps.5 It is important to note that maps of this type have been extensively studied in connection with the problem of the stability of hadron beams in high energy accelerators, see6 and references therein. Our numerical results reported here verify the theoretically predicted behavior of GALIs obtained in' for Hamiltonian systems. In addition we study in more detail the behavior of chaotic orbits which visit different regions of chaoticity in the phase space of our system. 2. Definition and behavior of GALI Let us first briefly recall the definition of GALI and its behavior for regular and chaotic motion, adjusting the results obtained in' to symplectic maps. Considering a 2N-dimensional map, we follow the evolution of an orbit (using the equations of the map) together with k initially linearly independent deviation vectors of this orbit $1, $2, ..., 4 Y k with 2 5 k 5 2N (using the equations of the tangent map). The Generalized ALignment Index of order k is defined as the norm of the wedge or exterior product of the k unit deviation vectors: GALIk(n) =I1 fil(n) A fi2(n) A

... A &(n) 11

(1)

and corresponds to the volume of the generalized parallelepiped, whose edges are these k vectors. We note that the hat (A) over a vector denotes that it is of unit magnitude and that n is the discrete time. In the case of a chaotic orbit all deviation vectors tend to become linearly dependent, aligning in the direction of the eigenvector which corresponds to the maximal Lyapunov exponent and GALIk tends to zero exponentially following the law:' GALIk(n) 0: e -[(ul-uZ)+(ul-u3)+...+(u1-uk)]7L 1

(2)

where 01,. . . , (Tk are approximations of the first k largest Lyapunov exponents. In the case of regular motion on the other hand, all deviation vectors tend to fall on the N-dimensional tangent space of the torus on which the motion lies. Thus, if we start with k 5 N general deviation vectors they will remain linearly independent on the N-dimensional tangent space of the torus, since there is no particular reason for them to become aligned. As a consequence remains practically constant for k 5 N . On the other hand, GALIk tends to zero for k > N , since some deviation vectors

358

will eventually become linearly dependent, following a power law which depends on the dimensionality of the torus on which the motion lies and on the number m ( m 5 N and m 5 k) of deviation vectors initially tangent to this torus. So, the behavior of GALIk for regular orbits is given by1l7 constant if 2 5 k 5 N if N < k 5 2N and 0 5 m < k - N . if N < k 5 2N and m 2 k - N

(3)

3. Dynamical study of a 6D standard map

As a model for our study we consider the 6D map:

+ 2; + 6 sin(2.rrxl) &{sin[2.rr(xg - X I ) ] + sin[2.rr(x3- x1)]} + x& xi = 2 4 + 6 sin(2.rrx3) &{sin[2.rr(xl - x3)] + sin[2.rr(x5- xs)]} x; = 5 5 + x& x& = X6 + 6 sin(2.rrx5)- &{sin[2.rr(xl xg)] + sin[2.rr(x3- xs)]} x'1 = 21

xk

= x2 x'3 = 23

-

-

(4)

-

which consists of three coupled standard maps5 and is a typical nonlinear system, in which regions of chaotic and quasi-periodic dynamics are found t o coexist. Note that each coordinate is given modulo 1 and that in our study we fix the parameters of the map (4) to K = 3 and p = 0.1. In order to verify numerically the validity of equations (2) and (3), we shall consider two typical orbits of map (4), a chaotic one with initial condition x1 = x3 = x5 = 0.8, 2 2 = 0.05, x4 = 0.21, 2 6 = 0.01 (orbit C l ) and a regular one with initial condition X I = x3 = 2 5 = 0.55, x2 = 0.05, x4 = 0.01, 2 6 = 0 (orbit R l ) . In figure 1 we see the evolution of GALIk, k = 2 , . . . ,6, for these two orbits. It is well-known that in the case of symplectic maps the Lyapunov exponents are ordered in pairs of opposite signs8 Thus, for a chaotic orbit of the 6D map (4) we have u1 = - 0 6 , uz = -u5, u3 = -u4 with 01 2 oz L u3 2 0. So for the evolution of GALIk, equation (2) gives GALIz(t) 0: e - ( " ' - n ~ ) t , GAL13(t) 0; e-(2u1-u2-~3)t, GALI4(t) 0; e-(3u1-uz)t, GALI5(t) 0: e-4u1t, GALIG(t) cx ec6"lt.

(5)

The positive Lyapunov exponents of the chaotic orbit C1 were found to be 01 M 0.70, 02 M 0.57, u3 M 0.32. From the results of figure l a ) we see that the functions of equation (5) for u1 = 0.70, 02 = 0.57, u3 = 0.32 approximate quite accurately the computed values of GALIs.

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-4

-12

-16

1

n

2

4

3

5

6

Log@)

Fig. 1. The evolution of GALIk, k = 2 , . . . , 6 , with respect to the number of iteration n for a) the chaotic orbit C1 and b) the regular orbit R1. The plotted lines correspond to functions proportional to e - ( U l - " Z ) t ~ - ( ~ U I - U Z - U ~e )- (~3 u i - u z ) t e-4u1t e-6u1t for ui = 0.70, uz = 0.57, u3 = 0.32 in a) and proportional to n - 2 , n - 4 , n-6 in b).

For the regular orbit R1 we first considered the general case where no initial deviation vector is tangent to the torus where the orbit lies. Thus, for the behavior of GALIk, Ic = 2 , . . . , 6 , equation (3) yields for m = 0

GALI2(t) 0: constant, GALIs(t) 0: constant, GALI4(t) 0: $, GALI6(t) 0: $, GALI6(t) 0: $.

(6)

From the results of figure l b ) we see that the approximations appearing in (6) describe very well the evolution of GALIs. In order to verify the validity of equation (3) for 1 5 m 5 3, in the case of regular motion, we evolve orbit R1 and three random initial deviation vectors for a large number of iterations (in our case for 5 x l o 7 iterations), in order for the three deviation vectors to fall on the tangent space of the torus. Considering the current coordinates of the orbit as initial conditions and using m = 1or m = 2 or m = 3 of these vectors (that lie on the tangent space of the torus) as initial deviation vectors we start the computation of GALIs' evolution. We note that the rest 6 - m initial deviation vectors needed for our computation are randomly generated so that they do not lie on the tangent space of the torus. The results of these calculations are presented in figure 2, where the evolution of GALIk, Ic = 2 , . . . , 6 , for different values of m is plotted. Figure 2 clearly illustrates that equation (3) describes accurately the behavior of GALIs for regular motion also in the case where some of the initial deviation vectors are chosen in the tangent space of the torus.

360

0

-

0

-4

-4 10 -

m

ae

i

6 (3

z

-8 _I

-8

_I

-12

-12

-16

-16 1

2

3

4

5

6

7

I

2

3

4

Log(n)

5

6

7

Log(n)

0

-4

-8

-12

-1fi

1

2

3

4

5

6

7

Log@)

Fig. 2. Evolution of GALIk, k = 2, . . . , 6 , for the regular orbit R1 on a log-log scale, for different values of the number m of deviation vectors initially tangent on the torus on which the motion occurs: a) m = 1,b) m = 2, c)m = 3. In every panel lines corresponding to particular power laws are also plotted.

Let us now consider the case of a chaotic orbit which visits different regions of chaoticity in the phase space of the map. The orbit with initial conditions z1 = z3 = z5 = 0.55, 22 = 0.05, 2 4 = 0.21, z6 = 0.0 (orbit C2) exhibits this behavior as can be seen from the projections of its first 1000 successive consequents on different 2-dimensional planes plotted in figure 3. The projections look erratic, indicating that the orbit is chaotic. However, we also observe in all three projections that the C2 stays ‘trapped’ for many iterations in two oval-shaped regions and eventually escapes entering the big chaotic sea around these regions. This behavior is also

361

0,O

0.2

0.4

0.6

0.8

1.0

x5

Fig. 3. Projections on the planes a) ( 5 1 , I C Z ) , b) successive points of the chaotic orbit C2.

(13,24) and

c)

( 3 5 , ~ of ~ )the

first 1000

depicted in the evolution of the Lyapunov exponents of the orbit (see figure 4a). The three positive Lyapunov exponents are seen to fluctuate around 0 1 M 0.033, n2 M 0.02, 0 3 M 0.005 for about 1000 iterations exhibiting 'jumps' to higher values when the orbit enters the big chaotic sea, stabilizing around 0 1 M 0.793, n~ M 0.624, 0 3 M 0.365. Let us now study how the GALIs are influenced by the fact that orbit C2 visits two different regions of chaoticity characterized by different values of Lyapunov exponents. Since C2 is a chaotic orbit, its GALIs should tend exponentially to zero following the laws of equation (5). Thus, starting the computation of the GALIk, k = 2 , . . . , 6, from an initial point of C2 located in the first chaotic sea, we see that the slopes of the exponential decay of GALIs are well described by equation (5) using for 0 1 , 0 2 , 0 3 the

362 8

0,o 4 4 5

-1 ,o

0

?-. 0-

-3 cn

-1,s

4

-2.0

-8

-2,5

-12

-16 0

200

400

600

800

n

Fig. 4. a) The evolution of the three positive Lyapunov exponents of the chaotic orbit C2. The evolution of GALIk, k = 2,. . . ,6, with respect t o the number of iteration n for the same orbit when we use as initial condition of the orbit its coordinates at b) n = 0 and at c) n = lo6 iterations. The plotted lines in b) and c) correspond to e-(zOl-"-Q)t e - ( 3 u ~ - W ) t e - 4 u 1 t , e-6u1t for functions proportional to e-('1-u2)t, 01 = 0.033, uz = 0.02, u3 = 0.005 in b) and 01 = 0.793, u~ = 0.624, u3 = 0.365 in c).

approximate values of the Lyapunov exponents of the small chaotic region (figure 4b). On the other hand, using as initial condition for this chaotic orbit its coordinates after lo6 iterations, when the orbit has escaped in the second chaotic region, the evolution of GALIs is again well approximated by equation ( 5 ) but this time for ~1 = 0.793, ~2 = 0 . 6 2 4 , ~=~0.365, which are the approximations of the Lyapunov exponents of the big chaotic sea (figure 4c). Thus, we see that in this case also the C J ~ ,i = 1,.. . ,k which appear in equation ( 5 ) are good approximations of the first k Lyapunov

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exponents of the large chaotic region in which the orbit eventually wanders after about lo3 iterations. 4. Conclusions

In this paper we verified the theoretically predicted behavior of the Generalized Alignment Index (GALI) by considering some particular regular and chaotic orbits of a 2N-dimensional symplectic map with N = 3. In particular, we showed numerically that all GALIk, 2 I k I 2N, tend to zero exponentially for chaotic orbits, while for regular orbits they remain different from zero for 2 5 k 5 N and tend to zero, following particular power laws, for N < lc 5 2N. Thus, by using GALII, with sufficiently large lc, one can infer quickly the nature of the dynamics much faster than it is possible by using other methods. Also, the study of chaotic orbits which visit different regions of chaoticity in the phase space of the system, provides further evidence that the exponents of the exponential decay of GALIs are related to the local values of Lyapunov exponents. Thus, we have shown that the different behaviors of the GALIs for regular and chaotic orbits can be used for the fast and accurate identification of regions of chaoticity and regularity in the phase space of symplectic maps. We plan t o investigate this further in a future publication concerning maps with N >> 3, which describe arrays of conservative nonlinear oscillators.

Acknowledgments T. Manos was partially supported by the “Karatheodory” graduate student fellowship No B395 of the University of Patras, the program “Pythagoras 11” and the Marie Curie fellowship No HPMT-CT-2001-00338. Ch. Skokos was supported by the Marie Curie Intra-European Fellowship No MEIFCT-2006-025678. The first author (T. M.) would also like to express his gratitude t o the Institut de M6canique Celeste et de Calcul des Ephkmerides (IMCCE) of the Observatoire de Paris for its excellent hospitality during his visit in June 2006, when part of this work was completed. References 1. Ch. Skokos, T. Bountis and Ch. Antonopoulos, Physica D, 231, 30, (2007). 2. Ch. Skokos, J . Phys. A : Math. Gen., 34, 10029, (2001). 3. Ch. Skokos, Ch. Antonopoulos, T. Bountis and M. Vrahatis, Prog. Theor. Phys. Suppl., 150, 439, (2003).

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4. Ch. Skokos, Ch. Antonopoulos, T. Bountis and M. Vrahatis, J . Phys. A , 37, 6269, (2004). 5. H. Kantz and P. Grassberger, J. Phys. A : Math. Gen, 21 L127, (1988). 6. T. Bountis and Ch. Skokos, Nucl. Instr. Meth. Phys. Res. A, 561,173, (2006). 7. H. Christodoulidi and T. Bountis, ROMAI Journal, 2, 2, (2007). 8. M. A. Lieberman and A. J. Lictenberg, Regular and Chaotic Dynamics, Springer-Verlag, (1992).

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