Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media

NONLINEAR PHYSICAL SCIENCE

NONLINEAR PHYSICAL SCIENCE INonlinear Physical Science focuses on recent advances offundamental theories an~ iJnnciples, analytical and symbolic approaches, as well as computatiOnal techmque§ lin nonlinear physical science and nonlinear mathematics with engineering applica-I Ifiill:iSJ

[I'opiCS of mterest m Nonfznear PhysIcal SCIence mclude but are not limIted toj New findings and discoveries in nonlinear physics and mathematic§ Nonlinearity, complexity and mathematical structures in nonlinear physic§ Nonlinear phenomena and observations m nature and engmeenngj ComputatiOnal methods and theones in complex system§ LIe group analysIs, new theones and pnncipies m mathematical modelingj StabIlity, bdurcation, chaos and fractals m phySIcal SCIence and engmeenngj Nonlinear chemIcal and biOlogIcal phYSIC§ Discontmmty, synchromzation and natural complexIty m the phySIcal sCIence§

SERIES EDITORS ~Ibert

c.J. Luo

pepartment of Mechamcal and Industnal !=\ngmeermg ISouthern IllmOis University EdwardSVille !=\dwardsville, IL 62026- 1805, USA Email: [email protected]

Nail H. Ibragimov Department of MathematIcs and SClencel Blekmge InstItute of 'Iechnologyl S-371 79 Karlskfona, Swedeiil Email: [email protected]

IINTERNATIONAL ADVISORY BOARD rmg Ao, Umversily of Washmgton, USA; Email: [email protected] cran Awrejcewicz, The Technical University of Lodz, Poland; Email: [email protected]! ~ugene ~shel

Bemlov, UmvefSlty of Limenck, Ireland; Emml: [email protected]

Ben-Jacob, lel AVIV UmvefSlty,Israel; Emml: [email protected]!

,"aunce Courbage, UmvefSlte Pans 7, France; Email:

[email protected]~

,"anan Gldea, Northeastern IlhnOis UmvefSlty, USA; Emml: [email protected] crames A. Glazier, Indiana University, USA; Email: [email protected] ~hlJnn

L130, Shanghm Jlaotong Umversily, Chma; Email: s]hao@s]tu.edu.clJl

croseAntonio Tenreiro Machado, ISEP-Institute of Engineering of Porto, Portugal; Email: [email protected] "hkolaI A. Magmtskn, RUSSian Academy of SCiences, RussJa; Emml: [email protected] goser J. Masdemont, Umversltat Pohtecmca de Catalunya (UPe), Spam; Email: [email protected] pmitry E. Pelinovsky, McMaster University, Canada; Email: [email protected] ~ergey

Prants, V.I.Il'lChev PaCificOceanologlcal Institute of the RussJan Academy of SCiences, Russlaj

IOmail: [email protected] IVlctor r Shnra, Keele Umverslty, UK; Email: [email protected] crian Qiao Sun, University of California, USA; Email: [email protected] ~bdul-MaJld

Wazwaz, Samt Xavier Umverslty, USA; Email: [email protected]

rei Yu, Ihe Umversily of Western Ontano, Canada; Emml: [email protected]

Vasily E. Tarasov

Fractional Dynamics Applications of Fractional Calculus to Dynamics of Particles, Fields and Media

It ~ 1li..if 'J: fit. if1. •~t * ..

HIGHER EDUCATION PRESS

BEIJING

~ Springer

Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics Moscow State University 119992 Moscow, RUSSia Email: [email protected]

ISSN 1867-8440 Nonlmear Physical SCience

e-ISSN 1867-8459

ISBN 978-7-04-029473-6 Higher Education Press, Beijing e-ISBN 978-1-042-14001-7 ISBN 97R-1-042-14002-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010928902

© Higher Education

Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

This work IS subJect to copynght. All nghts are reserved, whether the whole or part of the matenal IS concerned, speCifically the nghts of translation, reprmtmg, reuse of IllustratIOns, reCitatIOn, broadcastmg, reproductIOn on microfilm or m any other way, and storage m data banks. Duplication of this publication or parts thereof IS permitted only under the proVISIOns of the German Copynght Law of September 9, 1965, m ItS current verSIOn, and permiSSIOn for use must always be obtamed from Spnnger. VIOlatIOns are liable to prosecution under the German Copynght Law. The use of general descnptlve names, registered names, trademarks, etc. m thiS publicatIOn does not Imply, even m the absence of a speCific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Fndo Stemen-Broo, EStudlo Calamar, Spam Pnnted on aCld-free paper Spnnger ISpart of Spnnger SClence+Busmess Media (www.sprmger.com)

IPreface

fractional calculus is a theory of integrals and derivatives of any arbitrary real (of ~omplex) order. It has a long history from 30 September 1695, when the derivativcj pf order a = 172 was mentioned by Leibniz. The fractional differentiation and frac~ ~IOnal mtegration go back to many great mathematiCians such as Leibmz, LIOuvilleJ prtinwald, Letmkov, Riemann, Abel, Riesz and WeyI. The mtegrals and denvative~ pf non-mteger order, and the fractIOnal mtegro-differential equatIOns have foun~ Imany apphcations m recent studies m theoretical physics, mechamcs and apphe~ Imathematics. W'Jew possibihties m mathematics and theoretical physics appear, when the orderl ~ of the differential operator DC: or the integral operator I~ becomes an arbitrarYI Iparameter. The fractIOnal calculus is a powerful tool to descnbe physical system~ ~hat have long-term memory and long-range spatial mteractIOns. In general, manYI lusual properties of the ordmary (first-order) denvative D x are not reahzed for fracj ~ional derivative operators DC:. For example, a product rule, chain rule and semH group property have strongly complicated analogs for the operators DC: 1 ~ost of the processes associated With complex systems have nonlocal dynamic~ landit can be charactenzed by long-term memory m time. The fractIOnal mtegratIOili landfractIOnal differentiatIOn operators allow one to conSider some of those charac-I ~enstics. Osmg fractIOnal calculus, it is pOSSible to obtam useful dynamical mod-I ~ls, where fractIOnal mtegro-dlfterential operators m the time and space vanablesl ~escnbe the long-term memory and nonlocal spatial properties of the complex me1 ~ia and processes. We should note that close connectIOns eXist between fractIOna~ khfferential and mtegral equatIOns, and the dynamiCs of many complex systemsJ lanomalous processes and fractal medial [I'here are many mterestmg books about fractIOnalcalculus, fractIOnal dlfterentia~ ~quatIOns, and their phYSical apphcatIOns. The first book dedicated speCifically tq ~he theory of fractIOnal mtegrals and denvatives, is the one by Oldham and Spame~ Ipubhshed in 1974. There exists the remarkably comprehenSive encyclopedic-typel monograph by Samko, Kilbus and Marichev, which was published in Russian inl ~ 987 and in English in 1993. The works devoted substantially to fractional differ-I ~ntial equations are the book by Miller and Ross (1993), and the book by Podlubn)j

Ivi

Preface

1(1999). In 2006, KIlbas, Snvastava and TrujIllo publIshed a very Important and rej rtarkable book, in which one can find a modem encyclopedic, detailed and rigorj pus theory of fractional differential equations. This book can be recommended a§ la mam modem mathematIcal handbook for graduate students and researchers, whol laremterested m thIS subject. There eXIstsome mathematIcal monographs devoted tg Ispecial questions of fractional calculus, including the book by McBride (1979), thel Iwork by KIryakova (1993), the monograph by Rubm (1996) and the volume edIted! Iby Srivastava and Owa (1989). The physical applications of fractional calculus t9 ~escnbe complex medIa and processes were conSIdered m the very mterestmg voIj lumes edIted by Carpmtery and Mamardl m 1997 and by HIlfer m 2000. The book byl IWest, Bologna, and GngolIm publIshed m 2003 IS devoted to phySIcal applIcatIoIlj pf fractional calculus to fractal processes. The first book devoted exclusively to thel [ractional dynamics and application of fractional calculus to chaos is the one by Zaj Islavskypublished in 2005. One of the most recent books on the subject of fractiona~ ~alculus IS the edIted volume of SabatIer, Agrawal and Tenrelro Machado publIshed! 1m 2007. In 2010, the book by Mamardl WIll be publIshed to devote to applIcatIon~ pf fractIOnal calculus m dynamICS of VIscoelastIc matenals. Note that there are mter1 ratIOnal Journals such as "Journal of FractIOnal Calculus" and "FractIOnal Calculu~ land ApplIed AnalYSIS", whIch are dedIcated entIrely to the fractIOnal calculusl [I'he content of the noted books and edIted volumes about applIcatIons of fracj ~IOnal calculus m phYSICS, mechamcs and applIed mathematIcs does not mclude alII pf modem fractIOnal theoretIcal models, methods and approaches. A lot of new re1 IsuIts, obtamed recently m the fractIOnal dynamICS, are not reflected m the booksl ~n thIS monograph, some modem applIcatIOns of fractIOnal calculus to compleX! IphysIcal systems and new results of last years are descnbed. Therefore the bookl lIS supposed to be useful for phYSICIStS and mathematIcIans, who are mterested ml ~he modem theones of complex processes and medIa. Some of mterestmg su(iject§ 1m the theoretIcal phYSICS are not descnbed m thIS monograph, smce It IS not POS1 ISIble to realIze m one book a complete descnptIOn of all fractIOnal dynamICS. Fo~ ~xample, the applIcatIOns of fractIOnal calculus to the VIscoelastIc medIa and thel ~ontmuous tIme random walk processes are not conSIdered m the book, smce then~ lare mterestmg monographs and reviews, where these su(ijects are dIscussed] [The text IS self-contamed and can be used WIthout prevIOUS courses m fractIOna~ ~alculus and theory of fractals. The necessary mformatIOn, whIch IS beyond to un1 ~ergraduate courses of the mathematIcs, IS suggested m the book. Therefore thI~ Ibook can be used m the courses for graduate students. In the book the modem apj Iproaches and new fundamental results of last years are descnbed. Therefore thel Imanuscnpt IS supposed to be useful for phYSICIStS and mathematIcIans who are m1 ~erested m the electrodynamICs, statIstIcal and condensed matter phYSICS, quantuml ~ynamIcs, complex medIa theones and kmetIcs, dIscrete maps and cham models] ronlmear dynamICS and chaos] [I'he book conSIsts of five parts. The first part IS devoted the fractIOnal contmu-I pus models of fractal distributions of particles. The fractional integral equations arel lused to descnbe fractal dIstnbutIOns of mass, charge and probabIlIty. In the second! Ipart, we conSIder the fractIOnal dynamICS that descnbes the medIa WIth long-rang~

Preface

viii

linteraction of particles. The close connection of discrete models with long-rang~ linteractions and continuous medium equations with fractional derivatives is provedj [The fractional coordinate derivatives are used to describe nonlocal properties of thel ~omplex media. In the third part, we suggest the tractional vector calculus, tractiona~ ~xtenor calculus, and tractional vanation calculus to descnbe generahzed dynami-I ~al systems, fractional statistical mechanics and kinetics, fractional electrodynamic§ pf complex media. The suggested generalizations of vector operations and variationsl lareconsidered with respect to coordinate variables. In the fourth part, we describ~ ~he tractional temporal dynamiCs, where denvatives With respect to time vanablel Ihave non-mteger orders. The nonholonomic systems With generahzed constramts t9 ~escnbe a long-term memory are conSidered. The electrodynamics of dielectnc mej klia is described as a fractional temporal electrodynamics. The discrete maps withl ~emory are obtained from the fractional differential equations of kicked dynam-I lical systems. In the fifth part, we conSider an apphcatiOn of tractiOnal denvative~ 1m quantum dynamiCs. These denvatives are defined as tractional powers of selfj ladjomt denvatives. A tractiOnal generahzatiOn of quantum MarkOVian dynamiCs iil Isuggested. The quantizatiOn of different fractiOnal denvatives and fractal functiOn~ lare suggested. The numerous recent pubhcatiOns Cited are hsted m the references a~ ~he end of each chapter.1 [I'he author is greatly mdebted to professor George M. ZaslavskY for hiS mvalu-I lable diSCUSSiOns and comments. The author Wishes to express hiS gratitude to pro-I ~essor Albert C.l. Luo for hiS support of the editiOn of thiS book. The author would! ~ike to express hiS appreCiatiOn for the kmd hospitahty by the Courant Institute o~ ~athematical Studies of New York Umversity dunng hiS visits m 2005, 2007, 200~ land2009 IVasl1y E. Tarasovl 112 October, 2009, Moscowl

Contents

IPart I Fractional (:ontinuous Models ofl

IFractal I)istributions ~

~

Fractional Integration and Fractals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 RIemann-LiOuvIlle fractional integrals . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Liouville fractional integrals 11.3 RIesz fractiOnal integrals 11.4 Metnc and measure spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Hausdorff measure 11.6 Hausdorff dimension and fractals. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11.7 Box-counting dimension 11.8 Mass dImenSiOn of fractal systems. . . . . . . . . . . . . . . . . . . . . . . . . . .. 11.9 Elementary models of fractal dIstnbutiOns . . . . . . . . . . . . . . . . . . . . .. 11.10 Functions and integrals on fractals 11.11 Properties of integrals on fractals 11.12 Integration over non-integer-dimensional space. . . . . . . . . . . . . . . . .. 11.13 Multl-vanable integratiOn on fractals 11.14 Mass dIstnbutiOn on fractals 11.15 DenSIty of states in Euchdean space 11.16 Fractional integral and measure on the real axis 11.17 FractiOnal integral and mass on the real aXIS .. . . . . . . . . . . . . . . . . .. 11.18 Mass of fractal media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11.19 Electnc charge of fractal dIstnbutiOn 11.20 Probablhty on fractals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11.21 Fractal dIstnbutiOn of partIcles IReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

31

41

G 71 ~

10 141 161 l~

2q 221 25] 261 2~

29 311 321 341 361 3~ 3~

411 441

Hydrodynamics of Fractal Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42 12 1 Introduction 4<] 12.2 Equation of balance of mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5q

Ix

( :ontentil

12.3 Total time denvatIve of fractIOnal Integral. . . . . . . . . . . . . . . . . . . . .. 12.4 EquatIOn of contInUIty for fractal media. . . . . . . . . . . . . . . . . . . . . . .. 12.5 FractIOnal Integral equatIOn of balance of momentum 12.6 Differential equations of balance of momentum . . . . . . . . . . . . . . . .. 12.7 Fractional integral equation of balance of energy 12.8 DIfferential equatIOn of balance of energy. . . . . . . . . . . . . . . . . . . . .. 12.9 Euler's equatIOns for fractal medIa. . . . . . . . . . . . . . . . . . . . . . . . . . .. 12.10 NavIer-Stokes equatIOns for fractal medIa . . . . . . . . . . . . . . . . . . . . .. 12.11 Equilibrium equation for fractal media. . . . . . . . . . . . . . . . . . . . . . . .. 12.12 Bernoulli integral forfractal media 12 13 Sound waves in fractal media 12.14 One-dImenSIOnalwave equatIOn In fractal medIa 12 15 ConclUSIOn IReferences ~

~

Fractal Rigid Body Dynamics t3 1 Introduction [3.2 FractIOnal equation for moment of Inertia [3.3 Moment of inertia of fractal rigid body ball. . . . . . . . . . . . . . . . . . . .. [3.4 Moment of InertIa for fractal ngId body cylInder [3.5 EquatIOns of motion for fractal ngId body. . . . . . . . . . . . . . . . . . . . .. [3.6 Pendulum WIth fractal ngId body [3.7 Fractal ngId body rollIng down an InclIned plane. . . . . . . . . . . . . . .. 3.8 Conclusion................................................ IReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Electrodynamics ofl IFractalDistributions of Charges and Fields . . . . . . . . . . . . . . . . . . . . . .. 14 1 Introduction 14.2 E1ectnc charge of fractal dIstnbutIOn 14.3 E1ectnc current for fractal dIstnbutIOn .. . . . . . . . . . . . . . . . . . . . . . .. 144 Gauss' theorem for fractal distribution 145 Stokes' theorem for fractal distribution 14.6 Charge conservation for fractal distribution. . . . . . . . . . . . . . . . . . . .. 14.7 Coulomb's and BIOt-Savart laws for fractal dIstnbutIOn . . . . . . . . . .. 14.8 Gauss' law for fractal dIstnbutIOn . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14.9 Ampere's law for fractal dIstnbutIOn. . . . . . . . . . . . . . . . . . . . . . . . . .. 14.10 Integral Maxwell equatIOns for fractal dIstnbutIOn . . . . . . . . . . . . . .. 14.11 Fractal dIstnbutIOn as an effective medIUm 14.12 Electnc multIpole expanSIOn for fractal dIstnbutIOn 14.13 Electnc dIpole moment of fractal dIstnbutIOn 14.14 Electnc quadrupole moment of fractal dIstnbutIOn 14.15 Magnetohydrodynamics of fractal distribution 14.16 Stationary states in magnetohydrodynamics of fractall Idistributions

511 541 5~

561 57] 5§ 6q 621 6~

641 661 67] 69 6<] 7~

71 741 761 7§ 811 821 841 851 861

8Q 8<] 9q 921 93 93 941 9S 961 97] 9~

100 1011 1O~

1041 107] 110

xii

Contents

~

14 17 Conclusion IReferences

1111 1121

Ginzburg-Landau Equation for Fractal Media

11=1 11 S 1161 117] 11§ 1211 1211

15 1 Introduction 15.2 FractIOnal generalIzatIOn of free energy functIOnal 15.3 Ginzburg-Landau equation from free energy functional 15.4 Fractional equations from variational equation 15 5 Conclusion IReferences ~

r7

Fokker-Planck Equation for Fractall IDistributions of Probability

12~

15 1 Introduction !'l.2 FractIOnal equatIOn for average values ~.3 Fractional Chapman-Kolmogorov equation ~.4 Fokker-Planck equation for fractal distribution ~.5 Stationary solutions of generalized Fokker-Planck equation 15 6 Conclusion IReferences

123 1241 12=1 127] 13q 1321 1321

Statistical Mechanics ofl IFractal Phase Space Distributions

13~

17 1 Introduction 13S 17.2 Fractal distribution in phase space 1361 17.3 Fractional phase volume for configuration space 1361 17.4 FractIOnal phase volume for phase space 1351 17.5 FractIOnal generalIzatIOn of normalIzatIOn condItIon 13~ 17.6 ContIllmty equatIOn for fractal dIstrIbutIOn III configuratIOn space .. 1411 17.7 ContIllmty equatIOn for fractal dIstrIbutIOn III phase space 1421 17.8 FractIOnal average values for configuratIOn space 1441 17.9 Fractional average values for phase space 14=1 17.10 Generalized Liouville equation 1461 17 11 Reduced distribution functions 1471 17 12 Conclusion 148 IReferences 150

rart II Fractional Dynamics and] ILong-Range InteractionS] ~

Fractional Dynamics of1 IMedia with Long-Range Interaction 81 ~.2 ~.3 ~.4

Introduction Equations of lattice vibrations and dispersion law Equations of motion for interacting particles Transform operation for discrete models

15~

153 15=1 16q 1621

Ixii

~

~o

Contents ~.5

16~

~.6

1661 17q 1741 17] 18q 1811 1841 18] 19q 1941

Founer senes transform of equatIOns of motion Alpha-mteractIOn of particles ~.7 Fractional spatial derivatives ~.8 Riesz fractional denvatIves and mtegrals ~.9 Contmuous hmits of discrete equations ~.1O Linear nearest-neighbor interaction ~.11 Linear integer long-range alpha-interaction ~.12 Linear fractional long-range alpha-interaction ~.13 Fractional reactIon-dtffusion equation ~.14 Nonhnear long-range alpha-mteractIon ~.15 Fractional 3-dimensional lattice equation ~.16 FractIOnal denvatIves from dispersIOn law ~.17 Fractal long-range interaction ~.18 Fractal disperSion law ~.19 Griinwald-Letnikov-Riesz long-range interaction 8.20 Conclusion IReferences

2061 20S 209

Fractional Ginzburg-Landau Equation 19 1 Introduction 19.2 Particular solutIOn of fractIOnal Gmzburg-Landau equatIOn 19.3 Stabihty of plane-wave solutIOn 19.4 Forced fractional equation 19 5 Conclusion IReferences

2151 2161 22q 2211 2221 223

Psi-Series Approach to Fractional Equations 110.1 IntroductIOn 110.2 Smgular behaVIOr of fractional equation 110.3 Resonance terms of fractIOnal equatIOn 110.4 Psi-series for fractIOnal equation of ratIOnalorder 110.5 Next to singular behavior 110.6 Conclusion IReferences

rart III Fractional Spatial [l

19~

19§ 20~

21~

22] 2271 22§ 22~

23q 23~

2351 2361

Dynamic~

Fractional Vector Calculus II I I Introduction II I 2 Generalization of vector calculus 111.3 Fundamental theorem of fractIOnal calculus 111.4 Fractional differential vector operators 111.5 FractIOnal mtegral vector operatIOns II I 6 Fractional Green's fODllula II I 7 Fractional Stokes' fODllula III 8 Fractional Gauss' formula

2411 2411 2421 2471 25q 25~

2541 2571 25<]

Contents

III 9 ConclUSIOn IReferences ~2

[3

[4

~5

xiiil

2611 2621

Fractional Exterior Calculus andl IFractional Differential Forms 112 1 Introduction 112.2 Differential forms of integer order 112 3 Fractional exterior derivative 112 4 Fractional differential forms 112.5 Hodge star operator 112.6 Vector operatIOns by dIfferentIal forms 112.7 Fractional Maxwell's equations in terms offractional forms 112.8 Caputo derIvatIve ill electrodynamics 112.9 Fractional nonloca1 Maxwell's equations 112 10 Fractional waves 112 11 Conclusion IReferences

2661 26<] 2741 272 2811 2821 2841 285] 2871 288 289

Fractional Dynamical Systems 113.1 IntroductIOn 113.2 FractIOnal generalIzatIOn of gradient systems 113.3 Examples of tractIonal gradient systems 113.4 Hamiltonian dynamical systems 113.5 FractIOnal generalIzatIOn of Hamtltoman systems 1136 Conclusion IReferences

293 2941 3011 305] 3071 3111 3121

26:9 2n~

29~

Fractional Calculus of' Variations in Dynamics 114 1 Introduction 114.2 Hamtlton's equatIOns and VarIatIOns of illteger order 114.3 FractIOnal VarIatIOns and Hamilton's equatIOns 114.4 Lagrange's equatIOns and VarIatIOns of illteger order 114.5 Fractional variations and Lagrange's equations 114.6 Helmholtz conditIons and non-Lagrangian equatIOns 114.7 FractIOnal VarIatIOns and non-Hamtltoman systems 114.8 FractIOnal stabilIty 114 9 Conclusion IReferences

311 31 j 3q 31] 312 3211

Fractional Statistical Mechanics 115.1 Introduction 115.2 Liouville equation with fractional derivatives 115.3 Bogolyubov equatIOn With tractIOnal derIvatIves 115.4 Vlasov equation With tractIOnal derIvatIves 115.5 Fokker-Planck equation with fractional derivatives 115 6 Conclusion

33:9 3351 3361 34Q

32~

3261 32§ 3311 3311

34~

345] 34<]

Ixiv

Contents

IReferences

3511

rart IV Fractional Temporal Dynamic!! [6

Fractional Temporal Electrodynamics 116 1 Introduction 116.2 Universal response laws 116.3 Linear electrodynamics of medium 116.4 FractIOnal equatIOns for laws of unIversal response 116.5 FractIOnal equatIOns of the CUrIe-von Schweldler law 1166 Fractional Gauss' laws for electric field 116.7 Universal fractional equation for electric field 116.8 Universal fractional equation for magnetic field 116.9 Fractional damping of magnetic field 116 10 Conclusion IReferences

[7 Fractional Nonholonomic Dynamics 117 1 117.2 117.3 117.4 117 5 117.6 117.7 117.8 117.9

Introduction Nonholonomic dynamIcs Fractional temporal derivatives FractIOnal dynamICS WIth nonholonomic constraInts Constraints with fractional derivatives EquatIOns of motion WIthfractIOnal nonholonomic constraInts Example of fractIOnal nonholonomic constraInts FractIOnal condItional extremum HamIlton's approach to fractIOnal nonholonomic constraInts

17.10Conclu~on

IReferences ~8

Fractional Dynamics and! IDiscrete Maps with Memory 118.1 IntroductIOn 118.2 DIscrete maps WIthout memory 118.3 Caputo and Riemann-Liouville fractional derivatives 118.4 FractIOnal derIvative In the kicked term and dIscrete maps 118.5 Fractional derivative in the kicked term and dissipativ~ Idiscrete maps 118.6 FractIOnal equatIOn WIth hIgher order derIvatives and! Idiscrete map Il~.7 FractIOnal generalIzatIOn of unIversal map for 1 < a :(: 2 118.8 FractIOnal unIversal map for a> 2 118.9 RIemann-LIOuvIlle derIvative and unIversal map WIth memory 118.10 Caputo fractIOnal derIvative and unIversal map WIth memory 118.11 FractIOnal kicked damped rotator map 118.12 Fractional dissipative standard map

357] 3571 35§ 36q 3621 3641 3661 362 37q 3721 373 3741 377] 3771 37~

385] 38§ 3941 3961 39~

4011 40~

40j 4061

40Q 409 41q 415] 41§ 4221 42~

422 4341 4361 4411 44~

447]

( :ontents

118.13 Fractional Henon map 118 14 Conclusion IReferences

xvi

442 4511 4511

rart V Fractional Quantum Dynamic~ ~9

~o

~1

Fractional Dynamics o~ IHamiitonian Quantum Systems

45]

119 1 IntroductIOn 119.2 FractIOnal power of denvatIve and HeIsenberg equatIOn 119.3 PropertIes of tractIOnal dynamIcs 119.4 Fractional quantum dynamics of free particle 119.5 Fractional quantum dynamics of harmonic oscillator 119 6 Conclusion IReferences

4571 45§ 46q 4621

Fractional Dynamics of Open Quantum Systems

4671

120 1 Introduction 120.2 FractIOnal power of superoperator 120.3 FractIOnal equatIOn for quantum observables 120.4 FractIOnal dynamIcal semigroup 120.5 Fractional equation for quantum states 120.6 Fractional non-Markovian quantum dynamics 120.7 FractIOnal equatIOns for quantum OSCIllator WIthfnctIOn 120.8 Quantum self-reproducmg and self-cIomng 120 9 Conclusion IReferences

4671

Quantum Analogs of Fractional Derivatives 121 1 Introduction 121.2 WeyI quantIzatIOn of dIfferentIal operators 121.3 QuantIzatIOn of RIemann-LIOuvIlle tractIOnal denvatIves 121.4 QuantIzatIon of LIOuvIlle tractIonal denvatIve 121.5 QuantIzatIon of nonddlerentIable functIons 121.6 Conclusion IReferences

Iindex

46~

4641 46:1

46~

4711 47~

475] 47] 47~

4821 4861 4871 4911 4911 4921 4941 4961 497] 500 5011 50]

IPart II

[Fractional Continuous Models 011 [FractaLDistributions

~hapter

11

!Fractional Integration and Fractals

[I'he theory of mtegration and ddferentiatlon of non-mteger order has a long hIStoryl ~rom 30 September 1695, when the derivatives of order a = 172 was described byl OC:eIbmz m a letter to L'HospItal (Oldham and Spamer, 1974; Samko et aI., 1993j ~oss, 1975). The earlIest theory of mtegrals and derIvatives of non-mteger orderl goes back to LIOuVIlle and RIemann (Ross, 1975). There are many mterestmg booksl laboutfractIOnal calculus and fractIOnal ddferential equatIons (Oldham and Spamerj [974; Samko et aI., 1993; MIller and Ross, 1993; Podlubny, 1999; KIlbas et al.J 12006; Nahushev, 2003; Pshu, 2005); see also (McBride, 1976, 1986; Srivastava and! IOwa, 1989; NIshImoto, 1989; KIryakova, 1994; Rubm, 1996). DerIvatives and mte1 grals of non-mteger order, and fractIOnal mtegro-dIflerential equatIOns have foun~ ~any applIcatIOns m recent studIes m phYSICS (for example, books (West et al.J 12003; Zaslavsky, 2005; Uchaikin, 2008; Mainardi, 2010), edited volumes (Carpin-I ~erI and MamardI, 1997; HIlfer, 2000; Sabatier et aI., 2007), and reVIews (Metzle~ land Klafter, 2000; Zaslavsky, 2002; Montroll and Shlesmger, 1984; Metzler an~ IKlafter, 2004)). Now we can state that fractIOnal dynamIcs form a new paradIgm ml ISClence [I'he fractIOnal mtegrals have ddferent defimtIOns (Samko et aI., 1993; KIlba§ ~t aI., 2006), and exploiting any of them depends on the boundary conditions and! ~he specIfics of the consIdered phySIcal systems and processes. We note that some ofj IprobabIlIty, geometrIc and phySIcal mterpretatIOns of fractIOnal mtegrals and denva-I ~Ives were dIscussed m (NIgmatullIn, 1992; Rutman, 1994, 1995; Ren et aI., 1996j IYu et aI., 1997; Ren et aI., 1997; Machado, 2003; Tatom, 1995; Stamslavsky, 2004j IStanislavsky and Weron, 2002; Tarasov, 2004) and (Heymans and Podlubny, 2006j ~en Adda, 1997; MamardI, 1998; Gorenflo, 1998; Kempfle and Schafer, 2000j ~onsrefi-Torbati and Hammond, 1998; Podlubny, 2002)1 ~n thISchapter, we consIder some baSIC concepts of fractIOnal contmuous modelsl pf fractal dIstnbutIOn of particles and fields. The fractIOnal mtegral equatIOns arel lused to descnbe fractal dIstnbutIOns of mass, charge and probabIlIty. EquatIOns ofj motion for the fractal distributions are considered in next chapters.1

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

I Fractional Integration and Fractals

11.1 Riemann-Liouville fractional

integral~

[I'woof the most known defimtions of fractIOnal mtegrals are the so-called Riemannj [Liouville and Liouville integrals. First, let us consider the definition of Riemannj [Liouville fractional integration on a finite interval of the real line. More detailedj [nformation can be found m (Samko et aL, 1993; KIlbas et aL, 2006).1 OCf j(x) is a continuous function on the real line, then we can form the definitel Imtegral from a to xj

Repeatmg this process givesl

landthis can be extended arbitranlyl

[The Cauchy formula for repeated mtegratron isl

k\ proof is given by mduction. Equation (1.2) can be generahzed for non-mteger nJ lOsing the Gamma function r(n) = (n - I)!, to remove the discrete nature of thel ~actonal,

we obtam a fractIOnal generahzatIOn of the mtegratIOnl

[I'his equatIOn can be conSidered as a defimtIOn of fractIOnal mtegraIj ~et [a,b] be a finite or infinite interval of the real axis R We denote by Lp(a,b)j Iwhere p ~ 1, the set of those Lebesgue real-valued measurable functions j = j(x) ~m la, bl for whichl

OCf j(x) is a function on a finite intervalla,bl such thatj E LI(a,b), then the leftj land nght-sided RIemann-LIOuvIlle fractIOnal mtegrals arel

raI;Xj)(x)

aI;X[Z]j(z)

=

l(xIgj)(x) =xIg[z]j(z)

=

=

r/a) l x(x - z)a- Ij (z)dz, r/a) l (z - x)a- Ij (z)dz.

(1.3)1

b

(1.4)1

11.1 Riemann-Liouville fractional integrals

[Thefractional integration operators aIr; and xl;: with a > 0 are bounded (Samko e~ lal., 1993) in L p ( a, b) for all p ?: 1. Let us note the connection between the operator~ bI~ and xl;; of the forml

'(laIC; = xIf:QJ QxI;: = aIr;QJ Iwhere Q is defined bYI

(Of)(x) =f(a+b-x)j k\n Important property of the tractIOnal mtegratIon operators IS gIven by the fo11 statement (see Theorem 2.6 in (Samko et al., 1993)). The fractional integra-I ~ion operators aIr; and xl;: form a semigroup. If a > 0 and f3 > 0, then the equation~ ~owing

laresatisfied at almost every point x E [a,b] for f(x) E Lp(a,b) withp?: 1. If a+f3?1 ~, then relations (1.5) holds at any point of [a,b]. These equations with a > 0 and! If3 > 0 are satisfied at any point of [a,b] for f E C(a,b), i.e., f is continuous on [a,bn [I'hIS IS the semIgroup property of tractIonal mtegratIonJ [ThefractIOnal mtegral can be gIven m terms of elementary functIOns for a smaIlI rumher of functIOns. For example, the RIemann-LIOuvIlle mtegrations gIV~

~or

for

f3 > -I and a ?: O. For the constant C, we havel

a - I, EquatIOn (1.6) leads us to the usual relatIOnl

r n+ I I II(x-a)n = (x-at+ 1 = --(x-at+ 1 x T(n+2) n+1 '

(LlO)

Iwhere we use r(z+ I) = zr(z). Other relations can be found in Table 9.1 of (Samkol ~t aL, 1993).

1 Fractional Integration and Fractals

11.2 Liouville fractional

integral~

Now, let us consider fractional integrals on the whole axis R The left- and rightj Isided Liouville fractional integrals arel

(1.11)

1

00

I~f)(x) = xI~[z]f(z) =

r 1a

x (z-x)a-If(z)dz.

[he Liouville fractional integration operators If; are defined for the l~p<17a.

O
lfELp(lR),

(1.12)

function~

(1.13)1

[These equations can be represented by convolution operationl

[TheFouner transform of the LlOuvlIIe fractIOnal mtegrals IS given by the foIIoW1 ling statement. If f(x) ELI (lR) and 0 < a < 1, then the following relations hold!

WfJ =

[he function sgn(x) is equal to

~-I

{

(=t=ix)-a ~ {f}},

(1.15)1

+1 for x > 0, and -1 for x < O. The operator ~ i§

la Fourier transforml

land ~

1

is an inverse Fourier transforml

rI~-I {f}

= _1_

r

(2n)n JJRn

f(z) e-ixzdz.

k\s a result, the LlOuvlIIe fractIOnal mtegral can be defined by these Founer transj IfOfiTIS: rI'he relatIon between (I. I I) and (I. 12) has the fori11l

DI'tf = I~Qf, Qf(x)

(1.18)1

= f( -x)J

[Thenext properties of the Liouville fractional integrals give the connection betweenl ~he translation operator Th and the dilation operator II). defined bYI

1

11.3 Riesz fractional integrals

IThf(x) =f(x+h),

(1.19)1

= f(Xx),

(1.20)1

IIIJ.J(x)

X > 0.

[he operators If; satisfy the commutation relation~

(1.21)1

II1.A, If- f = A. a If- tt, f. [I'he tractIOnal mtegrals satIsfy the

(1.22)1

equatIon~

(1.23)1 lif f E Lp(lR), wher~

~ > 0, 13 > 0,

p ? 1,

[I'hIS IS the semIgroup property of the LIOUVIlle tractIOnal mtegralsJ OC:et us gIve some examples of the LIOUVIlle tractIOnal mtegratIons. For a > ~he LIOUVIlle mtegratron gIve~

OJ

(1.24)1

II~

sin(bx) = b-asin (bx=f U

7r )·

2

for lett-sIded mtegral, we havel

II+a (b -

ax)f3 = r( -a - f3) (b _ ax)a+f3

r( -f3)

Iwhere a ? 0, b - ax > 0, and

a + f3 < 1. For the right-sided integralJ

~a (x f3) = r( -a - f3) xa+f3 r( -13)

-

Iwhere a + f3 < 1, and I(Samko et aL, 1993)J

a>

,

,

0. Other relations can be found in Tables 9.2-9.3 o~

11.3 Riesz fractional integralsl [The RIesz tractIOnal mtegral for the real aXIS lR IS defined byl 1

Iaf)(x)

= 2r(a)cos(anj2)

/+00 f z dz -00 Iz-xI 1- a'

(1.28)

1 Fractional Integration and Fractals

18

Iwhere a

> 0, a i ],3,5, .... This integra] is defined for the functionsl

]<» « ~·I a [TheRiesz integrals can be represented through the Riemann-Liouville integrals bYI

for the mterval a :(; x :(; b, the Rlesz tractIonal mtegral (see SectIOn 12.4 ofj I(Samko et aL, 1993» IS defined byl

a> 0.

(1.30)

for multi-variable case, the Riesz integral is defined bYI

ra I

Iwhere a

j)(x)

=

] f j(z)dz Yn(a) lJRn Iz-xln-a'

> 0, a i n,n+2,n+4, ..., andl 2ann/2r(aI2)/r(n - aI2),

a i=- n+2k, ], n = -2k, (-] )(n-a)/22 a- 1nn/2 Tt. a12) T'(I + [a - nJ/2), a=n+2k.

n(a) =

ni=--2k,

[The Riesz mtegral can be defined by the FOUrIer transformsj

Ifa j

=§

1

{Ixl

a§

{In,

Iwhere ,@ is a Fourier transforml

land §

1

is an inverse Fourier transforml

[I'he Rlesz mtegral for all

a > 0 can be defined by the convolutIOnj

Iwhere Ka(x) is the Riesz kernel such tha~

(1.33)1

91

11.4 Metric and measure spaces

1

a(x) = Yn(a)

Ixl a-n , a - n -I- 0,2,4, .. Ixl a-n In(I/lxl), a-n = 0,2,4, ..

[Here Yn(a) is defined by Eq. (1.32)j

~.4

Metric and measure spaces

IWe will often need a notion of distance (called a metric) between elements of the set.1 [t IS reasonable to define a notion of metnc that has the most Important propertie~ pf ordinary distance in jRn j

pefinition 1.1. A metric space is a set W, together with a real-valued function d( , ) pn W x W, such that the followmg conditions are satisfied:1 ~. d(x,y) ;?;: for all x,y E W (nonnegativity condition)j 12. d(x,y) = 0, if and only if x = y.1 13. d(x,y) = d(y,x) for all x,y E W (symmetry condition)J ~. d(x, z) ~ d(x,y) + d(y, z) for all x,y, z E W (triangle inequality)J [Thefunction d( , ) is called the metric on Wj

°

~f

EI and E2 are non-empty subsets of W, the distance between them IS defined!

!bY ~

pair (W,88) is called a sigma-ring if 88 is a family of subsets of a set W such thatj

10) We 88. 1(2) B C 88 implies (W - B) C 88J 1(3) Bk E 88 (k = 1,2, ...) implies Uk=IBk E 88j

pefinition 1.2. Let (W,88) be a sigma-ring of sets in W. Then a triple (W,88,,u) i§ ~aIIed a measure space If ,u IS a non-negative, sigma-additive measure defined onl ~

°

1(1) ,u(E) ;?;: for every E E 88J 1(2) The sIgma-addItiVIty of Jl j

[or any disjoint sequence {B;, i E N} of sets in 88J 1(3) W is expressible as a countable union of sets E; E W such that ,u(E;) < 00.1 [Thevalue ,u(W) is called the measure of the set Wj ~et W be a closed subset of the n-dimenSIOnal Euchdean space jRn. The Borell Isubsets of Ware the elements of the smaIIest sigma-nng of subsets of w, whlchl ~ontains every compact set of W. A non-negative Borel measure on W is a sigmaj ladditive measure defined for every Borel subset of W such that the measure of everYI

110

I Fractional Integration and Fractals

~ompact set is finite. The Borel measure J.1 is called regular if for each Borel set BJ Iwe have J1(B) = inf{J1(U) : Be U}, where the infimum is taken over all open set§ IE containing B. If J1 is a Borel measure, then J1(B) < 00 for every compact subset BI 0IJl7:: [.-ebesgue measure on IR n is an extension to a large class of sets of n-dimensionall Ivolumes. Let us consider the generalized parallelepipedl

[The n-dimensional volume of this parallelepiped is defined bYI

[I'hen n-dImenSIOnal Lebesgue measure IlL IS defined bY]

L(W,n) = inf{E Vn(Ei) : We UEi} 1=1

i=I

Iwherethe mfimum IS over all covermg of W by countable sets of paraIleleplpedsl

11.5 Hausdorfl' measure [The deeper study of the geometrIc propertIes of sets often reqUIres an analysIs tha~ goes beyond what can be expressed m terms of Lebesgue measure. There eXIst set~ pf zero Lebesgue measure, whIch are some sense large. Usmg Hausdofft measureJ landHausdorff dImenSIOn (Hausdorff, 1919), we can dISCrImmate between there set~ pf zero Lebesgue measure. The Hausdorff measure can be consIdered as a general-I IIzatIOn of Lebesgue measures. In thIS sectIOn, we consIder Hausdofft measure and! ~ts propertIes. More detaIled mformatIOn can be found m (Falconer, 1990, 1985j Rogers, 1998; Edgar, 1990)J IWe consIder a metrIc set W. The elements of Ware denoted by x,y,z, ..., and! Irepresented by n-tuples of real numbers x = (Xl ,X2, ... ,x n ) such that W is embeddedl ~n IR n . The set W IS restrIcted by the condItIons: (I) W IS closed; (2) W IS unboundedj 1(3) W IS regular (homogeneous, umform) wIth ItS points randomly dlstrIbutedl [.-et W be a non-empty subset of n-dimensional Euclidean space IR n . A diametefj pf E C W IS a greatest dIstance apart of any paIrs of pomts mE. The dIameter I§ k1efined by the equatIOnl ~liam(E) =

IFor the metrid

~he

diameter iil

sup{d(x,y) : x,y

E

E}l

~ (x,y) = Ix-yl = (~IXi-Yil) n 1/2j

11 5 Hausdorff mea slife

111

kiiam(E) =sup{lx-YI: x,yEE}j [Let us consider a countable set {E;} of subsets of diameter at most IW, I.e.,

We UE;,

diam(E;)";; e for all

£

that coved

i1

~

for a pOSItIve number D and each £ > 0, we conSIder covers of W by countablt:j ~amilies {E;} of (arbitrary) sets E; with diameter less than £, and takes the infimui11l pfthe sum of [diam(E;)]D. Then we havel

[Thuswe look at all covers of W by sets of diameter at most £ and seek to minimizel ~he sum of the Dth powers of the diameters. The quantity ~D (W) increases, whenl 1£ decreases. Therefore the follOWIng lImIt eXIstsl

rthis limit exists for any subset W of JRn. In general, the limit J't'D (W) can be infinit~ pr zero. The value J't'D(W) is called the D-dimensional Hausdorff measure. Notel ~hat In partIcular, one haij for all £ > 0. [I'heHausdoftl measures generalIze the notIons of the length, area, volume. It canl Ibe proved that the n-dimensional Hausdorff measure of Borel subsets of JRn is, tq IWIthIn a constant multIple, just n-dImensIOnal Lebesgue measure, I.e., the usual nj ~imensional volume. In general, n-dimensional Hausdorff measure J't'n(w) in JRnJ lup to a constant factor equal to Lebesgue measure ,udW, n). If W is a Borel subse~ bfJRn, then Iwhere ,udW,n) is a Lebesgue measure on W, and ro(n) is a constant that depend~ pnly on the dImenSIOn n, such thatl

[Theconstant ro(n) is the volume of n-dimensional ball of diameter 1.1 [Let us define a measure ,uH(W,D) such that this n-dimensional measure ofj !Borel subsets W of jRn IS Just equal to n-dImensIOnal Lebesgue measure. We con1 Isidern-dimensional Hausdorff measure ,uH (W,n) in JR n , equal to Lebesgue measurel luL (W, n). If W is a Borel subset of JR n, thenl I,uH(W,n)

= ,udW,n).

(1.37)1

1 Fractional Integration and Fractals

112

[To define this measure, we consider covers of W by countable families ofj I(arbitrary) sets with diameter less than £, and takes the infimum of the sum ofj fu(D) [diam(Ei)]D for a positive number D and each £ > O. Then we hav~ j;)(W,D)

= inf

[.W(D)[diam(Ei)f: We UEi, diam(Ei)

~e

i=1

[rhus we look at all covers of W by sets of diameter at most £ and seek to mlmj ~ize the sum of the Dth powers of the diameters with a constant factor w(D). Thel [ollowing limit exist§

[This limit exists for any subset W of JRn, although the limit )1H (W,D) can be infi~ Inite or zero. The value )1H(W,D) will be also called the D-dimensional Hausdorffl rIeasure. It IS easy to see tha~

Ks a result, Equation (1.37) allows us to consider the Hausdorff measure )1H(W,D) las a generalization of the Lebesgue measure )1dW, D) from the integer values D t9 ~ractionaI. For mteger values of D the D-dlmenslonal Hausdortl measure of Borell Isubsets IS equal to D-dlmenslonal Lebesgue measure such that Eq. (1.37) IS satisfied.1 IWe begm with a Itst of properties satisfied by the Hausdortl measure (Falconer J ~990, 1985; Rogers, 1998; Edgar, 1990; Federer, 1969)1 [. Monotonicity: If WI 12. Sub-addltlVltyj

C

W2, then )1H(WI,D) ~ )1H(W2,D)] )1H

(U lV;,D) < f )1H(Wi,D). '=

1=1

13. If the distance between sets WI and W2 is positive, i.e., d(WI, W2) > 0, theij lUH(WI UW2,D ) = )1H(WI,D)

+ )1H(W2,D)

·1

IWe shall restnct ourselves to Borel sets. Roughly speakmg, Borel sets are thel Isets that can be constructed from open or closed sets by repeatedly takmg countabl~ lunions and intersections. Note that )1H(W,D) is a countably additive measure, whenl Irestricted to the Borel sets. If {Ei} is a countable family of disjoint Borel sets, thenl

~et

us conSider some properties of the Hausdorff measure for Borel sets1

11 5 Hausdorff mea slife ~efinition 1.3. A functIOn j defined on a set W of IR" satisfies the Holder conditioill Iwith exponent m on W if there exists C > 0 such thatl

If(x) - f(y)1 < qx-yn ~or all x,y E W. ThiS condition imphes that Isatisfies the Lipschitz condition on W ifj

f is contllluous. For m = 1, a functioIlj

If(x)-f(y)1
W.I

[The following statements can be provedj [. Suppose a functIOn f defined on a compact set W satisfies the HOlder conditioIlj Iwith exponent m > 0 and constant C > o. Thenl

for all D.

12. If a functIOn f defined on a compact set W satisfies the Lipschitz condition withl ~onstant C > 0, thenl (1.39)1 ~or all D. Note that lllequahty (1.39) holds for any differentiable functIOn withl Ibounded denvatives, SlllcethiS functIOn satisfies the Lipschitz conditIOn as a con1 Isequence of the mean value theoremJ 13. If f is a siml1anty transformatIOn (also called dl1atIOn) of ratio C, thenl

If(x) - f(y)1

= qx- ylJ

land we have the equatIOnl

f'f.

[I'his relation means that the Hausdoffi measure has the scahng property and C i§ la scahng factorJ If f is an isometry transformatIOn, l.e.J

Ilf(x) - f(y) I = Ix~hen

ylJ

we have the equatIOill ~H(f(W),D) =

J1H(W,D)·1

[This relation gives that the Hausdorff measure is translation and rotation invari-I

ram:

1 Fractional Integration and Fractals

114

lAs a result, we can emphasize the following important properties of the Hausj kloffl measure for Borel sets (Falconer, 1990, 1985; Rogers, 1998; Edgar, 1990j federer, 1969)l The Hausdofff measure is translation invarianti

Iwhere W +h = {x+h: x E W}j 12 The Hallsdorffmeasure is rotation

invarian~

Iwhere r is the rotation transformation]

13. The Hausdoffl measure satIsfies the scalIng propertyl

Iwhere XW is the set W scaled by a factor ;q < 00, then ,uH(W,Dz) = 0 for Dz > Dd If ,uH(W,Dt} > 0, then ,uH(W,Dz) = 00 forO < o, < DIl

~. If ,uH(W,Dd

15.

fropertIes 1-3 can follow one we observe that the dIameter of a set W IS Invanantl lunder translations and rotations, and satisfies diam(XW) = X diam(W) for X > OJ ~ote that scalIng property IS fundamental to the theory of fractals. PropertIes 4-~ lallow us to define a numencal mvanant of the set W that IS called the dImensIOn.1

~.6

Hausdorff dimension and fractalsl

for a Borel set W of JRn, there exists a unique D such thatl

[or D'

< D, and]

[or D' > D. As a result, there exists a critical value of D at which ,uH(W,D) Ijumps from 00 to O. ThIS value IS called the Hausdorff dImenSIOn or the Hausdofflj lBesIcovItch dImenSIOn (Hausdoffl, 1919; BesIcovItch, 1929); see also (FalconerJ ~990, 1985; Mandelbrot, 1983)j [The Hausdorff dimension dimH(W) of a Borel set W of JRn is defined b~ ~=

dimH(W) = sup{ a E JR: ,uH(W, a) =

00 } ,

(1.40)1 (1.41)1

IWe shall write D = dimH(W)j

11 6 Hausdorff dimension and fractals

151

from (1.40) and (1.41), we obtainl

H(W,a)

=

00,

0,

a < D = dimH(W), if a> D = dimH(W), if

(1.42)

[The Hausdorff dimension and other dimensions provide distinguishing character-I listics between fractals. The term fractal is commonly applied to sets of fractiona~ klimension (Falconer, 1990, 1985; Mandelbrot, 1983)j [I'he Hausdoftl dimenSIOnsatisfies the followmg properties] OC. If W is a countable set, then dimH (W) = 0.1 12. If We JRn is an open set, then dimH(W) = nJ 13. If W C JRn is a smooth m-dimensional manifold of JRn, then dimH (W) = m. Inl Iparticular, smooth curves have dimensional 1. Smooth surfaces have dimensiona~ 12. ~. If WI C W2, then dimH(Wt} ~ dimH(W2).1 15. If W C JRn and f satisfies the HOlder condition with exponent m, then]

16. IfW C JRn and! satisfies the Lipschitz condition, then dimH(f(W)) ~ dimH(W).1 [7. If We JRn and f satisfies the bi-Lipschitz conditionj

Iwhere 0

< CI

~ C2 <

00,

then dimH(f(W))

= dimH(W)]

[I'he last statement descnbes the fundamental property of the Hausdorff dlmen-I ISIOn. The Hausdoftl dimenSIOn IS mvanant under transformations that satisfy thel Ibl-Llpschltz conditIOn. We can say that two fractal sets are the same If there eXlst~ la transformatIOn mappmg between them and satisfymg bl-Llpschltz conditIOn. Thel Wlausdortf dImenSIOn and other dImenSIOns proVIde dlstmgUlshmg charactensticsl Ibetween fractals] fractals can be conSidered as metnc sets With non-mteger Hausdorff dlmensIOnJ ISlmple examples of fractals (Mandelbrot, 1983; Frame et aL, 2006; Oryson, 1951)1 larethe Cantor ternary set, the Sierpmskt tnangle and the Menger sponge1 ~.

Cantor ternary set. The Cantor ternary set IS defined by repeatedly removmgj middle thirds of hne segments.1

~he

la. One starts by removmg the mIddle thIrd from the umt mterval [0, I], leavmgl [0, 113] and [273, 1]1 lb. Next, the "middle third" of all remammg mtervals IS removedJ ~. ThiS process IS contmued ad infinitum1 [The Cantor ternary set consists of all points in the interval [0, 1] that are Iremoved at any step in this infinite process.1

no~

1 Fractional Integration and Fractals

116

12. Sierpinski triangle. An algonthm for obtammg the Slerpmskl tnangle IS followj ~

la. Start with any triangle in a plane. The canonical Sierpinski triangle uses ani ~quilateral triangle with a base parallel to the horizontal axisj lb. ShrInk the trIangle by 172, make two copIes, and posItIon the three shrunkeIlj ItrIangles so that each trIangle touches the two other trIangles at a corner.1 ~. Repeat step b with all smaller trianglesj 13. Menger sponge. It is a 3-dimensional extension of the Cantor ternary set and! ISierpinski triangle. Construction of a Menger sponge can be visualized as fo11 lows: la. Begm WIth a cube] lb. ShrInk the cube to 1727of ItS orIgmal sIze and make 20 copIes of ItJ ~. Place the copies so they will form a new cube of the same size as the one but lacking the centre parts] ~. Repeat the process from step b for each remammg smaller cubesJ

origina~

k\fter an Infimte number of IteratIons, a Menger sponge WIll remaIn] OC:et us gIve some examples of Hausdorff dImenSIOnsof well-known fractal sets] I- The Cantor ternary set has Hausdorff dimension D = In2/1n3:::::o 0.6311 I- The Sierpinski triangle has Hausdorff dimension D = In 3/ In2 :::::0 1.585j I- The Menger sponge has Hausdorff dimension D = In 20/ In 3 :::::0 2.727 j

~. 7

Box-counting dimension!

IWe note that there are other classes of coverIng set that define measures leadIng tq Wlausdortf dImenSIOn. For example, we could use coverIngs by spherIcal balls B;.I IQill!g

~(W) = inf{EIBdD: 1=1

We

UBi, ;=1

Iwhere {B;} is an £-cover of W by balls, we obtain the measur~

IWe can define a dimension at which ,%'D(W) jumps from infinity to zero. Note tha~

Isince any £-cover of W by balls is a permissible covering in the definition of Jt;D j OCf fE;} is an £-cover of W, then fBi} is a 2£-cover of W by balls B; such tha~ each B; is a ball of radius lEd :( e containing E;. Then we havd

11.7 Box-counting dimension

[faking the infima, we obtainl

fassmg to the bmIt e

---+

0, we havel

[his implies that the values of D at which JrOD(W) and ,%'D(W) jump from infinitYI ~o zero are the same, so that the dimensions defined by the two measures are equalj k'\s a result, we obtam the same values for Hausdorff measure and dImensIOn, If wei luse ,%'f(W) and ~D(W)J [There are other types of fractal dImensIOn for subset of jRn. A box-countmg dI1 rIenSIOn IS one of the most WIdely used dImensIOns due to ItS relatIvely sImplel ~aIculatIOn and empmcal estImatIOn.1

pefinition 1.4. Let W be any non-empty bounded subset of jRn, and let Ne (W) bel ~he smallest number of sets of dIameter at most e, whIch can cover W. The box-I ~ountmg dImenSIOns of W IS defined bY]

G.

()

LImB W

.

= -

InNe(W)

~~ lnts) ,

Iwhere we assume that e > 0 such that In( e)

< OJ

[I'here are several eqUIvalent defimtIons of box-countmg dImensIOn that can bel rIore convement to use. We conSIder a countable famIly of e-cubes m jRn of thel [Qi]]:i;

IWe note that diam(EiJ...in ) ~ ey'n.1 for a pOSItIve number D and each e > 0, we conSIder covers of W by countabl~ ~amilies e-cubes {Eil ...in}, and takes the infimum of the sum of [diam(Eil ...iJ]D.1 [Then we havel

f(w) = inf{E[diam(Eil ...in)f: We i=I IWe define the measuresl

land

U Eil ...in, i .. .i

1 Fractional Integration and Fractals

118

[ r(D)

D 2

=

n / I 2nnD/2+ 1)"

for the box-counting dimension dimB(W) of a set W oflFtn , we hav~ ID = dimB(W)

= sup{ a

E 1Ft: )1B(W, a)

=

ID = dimB(W)

= inf{ a

E 1Ft: )1B(W, a)

= 0 }J

00

~J

IWe shall write D = dimB(W)J [This version of the definitions is widely used empirically. To find the box-I ~ounting dimension of a plane set W, we draw a mesh of squares or boxes of sidel 1£ and count the number Ne(W) that overlaps the set for various small £. The dij Imension is the logarithmic rate at which Ne(W) increases as £ -+ 0, and may bel ~stimated by the gradient of the graph of InNe(W) against In( £-1) j IWe can use a definition of box-counting dimension that is obtained by takingl INe(W) as to be the smallest number of arbitrary cubes of side £ required to coverl IW. SImIlarly, we obtam exactly the same values of fractal dImensIOn, If m Eq. (1.43)1 Iwe take Ne (W) as the smallest number of closed balls of radius £ that cover WJ [t IS Important to note the relatIonshIp between the box-countmg dImensIOn and! ~ausdorff dimension. If W can be covered by Ne(W) sets of diameter £, then, froi11l ~efinition of the ~D (W), we ge~

k\s a result, we

hav~

every W C IRn . In general, we do not have equalIty here. Although Hausdofffl land box-countmg dImensIOns are equal for many regular sets, there are examplesl Iwhere thIS mequalIty IS stnct.1 IWe note that a smooth m-dimensional submanifold of IFtn has dimB(W) = m. Th~ Ibox-counting dimension dimB(W) is bi-Lipschitz invariant. Similarly, box-counting khmensIOns behave just lIke Hausdofff dImensIOns under bI-LIpschItz and HOldefj hansformations I [The box-countmg dImensIOns are determmed by covenngs by sets of equal sIze.1 [Therefore box-countmg dImensIOns are eaSIer to calculate than the Hausdoftl dI1 fuensIOns. In addItIOn, the box-countmg dImensIOns can be more effectIve for set§ IW that may be covered by smaIl sets of equal SIze, where as the Hausdorff dImen-1 ISIOn mvolves covenngs by sets of smaIl but perhaps WIdely varying sIzel ~or

11.8 Mass dimension of fractal systems

1S1

11.8 Mass dimension of fractal systemsl [I'he Hausdorff and box-countmg dlmenStons are defined as a local property m thel Isense that its measure properties of sets of points in the limit of a vanishing diamete~ lusedto cover the set. It follows that these fractal dimensions can depend on positionj [The definition of the Hausdorff dimension of a set of particles require the diamej ~er of the coverIng sets to vamsh. In general, physIcal systems have a characterIstIcl Ismallest length scale such as the radIUs, Ro, of a partIcle (for example, an atom o~ ~olecule). In real physical objects the fractal structure cannot be observed on alII Iscales but only those for which Ro < R < Rm , where Ro is the characteristic scalel pf the particles, and R m is the macroscopic scale for uniformity of the investigatedl Istructure and processes. Therefore we need a phySIcal analog of Hausdoftl dlmen-I Iston. Another, the mass dlmenSton, IS based on the Idea of how the mass of a systeml Iscales WIth the systems' SIze, If we assume unchanged densltyJ for many cases, we can write the asymptotic form for the relation between thel fuass M(W) of a region W of medium, and the radius R containing this mass a§ follows'

[or R/ Ro » 1. The constant Mo depends on how the spheres of radius Ro are packedj [The parameter D, does not depend on the shape of the regton W, or on whether thel Ipackmg of spheres of radIUS Ro IS close packmg, a random packmg or a porou~ Ipackmg WIth a umform dIstrIbutIon of holes. The number D IS called the mass dlj llIlellsioll. [The fractal mass dlmenSton IS a measure of how the medIUm fills the space I~ pccuples. Note that the fact that a medIUmIS porous or random does not necessarIlyl lImply that the medIUm IS fractal. A fractal system has the property that the mas~ Imcreases as the system SIze mcreases m a way descrIbed by the exponent m thel fuass-radIUs relattonJ [The mass dlmenSton characterIzes a feature of the system, ItS propertIes to filII ~he space. Note that the shape of the system IS not descrIbed by the mass dlmenstonl [There are other features of a system that can be quantIfied as well. For example) IItS ramlficatton IS a measure of the number of bonds to be cut m order to Isolate ani larbltrarIly large part of the system.1 [I'hereal fractal structure of the medIUmIS characterIzed by an extremely compleX] land megular geometry. Although the fractal dlmenSton does not reflect completely! ~he geometrIc propertIes of the fractal, It nevertheless permIts a number of Importantl ~onclustons about the behavtor of fractal structures. For example, If It IS assumed! ~hat matter WIth a constant denSIty IS dIstrIbuted over the fractal, then the mass ofj ~he fractal enclosed m a volume of characterIstIc SIze R satIsfies the power-lawl

W'(R)

rv

R~

1 Fractional Integration and Fractals

120

Iwith non-integer D, whereas for a regular n-dimensional Euclidean object M(R)

"1

~

k\ fractal medium is a medium with non-integer mass dimension. We can find! ~he

fractal medIa among the porous matenal, polymers, collOIdaggregates, aerogelsl I(Mandelbrot, 1983; Frame et aI., 2006; Feder, 1988; Kulak, 2002; Potapov, 2005)j for example, there IS an expenmental eVIdence(Katz and Thompson, 1985) mdIcatj ling that the pore spaces of a set of sandstone samples are fractals in length extending! [mm I nanometer to 105 nanometers! W'Janostructured matenals may be defined as those matenals whose structurall ~lements-clusters, crystallItes, molecules and atoms- have SIzes m the I to lOq Inm range. In general, nano-systems as the dIstnbutIons of these structural elementsl ~an demonstrate fractal propertiesj IWe note that the first synthetic nanoscale fractal molecule has been created! I(Newkome et aI., 2006). A University of Akron research team led by Georgcj W'Jewkome used molecular self-assembly technIques to syntheSIze the molecule ml ~he laboratory. The molecule, bound WIth IOns of Iron and ruthenIum, forms ij Ihexagonal gasket. To confirm the creatIOn of the fractal, the phYSICIStS sprayed thel rIolecules onto a pIece of gold, chIlled them to mmus 449 degrees FahrenheIt tq Ikeepthem stahle, and then VIewed them WIth a scannmg tunnelIng mIcroscopel

~.9

Elementary models of fractal distributions

[I'he mIddle thIrd Cantor ternary set, the SIerpmskI tnangle and the Menger spongg ~an be conSIdered as elementary models of fractals. The constructIons of these fracj ~als reqUIre an mfinIte number of IteratIons, and the lImIt of a vanIshmg dlamete~ lused to cover the set. In general, phySIcal systems have a charactenstIc smalles~ ~ength scale such as the radIUS, Ro, of partIcles (for example, an atom or molecule )1 [n real phySIcal objects the fractal structure cannot be observed on all scales bu~ pnly those for WhICh R > Ro, where Ro IS the charactenstIc scale of the partIclesJ [I'herefore the mathematIcallme segments of Cantor ternary set must be replaced bYI ~mear cham of atoms or molecules (see Chapter} of (Feder, 1988))1 ~et us conSIder a lInear cham consisting of two types of partIcles A and B, wIthl ~he masses landthe radIUS ~A

=RB=Roj

[I'he cham IS created by followmg algonthrI

Fn+I =An-Bn-An, ~n+l =

B n -Bn -BnJ

Iwhere n ;;: 0, and AD = A, Bo = B. The line denotes a bond. For n = I J

2~

11.9 Elementary models of fractal distributions

IA -B-A

1

for n - 2, IA - B - A - B - B - B - A - B - A I

forn=3,

0-B-A-B-B-B-A-B-Ad ~B-B-B-B-B-B-B-B-Bd

I-A -B-A -B-B-B-A -B-AJ ~t

is easy to see that the length of the chain i§

(1.46)1 [!'he total mass of the chain isl

(1.47)1 ~quatIOns

(1.46) and (1.47) gIvel

Iln(~)

=n In2J

~n (~) = n In 3.1 [These equations can be represented

a~

ID = ~:~ ~ 0.63 .... k'\s a result, we obtaml

[The suggested lInear cham can be conSIdered as an example of fractal dIstnbutIOili pf particles on ]R with mass dimension (l.48). This chain is a physical analog of thel mIddle thIrd Cantor set. In general, the sequence of the gIven partIcles m the chaml ~an be stochastic. Examples for fractal distributions in ]Rz and ]R3 can be similarlyl ~onstructed. It IS easy to obtam analogues of the SIerpmskI tnangle and the Mengerl Isponge. The correspondent fractal dIstnbutIOns of partIcles A and B have the mas~ Himensions In20 1 I In8 Dsc = In3 ~ 1.89..., DMS = In3 ~ 2.73 ... [These distributions can be considered as elementary models of fractal distributionsj

122

1 Fractional Integration and Fractals

[Letus consider randomized fractal chains. In this case, we assume that the num-I Iberof A-particles IS equal to Nn - 2n for the segment with Rn - 3nRo. Note that thel ~1Umber of particles of each type is in strict agreement with the number in regularl ~ractal cham. As a result, relatlOn (1.49) holds for the randomized fractal chamsl [n general, we can conSider boxes of A and B mstead of particles A and B. Thel Iboxes B describe unpermitted places (positions) for particles. The boxes A describ~ Ipermitted places (positions) for particles. For example, A are open boxes and Barel closed. In this case, we have the relation!

Iwhere Nn is a number of open boxes, and Rb is a linear size of the box (RA ~ IRB = Rb), which much more than radius of particles (Rb » Ro). The randomizedl Ihnear cham of A and B boxes can be conSidered as a representatIon of denSity ofj Istates in the I-dimensional Euclidean space !R j . The density of states describes howl Ipermitted states (places) of particles are closely packed in the space !R I . In statisticall landcondensed matter phYSICS, the notlOn of the denSity of states IS well-known and! litIS usually conSidered as a number of states per umt of energy or wave vector (see) ~or example, (Kittel, 2004)). In thiS case, we conSider a dlstnbutlOn functIon tha~ ~escnbes dlstnbutlOn of phYSicalvalues (for example, the mass, probablhty, electrHj ~harge, number of particles) on a set of possible states in the Euclidean space !R I. Ifj Iwe have a dlstnbutlOn such that each A-box contams IdentIcal number of particles) ~hen the mass of the chain satisfies the relationl

Iwhere Mo IS a mass of particle, and No IS a number of partIcles m a box. In generalJ Iwe can consider particles that are distributed with density p (x) over the randomizedl Ilmearcham of boxes With fractal dimenSion D. In thiS case, we use the equatIonl

Iwhere Cj (D,x) is a density of states (permitted places) on We !R I . Here x is dimen-I Isionless coordinate variable such that x' = XRh is usual coordinate variable]

11.10 Functions and integrals on fractal~ IWe note that analYSIS on fractals as a generahzatlOn of calculus on smooth man1 ~folds to calculus on fractals can be formulated (Stnchartz, 2006; Kugaml, 2001 j ISchweizer and Frank, 2002; Blei, 2003). Usually the starting point for the theorYI pf analysis on fractals is a Laplacian on fractals (Strichartz, 2006; Kugami, 2001 j

2~

11.10 Functions and integrals on fractals

ISchweizer and Frank, 2002; Blei, 2003). This turns out not to be a differential operj lator in the usual sense but has many of the desired properties. There are a number ofj lapproaches to defining the Laplacian operator on fractals: probabilistic, analyticall pr measure theoretIc. We also can be mterested m mtegratIOn on fractals (see, for exj lample, (SvoziI, 1987; Zverev, 1996; Ren et aL, 2003)). The analysis on fractals and! [factal domain integration can be used to describe dynamical phenomena, which ocj ~urs on objects modelled by fractals (Mandelbrot, 1983; Feder, 1988; Kroger, 2000j ~ymk, 1989; Potapov, 2005; Kulak, 2002; Calcagm, 201O)j ~et W be a non-empty subset of n-dimensional Euclidean space jRn, and let {E;}I Ibe a countable famIly of subsets of dIameter at most e that covers W, l.e.J

We UE;,

diam(E;)";; e for all

i1

~

[I'he startmg pomt IS the notIOn of a characterIstIc functIOn of subsets E;, whIch Iij ~efined by

IXEi(X)

= { 1, 0,

~f

If

x E E;,

xif-E;.

V\ function f(x),

defined on a set W, is called a simple function if there exists a finitel r.umber of disjoint sets {Ek} such tha~ if

x E Ek'

if

x

if- Uk=l Ei,

(1.51)1

[Thesimple function f(x) on W can be represented byl

(1.52)1 Iwhere J; E lR are constants. Note that the product of two SImple functIOns and anYI lfimte lInear combmatIOn of SImple functIOns, are again SImple functIOns. For con1 ~inuous function f(x)j IIimf(x) = f(y) , (1.53)1 IX---+YI

Iwhenever

I@1d(x,y) = O.

(1.54)1

V\ simple function f = f(x) on a set W is integrable if )1H(E;) < 00 for every i fo~ Iwhich f; #- O. [The Lebesgue-StIeIt]es mtegral over the contmuous functIOns (1.52) can be de1 lfined (SvozIl, 1987) byl

1 Fractional Integration and Fractals

124

rrherefore

f(x)dj.lH(X) = w(D)

lim

[J(x;) [diam(E;)]D.

(1.56)

diam(Ei)->O Eo

W

IWe note that there are other classes of covering set {E;} that define measures leadingl ~o Hausdofff dimension] IWe can use covenngs by sets of equal SIzeto use box-countmg dImenSIOn. Therel lare several equivalent definitions of box-counting dimension that can be more conj Ivementto use. For example, the collectIOn of cubes m the e-coordmate mesh of jRnl ~an be used. Similarly, we obtain exactly the same values of box-counting dimen-I ISIOn, If use closed balls of radIUS e that cover WJ [t is always possible to divide the subset W C jRn into the collection of e-cubesl lin lI~n of the forml

dj.lB(X) =

lim

W(D) [diam(E;)] .

(1.57)

diam(Ei ...in)->O

[The set W can also be parameterized by polar coordinates with r = d(x,O) = Ixl land angle Q. Using this parametrization, we can consider a spherically symj Imetnc covenng by Er,Q around a centre at the ongm. In the hmIt, the functIoIlJ ~(D)[diam(Er,Q)]D give~

dj.lB(r,Q) =.

lim

w(D)[diam(Er,Q)] = dQ -

- dr.

(1.58)

dlam(Er,Q )->0

[ThIS restnctIOn of spheneally symmetnc covenng may not yIelds correct value o~ ~he Hausdorff measure and Hausdorff dImenSIOn. The mfimum m the equatIon fo~ ~he Hausdorff measure has to be taken over all pOSSIble covenng. Usmg a sphencallyl Isymmetric e-covering of W, we obtain a box-counting measure dj.lB(r,Q) and ~ Ibox-counting dimension dimB(W) ~ dimH(W)] [Let us consider f(x) that is symmetric with respect to some centre Xo E W, i.e.] If(x) = const for all x such that d(x,xo) = Ix-xol = r for arbitrary values of r. Thenl me transformatIOIJI

Iw ----> Wi = T- xoW :

x---->xI =x-xo

(1.59)1

~an be performed to shIft the centre of symmetry. Smce W IS not a hnear space] 1(1.59) need not be a map of W onto Itself, and (1.59) IS measure preservmg. Thenl ~he mtegral over aD-dImenSIOnal metnc space can be represented (SvozIl, 1987) ml me form

251

11.11 Properties of integrals on fractals

[his integral is known in the fractional calculus (Samko et aI., 1993; Kilbas et al.J 12006) as a nght-sIded RIemann-LIOuvIlle fractIOnal Integral that IS defined byl

(D> 0). for z = 0, Equation (1.61)

(1.61)

give~

rI~f)(O) = r(~)

1'' ~-I

f(x)dx.

[t is easy to see that Eq. (1.60) can be represented in the forrn

[This equation connects the integral on fractal with integral of fractional order. A~ la result, the fractIOnal Integral can be consIdered as an Integral on fractal up to thel rumerical factor r(D/2)/[2n D / 2r(D)]. This result permits to apply different tool§ pf the fractIOnal calculus for the fractal dIstrIbutIOn of partIclesJ

~.11

Properties of integrals on

fractal~

[he integral defined in Eq. (1.56) (see also Eq. (1.60)) satisfies the propertiesj [. LInearIty:

Iwhere It and 12 are arbItrary functIOns on W; a and b are arbItrary real numbers] 12 Translational invariancel (1.65)1 Isince df.1H(X - xo) = df.1H(X) as a consequence of homogeneity (uniformitY).1 13. ScalIng propertyj

fl.

~or any positive Ii-, since df.1H(X/Ii-) = Ii- Ddf.1H (x)1

We can also require rotatIOnal covanance of the IntegralsJ

IWe note that these propertIes are natural and necessary In applIcatIOns. By evaI1 luating the integral of the function f(x) = exp{ -axl + bx} it has been shown tha~ ~ondItIons (1.64)-( 1.66) define the Integral up to normalIzatIOn:1

1 Fractional Integration and Fractals

126

(1.67) W'lote that, for b = 0, EquatIOn (1.67) IS Identical wIth result from (1.63), whIch canl IbeobtaIned dIrectly wIthout condItions (1.64)-(1.66)j

~.12

Integration over non-integer-dimensional space

[ntegratIOn In D-dimenSIOnal spaces wIth non-Integer D IS used In quantum field] ~heory. DImensIOnal regulanzatIon IS a way to get InfimtIes that occur when on~ ~valuates Feynman dIagrams In quantum field theory (CollIns, 1984). One assume§ ~hat the space-time dimension is not four but D, which need not be an integer.ltoftenl ~urns out that the integrals in momentum space extrapolated to a general dimensionj eonverge:

[The defimtIOn of D-dimensIOnal mtegratron In terms of ordInary mtegration IS fol1 ~OWIng

)D ~

L2=:(

l->

j(x)

= (

r

1)D 2n JQ(D)

sa Joroo dx ~-l j(x),

Iwhere we can usel

V\s a result, we have (CollIns, 1984) the explIcIt defimtIOn of the contmuation IIntegratIOn from Integer n to arbItrary fractIOnal D In the forml

o~

(1.70) [IhiS equatIOn reduced D-dimenSIOnal IntegratIOn to ordInary IntegratIOn so lIneantYI land translatIOn Invanance follow from IIneanty and translatIOn Invanance of ordI1 Inary mtegranon. ScalIng and rotation covanance are explICIt properties of the defi1 Ii:l.i.ti.Oi:L [The integral defined in Eq. (1.69) satisfies the following properties:1 [. LIneanty:

Iwhere a and b are arbItrary real numbersJ 12 Translational invariancel

271

11.12 Integration over non-integer-dimensional space

[or any vector hj 13. Scaling propertyj

[or any positive XJ

t!. We also have rotational the covariance of the integralsj [These properties must be imposed on a functional of f(x) in order to regan:~ Itt as D-dImensIOnal IntegratIons (CollIns, 1984). These propertIes are natural and! Inecessary In applIcatIOn of dImenSIOnal regulanzatIon to quantum field theory (seel ISectIOn 4 In (CollIns, 1984)).1 k\n example of an application ofEq. (1.70) is given by choosing the function!

Iwhere a and b are numbers. The Integral can be explICItly computed to gIvel

[I'he other example IS the

~n

Integra~

quantum field theory, the dIvergences are parametenzed as quantItIes proportIOnal1

~o (4 - D) 1, whose coefficients must be canceled by renormalization to obtain finitel

IphysIcal quantItIes] IWe note that IntegratIOn and equatIons of motIon In spaces WIth nomnteger dIj ImensIOns were dIscussed In (StIllInger, 1977; SvozIl, 1987; He, 1990, 1991; Palme~ land Stavnnou, 2004).1 k\n InterpretatIOn of the fractIOnal IntegratIOn can be connected WIth fractIOna~ klImensIOn (Tarasov, 2004). ThIS InterpretatIOn follows from formula (1.70) of thel form I 2 nD/2

Vr f(x)dDx =

r

r(D/2) Jo f(x)~-l dx

lused for a dImenSIOnal regulanzatIOn. The LIOuvIlle fractIOnal Integrall

lat the POInt z - 0 can be conSIdered as an Integral In the fractIOnal-dImensIOnall Ispace by

1 Fractional Integration and Fractals

128

lup to the numerical factor r(D/2)/(2nD/2r(D))j k\s a result, the Liouville fractional integral can be considered as an integral in [ractional-dimensional space up to the numerical factor r(D/2)/(2nD / 2r(D)).1

~.13

aI

Multi-variable integration on fractals

[The integral in Eg. (1.60) is defined for a single variable. It is only useful for inj ~egratmg sphencally symmetnc functions. We consIder multIple vanabIes by usmgl ~he product spaces and product measures (StIllInger, 1977; Palmer and StavnnouJ 12004). [Letus consider a collection of n = 3 measurable sets (Wkl,ukl D) with k = 1,2, 3j landform a CartesIan product of the sets Wk producmg W - WI X W2 X W3. The defj Imltion of product measures and applIcation of Fubmls theorem proVIde a measur~ ~or the product set W - WI X W2 X W3 a§ (1.78)1 [Then mtegratron over a functIOn j on W

I~

[n this form, the single-variable measure from (1.60) may be used for each coordi-I Inate xs, whIch has an assocIated dImenSIOn ad k = 1,2,3.

[Then the total dImenSIOn of W - WI

X

W2

X

(1.80)

W3 lsi

(1.81)1 ~et us reproduce the result for the smgle-vanable mtegratIOn (1.60) from WI XI IW2 X W3. For spherically symmetric functions f(Xl ,X2,X3) = f(r), where r Z= xlz-R ~l + xl, we can perform the integration in spherical coordinates (r, q" B). In thi~ ~ase, Equation (1.79) become~

Ifw d,uB f(XI,X2, X3l = A(al, a2, a3)

1= A(al,a2,a3)

w

w

JJJ dr

dq,

w

dX3IxIIUj-Ilx2Iu2-1Ix3IUrlf(Xl,X2,X3)

dBh(r,q,) rUI+U2+U3-1

1'(COSq,)UI l(sinq,)U2+ u3 Z(sinB)U3 If(r),

(1.82)1

29

11.14 Mass distribution on fractals

(1.83) land h (r, 0,

V

> O. From (1.81), we obtaml (1.85)

[I'his equation descnbes the D-dimenslOnal mtegration of a sphencally symmetncl [unction, and reproduces the result (l.60)j

11.14 Mass distribution on fractals [he mass that is distributed on metric set W

C ffi.3

with the density rho' (r', t) i§

~efinedby I

M3(W ) =

fw p'(r',t)dV;,

~V£ = dx'dy'dz'l

[or Cartesian coordinates x', y', z' with dimension [x'] = [y'] = [z'] = meter. We notel ~hat SI unit of M3 (W) is kilogram, and SI unit of p' is kilogram· meter 3J [To generalize Eq. (1.86), we represent this equation through the dimensionlessl coordmate vanables. We mtroduce the dimenslOnless valud ~=x'/Ro,

y=y'/Ro,

z=z'/Ro,

r=r'/Rol

Iwhere Ro is a charactenstic scale, and the denSity!

r(r,t) Iwhere SI unit of

=

RbP'(rRo,t)J

e(r, t) is kilogram, i.e., Ie I = kilogram. As a result, we obtainl

Iwhere dV3 = dxdydz for the dimenslOnless Cartesian coordmates. ThiS representa-I ~lOn allows us to generaltze Eq. (1.87) to fractal media and fractal distnbutlOn o~ mass.

I Fractional Integration and Fractals

130

Let us consider the mass that is distributed on the metric set W with the fractiona] klimension D. Suppose that the density of this distribution is described by the func-I ~ion p(r,t) that is defined by (1.52), where SI unit of p is kilogram. In this case, thel Imass IS defined byl

Iwhere r, Xl = land

X, X2 =

Yand

X3 = Z

are dimensionless variables, D = at

+ a2 + a3J (1.89)1

Iwhere dV3 - dxdYdz for CarteSIan coordmates,

an~

(1.90)

IWe note that SI umt of MD IS kIlogram. As a result, we have the Rlemann-LIOuvIlI~ [ractional integral up to numerical factor 87T;D!2. Note that the final equations, whichl Irelate the phySIcal varIables, are mdependent of numerIcal factor m the functIOriI b (D, r). However the dependence on r is important to these equations. We notel ~hat the symmetry of the function C3 (D, r) must be the defined by the symmetry ofj [he medIUm. ~quatlon (1.88) was used to deSCrIbe fractal medIa m the framework of fracj ~IOnal contmuous model (Tarasov, 2005a,b). Usmg a generalIzatIOn of ChrIstensenl lapproach (ChrIstensen, 2005), we represent medIUm WIth fractal mass dlmensIOriI Iby continuous model that IS deSCrIbed by fractIOnal mtegrals. In thIS model, we usel [ractional integrals over a region of jRn instead of integrals over a fractal set. W~ [Iote that real fractal medIa cannot be conSIdered as fractal sets. The fractal structure pf the medIa cannot be observed on all scales. The equatIons that define the fractall ~ausdorff and box-countmg dImenSIOns have the passage to the lImIt. ThIS passagel Imakes dIfficult the practIcal applIcatIOn to the real fractal medIa. The other dlmen-I ISIOns, whIch can be calculated from the experImental data, are used m the empmcall ImvestIgatIOns. For example, the mass dImenSIOn can be easy measured. The mas§ pf fractal medIUm obeys the power-law relatIOnl (1.91)1

Iwhere MD IS the mass of fractal medIUm, R IS a box SIze (or a sphere radIUS), an~ 10 is a mass fractal dimension The total mass of medium inside a box of size RI Ihas a power-law relatIOn (1.91). The dImenSIOn D of fractal medIa can empmcall ~stImated by drawmg a box of SIzeRand countmg the mass mSlde. To estImate thel Imass fractal dImenSIOn, we take the logarIthm of both SIdes of equation (1.91)j ~n(MD) = D In(R)

+ lnkl

3~

11.15 Density of states in Euclidean space

[he log-log plot of MD and R gives us the slope D, the fractal mass dimension. Fo~ ~hese reasons, we can consider the dimension D in Eg. (1.88) as a mass dimensionj bfthe mediumJ

~.15

Density of states in Euclidean space

[n order to descnbe fractal medIa by fractIOnal contInUOUS model, we use two dIfj [erent notions such as density of states cn(D, r) and distribution function p(r)J [The function cn(D,r) is a density of states (DOS) in the n-dimensional Euj ~lIdean space jRn. The denSIty of states descnbes how permItted states of partIclesl lare closely packed in the space jRn. The expression cn(D,r)dVn represents the num-I Iberof states (permItted places) between Vn and Vn +dVnJ [The function p (r) is a distribution function for the n-dimensional Euclideaij Ispace jRn. The distribution function describes a distribution of physical values (fofj ~xample, the mass, probabIlIty, electnc charge, number of partIcles) on a set o~ IpossIble states In the space jRn. For the denSIty of number of partIcles, we use thel rotation n (r). The number of particles in the region dVn is defined by the equationl

klN(r)

=

n(r)cn(D, r) dVnJ

~n

general, we cannot consider the value n(r)cn(D, r) as a new distribution functioill pr a denSIty of number of partIclesJ ~n the general case, the notIOns of denSIty of states and dIstnbutIOn functIOn arel ~hfferent. We cannot reduce all propertIes of the system to the dIstnbutIOn func-I ~Ion. ThIs fact IS well-known In statIstIcal and condensed matter phYSICS (see, fo~ ~xample, (KIttel, 2004; Bonch-BruevIch et a1., 1981), where the denSIty of states I§ lusually conSIdered as a number of states per umt of energy or wave vector. DensItYI pf states IS a property that descnbes how permItted states are closely packed In en1 ~rgy or wave vector spaces. For fractal dIstnbutIOns of partIcles In a region W, wei fuust use a denSIty of states of the regIOnJ ~n the fractional continuous model of fractal media, the density of states cn(D, rJ lin jRn is chosen such thatl

I(1)1D(r,n) ~escnbes

= cn(D,r)dV~

the number of states In sv; We use the follOWIng

notatIOn~

~o descnbe numbers of states In n-dImenSIOnal EuclIdean spaces WIth n = 1,2,3 J IWe note that the symmetry of the density of states cn(D, r) must be the defined byl ~he symmetry properties of the mediumj

I Fractional Integration and Fractals

132

11.16 Fractional integral and measure on the real axisl [he phase volume of the region W = {x: x E [a,b]} in Euclidean space]RI i~

[I'hIS equatIOn can be represented a§

[I'he left and rIght-sIded RIemann-LIouvIlle tractIOnal Integrals are defined b5J

1 lb

Y a I j(x)dx aIy [x]j(x) = qa) a (y_x)l-a'

a yIb [x]j(x)

j(x)dx qa) y (x-y)l-a· I

=

(1.94)

lOSIng (1.94), we reWrIte the phase volume (1.92) In the foriTIj

l,ul (W) = aI; [x] 1 + yIl [x] 1.

(1.95)1

IWe define a fractional generalization of (1.95) by the equation] (1.96)1

ISubstItutIOn of Eqs. (1.94) Into (1.96)

g1Ve~

(1.97)

Iwhere a ~

Xl

~quatIOn

~ Y ~ Xl ~ band W = la,blJ (1.97) can be represented asl

lOsing Iy- xl = Ix- yl, we hav~

r/a) 1Ix-yla-Idx. 6

l,ua(W)

=

IWe can define d,ua(x - y) such tha~

3~

11.16 Fractional integral and measure on the real axis

~Ix - yla = a!x - y!a-l sgn(x - y) dx,1

land an a) =

n a + 1), we can represent (1.99) in the foriTIJ (1.100~

Iwhere the function sgn(x) is equal to + 1 for x > 0, and -1 for x ~,ua(x) can be considered as a differential ofthe functionl IxlU

,ua(x)=qa+l)'

< O. If x > 0, thenl

1 x>Ol

I

[ntegration (1.97) and ar( a) = r( a

+ 1), give~

(1.101~ Iwhere a :::; y :::; b. In orderto let ,ua(la, b I) be independent on y, we can use y = a. Ifj Iwe use y - a m Eq. (1.101), thenl

(1.102~ [Using a:::; x:::; b, we have sgn(x-a) y = a gives

= 1, and lx-a! =x-a. Then Eq. (1.100) withl

(1.103~ k'\s a result, Iwherex E [a,b].1 IOsmgthe denSIty of state§

~n

the I-dImenSIOnal space JR, we

hav~

Id,ua(x)

= cda,x)dxJ

[Letus consider a similarity transformation of ratio It > 0, and a translation transj the region W = [a,b]. Using the dilation operator II;., and the transla~ ~IOn operator Th such tha~ ~ormation for

IThf(x)

=

f(h),

Thf(x)

=

f(x+h)

I Fractional Integration and Fractals

134

[or the function f(x) = x, we obtainl IlIa x = AX,

ThX =x+h.

(1.104j

IWe can use these operators to describe the similarity and translation transformation§ pfthe intervalla,bl such tha~

[h [a,b] OCf a

~

X

~

=

[Aa,Ab],

Th [a,b]

=

[a+h,b+h]j

b, thenl

rh[a,x) U[x,b]

=

[a+h,x+h)U[x+h,b+h]J

li.e., for each X E la, bl relations (1.104) hold.1 k\s a result, the scaling property!

landthe translation invariancel

rrh)1a([a,b])

=

)1a([a + h,b + h]) = )1a([a,b]),1

lare satisfied for the measure )1a(W) with W = la,blJ ~et us consider the measure d)1a(x) that is defined by Eq. (1.103). This measurq liS translation mvanantl

IsmcetranslatiOn X Iscaling propertYI

----. X

+ h tor all x means that a ----. a + h. The measure satisfies thel

Isincetransformation x ----. Xx for all x means that a ----. XaJ

11.17 Fractional integral and mass on the real axisl ~et

us consider a distribution of mass with the density p (x) over the region W = {x j The mass of the region isl

ft E la, bl} in Euclidean space R

(1.105~

11.17 Fractional integral and mass on the real axis

351

Iwhere x is dimensionless variable and SI unit of p(x) is kilogram. Using the fracj ~ional integration§

(1.106~

Iwe represent (1.105) in the forml

IMI (W) = ali [x]p(x) + yli [x]p(x),

(1.107~

Iwhere y E [a,b]. A fractional generalization of (1.107) has the forml

~D(W)

= al~[x]p(x) + ylf [x]p (x).

Here we use dlmenslOnless vanables x and y m order to MD has the usual ~Imension. SubstltutlOn of (1.107) mto (1.108) glve§

(1.108~ physIca~

(1.109 Iwhere a :( y :( b. ThIs equation can be wntten a~

(1.11O~ ~n order to let MD([a,b]) not to be dependent on y, we use y = a. Then Eq. (1.11O~ Ihasthe form

IOsmgthe densIty of

state~

lin the I-dimensional space JR, we obtainl

FD-(-W-)-=-l=b-p-(x-)-C I-(D-,x---a-)-d-x·1 [t IS easy to prove that homogeneIty and fractahty propertIes can be reahzed. Let u§ ~onsider p(x) = Po, then homogeneity

1 Fractional Integration and Fractals

136

Iholds, if Ibl - all =

Ih - a21. The fractality means tha~

[f Ibl - all = A,ulb2 - a21. This allows us to use integrals of non-integer order D t9 ~escribe media that have these properties]

11.18 Mass of fractal medial [I'he cornerstone of fractal media IS the non-Integer mass dimenSIOn. The mass dlj ImenslOn of a medIUmcan be best calculated by box-counting method, which meansl klrawing a box of size R and counting the mass inside. The properties of the fractall ~edium like mass can satisfy a power-law relation M rv R D , where M is the mas§ pf the box of size R (or the ball of radIUS R). The number D IS called the mass dl1 fuension. The power-law M rv RD can be naturally derived by using the fractiona~ IIntegral such that the mass dimenSIOn IS connected With the order of the fracttona~ IIntegraL ronsider the region W in 3-dimensional Euclidean space ffi.3. The mass of the re~ gion W in the fractal medium is denoted by MD(W). The fractality of medium meansl ~hat the mass of this medium in any region W C ffi.3 increases more slowly than thel 13-dlmenslonalvolume of thiS region. For the ball region of the fractal medIUm, thl§ Iproperty can be described by the power-law MD(W) rv RD, where R is the radius ofj ~he hall fractal medium is called homogeneous if the power-law MD(W) rv R D does no~ ~epend on the translatton of the regIOn. The homogeneity property of the medlUml ~an be formulated In the form: For all two regions WI and W2 of the homogeneou§ ~ractal medium with the equal volumes VD(WI) = VD(W2), the masses of thesel Iregions are equal MD(Wt} = MD(W2). A wide class of the fractal media satisfie§ ~he homogeneous property. Many porous media, polymers, collOId aggregates, and! laerogels can be conSidered as homogeneous fractal media. Note that the fact that ij Isystem IS porous or random does not necessarily Imply that the system IS fractaLI [1'0 deSCribe the fractal medIUm, we use a continUOUS medIUm model. In thl~ Imodel the fractahty and homogeneity properties can be reahzed In the follOWing rnJ.J:llS:: I- Homogeneity: The local denSity of homogeneous fractal medIUm can be de1 Iscribedby the constant density P (r) = Po = const. This property means that thel ~quatlOns With constant denSity must deSCribe the homogeneous media, I.e., If] r(r) = const and V (WI) = V(W2), then MD(WI) = MD(W2)1 I_ Fractahty: The mass of the ball regIOn W of fractal homogeneous medIUmobeysl la power-law relation M rv R D , where 0 < D < 3, and R is the radius of the ball. I~ IVn(Wt} = XnVn(W2) and p(r,t) = const, then thefractality means that MD(Wt} @ IA,uMD(W2)·1

371

11 18 Mass of fractal media

These two conditions cannot be satisfied if the mass of a medium is described! Iby integral of integer order. These requirements can be realized by the fractiona~ ~quation

MD(W,t)

=

h

p(r,t)dVD,

dVD

=

c3(D,r) dV3,

(1.111

Iwhere r is dimension less vector variable I IWe note that p(r,t) is considered as a distribution function, and c3(D, r) is a denj Isity of states in the Euclidean space ]R3. In general, these notions are different. W~ ~annot reduce all properties of the system to the dlstnbution function. In generalJ Iphyslcal values of a fractal mediUm cannot be descnbed by mtegration of mtegerl prder without a function cn(D,r). The form of function c3(D,r) is defined by thel Iproperties of fractal medium. Note that the final equations that relate the physi-I ~al variables have the form that is independent of numerical factor in the function] h(D,r). However the dependence on r is important to these equations. Note tha~ ~he symmetry of the density of states C3( D, r) must be the defined by the symmetr~ bf the mediUm] [I'he fractal mass dlmenslOn D IS an order of fractional mtegral m (1.111). Therg lare many dIflerent defimtlOns of fractlOnal mtegrals. For the Rlemann-LlOuvl1l~ ~ractlOnal mtegral, we havel (1.112 Iwhere x, y, z are Cartesian's coordmates, D - at + a2 + a3, and 0 < D ::::; 3. Wg ijote that for D = 2, we have the fractal mass dlstnbutlOn m the 3-dlmenslOnal Eu~ ~lidean space ]R3. In general, this case is not equivalent to the distribution on thel 12-dtmenslOnal surface] for p(r) = p(lrl), we can use the Riesz definition of the fractional integrals upl ~o numerical factor If we use the functionl

(1.113~ (1.114~ Therefore we can usel

f( ) 3 D,r

=

3

2 Dr(3/2) I ID- 3

r(D/2)

r

.

(1.115~

[The factor (1.115) allows us to derive the usual integral in the limit D ----+ (3 - 0)1 ~ote that the final equatlOns that relate mass, moment of inertia, and radiUS arel Imdependent of the numencal factorl for the homogeneous medium (p (r) = po = const) and the ball region W = {r j [r] ::::; R}, Equation (1.111) give~

I Fractional Integration and Fractals

138

[Usingthe spherical coordinates, we obtainl

Ks a result, we have M(W) rv RD , i.e., we derive equation M rv RD up to the nuj Imerical factor. It allows us to describe the fractal medIUm with non-lllteger mas§ klimension D by fractional integral of order Dj

11.19 Electric charge of fractal distributionl ~et us consider charged particles that are distributed with a denSity over a fractall IWith box-countlllg dimenslOn D. In the homogeneous case, the electric charge QI Isatisfies the scaling law Q(R) rv RD , whereas for a regular n-dimensional EuclideaIll pbject we have Q(R) rv Rn l [I'he total electric charge that is distributed on the metriCset W Withthe dimenslOIlj ID = 3 with the density p' (r', t) is defined byl

(1.116~ ~v{

= dx' dy' dz'l

~or Cartesian coordinates x', y', z' with dimension

~hat SI unit of Q3 is Coulomb, and SI unit of

[x'] = [y'] = [z'] = meter. We not~ p' is Coulomb- meter 31

[To generalize Eq. (1.116), we represent this equation through the dimensionlessl coordinate variables We can introduce the dimensionless valueS

~ =x'/Ro,

Y = y' /Ro,

z = z'/Ro,

r = r'/Rol

Iwhere Ro is a characteristic scale, and the charge densit5J

Ip(r,t) = R&p'(rRo,t)j IwhereSI unit of p is Coulomb, i.e., [p] = Coulomb. As a result, we obtainl

(1.117~ Iwhere dV3 = dxdydz for dimensionless Cartesian coordinates. This representationl lallows us to generalize Eq. (1.117) to fractal distribution of chargesj

3S1

11.20 Probability on fractals

[Let us consider a fractal distribution of electric charge. Suppose that the densitYI pf charge distribution is described by the function p(r,t) such that SI unit of p i§ ~oulomb. In fractional continuous model of fractal distribution, the total charge inl ~he regIOn W IS defined b5J

PD(W)

=

~VD =

lp(r,t)dvDj c3(D,r)dV3,

Iwhere dV3 = dxdydz for Cartesian coordinates, D = al ~he density of statesl

(1.1l8~

+ a2 + a3, and c3(D,r)

i§

(1.119 IWe note that SI umt of QD IS Coulomb. As a result, we have the RIemann-LtouvIlI~ [ractional integral up to numerical factor 87T;D!2. Note that the final equations, whichl Irelate the phySIcal varIables, are mdependent of numerIcal factor m the functIoIlj b (D, r). However the dependence on r is important to these equations.1 [The functions c3(D, r), which describe a density of states, is defined by the prop-I ~rtIes (for example, symmetry) of the dIstrIbutIOn. For example, If we consIder thel Iball region W = {r: Irl:( R}, and spherically symmetric distribution of chargedl Iparticles (p(r,t) = p(r,t», thelli

for the homogeneous case, p (r,t)

= po, and!

[fhe dIstrIbutIOn of charged partIcles IS called a homogeneous one If all regIOns W] land W2 with the equal volumes VD(Wd = VD(W2) have the equal total charges onl ~hese regions QD(WI) = QD(W2)]

~.20

Probability on

fractal~

~et us consIder a probabIlIty m the framework of fractIOnal contmuous model, ml Iwhlchwe use a fractIOnal mtegratton over a regIOn W mstead of an mtegratton overl la fractal set. [Theprobability, which is distributed on 3-dimensional Euclidean space objectsj ~an be defined byl

(1.120~

~o

Fractional Integration and Fractals

Iwhere p(r,t) is a density of probability distributionj

Ik3 p(r,t) dV3

=

I,

p(r,t)

~ oj

land dV3 = dxdydz for Cartesian's coordinates] [f we consider the probablhty that IS dlstnbuted on the metnc set W with a dlj ImensiOn D, then the probablhty IS defined by the mtegrall

rD(W,t) Iwhere D = aj

=

lp(r,t)dVD'

(1.121~

+ az + a3, andl (1.122]

[he function C3 (D, r) describes a density of state§

(1.123~ [Thedensity of probability distribution p (r, t) satisfies the condition~

~JR3 p(r,t)dVD =

1,

p(r,t)

~ 01

~ote that there are many dtfferent definitiOns of fractiOnal mtegrals. For the Rlemann1 !Liouville fractional integral, the function c3(D,r) i§

(1.124

z areCartesmn's coordmates, andD - al +az +a3, 0 < D ~ 3. As are1 Isult, we obtam Riemann-LiOuville fractiOnal mtegral m Eq. (1.121) up to numenca~ ~actor 8nD/z. Therefore Eq. (1.121) can be considered as a fractional generalization' pf (1.120). OCf P(r) = p (Irl)' then the Riesz definition of the fractional integrals can be used! Iai1d 3 = 2 Dr(3/2) I ID - 3 3 D,r r(D/2) r . Iwherex,y,

f( )

(1.125~

[The definition (1.125) allows us to derive the usual integral in the limit D --+ (3 - O)J for D - 2, EquatiOn (1.121) gives the fractal probablhty dlstnbutiOn m the 31 khmensiOnal space. In general, It IS not eqUivalent to the dlstnbutiOn on the 2j ~imensional surface. Equation (1.124) is equal (up to numerical factor 8n D / Z ) to thel lintegral on the metric set W with box-counting dimension dimB(W) = D. To havel ~he usual dimenSiOnsof the phySical values, we can use vector r, and coordmates xl Iy, z as dimensiOnless vanablesJ

11.21 Fractal distribution of particles

4~

[Equation (1.121) allows us to consider probabilistic processes in fractal media inl framework of fractional continuous model. In order to describe fractal media bYI [ractional continuous model, we use the notions of a density of states C3 (D, r) and! k{istribution function p (r, t). The density of states is a function that describes howl Ipermitted states are closely packed in the space. The function p (r, t) is a distributioij [unction that describes a distribution of probability on a set of permitted states in thel Ispace. To calculate probabilities of some processes in fractal media we can use ani lintegration of non-integer order that takes into account a density of states of fractall Imedium ~he

~.21

Fractal distribution of particles

k\ fractal dIstnbutton IS a dIstnbutIOn of parttcles WIth non-mteger-dImensIOn. Th~ Hausdorff and box-countmg dImenSIOns reqUIre the dIameter of the covenng setij ~o vamsh. In real dIstnbutIOn of partIcle the fractal structure cannot be observe~ pn all scales. In general, we have a charactensttc smaIlest length scale such as thel IradIUs, Ro, of a partIcle (for example, an atom or molecule). Therefore we need ~ IphysIcal analog of Hausdoftl and box-countmg dImenSIOns. To define thIS analogJ Iwe note that the number of partIcles of fractal dIstnbutIOn mcreases as the SIze ofj ~hstnbutIOn mcreases m a way descnbed by the exponent m the "number of partIcle1 IradIUs" relatIOn. For many cases, we can conSIder an asymptotIc form for the relatIOnl Ibetween the number of partIcles N, and the smaIlest baIl of radIUS R contammg thesel Iparttcles

(1.126~ ~or R!Ro » 1. The constant No depends on how the balls of radius Ro are packedl [The parameter D does not depend on whether the packmg of baIls of radIUS Ro I~ pose packmg, a random packmg or a porous packmg WIth a umform dIstnbutIOn o~ Iholes. Usmg relatIOn (1.126), we can define a "partIcle" dImenSIOn as a measure o~ Ihow the partIcles fiIls the spaceJ [rhe fractahty of the dIstnbutIOn of partIcles means that the number of partIcles ml lany regIOn W of EuclIdean space jRn mcreases more slowly than the n-dImensIOnall Ivolume of thIS region. For the baIl region of the fractal dIstnbutIOn, thIS propert)1 Fan be described by the power-law ND(W) rv RD, where R is the radius ofthe ballj fractal distribution will be called homogeneous if the power-law ND(W) rv R~ k10es not depend on the translatIOn of the regIOn. The homogeneIty property of thel khstnbutIOn can be formulated m the form: For all two regIOns WI and W2 of thel Ihomogeneous fractal distribution with the equal volumes VD(Wd = VD(W2), thel Inumbers of particles of these regions are equal: ND(WI) = ND(W2). To describe thel ~ractal dIstnbutIOn, we use a continuous model, in whIch the fractalIty and homo-I geneity properties are realized in the formj

~2

I Fractional Integration and Fractals

I_ The notion of homogeneity means that the local density of number of particlesl [or homogeneous fractal distribution can be described by the constant densitYI b(r) = no = const. This property means that the equations with constant densityl Imust describe the homogeneous distribution, i.e., if n(r) = const and V(Wd @ IV(W2), then ND(WI) = ND(W2)·1 I- The notion of fractality means that the number of particles in the ball regionj Wof fractal homogeneous distribution obeys a power-law relation N(W) rv RD J Iwhere 0 < D < n, and R is the radius of the ball. If Vn(Wd = AnVn(W2) and! &(r,t) = const, then the fractality means thatND(Wd = AD ND(W2)j [I'hese reqUIrements can be realIzed by the fractional equatIOns WIth mtegrals ofj bider D [I'he real fractal structure of the dlstnbutIOn IS charactenzed by an extremelY] ~omplex and irregular geometry. Although the "particle" dimension does not rej Iflect completely the geometric properties of the fractal distribution of particles, i~ Inevertheless permIts to descnbe features of the behaVIOr of these dlstnbutIOnsJ [The number of particles, which are distributed in the region W C jRn with thel ~ensity n' (r' ,t), is defined b~ En(W) =

IdV~

Lnl(r',t)dV~,

(1.l27~

= dx~ ... dx~

~or Cartesian coordinates x~, k = 1, ... , n, with dimension [XI] = ... = [x~] = meter.1 IWe note that SI unit of n' (r', t) is meter n. To generalize Eq. (1.127), we representl ~hls equation through the dImenSIOnless coordmate vanable~

~k

= xURo,

r

= r'IRo,1

Iwhere Ro IS a charactenstlc scale, and the dImenSIOnless denSIty! &(r,t) = R3n'(rRo,t)j

V\s a result, we obtam Eq. (1.127) m the forml

(1.l28~ Iwhere aVn = aXI ...aXn for the dImenSIOnless CartesIan coordmates. ThIS represen-I ~atIOn can be generalIzed to fractal dlstnbutIOn of partlclesJ [n the fractIOnal contmuous model for fractal dlstnbutIOn of partIcles, we usel ~ractIOnal mtegrals over a regIOnof jRn mstead of mtegrals over a fractal set. In orderl ~o descnbe fractal dlstnbutIOn by fractIOnal contmuous model, we use two dIfferentl Inotions such as density of states cn(D, r) and the density of number of particles n(r).1 [Ihe function cn(D, r) is a density of states in the n-dimensional Euclidean space jRn.1 [The density of states describes how permitted states of particles are closely packe~

4~

11.21 Fractal distribution of particles

lin the space jRn. We note that cn(D, r) is a function of the coordinates r such that thel ~xpression cn(D,r)dVn represents the number of states (permitted places) betweenl IVn and Vn + dVn. The density of number of particles n(r,t) describes a distribution] pf number of particles on a set of permitted states in the Euclidean space jRn j IDsmg these notIons, the number of partIcles that correspond to the regIon dVn I§ klefined by the equationl [n the general case, the notIons of densIty of states and dlstnbutIon functIon are dIfj ~erent. We cannot reduce all propertIes of the dlstnbutlon of partIcles to the densltYI pf number of partIcles. ThIS fact IS well-known m statIstIcal and condensed matte~ Iphysics, where the density of states is usually considered as a number of states pe~ lunit of energy or wave vector. Density of states is a property that describes howl Ipermitted states are closely packed in energy or wave vector spaces. For fractal dis-j ~nbutIOns of partIcles m a regIon W, we must use a densIty of permItted states ofj ~he regIon. In the fractIonal contmuous model of fractal dlstnbutIon of partIcles, thel k{ensity of states cn(D, r) in the space jRn must be chosen such tha~

I(l)1D(r,n) ~escnbes

= cn(D,r)dV~

the number of states m the region dVn. For n - 3, we use the

notatIOn~

~o describe densities of states in 3-dimensional Euclidean space jR3.1

IDsmg the fractIonal contmuous model, we can consIder a dlstnbutlon of partIj m the regIOn W C IRn , such that the "partIcle" dImenSIOn of the dlstnbutIOn I~ ~qual to D. We suppose that the densIty of the dlstnbutIOn IS descnbed by the dI1 Imensionless function n (r, t). The number of particles in the region W of jRn will bel k{enoted by ND(W), In the fractional continuous model, the total number of particlesl lIS defined by ~les

IND(W)

=

1

n(r,t)dVD,

(1.l29~

Iwhere r, xi, k - I, ... , n, are dImenSIOnless vanables, andl (1.130~

IWe note that n(r,t) is considered as a density of number of particles, and cn(D,r) i§ la density of states in the region W C jRn. The function cn(D, r) defines a kernel o~ ~ractIOnal mtegral of order D. In general, average physIcal values of fractal dlstnbuj ~ions cannot be described by integration of integer order without functions cn(D, r).1 [The form of function cn(D, r) is defined by the properties of fractal distribution.1 rrhe fractal dImenSIOn D IS an order of fractIOnal mtegral m (1. 129). There arel Imany dIfferent defimtIOns of fractIOnal mtegrals. For the RIemann-LIOuvIlle frac1 ~ional integral, we havel

I Fractional Integration and Fractals

Iwhere xk, k = 1, ... , n, are dimensionless Cartesian's coordinates, and 0 < D :(: n~ IWe note that for D = n - 1, we have the fractal distribution in the n-dimensionall [Euclidean space jRn. In general, this case is not equivalent to the distribution on thel I(n - 1)-dimensional hypersurfaceJ [The symmetries of the density of states C3 (D, r) must be the connected with thel Isymmetries ofthe medium. For n(r,t) = n(lrl), we can use the Riesz definition ofj OChe fractIOnal mtegrals (Samko et aL, 1993; KIlbas et aL, 2006). Thenl

(1.131~ Iwhere D < n. For the fractal homogeneous distribution (n(r,t) = no = const) ofj Iparticles, and the ball region W = {r: Irl:(: R}, Equation (1.129) with (1.131) givesl

IOsmg the sphencal coordmates, we obtaml

k\s a result, we obtain the relation ND(W)

rv

RD up to the numerical factor.1

Referencesl

f. Ben Adda, 1997, Geometnc mterpretatiOn of the fractiOnal denvatIve, Journal of! IFractional Calculus, 11, 21-52j k\.S. Beslcovltch, 1929, On lmear sets of pomts of fractIOnal dImenSIOns, Mathema-I ~ische Annalen, 101, 161-1931 ~.c. Blel, 2003, Analysis in Integer and Fractional Dimensions, Cambndge Um1 Iverslty Press, Cambndgel IV.L. Bonch-Bruevlch, J.P. Zvyagm, R. Kmper, A.G. Mlronov, R. ElderIam, B. EsserJ 11981, Electron Theory of Disordered Semiconductors, Nauka, Moscow, SectiOnl 11.5, 1.6, 2.5. In Russlan1 p. CaIcagm, 2010, Quantum Field Theory, Gravity and Cosmology in a Fracta~ IUniverse, E-pnnt: arXlV:1001.05711 k\. Carpmten, F. Mamardl (Eds.), 1997, Fractals and FractIOnal Calculus Tn Conj ~Tnuum Mechamcs, Spnnger, New York] R-M. Christensen, 2005, Mechanics of Composite Materials, Dover, New York.1 ~.c. Colhns, 1984, Renormallzatwn, Cambndge Omverslty Press, Cambndge. Sec1 Iti.Oi:i::'[

References

451

p.A. Edgar, 1990, Measure, Topology, and Fractal Geometry, Springer, New Yorkj p. Eyink, 1989, Quantum field theory models on fractal space-time. I: Introductionl ~nd overVIew, Communication in Mathematical Physics, 125, 613-636J IK.E Falconer, 1985, The Geometry of Fractal Sets, Cambridge University PressJ K=ambndgeJ IK.E Falconer, 1990, Fractal Geometry. Mathematical Foundations and Applica-I rions, Wiley, Chichester, New Yorkj ~. Feder, 1988, Fractals, Plenum Press, New York, London] R Federer, 1969, Geometric Measure Theory, Springer, Berlinj M. Frame, B. Mandelbrot, N. Neger, 2006, Fractal Geometry,http://classes.yale.eduj ILfractals IR. Gorenflo, 1998, Afterthoughts on interpretation of fractional derivatives and inj Itegrals, in: P. Rusev, I. Dimovski, V. Kiryakova, (Eds.), Transform Methods ancA ISpeClal FunctIOns, Varna, InstItute of MathematIcs and InformatIcs, BulganaIlj IAcademy of Sciences, Sofia, 589-591j f. Hausdorff, 1919, Dimension und ausseres Mass, Mathematische Annalen, 79 J 1157-179. In German] p(mg-Fel He, 1990, FractIOnal dImensIOnahty and fractIOnal denvatIve spectra o~ Imterband optIcal transitions, Physical Review B, 42, 11751-11756j p(mg-Fel He, 1991, ExcItons m anISotropIC solIds: The model of fractIonal-I dimensional space, Physical Review B, 43, 2063-2069j ~. Heymans, I. Podlubny, 2006, PhySIcal mterpretatIOn of InItIal condItIons for frac1 ItIOnal dIfferentIal equatIOns WIthRIemann-LIOuvIlle fractIOnal denvatIves, Rheo-I ~ogicaActa, 45, 765-771. E-pnnt: math-phL0512028J K HIlfer (Ed.), 2000, ApplicatIOns of FractIOnal Calculus In PhySICS, World SCIenj ItIfic PublIshmg, SmgaporeJ k\.J. Katz, A.H. Thompson, 1985, Fractal sandstone pores: implications for conduc-I ItlVlty and pore formatIOn, Physical Review Lettetrs, 54, 1325-1328.1 IS. Kemptle, I. Schafer, 2000, FractIOnal dIfferentIal equations and mItIal condItIOns) IFractional Calculus and Applied Analysis, 3, 387-400.1 k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj kwnal Dijjerentwl EquatIOns, ElseVIer, AmsterdamJ IV. KIryakova, 1994, Generalized Fractional Calculus and Applications, Longman,1 IHarlow and WIley, New Yorkj K=. KIttel, 2004, Introduction to Solid State Physics, 8th ed., WIley, New YorkJ WI. Kroger, 2000, Fractal geometry m quantum mechanICS, field theory and spml Isystems, Physics Reports, 323, 81-181J ~. KugamI, 2001, Analysis on Fractals, Cambndge UnIVerSIty Press, Cambndgej ~.I. Kulak, 2002, Fractal Mechanics oj Materials, VIsheIshaya Shkola, Mmsk. Inl IRussian ~.A.T. Machado, 2003, A probabIlIstIc mterpretatIOn of the fractIOnal-order dIffer-I entiation, Fractional Calculus and Applied Analysis, 6, 73-80.1 f. Mainardi, 2010, Fractional Calculus and Waves in Linear Viscoelasticity: Ani IIntroductwn to Mathematical Models, World SCIentIficPublIshmg, Smgaporej

f'l-6

1 Fractional Integration and Fractals

r. Mainardi, 1998, Considerations on fractional calculus: interpretations and applij ~ations, in: P. Rusev, I. Dimovski, V. Kiryakova (Eds.), Transform Methods and] ISpecial Functions, Varna1996, Institute of Mathematics and Informatics, Bulgarj lian Academy of Sciences, Sofia, 594-597 j lB. Mandelbrot, 1983, The Fractal Geometry of Nature, Freeman, New YorkJ V\,.c. McBride, 1979, Fractional Calculus and Integral Transforms of Generalized] IFunctions, Pitman Press, San Francisco] k'\,.e McBnde, 1986, Fractional Calculus, Halsted Press, New York1 K Metzler, J. Klaner, 2000, The random walk's gmde to anomalous diffUSIOn: 3j Ifractional dynamics approach, Physics Reports, 339, 1-77 j R Metzler, J. Klaner, 2004, The restaurant at the end of the random walk: recen~ f:!evelopments in the description of anomalous transport by fractional dynamicsj flournal oj Physics A, 37, RI61-R208.1 IK.S. MIller, E. Ross, 1993, An IntroductIOn to the FractIOnal Calculus and FracJ, klOnal Differential EquatIOns, WIley, New YorkJ M. Monsrefi-TorbatI, J.K. Hammond, 1998, PhySIcal and geometrIcal InterpretatIOnl pf fractional operators, Journal oj The Franklin Institute B, 335, 1077-1086.1 ~.w. Montroll, M.E ShlesInger, 1984, The wonderful world of random walks, In:1 IStudies in Statistical Mechanics, Vo1.11, 1. LebOWItz, E. Montroll (Eds.), North], Holland, Amsterdam, 1-121) k'\,.M. Nahushev, 2003, FractIOnal Calculus ans Its ApplcatlOn, Nauka, Moscow. Inl IRussian p.R. Newkome, P. Wang, eN. Moorefield, T.l. eho, P.P. Mohapatra, S. LI, S.H.I IHwang, O. Lukoyanova, L. Echegoyen, J.A. Palagallo, V. Iancu, S.W. RIa, 2006J INanoassembly of a fractal polymer: A molecular SIerpInskI hexagonal gasket] IScience, 312, 1782-1785j KR. NIgmatulhn, 1992, Fractional Integral and ItS phySIcal Interpretation, Theoretj licaland Mathematical Physics, 90, 242-251.1 IK. NIshImoto, 1989, Fractional Calculus: Integrations and Dijjerentiations oj Ar1 IfJltrary Order, UmversIty of New Haven Press, New HavenJ IK.E. Oldham, J. Spamer, 1974, The FractIOnal Calculus: Theory and AppltcatlOn~ r.f DijjerentlatlOn and IntegratIOn to Arbitrary Order, AcademIC Press, New YorkJ Palmer, P.N. Stavnnou, 2004, EquatIOns of motion In a non-Integer-dImensIOnall Ispace, Journal oj Physics A, 37, 6987-7003J ~. Podlubny, 1996, Fractional Dijjerential Equations, AcademIC Press, New YorkJ [. Podlubny, 2002, Geometnc and phySIcal InterpretatIOn of fractIOnal IntegratIOi1j and fractional differentiation, Fractional Calculus and Applied Analysis, 5, 367-1

r.

s:sn: k'\.A. Potapov, 2005, Fractals in Radiophysics and Radiolocation, 2nd ed., Umver-I ISItetskaya Kniga, Moscow. In RussIanJ V\,.V. Pshu, 2005, Equations with Partial Derivatives of Fractional Order, NaukaJ IMoscow. In RUSSIan] r.y. Ren, 1.R. Liang, X.T. Wang, w.y. Qiu, 2003, Integrals and derivatives on ne~ Ifractals, Chaos, Solitons and Fractals, 16, 107-1171

IR eferences

471

f.- Y. Ren, Z.-G.

Yu, F. Su, 1996, FractIOnal mtegral associated to the self-sImIlar se~ pr the generalized self-similar set and its physical interpretation, Physics Lettersl lA, 219, 59-681 f.Y. Ren, Z.-G. Yu, J. Zhou, A. Le Mehaute, RR Nigmatulhn, 1997, The relatIon-I Iship between the fractIOnal mtegral and the fractal structure of a memory setJ IPhysicaA, 246, 419-429J ~.A. Rogers, 1998, Hausdoiff Measures, 2nd ed., Cambridge University PressJ K:ambridgej lB. Ross, 1975, A briefhistory and exposition ofthe fundamental theory offractionall ralculus, Lecture Notes in Mathematics, 457, 1-36J lB. Rubin, 1996, Fractional Integrals and Potentials, Longman, Harlow.1 IR.S. Rutman, 1994, On the paper by R.R. Nigmatullin Fractional integral and it§ Iphysical mterpretatIOn, Theoretical and Mathematical Physics, 100, 1154-1156J IR.S. Rutman, 1995, On physical interpretations of fractional integration and differ-I entiation, Theoretical and Mathematical Physics, 105, 1509-1519J ~. SabatIer, O.P. Agrawal, J.A. Tenrelro Machado, (Eds.), 2007, Advances In Fracj rional Calculus. Theoretical Developments and Applications in Physics and Enj Igineering, Spnnger, DordrechtJ IS.G. Samko, A.A. KIlbas, 0.1. Manchev, 1993, Integrals and Derivatives oj Frac1 rwnal Order and Applzcatwns, Nauka I Tehmka, Mmsk, 1987, m Russianj ~nd FractIOnal Integrals and Derivatives Theory and Applzcatwns, Gordon and! IBreach, New York, 19931 ~. SchweIzer, B. Frank, 2002, Calculus on hnear Cantor sets, Archiv der MathematikJ 179, 46-50. IH.M. Snvastava, S. Owa, (Eds.), 1989, Univalent Functions, FractIOnal Calculu~ land Thelr Applzcatwns, PrentIce Hall, New JerseyJ k\.A. Stamslavsky, 2004, ProbabIhty mterpretatIOn of the mtegral of fractIOnal orderJ ITheoretical and Mathematical Physics, 138, 418-431J k\.A. Stams1avsky, K. Weron, 2002, Exact solutIOn of averagmg procedure over thel Cantor set, Physica A, 303, 57-66j f.R. StIllmger, 1977, AXIOmatIc baSIS for spaces WIthnonmteger dImenSIOns, Jourj Inal ofMathematical Physics, 18, 1224-1234J ~.S. Stnchartz, 1999, AnalYSIS on fractals, Notices oj the American Mathematicall ISociety, 46, 1199-1208J IRS. Stnchartz, 2006, Difjerential Equations on Fractals, Pnnceton Press, Pnnce1

rron: IK. SvozIl, 1987, Quantum field theory on fractal spacetIme: a new regulanzatIOnl Imethod, Journal oj Physics A, 20, 3861-3875.1 IV.B. Tarasov, 2004, FractIOnal generahzatIOn of LIOuvIlle equations, Chaos, 14J 1123-127 ~.E. Tarasov, 2005a, Continuous medium model for fractal media, Physics Letterss IA,336,167-174j ~E. Tarasov, 2005b, Fractional hydrodynamic equations for fractal media, Annal~ pi Physics, 318, 286-307J

~8

I Fractional Integration and Fractals

FE. Tatom,

1995, The relatIOnshIp between fractIOnal calculus and fractals, Fracj

tst; 3, 217-229l ~.v.

Uchaikin, 2008, Method of Fractional Derivatives, Artishok, Ulyanovsk. Inl IRussian f.S. Uryson, 1951, Works on Topology and Other Fields of Mathematics VoI.1-2j Gostehizdat, Moscow. In Russian] 1B.1. West, M. Bologna, P. Grigolini, 2003, Physics of Fractal Operators, Springerj INew York IZ.-G. Yu, F.-Y. Ren, J. Zhou, 1997, FractIOnal Integral assocIated to generahze~ ~ookie-cutter set and its physical interpretation, Journal of Physics A, 30, 5569-1

ssn.

p.M. Zaslavsky, 2002, Chaos, fractional kinetics, and anomalous transport, Physicsl IReports, 371, 461-580J p.M. Zaslavsky, 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford Unij IversIty Press, Oxford.1 ~v. Zverev, 1996, On conditions of existence of fractal domain integrals, Theoretij fal and Mathematical Physics, 107, 419-426J

~hapter21

IHydrodynamics of Fractal Medi~

~.1

Introduction

~n the general case, real medIa are charactenzed by an extremely complex and Ir1 Iregular geometry. Because methods of EuclIdean geometry, whIch ordmanly deal~ IWIth regular sets, are used to descnbe real medIa, stochastIc models m hydrodY1 InamIcs are taken mto account (Momn et aI., 2007a,b; Ostoja-StarzewskI, 2007aj ~ishik et aI., 1979; Vishik and Fursikov, 1988; Shwidler, 1985). Another possi-j Ible way of descnbmg a complex structure of the medIa IS to use fractal theory ofj Isets of non-mteger-dImensIOnalIty (Mandelbrot, 1983; Frame et aI., 2006; FederJ ~988). Although, the non-mteger-dImensIOn does not reflect completely the ge01 Imetnc and dynamIC propertIes of the fractal medIa, It however permIts some Imj Iportant conclusIOns about the behavIOr of the medIa. For example, the mass of thel OCractal medIa enclosed m a volume of charactenstIc SIze R satIsfies the scalmg lawl IM(R) rv R D , whereas for a regular n-dimensional Euclidean object M(R) rv R n . W~ ~efine a fractal medIUm as a medIUm wIth non-mteger mass dImensIOn. In general) fractal medium cannot be defined as a medium that is distributed over a fractal I W'Jaturally, m real medIa the fractal structure cannot be observed on all scales bu~ pnly those for WhICh R[ < R < R2, where R[ IS the charactenstIc scale of the partIj pes (molecules), and R2 IS the macroscopic scale for umformIty of the mvestIgatedl Istructure and processes.1 ~n general, the fractal medIa cannot be conSIdered as contmuous medIa. Therel lare pomts and domams that are not filled of the medIUm partIcles. We suggest dej Iscribing the fractal media by special continuous model (Tarasov, 2005a,b). We usel ~he procedure of replacement of the fractal medIUm wIth fractal mass dImenSIOn bYI Isome contmuous medIUm model that IS descnbed by fractIOnal mtegrals. ThIS pro-I ~edure IS a generalIzatIOn of Chnstensen approach (Chnstensen, 2005) and It lead~ Ius to the fractional integration (Samko et aI., 1993; Kilbas et aI., 2006) to describ~ [factal media. The integrals of non- integer orders allow us to take into account fracj ~al properties of the mediaj

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

fSO

2 Hydrodynamics of Fractal Media

[n many problems the real fractal structure of matter can be disregarded and thel can be represented by a fractional continuous model. In order to describ~ ~he medium with non-integer mass dimension, we use the fractional integrationj [I'heorder of these mtegrals is defined by the fractal mass dimenSiOn of the medmm] [I'he fractional contmuous model allows us to descnbe dynamiCs of fractal media] fractional integrals are used to derive the generalizations of the equations of balanc~ I(Vallander, 2005) for the fractal mediaj [n Sections 2.2-2.8, we derive the fractional generalizations of the integral equaj ~ions of balance of mass denSity, momentum denSity, and mternal energy. Osmgj ~he fractional contmuous model, we obtam the correspondent differential equation~ IWith denvatives of mteger orders to descnbe balance of mass denSity, momentuml klensity, and internal energy of fractal media. In Sections 2.9-2.10, the generaliza-I ~ions of Navier-Stokes and Euler equations for fractal media are considered. In Secj ~ions 2.11-2.12, the eqmhbnum equation for fractal media and generahzation ofj ~ernoulh mtegral are suggested. In Sections 2.13-2.14, the sound waves m the fracj ~ional contmuous model for fractal media are considered. Fmally, a short conclusiOIlj ~s grven m SectiOn 2.15J ~edium

~.2

Equation of balance of mas~

[The fractiOnal contmuous model can be used not only to calculate the total physi-I ~al values such as the mass, electnc charge, number of particles. It can be used tq ~escnbe the dynamiCs of fractal media. We consider fractional generahzations of mj ~egral balance equations for fractal medmm. Let W be a regiOn of the medmm. Thg Iboundary of this region is denoted by aw. Suppose that the medium in the region ill Ihas the mass dimension D, and the medium on the boundary aw has the dimensioili ~. In general, the dimension d is not equal to 2 and is not equal to (D - 1)J IWe descnbe fractal media by fractional contmuous model, where we use notionsl pf a density of states cn(D, r) and distribution function p(r,t). The density of state~ ~n(D,r) is a function in the n-dimensional Euclidean space JRn, that describes howl Ipermitted states of particles are closely packed m the space JRn. The expressiOlll rn(D, r) dVn represents the number of permitted places (states) between Vn and Vn-a KlVn. We use the notatiOn~

~o descnbe denSities of states m n-dimensiOnal Euchdean spaces With n = 1,2,3] [The density of mass p (r, t) is a function that describes a distribution of the mass onl la set of penllltted states m the Euchdean space JRnJ ~et us consider the mass of a regiOn W of a fractal medmm With the mass di1 rIension D. The density of the mass distribution is described by the function p (r, t) such that r is dimensionless vector variable, and SI unit of p (r, t) is kilogram. In thel [ractional continuous model of fractal media, the mass is defined bYI

5~

12.3 Total time derivative of fractional integral

Iwhere we can usel

IWe note that SI umt of MD is ktlogram. The balance of the mass is descnbed by thel ~quation

~MD(W) ~hat can

=0

be represented m the forml d I

( p(r,t)dVD =

dtlw

o.

[The mtegral (2.3) is considered for the reglOn W, which moves With the medmml [The field of the velocity is denoted by u = u(r,t). We note that SI unit of u i~ Isecond 1. [n the fractional contmuous model, we use fractional mtegrals (Samko et aI.J ~993; Ktlbas et aI., 2006) over a reglOn of jRn mstead of mtegrals over a fractal set.1 [The dimenslOn D of fractal media can empmcal estimated by drawmg a box of sizel IR and countmg the mass mSide. The dimenSion D m the fractional mtegral equation~ liS a mass dimenSion of the fractal medmm] IWe can denve dtfferential equations, which are connected With fractional mtegrall Ibalance equatlOns (Tarasov, 2005b). To obtam these differential equatlOns, we mus~ Ihave a formula for the total time denvative of fractlOnal volume mtegrals and ~ generahzatlOn of the Gauss theoreml

~.3

Total time derivative of fractional integra]

[n the fractlOnal contmuous model of fractal media, the total time denvative of thel ~ractional volume integral for the value A = A (r, u, t) is defined byl

=i

Iwhere u = ukek is a vector velocity field, Un = un(r,t) is defined by Un = (u,o) IUknb and 0 = nkek is a vector of normal. The surface integral for the boundary aWl ~an

be represented as a volume mtegral for the region W by usmg a generahzatlOnl bf the Gauss theorem] for contmuous media, the Gauss theorem is wntten m the forllll

f52

2 Hydrodynamics of Fractal Media

= d(Auk)

div(AU)

I

dXk 1

I

Here, and later we mean the sum on the repeated mdex k and I from 1 to 3. Usmgj (2.6) Iwe

obtain

for d = 2, we have c2(2, r) = 1. From the Gauss theorem (2.5), we hav~ (2.7)

ISubstItutiOn of the equatIoIlj

) dV3, dVD = C3( D,r

-DT(3/2) I ID-3 ( ) -_ 2 T(D/2) D,r r

C3

(2.8)

lin the form dV3 = c 3 I (D,r)dVD into Eq. (2.7) give~

[ThIS equation can be consIdered as a generalIzatiOn of the Gauss theorem for fractall

mema: k\s a result, the total tIme derIvatIve (2.4) of the fractiOnal volume mtegral Iil Irepresented b5J

(2.10)1 for the mteger dImensiOns d - 2 and D - 3, we have the usual equatiOn. To sImplIfYI ~he form of equations, we mtroduce the notatiOn~

(2.11)1

c(D,d,r)

=

c}1(D,r)c2(d,r)

=

2D-d-lT(D/2) I T(3/2)r(d/2) Ir1d+I-Dj

1

~ote

that the rule of the term-by-term dIfferentIatiOn for the operator (2.11) IS reaJ1

Ii.Zea:

(2.12)1

5~

12.3 Total time derivative of fractional integral

[Equation (2.11) defines a generalization of the total time derivative for fractionall ~ontinuous models. We note that the media with integer dimensions (D = 3, d = 2) Ihave c(D,d,r) = Ij [..:et us define the generalIzed divergencfj (2.13)1 IWe note that operator (2.13) is not a fractional divergence. The fractional differentia~ land integral vector operations were discussed in (Tarasov, 2008). Substitution ofj l(D,r) andc2(d,r) into(2.13)give~

b

(2.14) [Using the derivative of function A = A (r, t) with respect to the coordinate§ (2.15)1 ~he

generalIzed dIvergence (2.13) can be represented byl

for D - 3 and d - 2, we hav(j

IWe note that the rule of term-by-term differentiation for the operator Isatisfied:

Vp

is no~

IVf(AB) # AVf(B) +BVf(A)j rt'he operator

Vp satisfies the rulej [VP(AB) = AVP(B) +c(D,d,r)BVlA.

~n the general case,

(2.16)1

Vp (1) # 0, and we hav~

ff(l) =

c(D,d,r) (d - 2);~ 1

psmg (2.11) and (2.13), Equation (2.10) can be rewntten in the fori11l

d t

rAdVD Jwr (!!.-dt

Jw

=

A +A DivD(U)) dVD.

(2.17)

[I'hIS equatiOn descnbes a total tIme denvative of the fractiOnal volume mtegral ofj ~he value A = A(r, u, t) for fractal media]

f54

2 Hydrodynamics of Fractal Media

~.4

Equation of continuity for fractal medi31

[..:et us obtaIn dIfferentIal equatIOn that IS connected wIth the fractIonal Integral baH lance equationl

I~

{

dtlw

p(r,t)dVD

=

o.

(2.18)1

[To derive this differential equation, we consider A = p(r,t) in Eq. (2.17). Substitutj ling A = p(r,t) into Eq. (2.17), we getl

[I'hen Eq. (2.18) has the forml

IL

((:t) DP +pDiVD(U)) dVD

(2.19)1

= O.

IWIthout loss of generalIty It can be assumed that Eq. (2.19) IS satIsfied for all regIOn~ IW. As a result, we obtaInI [

~) D p+pDivD(u) =0.

(2.20)1

dt

~quatIOn

(2.20) can be consIdered as a contInUIty equatIon for fractal medIa. Equa-I (2.20), whIch IS obtaIned from the fractIOnal Integral balance equatIOn, IS no~ la fractIOnal dIfferentIal equatIOn. We note that fractIOnal dIfferentIal equation o~ k:onservatIOn of mass was dIscussed In (Wheatcraft and Meerschaert, 2008).1 lOSIng Eq. (2.14), we can reWrItten (2.20) asl ~Ion

(2.21) lOSIng the relatIOnl

~quatIOn

(2.21) IS represented byl

-£- +c(D,d,r)(u,gradp) + c(D,d,r)p (diV(U) + (d - 2 ) *)

= 0,

(2.22)

Iwhere (u,r) = ukxkl fractal medium is called homogeneous if the power-law MD(W) rv RD doesl Inot depend on the translatIOn of the regIOn. The homogeneIty of the medIUml Imeans that for all two regions WI and W2 of the homogeneous fractal medIUml Iwith the equal volumes VD(WI) = VD(W2), the masses of these regions are equa]

551

12.5 Fractional integral equation of balance of momentum

IMD(Wd = MD(W2). In the fractional continuous model, the local density of homo-I geneous fractal medium is described by the constant density p (r) = po = const. Fo~ ~he homogeneous media, Equation (2.22) give§

[I'herefore the velOCIty of fractal homogeneous medIUm IS the non-solenOIdal field] li.e., div(u) i' OJ IWe note that the continuity equation includes the density of momentuml r(r,t) u(r,t). To obtain the equation for the density of momentum, we considerl ~he mass force and sufface force]

~.5

Fractional integral equation of balance of momentuns

OC:ookIng at a regIOn W WIth a finIte volume, we see that rate of change In momen-I ~um In the regIOn must be equal to the momentum flux croSSIng the boundaneij I(Vallander, 2005), I.e., the rate of momentum entenng the volume mInUS the rate o~ rIomentum eXItIng. USIng the fractIOnal contInUOUS model for fractal medIa, we canl pbtaIn (Tarasov, 2005b) an equatIOn for the denSIty of momentum of the medIa. Le~ ~he force f = 1kek be a functIOn of the dImenSIOnless vector r and tIme t. The forc~ ~M (W), which acts on the mass MD(W) of the medium region W, is defined b~

(2.23)1 [he force FS(W), which acts on the surface of the boundary aw, is defined b~

(2.24)1 Iwhere p = p(r,t) is a density of the surface force, and Pn = Pktnket. Here, n = nkekl lIS the vector of normaU ~fthe mass dMD(W) = p(r,t)dVD moves with the velocity u, then the momen-I ~um of thIS mass I~

IdP(M)

=

dMD(W)u(r,t)

=

p(r,t)u(r,t)dVD.1

[The momentum P of the medium mass MD(W) that is situated in the region W i§ ~efined by

Ip(W) = kp(r,t)u(r,t)dVD' [Theequation of balance of momentum isl

(2.25)1

f56

2 Hydrodynamics of Fractal Media

(2.26)1 ISubstItutmg Eqs. (2.23)-(2.25) mto Eq. (2.26), we obtaml

(2.27)1 [This fractional integral equation describes the balance of momentum of fractall Imedmm. For D - 3 and d - 2, EquatIon (2.27) gIves the usual mtegral equatIoIlj I(Vallander, 2005) of the momentum balance for continuous medium]

~.6

Difl'erential equations of balance of momentuIij

OC:et us derIve dIfferentIal equatIons, whIch follow from the tractIonal mtegral equaj ~IOn of balance of momentum (2.27). Usmg the generalIzed Gauss theorem (2.9)J ~he suftace mtegral (2.27) can be represented a~

[wPndSd

=

j~w C2(d,r)PndS~

1=

fw a(C2~~lr)Pl) c3

I(D,r)dVD

= fw VfPldVDJ

[ThenEq. (2.27) has the forml

~ r PUdVD= Jwr (pf+vfpI)dVD. dt Jw

(2.28)1

for the components of vectors u - Ukek. f' -ikek. and PI - Pklek. Equation (2.28)1 gives

(2.29)

IVsmg the total tIme derIVatIve of tractIOnal volume mtegral (2. I 7) wIth A = PUk. wei 6liliW:i

[ThenEq. (2.29) can be represented a~

(PUk) + (PUk) DivD(u)-p/k-Vfpkl D

rrhISequation IS satIsfied for all regions W. As a result, we

hav~

571

12.7 Fractional integral equation of balance of energy

d dt

(2.30)

IOsmg the rule of the term-by-term differentiatIOn (2.12) for the generahzed totall ~Ime derIvative (2.11), Equation (2.30) can be represented asl (2.31) [I'he contmmty equatIon (2.20) for fractal medIUm reduces (2.31) to the foriTIj

f(:t)

D

Uk -

p/k -

Vfpkl = 0,

k = 1,2,3.

(2.32)1

[These differential equations describe the balance of density of momentum for fractall medium

~. 7

Fractional integral equation of balance of energy

OC=et W be a regIOn of fractal medIUm wIth fractal mass dImensIOn D. The rate ofj ~hange m the energy wlthm the regIOn of fractal medIa IS dIrectly related to the ratel pf energy conducted mto the regIOn W. In the general case, the denSIty of mternall ~nergy for inhomogeneous medium depends on the space-time point (r,t), i.e., e @ r(r,t). The internal energy dE(W) ofthe mass dMD(W) is equal tal ~E(W) =e(r,t)p(r,t)dVDl

for the regIOn W of fractal medIa, the total mternal energy lsi

IE(W) =

L

e(r,t)p(r,t)dvDj

[The kinetic energy dT(W) of the mass dMD(W) ~he velocity u = u(r,t), is equal t9

= p(r,t)dVD,

which moves withl

for the regIOn W, the kmetic energy I§

IT(W) =

fw u2~,t) p(r,t)dVD·1

V\s a result, the total energy IS a sum of the kmetic and mternal energle~

f58

2 Hydrodynamics of Fractal Media

[he change of the energy is described bYI

(2.33)1 Iwhere AM(W) is the work of mass forces, AM(W) is the work of surface forcesj Qs(W) is the heat that is influx into the region for the time interval dt J IOsmg the fractional contmuous model for fractal medIa, we state that the mas§ ~MD(W) = P dVD is subjected to the force fp dVD. Then the work of this force i§ I(u, f)p dVDdt, where (u, f) = Udk. As a result, the work ofthe mass forces for thel Iregion W is defined bYI

~AM(W)

= dt

1

(u,f) p(r,t)dVD.

(2.34)1

[he fractional continuous model allows us to consider a surface element dSd subj the force PndSd. The work of this force is (Pn,U)dSddt. Then the work ofj ~he suftace forces for the regIOn W I§

~ected to

(2.35)1 [The heat that is influx into the region W through the surface aw i~

(2.36)1 Iwhere qn = (0, q) = nkqk is the density of heat flow. Here, 0 is the vector of normall ISubstitutmg Eqs. (2.34), (2.35) and (2.36) mto (2.33), we obtaml

~ L(~+e) p(r,t)dV~ (u,f) p(r,t) dVD +

(2.37)

k\s a result, the velocIty of the total energy change IS equal to the sum of power ofj Imass force and the power of suftace forces, and the energy flow from through thel Isuftace. EquatIOn (2.37) IS as fractIOnal mtegral equatIOn that deSCrIbes the balancel pf energy of fractal medIUm. For D = 3 and d = 2, Equation (2.37) gives the usuall lintegral equation (Vallander, 2005) of the energy balance for continuous mediumj

~.8

Differential equation of balance of energ)j

!Letus derIve dIfferentIal equatIOn, WhICh follows from the fractIOnal mtegral equaj ~ion of balance of energy. Using Eq. (2.17) for A = p(u 2 / 2 + e), we rewrite thel Ileft-hand SIdeof Eq. (2.37) asl

5~

12.8 Differential equation of balance of energy

I~ 1 (~+e) p(r,t)dvJ

t1((~) DP(~

+e) +p(~ +e)

DiVDU) dVD·1

[Using the rule of term-by-term differentiation, we obtainl

~1 (~+e)p(r,t)dvJ

t1

(p

(~ ) (~+ e) +((~ ) D

D

+

P P DivD

u) (~+ e) ) dVD·1

[I'he equatIon of contInUIty gIvesl

~ 1 (~+e) p(r,t)dVD = 1(p(r,t) (~) (~+e)) dV4 D

1=

1(pu (~)

D

U+P

(~) /(r,t)) dVD.

(2.38)1

lOSIng the generalIzed Gauss theorem for fractal medIa, the surface Integrals In thel IfIght-hand sIde of Eq. (2.37) are represented by the fractIonal volume Integralij

(2.39)1 (2.40)1 ISubstItutIOn of Eqs. (2.38)-(2.40) Into Eq. (2.37) gIveij

K(PU (~)

D

u+P (~) D e(r,t)) dV4

Ej~ ((u,f)p + Vf(Pi'U) + Vfqk) dVD. for the components of vectors U -

Ukeb

f - ikeb and Pi -

Uk+P

(~)

Pkiek,

~quatIOn

11 (PUk (~)

r1

D

D

e)dvDI

(PUdk + Vf(PkiUk) + Vfqk) dVD.

10 SIng the rule (2.16) In the forml

we have thel

~o

2 Hydrodynamics of Fractal Media

Iwe represent Eq. (2.42) a§

(2.43) ~quatlon

(2.43) can be rewntten m the forml

~ (p (~) /-c(D,d,r)pkIV}Uk -

Vfqk) dV4

1= - fw Uk (p (~ )D Uk + Pfk + vfPkl) dVD·1 [Using the momentum balance equations (2.32), we

ge~

(2.44) [I'hIS equatIon holds for all regIOns WJ k'\s a result, we obtaml

pIfferentIal equation (2.45) descnbes the balance of densIty of energy for fractall Imedium

~.9

Euler's equations for fractal medial

lIn the framework of the fractIOnal contmuous model, we denve the fractIOnal mte~ gral balance equatIOns for fractal medIa. The correspondmg dIfferentIal equatIOn~ lareequatIOns wIth denvatIves of mteger ordersj [. The equatIOn of contmUItyl

12. The equation of balance of densIty of momentuml

13. The equation of balance of densIty of energyl

12.9 Euler's equations for fractal media

611

[n Eqs. (2.46), (2.47) and (2.48), we mean the sum on the repeated mdex k and [rom 1 to 3. The generalized total time derivative is defined bYI

IWe also use the generalIzed nabla

~

operato~

(2.50)1 Iwhere r =

Irl, Xb k = c(D,d,r)

1,2,3, are dimensionless variables, an~ =

a(D,d)rd + 1- D ,

[I'he differential equations of balance of denSIty of mass, denSIty of momentum and! ~ensIty of mtemal energy make up a set of five equatIOns, whIch are not c10sedl [n addition to the hydrodynamic fields e(r,t), u(r,t), e(r,t), Equations (2.47) and! 1(2.48) include the tensor of viscous stress pkl(r,t) and the vector of thermal fluxl m;cr,t). IWe can consider a special case of the set of Eqs. (2.46)-(2.48). Assume that aI ~ractal medIUm IS defined b5J

IPkl

= -pOkl' qk = 0,

(2.51)1

Iwhere P = p(r,t) is the pressure. Then Eqs. (2.46)-(2.48) for this medium arel

(f) De = -eVPUk, (f) =!k - ~vpp, D Uk

(~) D e = -c(D,d,r) ~ Vk Uk .

(2.52)1

(2.53)1 (2.54)1

[Now we have a closed set of Eqs. (2.52)-(2.54) for the fields e (r, t), uk(r,t), p(r, t] ~hat descnbe the hydrodynamICs of the fractal medIa WIth (2.51). ThIS set of equa1 ~IOns IS generalIzatIOn of the Euler equations for fractal medIa.1

~2

2 Hydrodynamics of Fractal Media

~.10

Navier-Stokes equations for fractal medial

[The equations of balance (2.46)-(2.48), besides the hydrodynamic fields p(r,t)J lu(r,t), e(r,t), include the tensor of viscous stress pkl(r,t) and the vector ofthermall Iflux qk(r,t). Let us consider a special case of the tensor Pkl = Pkl(r,t). According ~o Newton's law, the force of viscous friction is proportional to the relative velocitYI pf motion of medmm layers, that is to the gradient of the relevant component ofj Ivelocity. It can be assumed that tensor pkl(r,t) is symmetrical, and characterize~ ~he dissipation due to viscous friction. The general form of tensor of viscous stressJ Iwhich satisfies the above requirements, is determined by two constants J.l and gJ Isuch that (2.55)

rrhe coefficient ~ is called the coefficient of internal viscosity because it reflects thel ~xistence of internal structure of particles. In case of structureless particles g = OJ for fractal media, we can conSider a generahzatlOn of the Stokes law (2.55) ml [he form (2.56)

Iwhere V~ is defined by (2.50). In the general case, the generalization of the Stoke~ Ilaw can be descnbed by fractional denvatives With respect to coordmates mstead ofj

~

ir~n:cLfth::-:e:-s::Ltu=-=-dlyC:-:Co~fLh-=-ea:::-:;t:-:;t-=-ra::-:n:-::s~fe::-:r"i-=-nLflcu:-'-id""dCC:y-=-n-=-amcc::-,-ic::-:s:-,TIfl=-=-u=-=-x-'-is=-:Tde:::-:;fiCC:n::-:e::-:d'a::-:s::-:tCLh--:Ce--=a--=m::-:o::-:u::-:n:LtL1thc-a::-oJ~

Iflows through a unit area per unit time. The vector of heat flux qk = qk(r,t) can bel klescnbed by the empmcal Founer lawl (2.57)1

Iwhere T = T (r, t) is the field of temperature. The value of heat conductivity X canl Ibe found expenmentally. In the general case, we can conSider the generahzatlOn~ pf the Founer law (2.57) for fractal media. For example, we can assume that thel Founer law for fractal media has the forml (2.58)1

[n general, the generahzatlOn of Founer law can be descnbed by fractlOnal coordi-I Inate derivatives] V\s a result, we have a closed set of Eqs. (2.46)-(2.48), (2.55) and (2.57) fo~ ~he fields p(r,t), uk(r,t), T(r,t). These equations can be considered as a set o~ Ihydrodynamics equatlOns for fractal media. A generahzatlOn of the equatlOns ofj ~heory of elasticity for fractal solids can be obtained in a similar wayj IWe can conSider a speCialcase of the hydrodynamics equations for fractal medi~ 1(2.46)-(2.48), (2.55) and (2.57), where the coefficients J.l, S and A are constantsj

6~

12.11 Equilibrium equation for fractal media

for homogeneous VISCOUS fractal media, EquatIOn (2.46) gIve§

Ks a result, we have a non-solenoidal field of the velocity (div( u) ~he relation

Id'IV (U )--

~quation ~quations

nl vmU

_

m -

(2 - d)XkUk • XIXI

i- 0), that satisfie§ (2.59)1

(2.47) with qk = 0 and (2.55), gives a generalization of Navier-Stokesl for fractal media in the forml

~quations ~=

(2.59) and (2.60) form a closed system of 4 equations for 4 fields uk(r,t)l 1,2,3, and p(r,t). Equations (2.60) can be rewritten in an equivalent forml

1= pfk -

c(D,d, r)VkP + J1 c(D,d, r)V;ukl (2.61)1

~f c(D,d,r) = 1 and VP(I) = 0, then Eqs. (2.61) have the usualform of the Navier~ IStokes equatIOns]

~.11

Equilibriumequation for fractal medial

~qmhbnum IS descnbed by condItIOns when neIther ItS state of motIon nor ItSmterj Inal energy state tends to change wIth time. The eqmhbnum state of medIa IS defined! Iby the conditIon~

~=o,

2 Hydrodynamics of Fractal Media

[or the hydrodynamic fields A larerepresented in the forml

= {p, Uk, e}. In this case, the hydrodynamic equation~

~quation

= 0 gives pn = -pOkl' Using Eq. (2.47), we obtainl

(2.55) with JUJ!Jxk

(2.62)1 from the FOUrIer law, we hav¢1

(2.63)1 ~quatIOns

(2.62) and (2.63) are generalIzatIOns of the eqUIlIbrIum equatIOns on thel

fractal mediaJ

IWe note that Eq. (2.62) can be rewntten m the foririi

for the homogeneous medium, p(r,t) = Po = const, an~

[f the force

fk IS a non-potentIal force such thatl (2.64)1

h(d,r)p+ poU

= const.

(2.65)1

[This equation describes equilibrium of the fractal media in the force field (2.64). Ifj ~he force fk IS potentIal, then the eqUIlIbrIum does not eXIsts.1

~.12

Bernoulli integral for fractal medial

[3emoullI mtegral of the equatIOns of hydrodynamICs IS an mtegral, WhIch deterj Immesthe pressure at each pomt of a statIOnary flow of an Ideal homogeneous flUId! pr a barotropIc gas m terms of the velOCIty of the flow at that pomt and the potentIall ~nergy per umt mass. To derIve an mtegral for fractal medIa, we conSIder the equaj ~ion of balance of momentum density (2.47) with the tensor Pkl = - pOkl. Using thel Irelation

651

12.12 Bernoulli integral for fractal media

land Eg. (2.47), we obtainl

(2.66)1 [f the potentIal energy U and pressure pare tIme-mdependent field~

~=o,

(:t)D =C(D,d,r)~.

(2.67)1

11k = -c(D,d,r)VkU,

(2.68)1

for the non-potentIal forc~

~quatIOns

(2.68) and (2.66) gIvel

k'\s a result, we obtaml

~3 L -u2 +U +P(d) = const. =1

(2.69)1

2

[ThIS mtegral of motIon can be consIdered as a generahzatIOn of Bernoulh mtegrall pn fractal medIa. If the forces lk are potentIal, then thISgenerahzatIOn does not eXIst.1 for the densIty! -1 nd/2) 2-d P = Poc2 (d,r) = Po 2 2- d r: ,

(2.70)1

1

~he

mtegral (2.69)

g1Ve~

~ + PoU(D,d) +c2(d,r)p =

const.

(2.71)1

IWe note that Eq. (2.71) wIth u = 0 leads to the eqmhbnum equation (2.65) ron-potentIal force (2.68) and densIty (2.70)1

fo~

~6

2 Hydrodynamics of Fractal Media

12.13 Sound waves in fractal medial ISoundwaves eXist as vanations of pressure and denSity m media. They are created! Iby the vibration of an object, which causes the medium surrounding it to vibratej IWe can consider the small perturbations of density and pressur~ Ip Iwhere p'

«

= Po+p',

P = po+ p',

(2.72)1

Po, and p' « Po. The values Po and Po describe the steady statel lapo = caL

0,

1

Vkpo =0,

apo =0

at

1 VkPO = 0.

1

'

for fractal media, EquatiOns (2.52) and (2.53) With fk =

a have the forml (2.73)1

(2.74)1

ISubstitutiOn of (2.72) mto Eqs. (2.73) and (2.74) gives the equations for the prder of the perturbatiOnsj

firs~

(2.75)1 (2.76)1

[0 obtain the independent equations for perturbations p' and p', we consider thel Ipartial derivative of Eq. (2.75) with respect to timej

!J2P' au' IJt2 = -pvr a/' ISubstituting (2.76) into (2.77), we getl

1!J2P' t:atI = for adiabatiC processe~

~ = p(p,s),

nDnD ,

vkvkP,

(2.78)1

p' = vZp'J

k\s a result, we obtaml (2.79)1

12.14 One-dimensional wave equation in fractal media

67]

I~ = V2 VPVp p'.

(2.80)1

[These equations describe the waves in the fractal medium. For D = 3, we have thel lusual wave equationsj

~.14

One-dimensional wave equation in fractal

for I-dImenSIOnal case (n - 1), where D Ihave the forml

< 1 and C2 -

medi~

1, EquatIons (2.79) and (2.80)1

(2.81)1 Iwhere u(x, t) denotes the perturbations pi and pi, and CI(D,x) is a density of state§ bn a line such tha~

~quatlon

(2.81) deSCrIbes a wave that moves along a fractal medIUm hne. Let u§ a solutIOn for the wave equatIon (2.81). We wIll conSIder the regIOn 0 ~ x ~ ~ land the condItIonsJ ~erIve

au ~(x,o) =a(x), ar(x,O) = b(x),

I

~(O,t) = 0,

u(l,t)

=

01

[The solutIOn of Eq. (2.81) has the forml

!Here, an and bn are the Fourier coefficients for the functions a(x) and b(x)j

Ian =

IIYnll-2l a(x)Yn (x)dID = IIYnll-2l Cj(D,x)a(x)Yn(x)dxj

Ibn =

IIYnll-2l b(x)Yn(x)dID = IIYnll-2l Cj(D,X)b(x)Yn(x)dxj

Iwhere diD = Cj (D,x)dlj, dlj

= dx, andl

[The eigenfunctions Yn (x) satisfy the conditionl

~8

2 Hydrodynamics of Fractal Media

[he eigenvalues An and the eigenfunctions Yn(x) are defined as solutions of thel ~quation

Iwhere D~

= d n 7dx n . This equation can be rewritten a§ (2.82)1

[he solution of (2.82) has the forml

~(x) = Clx l-D/2t; (XVX) +C2xl-D/2yy (XVX) ,I Iwhere Iy(x) are the Bessel functions of the first kind, Yy(x) are the Bessel function§ pfthe second kind, and v = 11 - D/211 V\s an example, we consider the case that is defined bYI

11=1,

v=l,

O:S;x:S;l,

a(x)=x(l-x),

b(x)=OJ

rI'he usual wave has D - I and the solutIon IS ----'-------,,;'---;;-----'-- sin( nnx) cos( nnt) Ilf D = 0.5, thenl

[The eigenvalues Xn are the zeros of the Bessel functionl

for example,

IXI

~

4.937,

X2 ~ 9.482, X3 ~ 13.862, X4 ~ 18.310, AS ~ 22.756J

[The approximate values of the eIgenfunctIons an are followmg]

lal ~ 1.376,

a2 ~ -0.451,

a3 ~ 0.416,

a4 ~ -0.248,

as

~ 0.243~

[I'he solutIOn of the wave equatIOn l§

[Thisfunction describes the waves in I-dimensional fractal media with D - 0 5J

References

69

12.15 Conclusionl [n thIS chapter, we consIder hydrodynamICs of fractal medIa that are descnbed by 3j OCractIOnal contmuous model (Tarasov, 2005a,b). In general, the fractal medIUmcanj rot be considered as a continuous medium. There are points and domains that arel rot filled of partIcles. We suggest (Tarasov, 2005a,b) to consIder the fractal medIij las specIal contmuous medIa. We use the procedure of replacement of the medIUml IWIth fractal mass dImensIOn by some contmuous model that uses the fractIOnal mtej 19rals. This procedure can be considered as a generalization of Christensen approachl I(Chnstensen, 2005) that leads us to the fractIOnal mtegratIOn to descnbe fractal mej klia. Note that fractional integrals can be considered as integrals over the space withl [ractional dimension up to numerical factor (Tarasov, 2004, 2005c,d). The fractiona~ Imtegrals are used to take mto account the fractahty of the medIaj [I'he fractIonal contmuous models of fractal medIa can have a WIde apphcatIOnj [I'his is due in part to the relatively small numbers of parameters that define a fractall rIedIUm of great complexIty and nch structure. In many cases, the real fractal struc1 ~ure of matter can be dIsregarded and we can descnbe the medIUm by a fractIOna~ ~ontmuous model, m whIch the fractIOnal mtegratIon IS used. The order of fractIona~ Imtegral IS equal to the fractal mass dImensIOn of the medIUm. The fractIOnal conj ~muous model allows us to descnbe dynamICS of fractal medIa (Tarasov, 2005b)1 Fractional continuous models can be formulated to describe fractal media in thel ~ramework of the theory of elastICIty (Sokolmkoff, 1956) and the non eqUIhbnuml ~hermodynamics (De Groot and Mazur, 1962; Gyarmati, 1970). We note applicaj ~Ions of fractIOnal contmuous models by Ostoja-StarzewskI to the thermoelastIcItYJ I(Ostoja-StarzewskI, 2007b), and the thermomechamcs (Ostoja-StarzewskI, 2007c),1 ~he turbulence of fractal medIa (OstoJa-StarzewskI, 2008), the elastIc and melastIC1 ImedIa WIth fractal geometnes (OstoJa-StarzewskI, 2009a), the fractal porous medIa] I(Ostoja-StarzewskI, 200%) and the fractal sohds (LI and Ostoja-StarzewskI, 2009)j [I'he hydrodynamIC accretIOn m fractal medIa (Roy, 2007; Roy and Ray, 2007,2009)1 Iwas conSIdered by Roy and Ray by usmg a fractIOnal contmuous model. We notg ~hat graVItatIOnal field of fractal dIstnbutIOn of partIcles and fields can also be con1 ISIdered m the framework of fractIonal contmuous models (Tarasov, 2006); see alsCl I(CaIcagm, 2010). ApphcatIOns of fractIOnal contmuous models to descnbe fractall khstnbutIOns of charges and probabIhty are conSIdered m the next chaptersj

lReferencesl

p.

CaIcagm, 2010, Quantum FIeld Theory, GravIty and Cosmology In a Fracta~ IUniverse, E-print: arXiv: 1001.0571 j IR.M. Chnstensen, 2005, Mechanics oj Composite Materials, Dover, New York.1 IS.R. De Groot, P. Mazur, 1962, Non-Equilibrium Thermodynamics, North-Holland,1 IAmsterdamJ [. Feder, 1988, Fractals, Plenum Press, New York, Londonj

[70

2 Hydrodynamics of Fractal Media

M. Frame, B. Mandelbrot, N. Neger, 2006, Fractal GeometryJ Ihttp://classes. yale.edu/fractal~ Gyarmati, 1970, Non-Equilibrium Thermodynamics: Field Theory and Variaj klOnal PrincIples, Spnnger, BerlIn] k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj tional Differential Equations, Elsevier, Amsterdam] [. Li, M. Ostoja-Starzewski, 2009, Fractal solids, product measures and fractiona~ Iwaveequations, Proceedings ofthe Royal Society A: Mathematical, Physical ancA IEngineering Sciences, 465, 2521-2536J lB. Mandelbrot, 1983, The Fractal Geometry of Nature, Freeman, New York] k\.S. MOllIn, A.M. Yaglom, 2007a, StatIstIcal FLUId MechanIcs: MechanIcs of Tur-I bulence, Volume 1, Dover, New York; Translated from Russian: Nauka, Moscow] ~.

~

k\.S. MOllIn, A.M. Yaglom, 2007b, StatIstIcal FLUId MechanIcs: MechanIcs of Tur-I lfJulence, Volume 2, Dover, New York; Translated from RUSSIan: Nauka, MoscowJ Il262. ~. OstoJa-StarzewskI, 2007a, Microstructural Randomness and Scaling in Mechan-I lics oj Materials, Chapman and Hall, CRC, Taylor and FranCIS, Boca Raton, Lon1 klon, New YorkJ M. Ostoja-StarzewskI, 2007b, Towards thermoelastIcIty of fractal medIa, Journal of! IThermal Stresses, 30, 889-896j ~. OstoJa-StarzewskI, 2007c, Towards thermomechallIcs of fractal medIa, Zeitsch1 Irift jur angewandte Mathematik und Physik, 58, 1085-1 096J ~. OstoJa-StarzewskI, 2008, On turbulence III fractal porous medIa, Zeitschriftjurl 'angewandte Mathematik und Physik, 59, 1111-1118~ M. Ostoja-StarzewskI, 2009a, Extremum and vanatIonal pnnCIples for elastIc and! linelastic media with fractal geometries, Acta Mechanica, 205, 161-170.1 ~. OstoJa-StarzewskI, 20095, ContIlluum mechallIcs models of fractal porous me1 ~ha: Integral relatIOns and extremum pnncIples, Journal oj Mechanics oj Materi1 lals and Structures, 4, 901-912j W'J. Roy, 2007, On sphencally symmetncal accretIOn III fractal medIa, Monthly Noj kices ofthe Royal Astronomical Society, 378, L34-L38J W'J. Roy, AK. Ray, 2007, CntIcal propertIes of sphencally symmetnc accretIOn Illi ~ fractal medIUm, Monthly Notices oj the Royal Astronomical Society, 380, 733-1 l14Q.

W'J. Roy, AK. Ray, 2009, Fractal features

III accretIOn dISCS, Monthly NotIces of the, IRoyalAstronomial Society, 397, 1374-1385~ IS.G. Samko, AA KIlbas, 0.1. Manchev, 1993, Integrals and Derivatives oj Frac1 kional Order and Applications, Nauka I TehllIka, MIllSk, 1987, III RussIanj ~nd Fractional Integrals and Derivatives Theory and Applications, Gordon and! IBreach, New York, 1993J M.I. Shwidler, 1985, Statistical Hydrodynamics of Porous Media, Nedra, MoscowJ lIn RussianJ ItS. SokolllIkoff, 1956, Mathematical Theory of Elasticity, 2nd ed., McGraw-HIIIJ INew YOrk

IR eferences

711

IVB. Tarasov, 2004, FractIOnal generalIzatIOn of LIOuvIlle equatIOns, Chaos, 14J 1123-127 IVB. Tarasov, 2005a, Contmuous medIUm model for fractal medIa, Physics Lettersl lA, 336, 167-174j ~E. Tarasov, 2005b, Fractional hydrodynamic equations for fractal media, Annal~ 'Pi Physics, 318, 286-3071 IVB. Tarasov, 2005c, FractIOnal systems and fractIOnal BogolIubov hIerarchy equaj ItIOns, Physical Review E, 71, 011102J ~E. Tarasov, 2005d, Fractional Liouville and BBGKI equations, Journal ofPhysics.j IConferenceSeries, 7, 17-33.1 ~E. Tarasov, 2006, Gravitational field of fractal distribution of particles, Celestia~ Mechanics and Dynamical Astronomy, 94, 1-15J ~.E. Tarasov, 2008, Fractional vector calculus and fractional Maxwell's equationsj IAnnals ofPhysics, 323, 2756-2778j IS. v. Vallander, 2005, Lectures on Hydroaeromechanics, 2nd ed., St. Petersburg Statel IUmversIty. In RussIan.1 M.1. VIshIk, A.v. FurSIkov, 1988, Mathematical Problems oj Statistical Hydrome1 f:;hanics, Kluver, Dordrecht; Translated from RUSSIan: Nauka, Moscow, 1980j M.I. VIshIk, A.1. Komech, A.v. FursIkov, 1979, Some mathematIcal problems o~ Istatistical hydromechanics, Uspekhi Matematicheskikh Nauk, 34, 135-21Oj IS. W. Wheatcraft, M.M. Meerschaert, 2008, FractIOnal conservatIOn of mass, Adj Ivances in Water Resources, 31, 1377-1381.1

~hapter~

~ractal

~.1

Rigid Body Dynamics

Introduction

V\ ngId body IS an IdealIzatIOn of a solId body of fimte SIze m whIch defonnatIOnl lIS neglected. RIgId bodIes are charactenzed as bemg non-defonnable, as oppose~ ~o deformable bodIes. The dIstance between any two gIven pomts of a ngId bodyl Iremams constant m tIme regardless of external forces exerted on It. We can use thel Iproperty that the body IS ngId, If all ItS partIcles mamtam the same dIstance relaj ~Ive to each other. Therefore It IS sufficIent to descnbe the posItIon of at least threg ron-collInear partIcles. The ngId body dynamIcs IS the study of the motIon of ngI~ Ibodles. UnlIke pomt partIcles, whIch move only m three degrees of freedom (trans1 Ilatlon m three dIrectIOns), ngId bodIes occupy a regIOn of space and have spatIall IpropertIes. The mam propertIes of a ngId body are a center of mass and momentsl pf mertIa, that charactenze motIon m SIX degrees of freedom such as translatIOns ml ~hree directions and rotations in three directions I ~n claSSIcal dynamICS a ngId body IS usually conSIdered as a contmuous mas~ ~hstnbutIOn, whIle a ngId body IS a set of pomt partIcles such as atomIC nucleII land electrons. In the general case, the ngId bodIes can be charactenzed by fractall mass dImenSIOns. We define a fractal ngId body as a dIstnbutIOn of pomt partIclesl l(atomIc nucleI and electrons) that can be conSIdered as a mass fractal m a wldel Iscale range. We conSIder fractal ngId bodIes by usmg a generalIzatIOn of Chns1 ~ensen approach (Chnstensen, 2005), whIch allows us to represent the fractal bodyl las a contmuous medIUm. In many problems the real fractal structure of ngId bodlesl ~an be dIsregarded, and we can replace by some speCIal contmuous mathematIca~ model. Smoothmg of the mIcroSCOpIC charactenstIcs over the physIcally mfimtesIj Imal volume transforms the ImtIal fractal ngId body mto speCIal contmuous modell I(Tarasov, 2005a,e,b,d) that uses the fractIOnal mtegrals. The order of fractIOnal mte1 gral IS equal to the fractal mass dImenSIOn of the body. The fractIOnalmtegrals allowl Ius to take into account the fractal properties of the media. In the framework of fracj ~ional continuous model, we describe the fractal rigid bodies by using the fractiona~ Imtegrals. Note that the fractIOnal mtegrals can be conSIdered as an approximation o~ V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

[74

3 Fractal Rigid Body Dynamics

lintegrals on fractals (Ren et a1., 2003). The fractional integrals also can be considj ~red (Tarasov, 2004, 2005c,g) as mtegrals over the space wIth fractional dimensIOi1j lup to numerIcal factor. In the fractIOnal contmuous model (Tarasov, 2005a,b,d) ofj ~factal ngld body all charactenstIcs and fields are defined everywhere m the volumel Ibut they follow some generalIzed equations, whIch are denved by usmg mtegrals ofj ron-integer order. The order of the integral is defined by the mass dimension of thel [factal rigid bodyj [The laws of motion for a rigid body are known as Euler's laws (Goldstein, 2002)j [I'he first of Euler's laws descnbes "translatIonal" motion of the ngld body, l.e., thel ~hange of the velocIty of the center of mass. The second of Euler's laws descnbesl Ihow the change of angular momentum of the ngld body IS controlled by the momentl pf forces and couples applied on the body. The laws of Euler are written relative t9 Ian mertIal reference frame. In Ret. (Tarasov, 2005d), we proved that equations ofj ImotIon for fractal ngld body have the same form as the equations for usual ngHII Ibody. In the framework of fractional contmuous model of fractal medIa, we sugges~ ~he approach to compute the moments of mertIa for fractal ngld body, and conslderl ~he possIble expenmental testmg of the model for fractal ngld body. The waves onl ~he fractal solId strIng were consIdered m (Tarasov, 2005f)j IWe note that mtegrals and denvatlves of non-mteger order (Samko et aL, 1993)J land fractional integro-differential equations (Kilbas et al., 2006) have found manyl lapplIcatIons m recent studIes m mechamcs of contmuous medIa (MamardI, 2010j rarpmten and MamardI, 1997; Carpmten and CornettI, 2002; Uchmkm, 2008). Me1 ~hamcs of fractal matenals WIthoutapplIcatIOnof fractIOnal calculus was dIscussed! 1m (Ivanova et aL, 1994; SpecIal Issue, 1997; Vostovsky et aL, 2001; Kulak, 2002j fark, 2000; Zolotuhin, 2005; Tsujii, 2008)j [n Section 3.2, the fractional equation for moment of mertIa IS suggested. Inl ISectIOns 3.3-3.4, we consIder the moments of mertIa for fractal ngld body ball and! ~ylInder. In SectIOn 3.5, equatIOns of motion for fractal rIgId body are dlscussedj ~n SectIOn 3.6, we descrIbe the pendulum WIth fractal rIgId body. In SectIOn 3.7J la fractal ngld body rollmg down an mclIned plane are consIdered. Fmally, a shortj ~onc1usIOn IS gIven m SectIOn 3.8.1

~.2

Fractional equation for moment of inertial

Moment of mertla IS a measure of a body's reSIstance to changes m ItS rotatIOi1j Irate. It IS the rotational analog of mass, the mertIa of a ngld rotatmg body wlthl Irespect to its rotation. The moment of inertia has two forms, a scalar form I(t)J Iwhlch IS used when the aXIS of rotatIOn IS known, and a more general tensor fori11l ~hat does not reqUIre knowmg the aXIS of rotatIOn. The scalar moment of mertI@ I(often called SImply the "moment of mertla") allows an analysIs of many simplel Iproblems in rotational dynamics. The scalar moment of inertia of a rigid body withl ~ensity p' (r', t) with respect to a given axis is defined by the volume integrall

13.2 Fractional equation for moment of inertia

75]

(3·1)1 Iwhere I(r'l I is the perpendicular distance from the axis of rotation, an~

!Letr' be a distance to a point (not the perpendicular distance) of the rigid body, suchl Iilllit

r'

t3I

=

Lx~ebl

~

Iwhere~, k = 1,2,3, are components oh'. Equation (3.1) can be broken into comj Iponents in the tensor form of the moment of inertial

(3.2) ~or

a continuous mass distribution. Here Oki is the Kronecker delta. Depending onl ~he context, /ki may be viewed either as a tensor or a matrix. We note that SI unit o~

1/£, is kg m2 , i.e., [/£,] = kg . m2 J [To generalIze Eq. (3.2), we represent thIS equatIOn through the dImensIOnlessl Eoordinate variables We can introduce the dimensionless valueil

kk = xU 1o ,

r = r' /loJ

Iwhere fo IS a characterIstIC scale, and the densIt5J

Ip(r,t) = 16P'(r1o,t)J lSI unit of

e is kg, i.e., IeI = kg. We define the following moments of inertiij

k'\s a result, we obtaml

Iwhere aV3 = aX] aX2aX3 for CarteSIan coordmates, and the values xb k = 1,2,3 arel ~imensionless. We note that SI unit of hi is kg, i.e., [hi] = kg. This representationl laIIows us to generalIze Eg. (3.3) to the fractal bodyJ fib deSCrIbe fractal rIgId bodIes, we can use the fractIOnal contmuous modell I(Tarasov, 200Sa,b,d), where fractIOnal mtegrals (Samko et aI., 1993; KI1bas et aLJ 12006) are considered. The fractional generalization ofEg. (3.3) has the forml (3.4)

176

3 Fractal Rigid Body Dynamics

[I'he scalar moment of mertIa for a fractal ngid body with respect to a given aXiS i§ klefined by the mtegrall

Iwhere D is a fractal mass dimension of the fractal bodyj [Thesefractional integral equations describe the moments of inertia of fractal rigid! Ibodies (Tarasov, 2005d). The moment of mertIa tensor is symmetnc, l.e.,1

lIn Eq. (3.4), l~f) denotes the moment of inertia around the k-axis when the ob~

~ects are rotated around the I-axis. The diagonal elements of l~f) with k = I arel ~alled the principal moments of inertia. The values l~f) with k i= I are called thel Iproducts of mertIa. The pnncipal moments are given by the entnes m the diagonal-I lized moment of mertia matnx. The pnncipal axes of a rotatmg body are defined byl lfinding values of X such thatl Iwhere W = Wkek is the angular velocity vector. The moment of merna tensor mayl Ibe diagonahzed by transformmg to appropnate coordmate systeml

~.3

Moment of inertia of fractal rigid body bal.

[The moment of mertia of a fractal body can be calculated by Eqs. (3.4) and (3.5). Le~ Ius consider a fractal ngid ball with radms R, and mass M. Note that the componentl pf the radms perpendicular to the z-aXiS m sphencal coordmates i§

Iwhere if) is the angle from the z-aXiS. Osmg the fractiOnal equatIon (3.5), we

l~ D ) = 2 -Dr(3/2) r(D/2)

lR12lrllr p(r,t) (r 0

0

0

hav~

sinq,? ~-lsinq,dq,d8dr.

[I'his equatiOn can be rewntten m the forml

l~ D ) = 2 - T(3/2) T(D/2)

10 sing the

vanable~

lR12lrllr p(r,t)~+l (1 0

0

0

cos 2 q,) sin q, dq,d8dr.

(3.6)

711,

13.3 Moment of inertia of fractal rigid body ball

lu

=

cosifJ,

du

= -

(3.7)1

SInifJ difJ,

OChe Integral (3.6) can be wntten simply and solved by quadrature. For homogeneou§ [factal rigid body ball (p(r,t) = Po), we ge~

[fhls equatIOn can be represented In the forml

[ntegrations with respect to u,

~(D) z

e, and r givel 6

n2 Dr(3/2) D+2 =3(D+2)r(D/2)poR .

~quation (3.8) defines I~D) through the density Po. We can represent I~D) throughl ~he

mass MD of the rIgid bodyl [fhe mass of the fractal ngld body ball IS defined byl

for spherical coordinates

l/J,

e, r, Equation (3.9) give~

K=hanging variables (3.7) for homogeneous rigid body ball (p(r,t)

= Po), we get thel

~quatIOn

[ThiS equation can be rewntten MD =

a~

2 - T(3/2)

T(D/2)

[ntegrations with respect to u,

t'

Po io

r ~+I dr io

71:

de

1+ -I

1

du

e, and r givel

I""Y1 D _ n2 S - Dr(3/2) r

2

-

Dr(D/2)

D

PoR.

IWe see that D IS a fractal mass dimenSIOnof the rIgid body balq ISubstItutIon of Po from (3.10) Into (3.8) gives the moment of InertIa for fractall Irigid body ball in the forml

[78

3 Fractal Rigid Body Dynamics

~(D)

=

z

2D M R2 3(D+2) D .

[f D - 3, then we have the usual relatiOij

for D = (2 + 0), Equation (3.11) give§

IWe note that fractal rigid body baIl with dimension D = (2 + 0) cannot be considere~ las a spherIcal sheIl that ha~

II~2) = ~MR2 j for fractal rigid body baIl, we have the homogeneous distribution of fractal matte~ 1m the volume. Because of the symmetry of the baIl, each prIncipal moment is thel Isame, so the moment of mertla of the ball taken about any dtameter is (3.11 )J from Eq. (3.11), we obtain that IJD) and IJ3) are connected b~

~t

can eastly be checked thatl

Is /D) ~ < I~3) ~ [or 2

I

1

< D ~ 3. For example, we have I~D) lIP) = 10/11 for D = 2.4. We note tha~

~he deviation liD) from IP) is no more than 17 percent]

~.4

Moment of inertia for fractal rigid body cylinderl

OC:et us conSider a homogeneous fractal rIgid body cylmder W With the aXiS z. Th~ Imomentof merna of the cyltnder Withinteger mass dimenSiOnis defined by equatiOili (3.12)1

!Equation (3.12) can be rewritten in the forml

~F) =

Po

h(x2+l)dS21 dz;

7S1

13.4 Moment of inertia for fractal rigid body cylinder

Iwhere dS2 = dxdy, and x = XI, Y = X2, Z = X3 are dimensionless Cartesian coordi-I rates. We define a fractional generalization of Eq. (3.13) in the forml (3.14)1

(3.15)1

IzlJ3-1

1-1

t J3

=

r(f3) dz,

ISubstitution ofEq. (3.15) into (3.14) give§ (3.16) IWe use the numerical factor c(a) in Eq. (3.14) such that the limits a --+ (2 - 0) land [3 --+ (1- 0) give usual integral equation (3.13). For a = 2 and [3 = 1, Equatioij 1(3.14) gives (3.13). The parameter a IS a fractal mass dimenSIOn of the cross-sectlOnl pf cylInder. ThiS parameter can be easy calculated from the experImental data. It canl Ibe computed by box-countIng method for the cross-sectIOn of the cylInder.1 OC=et us consider the cylIndrIcal region W that IS defined b5J (3.17)1 [n the cylindrical coordinates (4), r, z)J (3.18)1 ISubstituting (3.18) into (3.16), we obtainj li a ) = 2npoc(a) I

[ntegratlOns With respect to

r(~)

r

R

Jo

ra+ldr

t" zJ3-ldzl Jo 1

z and r glv~ (3.19)1

[I'hls equatIOn defines the moment of Inertia of fractal rIgid body cylInder. If land [3 = 1, we get the well-known equatioij

a - 21

3 Fractal Rigid Body Dynamics

180

[he mass of the homogeneous medium cylinder (3.17) with D lis defined by

= 3, a = 2, {3 = 11 (3.20)1

["hen we havel

OC'et us consIder the fractIonal generalIzatIon of Eq. (3.20). The mass of fractall ImedIUm cylInder (3.17) can be defined a§

r

D =

Po

1 1 dS a

(3.21)1

dl[3,

Iwhere dS a and dl[3 are defined by Eq. (3.15). Using the cylindrical coordinates, wei pbtam the mass of fractal rIgId body cylInder m the forml

lu

rH D

= 2npoc(a)

r(f3)

r

R

a-Id

Jo r

r

H

rJo

[3-1d Z

1

z.

[ntegratlons WIth respect to Z and r gIVi.j

(3.22)1 ISubstItutmg (3.22) mto (3.19), we getl

r~a) = a~2MDR2,

(3.23)1

Iwhere a IS a fractal mass dImensIOn of cross-sectIon of the cylmder (l < a ,,;; 2)J ~ote that Eq. (3.23) has not the parameter /3. If a = 2, we have the weIl-knowlJl lrelation

OCor the homogeneous cylInder that has the mteger mass dImensIOn D - 3, and a - 2J [f we consIder the fractal medIUm cylInder WIth the mass and radIUS that are equall ~o mass and radIUS of the cylInder WIth mteger mass dImenSIOn, then the momentsl pf inertia of these cylInders are connected by the equatIOnl

(3.24)1

~ere IF) is the moment of inertia for the homogeneous medium cylinder with inte~ ger mass dImensIOn D - 3, and a - 2. If 1 ,,;; a ,,;; 2, thenl

8~

13.5 Equations of motion for fractal rigid body

lAs a result, the moments of inertia of the rigid body cylinders, which have equall and radiuses, and the fractal mass dimensions of cross-section equal to oj land 2, are connected b)j ~asses

~ a-2 z -1+ --

e-

(3.25)1

a+2'

[Theparameter a can be calculated by box-counting method for the cross-section ofj ~he cylinder.

~.5

Equations of motion for fractal rigid bod)j

k\ngular momentum IS a physIcal vector quantIty that IS useful In descrIbIng thel IrotatIOnal state of a physIcal system. EquatIOns of motIon of rIgId body deSCrIbe thel langular momentum dL of the massl ~MD(W) =

p(r,t)dVDJ

IwhIchmoves wIth the velOCIty v, IS equal tol ~L =

[I'he angular momentum L -

Lkek

[r,v] dMDJ

of rIgId body wIth D

-

3 IS defined by the equatIoIlj

(3.26)1 Iwhere I , I is a vector product, and the vector r = xkek is a dimensionless radiusl Ivector. The tractIOnal generalIzatIOn of Eq. (3.26) has the foririi

(3.27)1 [ising v = [co, r], we obtain that the moment of inertia I~f) is related to the momentl pf momentum L byl Iwhere CO = COkek IS the angular velOCIty vectorl OC:et us conSIder a fractal rIgId body wIth one POInt fixed. If the angular momen-I ~um L IS measured In the frame of the rotatIng body, then we have the equatIOIlj

~ + [co,L] = N,

(3.28)1

Iwhere N - Nkek IS the torque (moment of force). For components, Equation (3.28)1 Ihasthe form

3 Fractal Rigid Body Dynamics

182

(3.29)1 Iwhere Cklm IS the permutatIOn symbol. It the prIncIpal body axes are chosen, thenl ILk

= I~D) Wk, and Eq. (3.29) give~ (3.30)1

Iwhere liD) = liD), liD) = I~D), and I~D) = I~D), are the principal moments of inertiaj k\s a result, we obtaInI (3.31)1 (3.32)1

~(D) dWz + (/D) z dt y

_ /D))W x

fIl.. -

X-y -

N

z·

(3.33)1

~quatIOns (3.31)-(3.33) are the Euler's equatIons of motIon for fractal rIgId bodY] I(Tarasov, 2005d). As a result, the equations of motion for fractal rigid body hav~ ~he same form as the equatIons for usual rIgId body WIth Integer mass dImensIOIlj I(Tarasov,200Sd)·1

~.6

Pendulum with fractal rigid bodj1

~n

Ref. (Tarasov, 200Se), we note that the Maxwell pendulum WIth fractal rIgId bodyl be used to test the fractIOnal contInUOUS model of fractal medIUm. Usually, thel Maxwell pendulum IS used to demonstrate transformatIOns between gravItatIOna~ Ipotentlal energy and rotatIOnal kInetIc energy] IWe consIder the Maxwell pendulum as a cylInder that IS suspended by strIng. Th~ IstrIng IS wound on the cylInder. Let the aXIS Z be a cylInder aXIS, then the equatIOn~ pf motion for the pendulum arel ~an

(3.34)1 (3.35)1 Iwhere ay(a) = dVyjdt is an acceleration of the fractal rigid body cylinder, Cz ~ ~wz/ dt, and g is the gravitational acceleration (g ~ 9.81m/s Z) . Using ay( a) = czRj ~quation (3.35) gives the string tensionl (3.36)1

8~

13.6 Pendulum with fractal rigid body

ISubstItutIng (3.36) Into (3.34), we obtaulJ

(3.37)1 ~guatIOn

(3.37) gIve§ (3.38)1

ISubstituting Eg. (3.23) into (3.38), we obtainl I~

rA a )

=

a+2 2a+2 g,

1< a

~

2.

(3.39)1

for the cylinder with integer mass dimension of the cross-section (a = 2), we havel

py(2) = (2/3)g:=:::: 6.54 m/s z. Using the equationl

lay(~)t6 Iwhere

=

Lj

Lis a string length, we obtain the period TJ a) of oscillation for the penduluml ~(a) (2L 1'0 = 4to =4 y~.

~quatIOns

IIf 1 <

a

~

(3.39) and (3.40) give the relatIOIlI

2, thenl

V\s a result, two ngId body cylInders wIth equal masses and radIUses have the m01 ~ents of inertia such that (3.25), where a is a fractal mass dimension of cross1 IsectIOn of the cylInder (0 < a ~ 2). The parameter a can be measured by box-I ~ountIng method for the cross-sectIOn of the cylInder. The penods of OSCIllatIOn fofj ~he pendulums are defined by Eq. (3.41). Note that the deviation a) from is nol more than 6 percent. Therefore the precision of the experiments must be high. Thes~ ~quatIOns allow us to use an expenmental determInatIOn of a fractal dImenSIOnal fo~ ~ractal ngId body by measurements of penods of oscIllatIOnsl

TJ

TJ2)

3 Fractal Rigid Body Dynamics

184

~. 7

Fractal rigid body roIling down an inclined

plan~

~onsIder a fractal rIgId ball of mass MD and radIUs R, rollIng down a plane InclIned! lat an angle a with the horizontal. Let us assume that the fractal ball rolls withou~ Islipping. The condition for rolling without slipping is that at each instant, the poin~ pf contact is momentarily at rest and the ball is rotating about that as axis. The centr~ pf mass of the cylInder moves In a straIght lIne] lOSIng the law of conservatIon of energy, we havel

~

_ MDV

Dgh-

2

+

2

I~D) ro 2

'

Iwhere the moment of inertia liD) is defined by (3.11). We assume that the body pnly rolling not gliding. Then it is valid:1

d

Iwhere R IS a radIUs of the rollIng ball. SubstItutIon of (3.44) and (3.11) Into (3.43)1 gives

lHere

'h = Lsina,

(3.46)1

Iwhere a IS an angle of InClInatIOn of the InclIned plane, and L IS a length of thel IInclIned plane. Because the rollIng ball has the constant acceleratIOn a, there IS ~ luniform accelerated motion Thenl

~= at

2

v

2 '

a= -. t

ISubstituting (3.46) and (3.47) into (3.45), we obtain the velocitYI

IV(D)

=

3(D + 2) gt sin a. 5D+6

for D

= 3, we obtaIij IV(3) =

~ gt sin a·1

k'\s a result, we have the relatIOriI

~ v 3

=

21(D+2~.

5(5D+6

IWe note that the deviation v(D) from v(3) is no more than 5 percent]

851

13 8 Conclusion

IWe can consider a fractal rigid cylinder of mass MD and radius R, rolling down aI Iplane inclined at an angle f3 with the horizontal. Using the moment of inertia (3.23)J Iwe obtain 1

V

a

( )

=

a+2

.

2( a + I) gt sm JJ. R

(3.S0)1

for a = 2, Equation (3.S0) givesl

IV(2) ~s

=

~ gt sin f3 j

a result, we have the relation!

~ = 3(a+2). v3

4(a+l)

(3.S1)1

IWe note that the deviation v(a) from v(2) for fractal rigid body cylinders is no mor~ ~han 8.4 percentJ [Therefore the preCISiOn of the expenments must be hIgh. These equatiOns allowl Ius to use an expenmental determmatiOn of a fractal dImenSiOnal for fractal ngICl1 Ibody by measurements of velocIty.1

~.8

Conclusionl

[n thIS chapter, we conSIder mechamcs of fractal ngId bodIes, whIch are descnbe@ Iby a fractional continuous model (Tarasov, 200Sa,b,d,f). In the general case, thel OCractal ngId body cannot be conSIdered as a contmuous medIUm. There are pomtsl land domams that are not filled of partIcles. We suggest (Tarasov, 2005a,b) descnb1 Img fractal ngId bodIes as speCIalfractiOnal contmuous medIa. We use the procedur~ pf replacement of the medIUm WIth fractal mass dImenSiOn by some contmuou~ fuodel that uses the fractiOnal mtegrals. ThIS procedure IS a generalIzatiOn of Chnsj ~ensen approach (Christensen, 200S). Suggested procedure leads us to the fractiona~ ImtegratiOn to descnbe fractal medIa. The fractiOnal mtegrals are used to take mtol laccount the fractalIty of the ngId bodlesl IWe suggest a method for computing the moments of inertia for fractal ngId body1 [The simple experiments (Tarasov, 200Se) to test the fractional continuous modell I(Tarasov, 200Sa,b,d) for fractal rigid bodies can be performed. These experiment~ lallow us to prove that the fractiOnal mtegrals can be used to descnbe fractal ngI@ Ibodles. We note that the suggested equatiOns allow us to use an expenmental deter1 ImmatiOn of a fractal dImenSiOnal for fractal ngId body by measurements of oscI1la1 ~iOn penods and velocIty.1

3 Fractal Rigid Body Dynamics

186

lReferencesl IA. Carpmten, P. Cornetti, 2002, A fractional calculus approach to the descnption ofj Istress and stram localIzatIOn m fractal medIa, Chaos, Solitons and Fractals, 13J IK2H: k\. Carpinteri, F. Mainardi (Eds.), 1997, Fractals and Fractional Calculus in Conj kmuum Mechanics, Spnnger, New YorkJ R-M. Christensen, 2005, Mechanics of Composite Materials, Dover, New York.1 R Goldstein, c.P. Poole, J.L. Safko, 2002, Classical Mechanics, 3nd ed., Addison-I IWesley, San Fransiscoj ~.S. Ivanova, A.S. Balankin, LZh. Bunin, A.A. Oksogoev, 1994, Synergetics ancA IFractals m Matenal SCiences, Nauka, Moscow. In RussIanJ k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj klOnal Dijjerentlal EquatIOns, ElseVIer,AmsterdamJ IM.L Kulak, 2002, Fractal Mechanics of Materials, Visheishaya Shkola, Minsk. Inl IRussian f. MamardI, 2010, Fractional Calculus and Waves in Linear Viscoelasticity: Ani IIntroductlOn to Mathematical Models, World SCIentificPublIshmg, SmgaporeJ IY. Park, 2000, On fractal theory for porous medIa, Journal of Statistical PhyslcsJ 1101, 987-9981 f.Y. Ren, J.R. LIang, X.T. Wang, W.Y. QIU, 2003, Integrals and denvatives on ne~ Ifractals, Chaos, Solitons and Fractals, 16, 107-117J IS.G. Samko, A.A. KIlbas, 0.1. Manchev, 1993, Integrals and Denvatlves of Fracj ~lOnal Order and ApplicatIOns, Nauka I Tehmka, Mmsk, 1987, m RUSSIan; an1 ~and FractIOnalIntegrals and Derivatives Theory and ApplicatIOns, Gordon and! IBreach, New York, 1993J ISpecIaI Issue, 1997, ApplIcatIOn of Fractals m Matenal SCIence and Engmeenng,1 IChaos, Solitons and Fractals, 8,135-3011 ~E. Tarasov, 2004, Fractional generalization of Liouville equations, Chaos 14, 123-1 [2T.

IVB. Tarasov, 2005a, Contmuous medIUm model for fractal medIa, Physics Lettersl lA, 336, 167-1741 IVB. Tarasov, 2005b, FractIOnal hydrodynamIC equations for fractal medIa, Annal~ pi Physics, 318, 286-307 j ~E. Tarasov, 2005c, Fractional systems and fractional Bogoliubov hierarchy equaj ItIOns, Physical Review E, 71, 011102J IVB. Tarasov, 2005d, DynamICS of fractal solId, International Journal oj Modernl IPhysics B, 19, 4103-4114J ~E. Tarasov, 2005e, Possible experimental test of continuous medium model fofj fractal media, Physics Letters A, 341, 467-472.1 ~E. Tarasov, 2005f, Wave equation for fractal solid string, Modern Physics Lettersl IB, 19, 721-7281 IV.E. Tarasov, 2005g, FractIOnal LIOuvIlleand BBGKI equations, Journal oj Physics.j IConferenceSeries, 7, 17-33.1

IR eferences

871

IK. Tsujii, 2008, Fractal materials and their functional properties, Polymer Journalj 140, 785-799l ~.v.

Uchaikin, 2008, Method of Fractional Derivatives, Artishok, Ulyanovsk. Inl IRussian p. V Vostovsky, A.G. Kolmakov, I.Zh. Bumn, 2001, IntroductIOn to Multllracta~ IParametrization ofMaterial Structure, RHD, Moscow. In Russian] [. V Zolotuhm, Yu.E. Kabmn, VI. Logmova, 2005, Sobd fractal structures, Interna-I ~ional Scientijic Journaljor Alternative Energy and Ecology, 9, 56-661

~hapter~

[Electrodynamics o~ ~ractal Distributions of Charges and Fields

'l.t Introduction ~oseph LIOuvIlle was a pIOneer m development of fractIOnal calculus to electrody-I ramics (Lutzen, 1985). The theory of fractIOnal denvatIves and mtegrals (KIlba~ ~t aI., 2006; Samko et aI., 1993) can be applIed to several specIfic electromagnetICj Iproblems (see, for example, (Engheta, 1997; ZelenYI and MIlovanov, 2004; MIloj Ivanov, 2009; Potapov, 2005; Tarasov, 2008, 2009; Bogolyubov et aI., 2009)). In thi§ ~hapter, we consIder electrodynamIcs of fractal dIstnbutIOn of charges and fields ml OChe framework of fractIonal contmuous model (Tarasov, 2005a,b, 2006a,b)l [The lInear, surface, or volume charge dIstnbutIOns of partIcles can be descnbe~ Iby the amount of electnc charge m a lme, surface, or volume, respectIvely (JacksonJ [998; De Groot and Suttorp, 1972). In general, these dIstnbutlOns can be fractaIJ Il.e., the charged partIcles fonn a set WIth non-mteger-dimenslOn. Therefore elecj ~nc and magnetIc fields of fractal dIstnbutlOn of charged partIcles and fields mus~ Ibe descnbed. Fractal dIstnbutlOn can be descnbed by fractIOnal contmuous modell ~n the general case, the fractal dIstnbutlOn of partIcles cannot be consIdered as ~ ~ontmuous dIstnbutlOn. There are pomts and domams that have no charges. In Refs.1 I(Tarasov, 2005a,b, 2006a,b), we suggested to consider fractal distribution of charge§ land fields as a speCIal contmuous dIstnbutlOn. We use the procedure of replace1 Iment of the dIstnbutlOn wIth fractal dImenSIOn by some contmuous model that use~ ~ractlOnal mtegrals. ThIS procedure can be consIdered as a generalIzatIOn of Chns1 ~ensen approach (Christensen, 2005), that leads us to use the fractional integrationj OCor fractal dIstnbutIOns. Osmg fractIOnal contmuous model for fractal dIstnbutIOnsl pf charged partIcles and fields, we consIder the electnc and magnetIc fields of thesel khstnbutIons (Tarasov, 2005a,b, 2006a,b)l ~n SectIOns4.2=4.3, the denSItIes of electnc charge and current for fractal dIstn1 Ibution are considered. In Sections 4.4-4.5, Gauss' and Stokes' theorems for frac.j ~al distributions in the framework of fractional continuous model are suggested. Inl ISections 4.6-4.9, we consider the simple examples of the fields of homogeneou§ ~ractal dIstnbutlOn. The Coulombs and Gauss' laws, the BIOt-Savart and Amperesl

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

~o

4 Electrodynamics of Fractal Distributions of Charges and Fields

~aws for fractal distribution in the framework of fractional continuous model arel Isuggested. In Section 4.10, we consider the fractional generalization for integrall Maxwell equation. In Section 4.11, we represent fractal distribution as an effecj ~Ive medIUm.In SectIons 4.12-4.14, the examples of electrICdIpole and quadrupol~ moments forfractal distribution are considered. In Sections 4.15-4.16, the magne-I ~ohydrodynamic equations for fractal distribution of charged particles are discussedj finally, a short conclusion is given in Section 4.17 j

f:J.2 Electric charge of fractal dlstributlors [The total electric charge that is distributed on the metric set W with the dimensionj 3 with the density p'(r',t) is defined byl

ID =

(4·1)1

~v{ =

dx'dy'dz'l

~or Cartesian coordinates x', y', z' with dimension

[x'] = [y'] = [z'] = meter. We notq

~hat SI unit of Q3 is Coulomb, and SI unit of p' is Coulomb- meter 3J [To generahze Eq. (4.1), we represent thIS equatIOn through the dImensIOnlessl

coordmate varIables. We can mtroduce the dImenSIOnless values:1 ~=x'/lo,

y=y'/lo,

z=z'/lo,

r=r'/loJ

Iwhere fo IS a characterIstIC scale, and the charge densIt5J

Ip(r,t) = 15P'( r 1o,t),1 Iwhere SI unit of p is Coulomb, Le., Ipl

= Coulomb.

As a result, we obtainl

Iwhere dV3 - dxdYdz for dImenSIOnless CarteSIan coordmates. ThIS representatIOnl lallowsus to generahze Eq. (4.2) to fractal dIstrIbutIOn of charges.1 ~n the fractIOnal contmuous model for fractal dIstrIbutIOn of charges and fields) Iwe use fractional integrals over a region of jRn instead of integrals over a fractal set.1 [n order to deSCrIbe fractal dIstrIbutIOn by fractIOnal contmuous model, we use thel rotion of density of states cn(D, r) and the density of charges p(r,t). The functioi1l kn (D, r) is a density of states in the n-dimensional Euclidean space jRn. The densityl pf states deSCrIbes how permItted states are closely packed m the space jRn. Thel ~ensity of charges p (r, t) describes a distribution of charges on a set of permitte~ Istates in the Euclidean space jRn. In the fractional continuous model of fractal mediaj

9~

~.2

Electric charge of fractal distribution

~he

density of states cn(D, r) in IFt n is chosen such thatl I{lJ1D(r,n)

= cn(D,r)dVJ

klescribes the number of states in dVn . We use the following notations:1

~o

descrIbe densItIes of states m n-dImenSIOnal EuclIdean space wIth n - 1,2, 3J [Let us consider a fractal distribution of electric charges. Suppose that the densitYI pf charge distribution is described by the function p (r, t) such that SI unit of p i§ ~oulomb. In fractional continuous model, the total charge is defined b)j

(4.3)1 Iwhere dV3 = dxdydz for Cartesian coordinates, D = al ~he denSIty of statesl

+ a2 + a3, and c3(D,r) i§ (4.4)

IWe note that SI UnItof QD IS Coulomb. DenSIty of states (4.4) defines the Rlemannj !Liouville fractional integral up to numerical factor 8n D / 2 . Note that the final equa~ ~IOns, whIch relate the phySIcal varIables, are mdependent of numerIcal factor m thel [unction C3 (D, r). However the dependence on r is important to these equations] fractional integral equation (4.3) describes the charge that is distributed in 1Ft} .1 for the Riemann-Liouville fractional integral, the function c3(D,r) is defined byl 1(4.4) without the factor 8 nD/2. Note that for D = 2+, we have the distributiod 1m the volume. In general, thIS case IS not eqUIvalent to the dIstrIbutIOn on the 21 ~imensional surface. For the case p(r) = p( Irl), we can use the fractional integralsl IWIth

~onsIder a SImple example of dIstrIbutIOn of charged partIcles. Let W be a baIlI Iregionsuch that W = {r: Irl ~ R}. For stationary spherically symmetric distributioij pf charged particles (p(r,t) = p(r», we havel

QD(R) = 4n ~n the homogeneous case,

p(r,t)

r

2 -Dr(3/2) R -1 r(D/2) Jo p(r)~ dr. =

Po, and we havel

~2

4 Electrodynamics of Fractal Distributions of Charges and Fields

[he distribution of charged particles is called a homogeneous one if all regions WI land W' with the equal volumes VD(W) = VD(W') have the equal total charges inl ~hese regions, QD(W) = QD(W').I for homogeneous fractal dIstrIbutIOn of charged partIcles, the electrIC charge QI Isatisfies the scaling law Q(R) rv RD , whereas for homogeneous regular n-dimensionall klistribution we have Q(R) rv Rn . This property can be used to measure the fractall klimension D of fractal distributions of charges. We consider this power-law relaj ~ion as a definition of a fractal charge dimension. If all particles of a distributionj lare IdentIcal, then the charge dImenSIOn IS equal to the mass dImenSIOn. In generalJ ~hese dimensions can be considered as different characteristics of distribution]

'l.3 Electric current for fractal distribution ~et

us consider charged particles with density p (r, t) flowing with velocity u ==i Then the current density J (r, t) is defined by the equationl

~(r, t).

~(r,t) =

p(r,t)u(r,t)l

IWe can define the electric current/(S) as the flux
Iwhere dS2 = dS2D IS a dImenSIOnless dIfferentIal umt of area pomtmg perpendIcula~ OCo the surface S, and the vector D - nkek IS a vector of normal. We can assume tha~ OChe field J(r, t) passmg only through a surtace WIth fractal dImenSIOn d < 2. Thenl Iwe must take mto account a denSIty of states on the surtace. Usmg the fractIOna~ ~ontmuous model, we define the fractIOnal generalIzatIOn of electrIC current (4.5)1

lOY

(4.7)1 landC2 (d, r) is the density of states on the surface S = aw. We can use the functionj

for d = 2, we have c2(2,r) = 1. The boundary aw has the dimension d. In general] ~he dimension d is not equal to 2 and (D - I) J

93

f:I..5 Stokes' theorem for fractal distribution

gJ.4 Gauss' theorem lor fractal distributionl II,et us consider Gauss' theorem for fractional continuous model of fractal distribud ~ion of charges and fields. Using (4.7), we obtainl

Iwherethe vector J(r,t) = lkek is a field. Using the well-known Gauss' formula, wei ~

[The relationlallows liS to obtainl

~w (J(r,t),dSd) =

fw c3

1(D,r)div(c2(d,r)J(r,t))

dVD.

(4.9)1

~quatIOn

(4.9) can be conSIdered as Gauss' theorem for fractIOnal continuous modell pf fractal dIstnbutIOn of charges and fieldsl

kJ..5 Stokes' theorem for fractal distributionl IStokes' theorem relates the surface Integral of the curl of a vector field E over a sur"1 [ace S in Euclidean space ]R3 to the line integral of the vector field over its boundar~

It-as· Iwhere E = E(r,t) is an electric field at a point r. Usingl

Iwhere cr ( Y, r) is the density of states on the boundarYI

Iwe obtain

II

(E,dly) =

I

Cl

(Y,r) (E,dlI).

[We note that cjf yr] = I fory= 1. Equations (4.10) and (4.1I)givel

4 Electrodynamics of Fractal Distributions of Charges and Fields

[Using relation (4.7) in the formj

Iwe obtain

(cur1(CI (y,r)E),dSz) =

1l c

l

(d, r)(cur1(CI (y, r)E),dSd)

k\s a result, we have the equationl C

l

l

(d, r) (curl (Cl (y, r)E), dSd).

~quatIOn (4.12) can be consIdered as Stokes' theorem for fractIonal Imodel of fractal dIstrIbutIon of charges and fieldsJ

(4.12) contInuou~

FI-.6 Chargeconservation for fractal distributionl [I'he law of charge conservatIon states that the velocIty of charge change In regIon WI Ibounded by the surface S = aw is equal to the flux of charge through this surfacel ~quatIOn of the law of charge conservation has the forml

~n

the fractIOnal contmuous model, thIS law IS deSCrIbedby the equatIOili

Iwhere QD(W) is defined by (4.3), and Id(S) is defined by (4.6). Substitution of (4.3)1 land (4.6) into (4.13) givesl (4.14)1 k\s a result, we obtaIn that the conservatIOn of electrIC charge IS deSCrIbed by fracj ~IOnal Integral equatIon In the framework of the fractIonal contInUOUS modelJ lOSIng the generalIzed Gauss' theorem, we can derIve dIfferentIal equatIon tha~ ~oIIows from the fractIOnal Integral equatIOn (4.14). Gauss' theorem (4.9) of fracj ~IOnal contInUOUS model, gIvesl (4.15) [£the surface S = dW is fixed, the left hand side ofEq. (4.14) can be written a§

951

f:I..7 Coulomb's and Biot-Savart laws for fractal distribution

~1 p(r,t)dVD t W

=

1 W

ap(r,t) a dVD.

ISubstItutIOn of Eqs. (4.16) and (4.15) mto (4.14)

t

(4.16)1

g1Ve~

(4.17)1 [I'hIS IS a fractIonal mtegral equatIOn that descnbes the conservatIon of electn<j ~harge. EquatIon (4.17) IS satIsfied for arbItrary regIOn W. Therefore we havel c3(D,r)

ap(r,t) at

a

+ aXk (c2(d,r)lk(r,t)) = O.

(4.18)1

1

k\s a result, we obtain the law of charge conservation as differential equation (4.18)j [I'hIS equatIon IS a contmUIty equatIon for fractal dIstnbutIOn m fractIOnal contmu-I pus model. We note that dIfferentIal equatIon (4.18) IS not a fractIOnal ddferentIa~ ~quatIOn. For D - 3 and d - 2, EquatIon (4.18) gIve§

IJP(r,t)

at

+ aJXk lk (r,t ) --

oj

IwhIch IS well-known contmuity equation for electnc chargej

kJ..7 Coulomb's and Biot-Savart laws for fractal distributionl II,et us consider Coulomb's law in fractional continuous model for fractal distribud ~Ion of electnc charges. The chargg

I(lQD = p(r')dVhl latposition r' creates the electric field dE(r) at a point r, which is defined b~

r

E(r)

dQD

= 4neo

r-r' Ir-r'13

j

Iwhere r, r' are dimensionless vectors, and Eo is a fundamental constant called thel Ipermittivity of free space. For Cartesian's coordinates dV£ = dX!dy'dz'l for the distribution p(r') of electric charge with the charge dimension D = 3, thel ~lectnc field at a pomt r IS descnbed byl

(4.19)1 Iwhere rand r' are dimensionless radius vectors. For the region W of fractal distrH IbutIOn, the electnc field at a pomt r IS equal tol

~6

4 Electrodynamics of Fractal Distributions of Charges and Fields

r

1 r-ri lw Irr' 3 p(r' )dVD, , E(r) = 4neo

1

(4.20)1

Iwhere dVb = c3(D,r')dV~. Equation (4.20) is a fractional generalization of (4.19)j [Equation (4.20) is considered as Coulomb's law for a stationary fractal distribution] pf electnc charges in the framework of fractIOnal continuous modeIJ [I'he BIOt-Savart law relates magnetIC fields to the currents, whIch are theI~ Isources (Jackson, 1998). For the distribution density J(r') of electric current withl ~he charge dimension D = 3, the magnetic field at a point r is equal tg

r(r)=}10

r [J(r'),r-r']dV' 3

4n Jw

[r - r'1

(4.21)1

3'

Iwhere r, r' are dimensionless vectors, [ , ] is a vector product, J is the current density j IJ10 is the permeability of free space. Equation (4.21) represents the Riot-Savar! lawj [I'he fractIOnal generalIzatIon of Eq. (4.21) I§

IB(r) = J10

r [J(r'),r-r']dV'.

4nJw

Ir-r'13

(4.22)1

D

[ThIS equatIOn IS RIOt-Savar! law wntten for a steady current of fractal dIstnbutIOnl ~n fractIOnal contmuous model, the law (4.22) descnbes the magnetic field producedl Iby fractal dIstnbutIOn of steady currentsl

kJ..8 Gauss' law for fractal distributionl [n claSSIcal electrodynamICs, Gauss' law, also known as Gauss' flux theorem, IS ij Ilaw relatmg the dIstnbutIon of electnc charge to the resultmg electnc field (JacksonJ ~ 998). Gauss' law states that the total flux
(4.23)1 [n fractional continuous model, the electric flux for the surface S =

aw i~ (4.24)1

Iwhere E(r, t) is the electric field vector, and dSd is a differential unit of area pointing Iperpendicular to the surface S. The total charge QD(W) of fractal distribution i~ k1efined by Eq. (4.3)j [n the fractIOnal contmuous model of fractal dIstnbutIOn of charges and fieldsJ pauss' law (4.23) has the formj

~.9

971

Ampere's law for fractal distribution

(4.25)1 Iwhere p(r,t) is the density of electric charge, dVD = c3(D,r)dV3, and eo is thel Ipermlttlvlty of free space] ~f p(r,t) = p(r), and W = {r: Irl:S;; R}, thenl Q(W) = 4n

l

R

o

p(r)c3(D,r)r2dr = 4n

for d = 2 and the sphere S =

23-Dr(3/2)

aw = {r: Irl I
r

ISubstttutlllg (4.26) and (4.27) III (4.23), we

0

p(r)~-Idr.

(4.26)

R}J

=

= 4nR

D 2

lR

2E(R).

(4.27)1

ge~

(4.28) for homogeneous (p(r) = Po) distribution, Equation (4.28) give~

for fractal dlstnbutlOn with the dimenSIOn D - 2+, we have the constant electn(j lfield E(R) = constl

FI-.9 Ampere's law for fractal distributionl IWe note that, LIOUVille, who was one of pioneers III the development of fractlOna~ ~aIculus, was Illsplred by the problem of fundamental force law III Ampere's elec1 ~rodynamlcs and used fractIOnal differential equatIOn III that problem. Ampere's lawl Istates (Jackson, 1998) that the magnetIc field III space around an electnc current I§ IproportlOnal to the electnc current, whIch serves as ItS source. For the statIc elecj ~nc field, the hne Illtegral of the magnettc field around a closed loop IS proportlOnall ~o the electnc current flowlllg through the loop. Ampere's law IS eqUIvalent to thel Isteady state of the Illtegral Maxwell equatIOn III free space, and relates the spattallyl Ivarying magnetic field B(r) to the current density J(r)J [n fractIOnal contllluous model, Ampere's law for fractal dlstnbutlOn of charge§ land fields IS represented by the equatlOnl

(4.29)1

~8

4 Electrodynamics of Fractal Distributions of Charges and Fields

Iwhere dl y is the differential length element. The electric current Id(S) is defined byl 1(4.6), and Ampere's law (4.29) takes the formj

(4.30)1 [I'hIS IS a fractIOnal mtegral equatIon that descnbes Ampere's law for fractal dIstnj IJ.:llili.ill:L [Let us consider a simple example of fractal distribution. For a cylindrically symj ~etric distribution with y = 1, we havel

Iwhere the density of states Cz (d, r) is defined by Eq. (4.8). For the circle L = dW I{r: Irl = R}, we ge~

[ (B,dlt)

=

~

2nR B(R)l

As a result, we obtaml (4.31)1

for the homogeneous case, l(r) = 10, Equation (4.31) give~

[ThIS equatIOn represents the magnetIc field for the cyhndncally symmetnc fractall ~istribution. For d = 1+, we have the constant magnetic field B(R) = const]

f:J.I0 Integral Maxwell equations for fractal distribution! [I'he Maxwell equatIOns are the set of fundamental equatIOns for electnc and magj r.etIc fields. Osmg the fractIOnal contmuous model, we obtam a generahzatIOn ofj ~he mtegral Maxwell equatIOns for electnc and magnetIc fields. These generahze~ ~quatIOns are fractIOnal mtegral equatIOnsl for fractal distribution of charged particles in lR3 , the integral Maxwell equation~ Ibecome

~(E,dSZ) = ~ kPdVDJ

Ii

(E,dlt)

=

-~ l(B,dSz)J

~.1O

9S1

Integral Maxwell equations for fractal distribution

[I'hese equations descrIbe fractal dIstrIbutIOnsof charged partIcles onlyj [Using the fractional continuous model, we can consider the other type of fracj ~al distribution. We can consider a fractal distribution of charges and fields. Let u§ ~onsIder the fields that are defined on fractal only. We assume that the electrIc field] ~(r,t) and magnetic fields B(r,t) are defined on fractal and do not exist outside ofj [mctal in Euclidean space JR.3. The fractal distribution in which the electromagnetiq lfields are defined on fractal is considered as an approximation of some real casel Iwith fractal medium. In this case, we use the fractional generalization of the integrall Maxwell equations in the formj

(4.32)1

~ (E,dl y) = -~ l(B, dSd)' li;(B,dSd)

=

0,

(4.33)1 (4.34)1 (4.35)1

[TheseequatIOns are the fractIOnal Integral Maxwell equatIOns that deSCrIbe electro1 Imagnetic fields of thIS generalIzed fractal dIstrIbutIOnof charges and fieldsl [n fractional contInUOUS model, we can derIve dIfferentIal equatIOns that corre-I Ispondto the fractional Integral Maxwell equations. USIng Stokes' and Gauss' theoj Iremsfor fractIOnal contInUOUS model, I.e., Eqs. (4.9) and (4.12), we can reWrIteEqsj 1(4.32)-(4.35) In the form:1

k'\s a result, we obtaInI

(4.36)1

4 Electrodynamics of Fractal Distributions of Charges and Fields

1100

(4.37)1

Idiv(c2(d,r)B) = 0,

(4.38)1 (4.39)

[I'hese differentIal equatIons descnbe electromagnetIc field of fractal dIstnbutIOn ofj Iparticles and fields in the framework of fractional continuous model. It is easy t9 ~lotIce that differentIal equatIOns (4.36)-(4.39) are not fractIOnal equatIOns.1 IWe note that the law of absence of magnetIc monopoles (4.38) can be rewnttenl

as ~ivB = -(B,gradc2(d,r)).1 ~n the general case (d =I- 2), the vector grad (c2(d,r)) is not equal to zero, and! kliv B =I- 0. For d = 2, we have div(B) =I- only for non-solenoidal field B. There-I ~ore the magnetIc field IS sImIlar to the non-solenOIdal field. As a result, the field] pf generalIzed fractal dIstnbutIOn can be consIdered as a field WIth some "fractIona~ ImagnetIcmonopole"l

°

a.n

Fractal distribution as an efl'ective medium

~n

fractIOnal contmuous model, the Maxwell equatIOns (4.36)-(4.39) for fractall of charges and fields can be consIdered as the equations for effectIv~ fuedmm: Irl' (D) -- Pfree' eff (4.40)1 ~IV ~hstnbutIOn

Icurl(Eeff)

=-

%t Bef f ,

~iv(Beff) = 0,

ICUr1(H)

= Jeff

(4.41)1 (4.42)1

+ ~~.

[I'he effectIve Maxwell equatIOns (4.40)-(4.43) prove that fractal dIstnbutIOn crej latessome polanzatIOn and magnetization. In these equations, we use some effectIv~ Ifi.iiliI:S:: IEefJ (r,t) = cI(y,r)E(r,t)]

WfJ (r,t)

=

c2(d,r)B(r,t)1

[The fields Eeff and Beff mean that electromagnetic fields E and B are changed byl Hensities of states of the fractal distribution] ~quatIOns (4.40)-(4.43) have the effectIve free charge and current densItIesj

~.12

Electric multipole expansion for fractal distribution

1O~

~eff(r,t) = c2(d,r)J(r,t)j

Pg:e

IWe can interpret the existence of and Jeff as an effect of change of the freel ~lectnc charge and current densities by fractal dlstnbution. This change eXists ml laddltion to the effect of appearance the dipole charges and polanzation or magneti-I Ization currents. The fractal dlstnbution of charges and fields can be conSidered as 3j ~edium that has the electrical and magnetic permittivities in the formj ~eU = c2(d, r)cjl (y, r)j ~eU = c2(d,r) cjl(y,r)j

[he fields D and H are related to Eejj and Bejj by the usual equationsJ

W'Jote, that the contmmty equation (4.18) for fractal dlstnbution can be represente~

IQY apeff(r,t) +div(Jeff(r,t)) =0.

V\s a result, the fractal dlstnbutlOn of charges and fields can be conSidered a~ la speCific medIUm that changes the fields, free charges and currents m additIOn tq ~he creation of polanzation and magnetization. We can mterpret the equations fo~ ~lectromagnetic fields for fractal dlstnbutlOn as an effect of creation of some polarj Ilzation and magnetization by the dlstnbution. Moreover, the electromagnetic field§ lare also changed by fractal dlstnbutlOn. From the generalIzed Maxwell equatIOns) Iwe see the effect of change of the free electnc charge and current densities by den1 ISlty of states of the fractal dlstnbutlOn. ThiS change eXists m additIOn to the effec~ pf appearance the dipole charges and polanzation or magnetizatIOn currents. Thg Fffective electncal permittivity feU and the magnetic pernllttlvlty JIeU of fractall ~ltstnbutlOn are defined by the denSity of states and the charge dimenSIOn of fractall Histribution

f:J.12 Electric multipole expansion for fractal dlstributlors ~et us conSider an electnc multipole expansIOn for fractal dlstnbutlOn of chargedl Iparticles in ]R3. To compute the multipole expansion we use fractional continuou~ model. Let R - Xkek be a vector from a fixed reference pomt to the observatlOnl Ipomt, r = Xkek be a vector from the reference pomt to a pomt m the fractal dlstnj IbutlOn. The potential of the electnc field for fractal dlstnbutlOn IS defined by thel

~quatlOn:

4 Electrodynamics of Fractal Distributions of Charges and Fields

1102

p(r) dVD 4neo Jw R-r j

q>(R) = _1_ ( 1

1

Iwhere R - r IS a vector from a pomt m the dlstnbutlon to the observation pomt] IOsmgthe theorem of cosmes, we hav~

IR - rl2 = ? + R2 Iwhere r =

2rR cos e,

(4.44)1

Irl, R = IRI, and e is the polar angle, defined a§

Eose=~ (r,R) j [Using Eg. (4.44), we obtainj

!R-rl =RVI-2icose+ ~: j INow define the variablesl Iwe get

[Theright hand side ofEg. (4.45) is the generating function for Legendre polynomij lals Pn(~):

~1-2e~+e2)-172= LenPn(~)l ~

rfhen we havel

V\s a result, the electnc potential can be represented as the multlpole expanslOnj (4.46)1

[I'he nth term IS usuaIIy named accordmg to the foIIowmg: n-multlpole, O-monopole) [-dipole, 2-quadrupole, 3-octupole. Using Po(x) = 1, the monopole term has thel form: L(R) = _1_~ ( p(r)dVD = QD(W) 1

r

4neo R Jw

4neoR 1

rfhe electnc dipole and quadrupole moments of fractal dlstnbutlOn are lin next sections I

consldere~

~.13

1O~

Electric dipole moment of fractal distribution

FI-.13 Electric dipole moment of fractal distributionl [fthe total charge QD(W) offractal distribution in the region W is equal to zero, thenl ~he monopole term Po(R) vanish. In this case, it is important to take into accountl ~he electric dipole moment of fractal distributionj IWe use the coordInate system such that measures the angle from the charge-I ~harge line with the midpoint of this line being the origin. Then the term of (4.46)1 Iwith n - 1 is 1 l
e

r

Iwe obtain


=

1 4ncoR3

r

Jw (r,R) p(r)dVD =

1 4neoR3

R,

L

rp(r)dVD

.

for a contInUOUS charge dIstrIbutIOn, the electrIC dIpole moment IS defined b5J

Iwhere r pOInts from posItIve to negatIve. Defimng the dIpole moment for the fractall ~hstrIbutIOn by the equatIOnj r(D) =

L

rp(r)dVD,

~ R __ 1_ (p(D),R) _ _ 1_p(D)cosa 1( ) - 4neo R3 - 4neo R2 '

~quatIOn

(4.50)1

(4.49) defines the dIpole moment of a fractal dIstrIbutIOn of electmj

~harges. ~xample. ~n

(Tarasov, 2005b), we conSIdered the example of electrIC dIpole moment fo~ = Po) fractal distribution of electric charges in the paraIj IlelepIped regIOn:1 ~he homogeneous (p (r)

k= 1,2,3}. for Riemann-Liouville density of states, we obtainl

(4.51)1

4 Electrodynamics of Fractal Distributions of Charges and Fields

1104 1

(D) -

Pk Iwhere a

~he

-

po(ala2 a3)(X ak r3(a) a2(a+ 1)'

(4.52)1

= D/3. Using the total electric charge in region (4.51) in the formj

dipole moment (4.52) can be represented b)j

HD) = ahQ(w)ak'

k=

1,2,3·1

[Using aj(a + 1) = Dj(D+ 3), we obtainl Ip(D) =

:~3Q(W),

(4.53)1

Iwhere L IS a length of parallelepIped dIagonal. EquatIOn (4.52) descnbes the dIpolel rIoment of fractal dIstnbutIOn of charged partIcles m the parallelepIped region. US1 Img the relatIOn 2 < D :s; 3, we obtam the mequahtyj

P.8 < p(3) :s; 1. p(D)

(4.54)1

Ks a result, we see that the deviation p(D) from p(3) for fractal distribution of chargedl IpartIcles m the parallelepIped regIOn IS no more than 20 percent]

FI-.14 Electric quadrupole moment of fractal distributionl ~et

us conSIder electnc quadrupole moment of fractal dIstnbutIOn m the frameworkl pf fractIOnal contmuous model. The electnc quadrupole term, whIch IS the thIrd! ~erm in an electnc multIpole expansion, m SI umts IS given b)1

Iwhere eo IS the permIttIvIty of free space, and R IS the dIstance from the fractall khstnbutIOn of charges. Osmgl

1 2 1 (3(R,r) 2(cos8) = -(3cos 8 -1) = 2 2 2 2 r R Iwe obtain

r

(R) =

1 -1- 3 4neo 2R

h( w

2 2) p(r)dVD.

3 2(R,r) - r

R

(4.55)1

1051

f'I-.14 Electric quadrupole moment of fractal distribution

[Usingthe dimensionless Cartesian's coordinates Xk of the vector R, and the dimen-I ISIOnless coordmates Xk of the vector r, the electnc quadrupole term (4.55) can bel Irepresented a§

(4.56)1 Iwherethe tensor Qkl is the electric quadrupole moment, which is defined b)j

(4.57)1 Iwhere Xk - X, y, or z . It is easy to see that the tensor of electnc quadrupole momentl Isatisfies the conditions BI

Rkl = Qlk,

L Qkk =

O.

(4.58)1

~

!Example 1~ [n (Tarasov, 2005b), we considered the example of quadrupole moment for the hoj fuogeneous (p(r) = Po) fractal distribution of electric charges in the parallelepipedl IregIOn (4.51). We use the Riemann-LIOuville denSity of states of the forml

a =D/3.1 ~he electric quadrupole moments Qi~) of fractal distribution ar~

In (D)

r

kk

=

3D D+6

(3) Q kk '

Iwhere Qg) are moments for the usual homogeneous distribution with D rondiagonal elements of

Qif), i.e., for the case k #- I, we obtainl k#- I.

IOsmgmequaltty 2 < D

~

3, we get the relatIOns for diagonal elementsj

landthe nondiagonal elements:1

= 3. Th~

(4.59)1

4 Electrodynamics of Fractal Distributions of Charges and Fields

1106

Qif)

Qii)

~n general, the deviation from for fractal distribution of charged particlesl lin the parallelepiped region is no more than 36 percent for the nondiagonal elementsj OO:xample 2J [n (Tarasov, 20056), we consIdered the example of quadrupole moment for the hoj Imogeneous (p (r) = Po) fractal distribution of electric charges in the ellipsoid regioij Ill7:: (4.60)

[The total charge of the ellipsoid region W is equal tg

Using the relation]

Iwe obtam the dIagonal elements of the electrIc quadrupole moments m the forml (4.61)1

Iwhere we lise the notationl (4.62)1

ISllch that ~(D) 11

(D)

Q22 =Q(-1,2,-1),

=Q(2,-1,-1),

(D) = Q(-1,-1,2) 1 Q33

k'\s a result, we havel (D) -

1

Q kk

-

5D

(3)

3D±6Q kk

'

k = 1,2,3j

for the nondiagonal elements of electrIc quadrupole momentl h(D) rkl

= 3po

r XkXI lw

dVD,

Iwe obtain the relationl Iwhere

I~(a) = _6_T

f

3a±2

2(a/2±

1/2) I T2(a/2) 1

1071

f'I-.15 Magnetohydrodynamics of fractal distribution ~s

a result, we havd

Iwhere k -=/=l. Here we use r(1/2) = fiJ for 2 < D < 3, we get the inequalitYI

k-=/=l.

Qif)

Qii)

~n general, the deviation from for fractal distribution of charged particlesl lin the ellipsoid region is no more than 31 percent for the nondiagonal elementsj

FI-.15 MagnetollYdrodynamics of fractal distributionl ~agnetohydrodynamIcs

IS the theory, whIch studIes the dynamIcs of electncallyl lIqUId medIa (KulIkovskIy and LyubImov, 1965). Examples of such me1 ~ha mclude plasmas, lIqUId metals, and salt water. TypIcal turbulent medIa could bel pf fractal structure and the correspondmg equatIons should be changed to mcludg ~he fractal features of the medIa. The set of equatIons, whIch descnbes magneto-I IhydrodynamIcs IS a combmatIOn of the hydrodynamICs equations and Maxwell'~ ~quatIOns (KulIkovskIy and LyubImov, 1965)j IWe use the fractIOnal contmuous model, to conSIder magnetohydrodynamIcs o~ OCractal dIstnbutIOns of charges and fields. We assume that the fields are defined onl [ractal only and do not exist outside of fractal in Euclidean space JR3. The fractall ~hstnbutIOn m whIch the fields are defined on fractal IS conSIdered as an approxIma1 ~ion of some real case with fractal medium In order to describe fractal distributionl Iby fractional continuous model, we use the notion of density of states cn(D, r) and! ~he density of charges p(r,t). The density of states cn(D, r) describes how permit~ ~ed states are closely packed in the space JRn. In the fractional continuous model ofj OCractal medIa, we use the followmg notatIOn§ ~onductmg

~o descnbe number of states m n-dImensIOnal EuclIdean spaces WIth n = 1,2,3. Thel IhydrodynamIc and Maxwell equations for fractal dIstnbutIOn of partIcles and field~ larefollowmg]

[. The equatIOn of contmUItyj

(4.63)1

4 Electrodynamics of Fractal Distributions of Charges and Fields

1108

12. The equation of balance of momentumj (4.64)1 13. The absence of magnettc monopolesj ~iv(c2(d,r)B) =

o.

(4.65)1

t!. Ampere's law:1 (4.66)1 Iwhere the dtsplacement current IS neglected.1 15. Faraday's lawj (4.67)1 [n Eqs. (4.63) and (4.64), we use the notationsj

(4.68)1

~DA _ -I( ) JC2(d,r)A k - c3 D,r

aXk .

(4.69)1

[The Idea of magnetohydrodynamIcs IS that magnettc fields can mduce currentsl 1m a movmg conductIve flUId, whIch create forces on the flUId, and also change thel ~agnettc field Itself. Let us obtam a set of equatIOns, whIch descrIbe magnetohY1 ~rodynamIcs of fractal medIa. For lInear electrodynamICs, we have J and E* j IJ(r,t)

= O"E*(r,t),

(4.70)1

Iwhere 0" IS the electrIC conductIvIty, and E* IS an electrIC field m the moved COordI-1 Inate system. If lui «c, theij (4.71)1 IUsmg (4.71), Faraday's law (4.67) gIvesl

V\mpere's law (4.66) and lInear relatIOn g1V~ E* =

1

1( ) curl(cI(y,r)B). d, r

0" J.1oC2

ISubstituting (4.73) into (4.72), we havel

(4.73)1

1O~

f'I-.15 Magnetohydrodynamics of fractal distribution

(4.74)1 [Thisequation describes the diffusion of magnetic fieldj ISubstItutIon of the Lorenz force densIty:1

rf= [J,B],I linto Eq. (4.64) of balance of momentum, we obtainl

r(*t)

D

U+ \lD p =

[J,B].

(4.75)1

[I'hIS equatIon descrIbes the balance of densIty of momentum, where we take mtol laccount the Lorenz forces] ~quatIOns (4.74) and (4.75) allow us to replace Faraday's law (4.67) and equatIOIll pf momentum balance (4.64). As a result, we obtam (Tarasov, 2006a) the followmg Iset of equations to descrIbe magnetohydrodynamIcs of fractal medIal ~.

The equation of contmUltYl

(4.76)1 12. The equation of balance of momentumj

13. The absence of magnetIc monopolesj ~iv(c2(d,r)B) =

fl.

o.

(4.78)1

Ampere's law:1 ~url(cI(r,r)B)

=J.loc2(d,r)J.

(4.79)1

15. The diffusion equation for magnetic fieldj

(4.80)1 IWe have a closed system of eleven equatIOns for eleven varIables p, p, u, JJ land B. These equations can be considered as magnetohydrodynamic equations fofj ~ractal dIstrIbutIOn of charged partIcles and fieldsl

1110

4 Electrodynamics of Fractal Distributions of Charges and Fields

FI-.16 Stationary states in magnetohydrodynamics of fracta. IlistributiomJ [..:et us consIder the statIonary states for fractal dIstrIbutIon m magnetohydrodynam-I IICS. The statIOnary states are defined by the condItIons:1

(4.81)1 [n thIS case, EquatIon (4.77) has the form:1

(4.82)1 ISubstituting Amperes' law (4.79) in the form:1

(4.83)1 Imto Eq. (4.82), we obtaml

(4.84)1 IWe simplify our transformations by assumption B ~IOn (4.84) g1Ve~

= {O, 0, Bz }. In this case, Equa-I k= 1,2.

(4.85)

Rsing (4.69) and (4.81), Equations (4.85) take the formj

[The rule of term-by-term dIfferentIatIOn allows us to represent these equations ml ~he form·

k= 1,2,

(4.86)

k\s a result, we obtaml

(4.87)

bI- 17 Conclusion

1111

[Equation (4.87) describes stationary states magnetohydrodynamic equations fo~ OCractal dIstrIbutIOn (Tarasov, 2006a). ThIS equatIOn for statIOnary states eXIsts onlyl lif

[t IS easy to see that we don't have the usual InvarIants for fractal dIstrIbutIon ofj ~harged particles. Therefore equilibrium for fractal media exists for the magneticl lfieldthat satisfies the power-law relationj

for the case D - 3 and d - 2, we have the usual relatIon for statIonary states ofj Imagnetohydrodynamlcs·1

~.17

Conclusioril

for homogeneous fractal dIstrIbutIOn of charged partIcles, the electrIC charge Q satj ~sfies the scaling law Q(R) rv RD , whereas for homogeneous regular n-dimensionall k{istribution we have Q(R) rv Rn • This property can be used to measure the fractall ~hmensIOn D of fractal dIstrIbutIOns of charges. We conSIder thIS power-law relatIOnl las a defimtIOn of a fractal charge dImensIOn. If all partIcles of dIstrIbutIOn are Idenj ~Ical, then the charge dImensIOn IS equal to the mass dImensIOn. In general, thesel Inotions can be considered as different characteristics of fractal distribution] lIn the fractIOnal contInUOUS model for fractal dIstrIbutIOns, the fractIOnal In~ ~egrals are used to deSCrIbe electromagnetIc fields for these dIstrIbutIOns. In thI~ fuodel, a fractIonal generalIzatIon of Integral Maxwell equatIOns IS derIved. Thg fuagnetIc field on fractal can be conSIdered as a field WIth some "fractIOnal magj ~etic monopole" (Tarasov, 2006b). We can interpret the equations for electromag-I Inetlc fields of fractal dIstrIbutIOn as an effect of creatIOn of some polarIZatIOn and! ImagnetIzatIOn by fractal dIstrIbutIOn. Moreover, the electromagnetIc fields are alsq ~hanged by fractal dIstrIbutIOn. From the generalIzed Maxwell equatIOns, we canl Isee the effect of change of the free electrIC charge and current denSItIes by fractall khstrIbutIOn. ThIS change eXIsts In addItIOn to the effect of appearance of the dlpolg ~harges and polarIZatIOn or magnetIzatIOn currents. The electrIcal permIttIVIty £ and! ~he magnetIc permlttlVlty J1 of fractal dIstrIbutIOnare defined by the denSIty of state~ land the charge dImenSIOn of thIS dIstrIbutIOn. Note that the fractIOnal contInuou~ fuodel of fractal dIstrIbutIOn has been used to deSCrIbe magnetIC reconnectIOn ratel pf space plasmas In (Materassl and ConsolIm, 2007). The graVItatIOnal field of fracj ~al dIstrIbutIOn of partIcles and fields can also be conSIdered In the framework o~ ~ractIOnal contmuous models (Tarasov, 2006c); see also (Calcagm, 2010)1

1112

4 Electrodynamics of Fractal Distributions of Charges and Fields

lReferencesl IA.N. Bogolyubov, A.A Potapov, S.Sh.RehvIashvIlI, 2009, An approach to mtroduc-I ling fractional integro-differentiation in classical electrodynamics, Moscow Unij Iversity Physics Bulletin, 64, 365-368J p. Calcagni, 2010, Quantum Field Theory, Gravity and Cosmology in a Fracta~ IUniverse, E-print: arXiv: 1001.0571 j R-M. Christensen, 2005, Mechanics of Composite Materials, Dover, New York.1 IS.R. De Groot, L.G. Suttorp, 1972, Foundation ofElectrodynamics, North-Holland,1 IAmsterdam I IN. Engheta, 1997, On the role of fractional calculus in electromagnetic theory, Anj kennas and Propagation Magazine, 39, 35-46j ~.D. Jackson, 1998, ClaSSIcal ElectrodynamICs, 3rd ed., WIley, New YorkJ k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj tional Differential Equations, Elsevier, Amsterdam] IA.G. KulIkovskIy, G.A LyubImov, 1965, Magnetohydrodynamics, AddIson WesleyJ IMassachusetts; Translated from RUSSIan: Nauka, Moscow, 19641 [. Lutzen, 1985, Liouville's differential calculus of arbitrary order and its electrody-I InamIcal ongm, m Proc. 19th NordIC Congress Mathenzatlclans, IcelandIc Mathj ~matIcal SOCIety, ReykJaVIk, 149-1601 ~. MaterassI, G. ConsolIm, 2007, MagnetIc reconnection rate m space plasmas: AI Ifractal approach, Physical Review Letters, 99, 1750021 k\. V MIlovanov, 2009, Pseudochaos and low-frequency percolatIon scalIng for turj Ibulent diffusion in magnetized plasma, Physical Rewiew E, 79, 046403j k\.A. Potapov, 2005, Fractals in Radiophysics and Radiolocation, 2nd ed., Univer-I ISItetskaya Kmga, Moscow. In RussIanJ IS.G. Samko, AA KIlbas, 0.1. Manchev, 1993, Integrals and Derivatives oj Frac1 ~wnal Order and Appftcatwns, Nauka I Tehmka, Mmsk, 1987, m RussIanj ~nd FractIOnal Integrals and Derivatives Theory and Appftcatwns, Gordon and! IBreach, New York, 19931 IVB. Tarasov, 2005a, ElectromagnetIc field of fractal dIstnbutIOn of charged partI1 ~Ies, Physics oj Plasmas, 12, 082106J IVB. Tarasov, 2005b, MultIpole moments of fractal dIstnbutIOn of charges, Modernl IPhysics Letters B, 19, 1107-1118~ ~E. Tarasov, 2006a, Magnetohydrodynamics of fractal media, Physics ofPlasmasj 113, 052107J IVB. Tarasov, 2006b, ElectromagnetIc fields on fractals, Modern Physics Letters AJ 121, 1587-1600J ~E. Tarasov, 2006c, Gravitational field of fractal distribution of particles, Celestia~ I/vfechanics and Dynamical Astronomy, 94, 1-15~ IVB. 'I'arasov, 2008, FractIOnal vector calculus and fractIOnal Maxwell's equatIOnsJ IAnnals oj Physics, 323, 2756-2778J IV.B. Tarasov, 2009, FractIOnal mtegro-dIfferentIal equations for electromagnetIq Iwaves in dielectric media, Theoretical and Mathematical Physics, 158, 355-359~

IR eferences

113

[L.M. Zelenyi, A.V. Milovanov, 2004, Fractal topology and strange kinetics: froml IpercolatIOn theory to problems In cosmIC electrodynamIcs, PhYsics Uspekhi, 47J 1749-788

~hapter

5]

~inzburg- Landau

Equation for Fractal Medial

S.l Introduction [The Gmzburg-Landau equatIOn (Gmzburg and Landau, 1950; Gmzburg, 2004) I~ pne of the most-studIed nonhnear equatIOns m phYSICS (Aranson and Kramer, 2002)1 ~t descrIbes a vast varIety of phenomena from nonhnear waves to second-order phasel ~ransItIons, from superconductIvIty, supertlUldIty, and Bose-Emstem condensatIOnl ~o hqUld crystals and strIngs m field theory. The Gmzburg-Landau equatIon can bel ~erIved (LIfshItz and PItaevskY, 1980) from free energy functIonal. We can defing ~he free energy functIOnalm the formj

Iwhere P = P(x) is a real-valued function. In Eq. (5.1) the integration is over 3~ ~ImensIOnal regIOn W of contmuous medIa. Here Fo IS a free energy of the norma~ Istate, Le., F {P(x)} for P(x) = 0. Using the variational Euler-Lagrange equatioij

IOF{P(X)} _ 8P(x)

- 0,

Iwe obtam the Gmzburg-Landau equatIOnj

psmg the fractIOnal contmuous model, we consIder a fractIOnal generahzatIOnl pf (5.1) that can appear from two places: fractional generalization of the integrall lin Eq. (5.1) and fractional generalization of the derivatives in (5.1). The Ginzburgj ~andau equatIOn wIth fractIOnal derIvatIves was suggested m Ref. (WeItzner and Za1 Islavsky, 2003) and It was conSIdered m (MIIovanov and Rasmussen, 2005; Tarasovl land Zaslavsky, 2005, 2006; Tarasov, 2006). ThIS equation can be used to descnbel ~he dynamical processes in media with fractal dispersion. Since the fractals can bel Irealized in nature as a fractal process or fractal media, it is possible to derive aI V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

1116

5 Ginzburg-Landau Equation for Fractal Media

[generalized Ginzburg-Landau equation by using a corresponding generalization ofj OChe free energy functIOnal (Tarasov and Zaslavsk)', 2005). The Agrawal vanatIOna~ ~quation (Agrawal, 2002, 2007b) and its generalization have been applied to obtainl OChe fractIonal GInzburg-Landau equatIOn] [n Section 5.2, the fractional generalization of free energy functional is sugj gested. In SectIon 5.3, we obtaIn the GInzburg-Landau equatIon from the free enj ~rgy functIOnal. In SectIon 5.4, the fractIOnal GInzburg-Landau equatIOns are dej Irivedfrom the variational equation. A simplest solution of I-dimensional Ginzburgj [.:andau equatIon for fractal medIa IS consIdered. FInally, a short conclusIon IS gIveIlj lin Section 5 5J

~.2

Fractional generalization of free energy functionall

~et

us consider the free energy functional F{P(x)} for the non equilibrium statel pf the fractal medium, where P(x) is a scalar real-valued field. We define the freel ~nergy functIonal by the equatIOnj

Iwhere § (P (x), D 1P (x)) is the free energy density of the formJ

IHere we use the notationl

~quatIOn (5.3) can be conSIdered as terms of the expansion In senes over smalll Ivalues P (x) and the integer derivatives D 1 P 1 [n order to descnbe fractal medIa wIth a fractal dImenSIOn D, we conSIder thel ~ractional continuous model, where the notion of density of states c3(D,x) is usedl [The density of states C3 (D, x) describes how permitted states are closely packed inl ~he space ]R3. In this model, the free energy functional for fractal media is define~

!bY

(5.5)1 rrhe modIfied Riesz denSIty of states can be defined In the form:1 3

c3(D,x) =

1

2 Dr(3/2) D-3 r(D/2) [x] .

(5.6)1

1171

f'j.3 Ginzburg-Landau equation from free energy functional

[he initial point of the Riesz fractional integral is set to zero. The numerical facto~ in Eq. (5.6) gives the usual integral in the limit D ----+ 3 - 0 . The usual numerical actor y I (D) = r(1/2)/2Dn3 2r(DI2) of the Riesz fractional integral (Samko e la1., 1993; Kilbas et a1., 2006) gives Y3 1 (3 - 0) #- 1 in the limit D ----+ (3 - 0). Wg ~efine the RIemann-LIOuville densIty of states by the eguatIonj

(5.7)1 Iwhere Xk are dimensionless Cartesian's coordinatesJ for free energy denSIty (5.3), functIonal (5.4) has the formj

~guation

(5.8) is a fractional generalization of (5.1) for fractal mediaj us consIder a homogeneous fractal medIUm WIthout external fields. Then thel Iparameter 'P(x) = 'P does not depend on coordinates and Eq. (5.8) give~ ~et

Iwhere VD IS a D-dlmensIOnal volume of fractal medIUm In the region W. The sta"1 ~ionary value of 'P corresponds to the minimum for (5.9). If alb> 0, then the freel ~nergy potential has the single minimum at 'P = O. If a/b < 0, then there are twq ~inima at 'P = ± ECJbj

~.3

Ginzburg-Landau equation from free energy functionall

~n general, the equilibrium value of 'P( x) is defined by the condition of the stationaryl Istate of functIonal (5.4), whIch has the form of vanatIOnal Euler-Lagrange equatIOnj

(5.10)

[or the free energy density § 1(5.10) gives

= §('P(x),D 1'P(x)). For the density

~C31(D,x)D;k h(D,x)D;k'P) -a'P-b'P3 =0.

(5.3), Equatioq

(5.11)1

Note that Eg. (5.11) can be rewritten in the formj

~D2'P + qk(D,x)D;k 'P - alp - b'P 3 = 0,

(5.12)1

1118

5 Ginzburg-Landau Equation for Fractal Media

~quatIOn (5.11) can be consIdered as the GIllzburg-Landau equatIOn for fractal mej Hia in the framework of fractional continuous model I ~n order to describe fractal distribution in ]R1 with dimension 0 < D < 1, we con~ Isider the fractional continuous model, where the notion of density of states CI( Y, a] lis used. The density of states CI ( Y, x) describes how permitted states are closelyl Ipacked in the space ]RI. In this case, the Ginzburg-Landau equation for fractal disj OCnbutIOn (5.12) can be represented III the formj

(5.13)

IWe note that Eq. (5.13) is analogous to equation of a nonlinear oscillator with fricj ~ion. It allows us to conclude that the Ginzburg-Landau equation for fractal medi~ klescribes dissipative nonlinear oscillations of the field 'P (x) .1

~.4

Fractional equations from variational equation

~n the general case, the free energy denSIty also depends on fractIOnal denvatIve~ I(Samko et aL, 1993; Kilbas et aL, 2006) of 'P. In this case, we can use the frac1 OCIOnal generalIzatIon of vanatIonal equatIon suggested III (Agrawal, 2002); see alsq I(Agrawal, 2006, 2007a,b, 2008). In (Tarasov and Zaslavsky, 2005), we extended thel ~ractIOnal vanatIOnal equations (Agrawal, 2002) for the case of fractal medIa wIthl ~he fractal dimension DJ OC=et us conSIder the free energy functIOnal WIthfractIOnal Illtegrals and denvatIve~ I(KiIbas et al., 2006) in the form]

IF {'P(x)} = Fo + j~ g;('P(x),DU'P(x))dVD,

(5.14)1

Iwhere D U is a fractional derivative (Samko et al., 1993; Kilbas et al., 2006) of non-I IIlltegerorder a, and] [The density of states c3(D,x) that can be defined by relations (5.6) and (5.7). Inl general, the dImenSIOn of fractal medIUm IS not connected WIthorder a of fractIOna~ k1erivative, i.e., D i 3a.1 [The stationary states of free energy potential (5.14) give the fractional Eulerj OC=agrange equatIOn:1 (5.15) [Letus consider the potential density in the formj

ll~

15.4 Fractional equations from variational equation

(5.16)1 [Then Eq. (5.15) gIve§ BJ

ED~(C3(D,x)n~/I') +alI' +blI'

~C31(D,x)

3

= 0.

(5.17)1

~quation (5.17) is fractional Ginzburg-Landau equation for fractal media in thel !framework of fractional continuous modelJ

[Example Ij [n the I-dimensional case lI' = lI'(x), the coordinate fractional derivative is D~, i.e.J

[ThepotentIal denSIty has the form:1 (5.18)1 psmg the formulas for fractIOnal integration by partsj j(x)D~ g(x)dx

=

g(x)D~ j(x)dx,

(5.19)

+ Cl ()dff r,x dp = 0,

(5.20)

Iwe obtam the Euler-Lagrange equatIOnj a

x

Cl

Iwhere the density of states

(

Cl

) dff r,x dDalI'

(r,x) can be defined byl (5.21)1

[Using for Jj; (5.18), we arrive atl

for the case

r-

1, we have

ci -

1 and (5.21) transforms mtq (5.22)1

Iwhere D~ is the Riesz derivative.1

[Example z, [n the general case, the free energy density functional depends on lI' = lI'(x), and! ~erivatives D~:lI' of fractional orders ak with respect to coordinates Xk, i.e.,1

1120

5 Ginzburg-Landau Equation for Fractal Media

(5.23)1

[I'he potentIal densIty has the form:1 (5.24)1

IOsmg (5.19), we obtam the Euler-Lagrange equatIOnj

for the density (5.24), we obtain the equation for fractal mediaj

E3!(D,x)

L gkD~:(C3(D,x)D~tI') +aP +bp3 = oj ~

[The sum of orders ak can be equal to the fractal mass dImensIOn D of the medmllll

Ibutin the general case it can be that al

+ a2+ a3 i D.I

!Letus consider for fractal distribution on ffi.! with the dimension 0 pmzburg-Landau equation for thIS dIstnbutIOn has the formj

< Y < 1. Thel

(5.25)1

Iwherex=xl, andcl(Y,X) is defined by (5.21). We rewrite Eq. (5.25) a~ (5.26) ~quatIOn (5.26) IS analogous to the equatIOn for a nonhnear oscIllator WIthfnctIOnl V\s a result, EquatIOn (5.25) for fractal medIa descnbes nonhnear oscIllatIOns wIth ~ ~ISSIpatIve-hke term (Tarasov, 2005; Tarasov and Zaslavsky, 2005). Let us consIderl Isolution of (5.25) with b = O. Then P(x) satisfies the equationj

Iwhere x E (0,00), that can be rewritten a§

IgxD;p(x)

+ (y-1)D;P(x) - axP(x) = O.

[The solutIOn of (5.27) can be represented m the formj

(5.27)1

IR eferences

1211

Iwhere v = 11 - r/21, Jy(x), and Yy(x) are the Bessel functions ofthe first and second! lkind, and A, B are constants]

0.5 Conclusionl [The fractional generalization of the Ginzburg-Landau equation (Weitzner and Zaj Islavsky, 2003; Milovanov and Rasmussen, 2005; Tarasov and Zaslavsky, 2005) canl Ibe used to descnbe the dynamical processes m media Withfractal disperSiOn. Smcel ~he fractals can be realized m nature as fractal processes or fractal media, we obj ~ain a generalized Ginzburg-Landau equation by using a corresponding generaliza-I OCiOn of the free energy functiOnal (Tarasov and Zaslavsk)', 2005). The fractiOna~ pmzburg-Landau equatiOn and the Gmzburg-Landau equatiOn for fractal medi~ lare denved from the correspondmg generalizatiOn of free energy functiOnal and! Ivanational Euler-Lagrange equations. A generalizatiOn of the vanatiOnal equatioIlj I(Agrawal, 2002, 2007a; Tarasov and Zaslavsky, 2005) for the functiOnal With frac1 ~iOnal mtegro-dlfferentiatiOn (Samko et aI., 1993; Kdbas et aI., 2006) can be used! ~o descnbe complex media. We note that an applicatiOn of the Gmzburg-LandalJl ~quatiOn to phase tranSitions m fractal systems was discussed m (Bak, 2007).1

IReferencesl P.P. Agrawal, 2002, Formulation of Euler-Lagrange equatiOns for fractiOnal vanaj Itional problems, Journal of Mathematical Analysis and Applications, 272, 368-1 1379. P.P. Agrawal, 2006, FractiOnal vanatiOnal calculus and the transversality condi-I Itions, Journal ofPhysics A, 39, 10375-10384j P.P. Agrawal, 2007a, FractiOnal vanatiOnal calculus m terms of Riesz fractiOna~ derivatives, Journal ofPhysics A, 40,6287-6303.1 P.P. Agrawal, 2007b, Generalized Euler-Lagrange equatiOns and transversality con1 ~itions for FVPs m terms of the Caputo denvative, Journal oj Vibration and Con1 ~rol, 13, 1217-12371 P.P. Agrawal, 2008, A general timte element formulatiOn for fractiOnal vanatiOna~ Iproblems, Journal ofMathematical Analysis and Applications, 337, 1-12.1 ~.S. Aranson, L. Kramer, 2002, The word of the complex Gmzburg-Landau equa1 ItiOn, Reviews oj Modern Physics, 74, 99-143. E-pnnt: cond-maUOl061151 IZ. Bak, 2007, Landau-Gmzburg theory of phase transitions in fractal systems, Phasel ITransitions, 80, 79-87 ~ IY.L. Gmzburg, 2004, Nobel Lecture: On superconductiVity and supertImdity (wha~ II have and have not managed to do) as well as on the "physical minimum" at thel Ibegmmng of the XXI century, Reviews oj Modern Physics, 76, 981-998.1

1122

5 Ginzburg-Landau Equation for Fractal Media

IV.L. Gmzburg, L.D. Landau, 1950, To the theory of superconductIvIty, Zhurna~ IEksperimental'noi i Teoreticheskoi Fiziki 20, 1064-1082J k\.A. KIlbas, H.M. Snvastava, J.J. TrujIllo, 2006, Theory and Applications of Fracj klOnal Dijjerentwl EquatIOns, ElsevIer, AmsterdamJ ~.M. LIfshItz, L.P. Pitaevsky, 1980, Statistical PhysIcs, Landau Course on Theoretj licalPhysics, Vo1.9, Pergamon Press, Oxford, New Yorkj k\.v. Mllovanov, J.J. Rasmussen, 2005, FractIOnal generabzatIOn of the Gmzburgj ILandau equation: an unconventional approach to critical phenomena in complex] Imedia,Physics Letters A, 337, 75-80.1 IS.G. Samko, A.A. KIlbas, 0.1. Manchev, 1993, Integrals and DerIvatives of Fracj klOnal Order and ApplicatIOns, Nauka I Tehmka, Mmsk, 1987, m Russianj !Ind Fractional Integrals and Derivatives Theory and Applications, Gordon and! Breach, New York, 1993~ ~E. Tarasov, 2005, Wave equation for fractal solid string, Modern Physics Lettersl IB, 19, 721-728J ~E. Tarasov, 2006, Psi-series solution of fractional Ginzburg-Landau equationj Vournal oj Physics A, 39, 8395-84071 IV.E. Tarasov, G.M. Zaslavsky, 2005, FractIOnal Gmzburg-Landau equation for frac1 Ital medIa, Physica A, 354, 249-261 J ~E. Tarasov, G.M. Zas1avsky,2006, Fractional dynamics of coupled oscillators withl 110ng-range interaction, Chaos, 16, 02311 OJ ~. WeItzner, G.M. Zaslavsky, 2003, Some appbcatIOns of fractIOnal denvatIvesJ ICommunications in Nonlinear Science and Numerical Simulation, 8, 273-2811

~hapter~

[Fokker-Planck Equation for Fractall pistributions of Probabilit~

16.1 Introduction [The Fokker-Planck equatIOn descrIbes the tIme evolutIOn of the probabIhty densItyl ~unctIOn. It IS also known as the Kolmogorov forward equatIOn. The first use o~ ~he Fokker-Planck equatIOn was the statIstIcal deSCrIptIOn of Browman motIon of ~ IpartIcle m a flUId. It IS known that the Fokker-Planck equatIon can be derIved froiTI] ~he Chapman-Kolmogorov equation (Gardiner, 1985). We note that the Chapman-I IKolmogorov equatIon IS an mtegral IdentIty relatmg the jomt probabIhty dIStrIj IbutIOns of dIfferent sets of coordmates on a stochastIc process. In Ret. (Tarasov J 12007), we obtamed a fractIOnal generahzatIOn of the Chapman-Kolmogorov equa1 ~ion, where integrals of non-integer order (KUbas et aI., 2006) were used. The sugj gested equatIon IS fractIonal mtegral equatIon (Samko et aI., 1993). The fractIona~ ~hapman-Kolmogorov equatIOn can be apphed to descrIbe fractal dIstrIbutIOns ofj IprobabIhty m framework of the fractIOnal contmuous model. The mtegrals of frac1 ~IOnal order are a powerful tool to study processes m the fractal dIstrIbutIOns. Gen1 ~rahzed Fokker-Planck equatIOn can be derIved (Tarasov, 2005a, 2007) from thel OCractIOnal Chapman-Kolmogorov equatIOn. The suggested Fokker-Planck equatIOn~ lallowus to descrIbe dynamIcs of fractal dIstrIbutIOnsof probabIhty m framework ofj ~he fractional continuous model ~n SectIOn6.2, we obtam the fractIOnal generahzatIOn of the average value equa1 ~IOn. In SectIOn 6.3, the fractIOnal Chapman-Kolmogorov equation IS derIved byl lusing the integration of non-integer order. In Section 6.4, the Fokker-Planck equaj ~IOn for the fractal dIstrIbutIOnsIS obtamed from the suggested fractIOnal Chapman-I IKolmogorov equatIOn. The statIOnary solutIOns of the Fokker-Planck equatIOn fofj ~ractal dIstrIbutIOns are derIved m SectIOn 6.5. Fmally, a short conclUSIOn IS glVelll ~n Section 6 61 1

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

1124

6 Fokker-P1anck Equation for Fractal Distributions of Probability

1

16.2 Fractional equation for average value~ [.Jet p' (X' ,t) be a probability density function on lR at time t such thatl

IL~= p'(x',t)dx' =

p'(x',t)

1,

~ O.

(6·1)1

[hen the usual average value of the physical value A' (x') is defined by the equationj

I1=

£~= A'(x')p'(x',t)dx'.

[To generalIze Eq. (6.2), we represent thISequatIOn through the dImensIOnless coorj dinate:

Ix = x' /101 Iwhere fo IS a charactenstIc scale. USIngthe dIstnbutIOn functIOnj

r(x,t) Isuch that

~quatIOn

= lop'(xlo,t~

l£~= p(x,t)dx =

1,

p(x,t)

~ oj

(6.2) can be rewntten In the formj

I1=

£~= A(x)p(x,t)dx,

IA(x)

=

A'(lox) 1

[I'hiS representatIOn allows us to generalIze the defimtIOn of average values to thel [factal dIstnbutIOns In framework of the fractIOnal contInUOUS model (Tarasov J ~ ~quatIOn

(6.3) can be wntten

a~

fA >j= T. A(x)p(x,t)dx+ [" A(x)p(x,t)dx. Rsing the fractional Liouville integrals (KUbas et al., 2006) ofthe formj

t a lqa) j'y (y-x)l-a' A(x)dx =Iy [x]A(x) =

-c-co

1=

a 1 A(x)dx yI=[x]A(x) = qa) y (x_y)l-a'

I

~he

average value (6.4) IS represent byl

~.3

Fractional Chapman-Kolmogorov equation

1< A >1 =

1251

_001; [xlA(x)p(x,t) + yIl [xlA (x)p (x,t).

IWe may assume that a fractional generalization ofEq. (6.7)

i~

k A >a (y) = _ooI~[x]A(x)p(x,t) + yI~[xlA(x)p(x,t). ~quatlOn

(6.7)1

(6.8)1

(6.8) can be rewntten a§

Ia (y) = l""((AP)(y-x,t) + (Ap)(y+x,t))dJla(x),

(6.10)1 [To have the symmetnc hmlts of the mtegral, we consider (6.9) m the fonnl

r

A >a (y) =

~f

1 £+00 ((Ap)(y-x) + (Ap)(y+x))dJla(x). 200

a

= 1, then we have usual Eq. (6.3) for the average value of A(x)l [To simplify Eq. (6.11), we can define the fractional integral operatorl

la[xlf(x)

=

211+00 -00 [f(x) + f( -x)]dJla(x).

(6.12)1

1

[hen (6.11) has the forlll]

f2 A >a= la [xlA(x)p (x),

(6.13)1

Iwherethe mltIal pomt YIS set to zero (y - 0). We note that the tractional normahza-I ~Ion condition IS a special case of thiS defimtIon of average values: < 1 >a- 1.1

16.3 Fractional Chapman-Kolmogorov eguatioij rIhe Chapman-Kolmogorov equatIOn (Kolmogoroff, 1931; Kolmogorov, 1938; Chapj man, 1928); see also (Gnedenko, 1997; Gardmer, 1985; Tlhonov and Mlronov,1 11977; van Kampen, 1984) may be mterpreted as a conditIOn of consistency of dlstn"1 IbutlOn functIOns of different orders. Kolmogorov (Kolmogoroff, 1931; Kolmogorovj [938) denved a kmetlc equatIOn usmg a speCial scheme and conditIOns that are Imj Iportant for kmetIcsJ !Let P(x,tlx',t') be a probability density of finding a particle at the position x a~ ~ime t if the particle was at the position X at time t' ~ t. We denote by p(x,t) thel probability density function for t > t'. In probability theory (Gnedenko, 1997), wei Ihavethe equationsl

1126

6 Fokker-P1anck Equation for Fractal Distributions of Probability

dx' P(x,tlx',t')p(x',t'), +00

1

(6.14)

+00

p(x,t)dx = 1,

P(x,tlx',t')dx = 1.

(6.15)

IOsmg notatIOns (6.12), we rewnte Eqs. (6.14) and (6.15) a§

r(x,t) = II [x'] P(x,tlx',t')p(x',t'),1

[I [x] p(x,t)

=

1, II [x] P(x,tlx',t') = Ij

[To describe a fractal distribution of probability, we use fractional continuou~ We assume that the probability density P(x,tlx',t') and distribution func-I ~ions P(x, t) are defined on fractal only and do not exist outside of fractal in Euj Flidean space. The fractal distribution of probability in which the fields P(x,tlX',t') land p(x,t) are defined on fractal is considered as an approximation of some reall ~ase WIth fractal medIUm. In order to descnbe fractal dlstnbutIOn by fractIOnal con1 ~muous model, we use the notIon of denSIty of states that descnbes how permltte~ Istates are closely packed m the Euchdean space. In fractIOnal contmuous model, wei luse a fractional generalization of (6.14) in the formj ~odel.

r(x,t)

=

l a [x' ]P(x,tlx',t')p(x',t'),

(6.16)1

Iwhere fractional integration (KUbas et al., 2006) is used. Equation (6.16) is the defj linition of conditional distribution function P(x, t lx', t') referring to different timel Imstants. For the fractIOnalcontmuous model, normahzatIOn condItIOnsfor the func-I ~ions P(x,tlx',t') and p(x,t) are given by the fractional equations:1

[a [x] p(x,t) ~quations

=

1,

l a [x]P(x,tlx',t') =

1.

(6.17)1

(6.17) are fractional integral equations (Samko et al., 1993; KUbas et al.J

~

[he function p (X', t') can be expressed via the distribution p (xo, to) at an earlie~ ~Ime by (6.18)1 Ip(x',t') = 1a[xo] P(x',t'lxo,to)p(xo,to). ISubstitution of (6.18) into (6.16) give~

Ip(x,t) = 1a[x'] 1a[xo] P(x,tlx',t')P(x',t'lxo,to)p(xo,to). [This fractional integral equation includes the intermediate point 1(6.19) and (6.16) m the form:1

(6.19)1

x'. Using Eqsl (6.20)1

Iwe obtain the closed equationl IAa

I

[xo] P(x,tlxo,to)p(xo,to)

Aa

= I

, , [x], IAa [xo] P(x,tlx" ,t )P(x ,t Ixo,to)p(xo,to) j

~.4

Fokker-Planck equation for fractal distribution

127]

ISince the equation holds for arbitrary p(xo,to), we hav~

Ip(x, t Ixo, to) = la [x'] P(x, t lx', t')P(x', t' Ixo, to).

(6.21)1

~quatIOn (6.21) IS the tractIonal Chapman-Kolmogorov equatIOn (Tarasov, 2005aJ 12007). The suggested equatIon IS a tractIonal Integral equatIon of order a. EquatIoIlj 1(6.21) can be used to describe the Markovian process in fractal medium, which i§ klescribed by the fractional continuous modelj

16.4 Fokker-Planck equation for fractal distributioIlj [The Fokker-Planck equation describes the time evolution of the probability densitYI [unction of the position of a particle. It is also known as the Kolmogorov forwar~ ~quatIOn (Gnedenko, 1997). The Fokker-Planck equatIon can be obtaIned (Gnej klenko, 1997; Gardiner, 1985; Tihonov and Mironov, 1977; van Kampen, 1984) from ~he Chapman-Kolmogorov equatIon. USIng tractIonal contInUOUS model, we denvg la generalIzed Fokker-P1anck equation from the fractIOnal Chapman-Ko1mogorovl ~quatIOn.

ISubstItutIOn of Eq. (6.16) In the formj (6.22)1 lintoEq. (6.13) give§ I~

f-

A >a= I a [x] A(x) I a [xo] P(x,tlxo,to)p(xo,to).

IWe can rewnte Eq. (6.23)

A

A

(6.23)1

a~

(6.24)1 [t IS known that any real vanable x can be expressed as the product of ItS absolutg Ivalue Ixl and its sign function sgn(x)j ~

= sgn(x)Ix[.1

~n order to denve a generalIzed Fokker-P1anck equation, we Introduce the follOWIng Inotations (6.25)1

[Let us consider the function A = A(x U ) . The Taylor expansion for this function withl Irespect to x a is represented in the form]

1128

6 Fokker-P1anck Equation for Fractal Distributions of ProbabilityI

Iwhere we lise the notationj (6.27)1 IWe use the integration by parts in the formj

for the case, Ilim (A(x)B(x)) = OJ xl--+oo

Iwe have (6.28)1 IWe note that if the usual Taylor expansion is used instead of (6.26), then the intej gration by parts in Eg. (6.24) is more complicatedj ISubstituting Eg. (6.26) into Eg. (6.24), we obtainl

I+[a[xo] (D;aA(x a)) XQ p(xo,to)[a [x] LlxaP(x,tlxo,to)1

~[a[xo] (D;aA(x a)) p(xo,to)[a[x] (Llxafp(x,tlxo,to) +.... (6.29) XQ

Let us mtroduce the functiOns:1 (6.30)1 IUsmg (6.30) and (6.17), EquatiOn (6.29) g1Ve~

PA >a = [a [xo] A(xg)p(xo,to) + [a[xo] (D;aA(x a)) XQ p(xO,tO)Ml (xo,t,to)1 2"[a[xo] (D;aA(x a))XQp(xo,to)M2(Xo,t,to) +... ISubstltutmg the fractiOnal average value in the formj

linto Eg. (6.31), we obtainl

.

(6.31)

~.4

12~

Fokker-Planck equation for fractal distribution

11a[xo] A(xo) (p(xo,t) - p(xo,to))1 1= fa [xo] (D;aA(x))xo p(xO,tO)Ml (xo,t,to~ _fa [xo] (D;aA(x))xo p(xo,to)M2(Xo,t,to) +...

.

(6.32)

[Thenwe assume that the following finite limits existj

11.Hfl Ml(X,t,to) =a (x,to)1, ;\ t

iIt--+o

I

lim Mn(x,t,to) = OJ At

AHO

Iwhere n = 3,4, ... , and ziz = t - to. We multiply both sides of Eq. (6.32) by (.M)-I.1 [n the limit ,1t --+ 0, we obtain!

IUsmgEq. (6.28), and the hm1tj 1

lim p(x,t)

x---+±oo

ImtegratIOn by parts

= OJ

glve~

Ifa [x] p(x,t)a(x,t) D;aA(xa)

= _fa [x] A(xa)D;a (p(x,t)a(x,t)),

Ifa[x] p(x,t)b(x,t) D;aA(xa) = fa [x] A(Xa)D;a (p(x,t)b(x,t)). ISubstItutIOn of (6.34) and (6.35) mto (6.33)

(6.34)1 (6.35)1

g1Ve~

=0 [The function A = A(x a) is an arbitrary function. As a result, we obtainl

P x,t

f---'--::-----'--

+ Dxa1 (p (x,t )a(x,t)) -

1 2

-Dxa (p (x,t )b(x,t))

=

O.

(6.36)

[n the fractional continuous model, Equation (6.36) is the Fokker-Planck equaj ~ion for fractal distribution of probability (Tarasov, 2005a, 2007). We note that Eqj 1(6.36) is not fractional. At the same time this equation is derived from the fractiona~

1130

6 Fokker-Planck Equation for Fractal Distributions of ProbabilityI

lintegral Chapman-Kolmogorov equations and the fractional integral equation of av-j ~rage value. EquatIOn (6.36) descnbes the tIme evolutIOn of the probabIlIty densitYI [unction of the position of a particle in fractal media and distributions] IOsmg the fractIonal contmuous model, we can obtam a generalIzatIOn of thel IKramer-Moyal equatIOn (van Kampen, 1984) to descnbe fractal dIstnbutIons ofj Iprobability.

16.5 Stationary solutions of generalized Fokker-Planck eguatioij IStatIOnary solutIOns of Eq. (6.36) descnbe statIOnary probabIlIty dIstnbutIOn of thel Iposition of a particle in fractal media. For the stationary case, we have Df p (x, t) = oj [Then the Fokker-Planck equation (6.36) give§ I

1

2

xa(p(x,t)a(x,t)) - -Dxa(p(x,t)b(x,t))

=

O.

(6.37)

IWe rewrite Eq. (6.37) a§ I

1

1

Dxa p(x,t)a(x,t) - "2Dxa(p(x,t)b(x,t))

=

O.

(6.38)

k'\s a result, we obtaml I

1

(x,t)a(x,t) - -Dxa(p(x,t)b(x,t))

= const.

[f we assume that the constant is equal to zero, then Eq. (6.39) can be

as 1

xa (p(x,t)b(x,t)) [The solution of (6.40)

=

(6.39) represente~

2a x,t --p(x,t)b(x,t). b x,t

(6.40)

2a x,t a b(x,t) dx +const.

(6.41)

i~

n(p(x,t)b(x,t)) = ~ere we use the notation dx a

J

= alxl a Idx. As a result, we obtainl (6.42)

Iwhere the coefficient N is defined by the normalization condition (Tarasov, 2005aj i2OO7). ~quatIOn (6.42) descnbes statIOnary probabIlIty dIstnbutIOn that IS a solutIOn o~ ~he Fokker-Planck equation (6.36) for fractal medIal

~.5

Stationary solutions of generalized Fokker-Planck equation

13~

[n (Tarasov, 2005a), we obtallled the folIowlllg special cases of the solutIOnl

Illim ~.

If a(x) = k and b(x) = -D, then the Fokker-Planckequation (6.36) has the form:1

(6.43) [The stationary solution (6.42) has the formj

12. For a(x)

= klxl J3 and b(x) = -D, the Fokker-Planck equation (6.36) give§

[The stationary solutIOn of thIS equation has the formj

r

{

(x,t) =N2 exP - 2aklxla+J3} (a+f3)D .

~f

a + f3 = 2, we hav~

13. For the functions b(x) = -D,

an~

Iwe obtalll the folIoWlllg statIOnary dIstnbutIOnl

consider Eq. (6.37) with a(x) = klxlu and b(x) = -D, then the stationaryl dIstnbutIOn has the form:1

~. If we

IwhlCh lllterpreted as a generahzatIOn of Gauss probabIhty dIstnbutIOnJ

1132

6 Fokker-P1anck Equation for Fractal Distributions of Probability

1

16.6 Conclusionl IWe descnbe the fractal dlstnbutIOns of probabIlIty by usmg the fractIOnal contm-I luous model. The fractional integrals are used in order to formulate the fractiona~ klynamics for the fractal distributions in framework of the model. The Chapman-I IKolmogorov equation is an integral equation for the probability distributions of difj ~erent sets of coordmates on a stochastIc process. Usmg fractIonal mtegrals, wei pbtam a fractIOnal generalIzatIOn of the Chapman-Kolmogorov equatIOn (Tarasov J 12007). This fractional integral equation can be used to describe Markovian pro-I cess for the fractal distributions in framework of the fractional continuous modelJ IWe hope that the suggested fractional Chapman-Kolmogorov equation has a widel lapplIcatIon to descnbe processes m fractal dlstnbutIOns smce It uses a relatIvely] IsmaIl number of parameters that define a fractal dlstnbutIOn. Usmg the fracj ~IOnal Chapman-Kolmogorov equatIOn, we obtam a generalIzatIon of the Fokker-I flanck equatIOn (Tarasov, 2005a, 2007) on the fractal dlstnbutIOns. The general-I IIzed Fokker-Planck equatIOn descnbes the tIme evolutIOn of the probabIlIty densltYI ~unctIOn of the pOSItIOn of a partIcle m fractal medIa. In the framework of fractIOna~ ~ontmuous model, a generalIzatIOn of the Kramer-Moyal equatIon (van KampenJ ~984) for fractal dlstnbutIOn can be denved1

lReferencesl IS. Chapman, 1928, On the BroWnIan dIsplacements and thermal dIffUSIOn of gramsl Isuspended m non-UnIform flUId, Proceedings oj the Royal Society A, 119, 34-541 ~. W. Gardmer, 1985, Handbook oj Stochastic Methods jor Physics, Chemistry andl lNatural SCIences, 2nd ed., Spnnger, BerlInJ !B.y. Gnedenko, 1997, Theory of Probability, 6th ed., Gordon and Breach, Amsj Iterdam; Translated from RUSSIan: Course of Probablltty Theory, 4th ed., NaukaJ IMoscow, 1965.1 V\.A. KIlbas, H.M. Snvastava, J.J. TruJIllo, 2006, Theory and Applications oj Frac1 ~ional Dijjerential Equations, ElseVIer, Amsterdaml V\. Kolmogorotl, 1931, Uber dIe analytIschen Methoden m der WahrschemlIchkelt-1 Isrechnung, Mathematische Annalen, 104,415-458. In Germanj V\.N. Kolmogorov, 1938, On analytIC methods in probabIlIty theory, Uspehi Matem1 laticheskih Nauk, 5, 5-4U IS.G. Samko, A.A. KIlbas, 0.1. Manchev, 1993, Integrals and Derivatives oj Frac1 ~wnal Order and ApplIcatIOns, Nauka 1 TehnIka, Mmsk, 1987, m Russlanj ~nd FractIOnal Integrals and Derivatives Iheory and Appltcatwns, Gordon and! IBreach, New York, 1993J IV.B. Tarasov, 2005a, FractIOnal Fokker-Planck equation for fractal medIa, ChaosJ 115, 0231021 ~.E. Tarasov, 2005b, Continuous medium model for fractal media, Physics Letterss IA,336,167-174j

IR eferences

133

~.E.

Tarasov, 2007, Fractional Chapman-Kolmogorov equation, Modern Physicsl ILetters B, 21, 163-174l ~.I. Tihonov, M.A. Mironov, 1977, Markov processes, Sovietskoe Radio. In Rus-j sian;

W'].G. Van Kampen, 1984, Stochastic Processes !Holland, Amsterdam]

In

PhySICS and Chemistry, Northj

~hapter71

IStatistical Mechanics ofl ~ractal Phase Space Distributions

r?l Introduction ~n thiS chapter, we conSider fractal distnbutiOns of states m the phase space. We usel la contmuous phase space model to descnbe those distnbutiOns. In thiS model, thel ~ractal distnbutiOns of states are descnbed by fractiOnal generahzatiOns of expec-I ~atiOn values and normahzatiOn conditions. These generahzatIons use mtegrals ofj Inon-mteger orderJ IWe define the fractional analog of the average value and reduced distnbutiOnsl I(Tarasov, 2004, 2005a,b, 2006, 2007). The LiOuvdle equatiOn for fractal distnbu1 ~ions is derived from the fractional normalization condition It is known that Bod golyubov equations can be denved from the LiOuvdle equation and the defimtiOn ofj ~he average value (Bogolyubov, 1970, 1946; Gurov, 1966; Petrina et aI., 2002; Bo-j golyubov, 2005a,b; Uhlenbeck and Ford, 1963; Martynov, 1997). The Bogolyubovl ~quatiOns for fractal distnbutiOns also can be obtamed from the LiOuvdle equatiOili land the defimtiOn of the fractiOnal average value (Tarasov, 2004, 2005a,b, 2006J

~

[n SectiOn 7.2, the fractal distnbutiOn of states m the phase space is defined. Inl ISections 7.3-7.4, we consider the fractional phase space volume. In Section 7.5j ~he fractiOnal generahzatiOn of normahzatiOn conditiOn and some notatiOns are sug1 gested. In SectiOns 7.6-7.7, the contmmty equatiOns for fractal distnbutiOn of par1 ~icles for the configuratiOn and phase spaces are obtamed. In SectiOns 7.8-7.9, thel OCractiOnal average values for the configuratiOn and phase spaces and some notatiOnsl lare conSidered. In SectiOn 7.10, a generahzatiOn of the LiOuvdle equatiOn is sugj gested m the framework of fractiOnal contmuous model. In SectiOn 7.11, we defin~ ~he fractiOnal generahzatiOns of the reduced one-particle and two-particle distnbu1 ~iOn functiOns by usmg fractiOnal integration. FmaIIy, a short conclUSiOn is given ml ISectiOn 7.12.

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

1136

7 Statistical Mechanics of Fractal Phase Space Distributions

rJ.2 Fractal distribution in phase spacel ~et W be a region in a 2n-dimensional phase space JR2n. We assume that states ofj klynamical system form a subset Ws of the set W. In general, the Hausdorff andj Ibox-counting dimensions ofthe set Ws are non-intege~

[I'he fractal dimensIOn is defined as a local property m the sense that it measuresl Iproperttes of subset Ws of phase space pomts m the hmit of a vamshmg diamete~ lused to cover the subset The definition of the fractal dimension of a set of states inl Iphasespace requires the diameter of the covering sets to vanish. In general, physica~ Isystemshave a characteristic smallest volume scale in the phase space. For examplej 1(2nn)n can be considered as a smallest volume in the phase space JR2n. In this casej ~he characteristic smallest length scale is the radius Ro = V2nn.1 [1'0 use a fractal dimenSion, we can conSider the asymptottc form for the relattonl Ibetween the number of states m phase space and the Size of the state region Wsl ~easured by Ro = v2pin the smallest sphere of radius R containing the region las follows·

w,

(7·1)1 [or R/Ro » I or N ---+ 00. The constant No depends on how the spheres of radius RcJ lare packed. The parameter D, on the other hand, does not depend on whether thel Ipackmg of spheres of radms Ro is a close packmg, a random packmg or a packmg IWith a umform distnbutton of holes] [I'he fractal state-number dimenSion is a measure of how the states fills the phasg Ispace it occupies. A fractal distnbutIOn of states has the property that the number o~ Istates mcreases as the regIOn Size mcreases m a way descnbed by the exponent ml IrelatIOn (7.1). The state-number dimensIOn charactenzes a feature of the states, itij Iproperties to fill the phase space. Note that the shape of the system is not descnbe~ Iby the mass dimensIOn]

rJ.3 Fractional phase volume for configuration spac~ ~et us consider the phase volume for the region such that x E la, b]. The usual Ivolume ofthe region W = {x: x E [a;b]} in Euclidean space JRI i§

[This equation can be represented a§

phas~

[7.3 Fractional phase volume for configuration space

1371

(7.3)1 IOsmgthe left- and fight-sided Rlemann-Llouvtlle fractional mtegrals (Samko et al.J ~993; Kilbas et al., 2006) ofthe form]

~

11Y (y_x)l-a' dx 1 jb dx 1~[x]l = T(a) y (x_y)l-a' 1~[x]l = T(a)

a

(7.4)

Iwe represent the phase volume (7.2) a§

l,ul (W) = ali [x] 1 + yll [x] 1.

(7.5)1

IOsmg non-mteger parameter a, we may assume that a fractional generalIzation ofj ICUI!§ l,ua (W) = al~ [x] 1 + yl};[x] 1. (7.6)1 ISubstitution of Riemann-Liouville fractional integrals (7.4) into (7.6), give§ (7.7)

IWe can define d,ua(x - y) such tha~

[The relationsl land ar(a)

= r( a + 1), allow us to represent d,ua(x - y) in the forml

~ ,ua(x-y)=sgn(x-y)d { qa+1) Ix-Yla } . !Here the function sgn(x) is equal to + 1 for x > 0, and -1 for x l(1,ua (x) can be considered as the differential of the functionl

l,ua(x) [Using (7.6) and aT( a)

= T(~: 1)' x> oj

= T( a + 1), we obtainl

< O.

For x

> OJ

1138

7 Statistical Mechanics of Fractal Phase Space Distributions

Iwhere a :( y :( b. In order to make ,ua([a,b]) be not dependent on y, we can usel

lY = a. If we use y = a in Eg. (1.101), thenj

[Using a:( x:( b, we have sgn(x-a) = 1, and Ix-al =x-a. Then Eg. (7.8)

give~

(7.11)1 ~s

a result,

(7.12)1 Iwherex ~ a. [Usingthe density of state§

lin the I-dimensional space JR, we

hav~

Id,ua(x)

=

cdx, a)dxJ

[;et us consider a similarity transformation of ratio X > 0, and a translation transj [ormation forthe region W = la,bl. Using the dilation operator II;., and the translaj ~IOn operator Th such tha~

rrr.d(x) = f(Xx) , ~or the function

f(x)

=

Thf(x) = f(x+h)

x, we obtainl (7.13)1

IWe can use these operators to deSCrIbe the sImIlarIty and translatIOn transtormatIOn§ pfthe intervalla,bl such tha~

rrr.da,b]

=

[Xa,Xb],

Th[a,b]

=

[a+h,b+hn

[t a :( x :( b, thenl

r.da,x)U[x,b]

=

[Aa,h)U[h,Ab]J

tzh[a,x)U[x,b] = [a+h,x+h)U[x+h,b+hjj ~ere we use that relations

(7.13) hold for each x E la,blJ V\s a result, the scaling property!

IIh,ua([a,b]) = ,ua([Aa,Ab]) = Aa,ua([a,b])j

r.5

13~

Fractional generalization of normalization condition

landthe translation invariancel

are satisfied for the measure J1a(W) with W = [a,b]j [Using (7.12), we can prove that the measure dJ1a(x), which is defined by EqJ 1(7.11), IS translation illvanant and It satisfies the scahng property]

rJ.4 Fractional phase volume for phase spac~ [I'he tractional measure for the region W of 2n-dlmenslOnal phase space can be dej lfined by

~a(W) =

L

dJ1a(q,p),

(7.14)1

IwheredJ1a (q,p) is a phase volume element,1

(7.15)1 for example, the phase volume for the 2-dlmenslOnal phase space has the formj

dJ1a(q) /\dJ1a(P) =

r 2 (a ) Iqpla-Idq /\dp,

(7.17)

Iwe present Eq. (7.16) ill the fonnj

~

a(W)

=

1

% k- Y 1 Pbk- Y

qak-Y

Iqpla Idq /\dp 2'

Pak-Y

T (a)

(7.18)1

[ThiS IS the measure of the region ill the 2-dlmenslOnal phase spacel

rJ.5 Fractional generalization of normalization condition! !Let us consider a distribution of probability with the density p (x, t) for x in Eu~ ~Iidean space ffi.I. Assume that p (x,t) E LI (lFr.I), where t is a parameter. Then thel nonnahzatlOn condltlOn lil I

roo

~-oo p(x,t)dx = 1.

(7.19)1

1140

7 Statistical Mechanics of Fractal Phase Space Distributions

!Let p (x, t) E Lp(!~l), where 1 < P < 1/ a. The Liouville fractional integrations (Kilj Ibas et al., 2006) on (-oo,y) and (y, +00) are defined byl

a)( ) I+p y,t

=

r(a)

-00

1+

a_I Lp)(y,t) - r(a) y

p x, x (y-x)l-a' (7.20)

00

p(x,t)dx (x_y)l-a·

IOsmg (7.20), we rewnte Eq. (7.19) m an eqmvalent form:1

tI~-p)(y,t)+(I~p)(y,t) = 1,

(7.21)1

Iwhere y E (-00, +00). The fractional analog of normalization condition (7.21) canl Ibe represented a§ (7.22)1 U%p )(y,t) + (I~p )(y,t) = 1. [I'he mtegrals (7.20) can be rewntten bY] (7.23)1 [Then Eq. (7.22) has the formj

IL~oo p(x,t)djIa(x) =

-(x,t)

=

Txp

1

= -2 (p(y-x,t)

+p(y+x,t)),

1,

djIa(x) =

~dx. ra

(7.25)

Rsing (7.24) and (7.25), the fractional normalization condition in the phase spacel ~

jIa(q,p) [The distribution function

=

djIa(q)I\djIa(P)

=

Iqpla

r 2 a dql\dp.

(7.27)

p(q,p, t) is defined byl Ip(q,p,t) = TqTpp(q,p,t),

(7.28)1

Iwhere the operators Tq and Tp are defined by the equatIOnj

TxJ( ... ,Xk, ... ) = 2(j(···,x~-Xk, ... )+ f(···,x~+Xk,... )). [I'he operator t; allows us to rewnte the distnbutIOn functIOnj

(7.29)

r.6

14~

Continuity equation for fractal distribution in configuration space

I+p(q' - q,p' + p,t) + p(q' +q,p' + p,t)) linthe simple form (7.28).1

rJ.6 Continuity equation for fractal distribution in configuratlon

space IWe consider fractal distributions in configuration space by fractional Imodel. In the Hamilton pIcture, we havel

continuou~

(7.30)1

Iwe get the equatIon:1

(7.31) [I'he derIvatIve of (7.31) wIth respect to t gIve§

k\s a result, we obtaml

Inra(Xt,t) = di d In ( Ixtl a-I ~. ')xt)

(7.33)I

[I'he functIOn (7.33) descrIbes velocIty of phase volume change. EquatIOn (7.32) I§ lacontmUIty equatIOn for configuratIOn space m the HamIlton pIcture. The functIOi1j Ina can be represented a~

d ( In IXtl a-I { ln - Xt) =a-I dx, Qa(xt,t)=--- + - dx, -. dt

axo

for the equation of motIOnj

~ = F((x) j

Xt

dt

aXt dt

(7.34)

1142

7 Statistical Mechanics of Fractal Phase Space Distributions

Iweobtain the relationj (7.35)1 for

a - 1, we have the well-known equationj

[I'hIS function deSCrIbes a velOCIty of the configuration volume change]

rJ.7 Continuity equation for fractal distribution in phase spac~ IWe deSCrIbe fractal dIstrIbutIOns In phase space by the tractIOnal contInUOUS model] [The phase space analog of Eq. (7.30) has the formj

(7.36)1

~quatIOn

(7 .36)

gIve~

(7.37)

lit is known

tha~

(7.38)1 Iwhere { ql, PI }o is Jacobian defined byl

aqktlaqto aqktlaPto apktlaqto apktlapto ISubstItutIOnof (7.38) Into (7 .37)

gIve~

(7.39)1

lIt we conSIder the total time derIvative of (7.39), then we obtaull

r£ + Iwhere

.Q aP= -

0,

fl.? Continuity equation for fractal distribution in phase space

14~

(7.41) [Usingthe well-known re1ationj ~n

Det A - Sp In AJ

[or the expressionj (7.42) Iwe obtain

Qa(q, p)

q

t } = (a - 1) ( -1 -dq, + -1 -d Pt ) + {d-,Pt qt dt Pt dt dt

[n the general case (a -11), the function Qa(q,p) is not equal to zero (Qa(q,p) -10] ~or the systems that are HamIItoman systems III the usual phase space varIables. l~ ~ = 1, we have Qa(q,p) -10 only for non-Hamiltonian systems. For the equation~ bf motIon:

IrelatIon (7.43) gIvesl (7.45)

~A,B} = aA aB _ aA aB1 ~ dqdp dpdql [ThIS relatIOn allows to derIve Q a for all dynamIcal systems (7.44). It IS easy to seel ~hat the usual nondIssIpatIve systeml

Ihasthe Omega functIOnl

landcan be consIdered as a dISSIpatIve system.1

1144

7 Statistical Mechanics of Fractal Phase Space Distributions

rJ.8 Fractional average values for configuration spac~ IOsmg LIOUVIlle tractIonal mtegrals, we obtam a tractIonal generalIzatIon of thel ~quation that defines the average value of the classical observable.1 OCn the configuration space lR 1 the average value is defined by equation:1

IOsmg the LIOuvIlle tractIOnal mtegralsj

~ a lLj)(y) ~quation (7.46)

1

qa)

=

roo

Jy

j(x)dx (x-y)l-a'

can be represented byl

1< A >1 = (lIAp )(y) + (I~Ap )(y).

(7.49)1

~n tractIOnal contmuous model, the average value of claSSIcal dynamIcal valuel I(observable) IS defined by the tractIOnal mtegrals:1

f2 A >a= (I~Ap )(y) + (I~Ap )(y).

(7.50)1

~quation (7.50) is a fractional generalization of (7.49). Liouville fractional integra1s1 1(7.47) and (7.48) can be represented by the equatIonsj

II~j= ~ lo'~ Sa-l j(y-S)d S

II~j= ~ [Then (7.50)

1=

,1

j(Y+Sga-l d S J

i~

fA >a= r/a) l=((AP)(y-S)+(AP)(y+s)ga-ld S.

(7.51)1

[.-etus rewrite Eq. (7.51) in the formj

fA >a= L'" ((Ap )(y -x) + (Ap )(y+x))dJla(x), Ixl a ldx

C

rJla(x) = r(a) .

~ote

that Eq. (7.52) can be wntten

a~

(7.52)1

[7.9 Fractional average values for phase space

b

A >a=

1451

t~ ((Ap )(y-x) + (Ap )(y+x))dJ1a(x).

(7.54)1

[To have the symmetric limits of the integral, we consider the sum of integrals (7.52)1 land (7.54) m the formj

r

A >a

=

~

.fo ((Ap)(y-x) + (AP)(y+x))dJ1a(x~

I+~

L:

((Ap)(y-x)

+

(Ap)(y+x))dJ1a(x).

(7.55)1

k\s a result, the fractional average value can be represented bY] (7.56) Iwhere dJ1a(x) is defined by (7.53). We note that Eq. (7.56) is a fractional integrall ~quatIOn (Samko et aL, 1993)1

rJ.9 Fractional average values for phase space IWe mtroduce the followmg notatIons to consIder the fractIOnal average value ofj passIcal observables on phase space.1 I- The operator TXk IS defined bYl

XJ( ... ,Xk,"')

=

~(f( ... ,X~-Xkl"')+ f(···,X~+Xkl···))

I_ For the phase space of n-particle system, we use the operatorj

ITII, ...,nl = TIII ..·TlnIJ Iwherethe operator Tiki is defined b~

I-

~ere qks are generalIzed coordmates and Pks generalIzed momenta of k-partIcleJ Iwhere s - 1, ... , m.1 The operator j~ is defined byl

(7.57)1 [The average value (7.56) can be represented m the form:1

k A >a= I~TxA(x)p(x)J

1146

7 Statistical Mechanics of Fractal Phase Space Distributions

I_ The phase-space integral operator fa [k] for k particle is defined byl Ifa [ ] ~ k

a

ta = Iqk/Pkl A

A

a

A

a

... IqkmIPkm'

(7.58)1

[This equation give§

[Here, dJ1a (qk, Pk) is an elementary 2m-dimensional phase volume. For the phasel Ispace of n-particle systemj

I_ We define the tractIOnal average value byl

(7.60)1

~.10

Generalized Liouville equatiors

IWe consider system with fixed number n of identical particles, such that k-particlg lis described by the generalized coordinate qk = (qkl,'" ,qkm) and generalized mo~ rJentum Pk = (Pkl,'" ,Pkm), where k = 1, ... , n. The Hamilton equations for thi~ r-particle system arel

Ldf

'(l{jkS

=

k Gs(q,p),

dPks -----;]1

k(

= Fs

q,P,t ) ,

(7.61)1

Iwhere G~ and Fsk are generalized foces. The state of this system can be describe~ Iby dimensionless n-particle distribution function Pn = Pn(t,q,p). The function Pnl ~escribes probability density to find a system in the phase volume dJ1a(q,p). Thel ~volution of Pn = Pn (t, q, p) is described by the Liouville equation:1

Iwhere Pn

= Til, ... ,nIPn. This equation can be derived from the fractional normal-I

lization conditionl

[a [I ,... ,njpn(q,p,t) ~n

=

1.

the Liouville equation djdt is the total time derivativ~

(7.63)1

rz 11

Reduced distribution functions

1471

t

t

d = _+ dqkS_+ dPks_ dt dt ks=1 dt dqks ks=1 dt dPks

[The a-omega function is defined bYI !b11l

pa(q,p)

=

L

((a-l)(qkslG~+PkslFsk)+{G~,Pks}+{qkSlFsk}),

(7.64)1

ne - dA dB) L --d Pks dqks .

(7.65)

V<,s=II

n,m (dA AB}= , k s=1 dqks d Pks

IUsmg (7.62), and (7.64), we present the LIOuvIlle equatIOn m the form:1 (7.66)1

Iwhere An is LIOUVille operator that is defined b5J

[ThiS is the LIOUVille equatIOn m operator form. We note that thiS equation is not ~racttonal mtegro-dtfferenttal equattonJ

~

r?ll Reduced distribution functions [1'0 descnbe dynamiCs of fractal distnbutIOn of particles, we define reduced distnj Ibution functions (Bogolyubov, 2005a,b; Uhlenbeck and Ford, 1963; Liboff, 1998j ~artynov, 1997). If we assume that the distnbutIOn functIOn is mvanant under thel IpermutatIOns of identtcal particlesj

~hen the average values for the claSSical observables are simphfied. For example, thel laverage value of the additive observablel

IAn = EAi(qk,Pklt~ ~

Ihasthe form:

1148

7 Statistical Mechanics of Fractal Phase Space Distributions

I< An > = la[1, ... ,n]T[I, ... ,n]AnPnl 1= ja[1, ... ,n]T[I, ... ,n] LAI(qk,Pk,t)pJ ~

1=

nla[1, ... ,n]T[I, ... ,n]AI(ql ,PI ,t)pJ

k\s a result, this equation is rewritten in the formj

!PI(q,p,t)

=

P(ql,PI,t)

=

ja[2, ... ,n]T[2, H,n]Pn(q,p,t).

(7.68)1

[Using the notationj

rn(q,p,t)

=

T[I, ... ,n]Pn(q,p,t),

(7.69)1

Iwe define (7.70)1 function (7.70) IS called one-particle reduced dlstnbution function, which IS defined! ~or the 2m-dimensIOnal phase space. ObVIOusly, that (7.70) satisfies the normalIza-1 ~ion condition-I (7.71)1 Wa[1]fh(q,p,t) = 1. IWe can define the dlstnbution functionJ (7.72)1 [hiS IS the two-particle reduced dlstnbutlOn functIOn 152 that IS defined by the fracj ~lOnal mtegratlOn of the n-particle dlstnbutlOn over all qk and Pk, except k - 1,21 IWe note that the Bogolyubov equations are equations for the reduced dlstnbutlOili [unctions (Bogolyubov, 2005a,b; Uhlenbeck and Ford, 1963; Petrina et al., 2002j OC:lbofi, 1998; Martynov, 1997). These equatIOns can be denved from the suggested! generalized Liouville equation (Tarasov, 2004, 2005b,a)j

r1.12 Conclusioril [I'he fractIOnal generalIzatIOns of the phase volume and the phase volume eIemen~ laresuggested. In these generalIzatIOns, we use mtegrals of non-mteger order. It lead~ Ius to a generalIzatIOn of the phase space that can be conSidered as a fractal dlmen-I ISlOnal space. [0 derive generalizations of the Liouville and Bogolyubov equations (Tarasov j 12004, 2005a,b), we use a fractIOnal generalIzatIOn of normalIzatIOn condition and ~ ~ractlOnal generalIzatIOn of the defimtlOn of average values. In these generalIzatIOns)

References

149

Iwe use the integrals of fractional order. The interpretation of the fractional integralsl lis connected with non-integer-dimensional space. The order of fractional integral i§ ~qual to the fractal state-number dimension.1 [n (Tarasov, 2004, 2005a), we suggested the second interpretation of the generj lahzed phase space. ThIs mterpretatIon follows from the fractIOnal measure of phasel Ispace that is used in the fractional integrals. The generalized phase space can bel ~onsidered as a space that is described by the fractional powers of coordinates and! ~omenta. A dynamical system in this space is called fractional power system. I~ ~an be consIdered as effectIve systems m mteger-dImensIOnal phase space. Osmg] ~he phase space of fractional powers of the form sgn(x) Ixl u , we can consider widel pass of non-Hamlltoman systems as generahzed Hamlltoman systems. In thIS caseJ ~he fractional normalization condition and the fractional average values are conj Isidered as a condition and values for the generalized Hamiltonian systems that arel Inon-HamIltoman systems in the usual phase spaceJ [I'he Bogolyubov hIerarchy equatIOns for fractIonal power systems (Tarasov J 12005a,b) are derived by using a generalized Liouville equation (Tarasov, 2004j 12005a), the fractIOnal average values, and the fractIOnal reduced dIstrIbutIOn func-I ~IOns. The generahzed Bogolyubov hIerarchy equatIOns are used to derIve the En1 Iskog transport equatIOn (Tarasov, 2006). The generahzed hydrodynamICs equatIOn~ larederIved from the first Bogolyubov equatIon. In (Tarasov, 2007) a generahzatIOnl pf the Fokker-Planck equatIon IS derIved from the Bogolyubov equatIon for fracj ~IOnal power-systemsJ IWe note that the Idea of fractal dImenSIOns of phase space varIables can be used tq ~evelop a concept of adaptIve resolutIOntreatment of a molecular hqUld (Praprotmkl ~t aI., 2007a,b). It allows to calculate statIstIcal averages of thermodynamIC quantI-I ~Ies m multIresolutIOncomputer SImulatIOn algOrIthms, where the molecular degree~ pf freedom change on the fly (Praprotmk et aI., 2007a,b).1

lReferencesl ~.

de Boer, G.B. Ohlenbeck (Bds.), 1962, Studies in Statistical Mechanics, Northj !Holland, Amsterdaml ~.N. Bogolyubov, 1946, KmetIc equations, Zhurnal Eksperimental'noi i Teoretich1 ~skoi Fiziki, 16,691-702. In Russian; and Journal ofPhysics USSR, 10, 265-274j ~.N. Bogolyubov, 1970, Selected Works, VoI.2, Naukova Dumka, KIev] ~.N. Bogolyubov, 2005a, Collection oj Scientific Works in 12 Volumes, Vol.5J INauka, Moscow. In Russlanj ~.N. Bogolyubov, 2005b, Collection oj Scientific Works in 12 Volumes, Vol.6J INauka,Moscow. In Russlanj IKP. Gurov, 1966, Foundation of Kinetic Theory. Method of N.N. Bogolyubov,1 INauka,Moscow. In Russlanj V\.A. KIlbas, H.M. SrIvastava, J.J. TruJIllo, 2006, Theory and Applications oj Frac1 klOnal Dijjerentwl EquatIOns, ElseVIer, Amsterdam]

1150

7 Statistical Mechanics of Fractal Phase Space Distributions

[R.L. Liboff, 1998, Kinetic Theory: Classical, Quantum and Relativistic Descriptionj gnd ed., Wiley, New Yorkj p.A. Martynov, 1997, Classical Statistical Mechanics, Kluwer, Dordrecht] p. Ya. Petrina, V.l. GerasImenko, P.V. Mahshev, 2002, Mathematical FoundatIOn of! IClasslcal Statistical Mechanics 2nd ed., Taylor and FranCIS, LondonJ 1M. Praprotnik, K. Kremer, L. Delle Site, 2007a, Fractional dimensions of phas~ Ispace variables: a tool for varying the degrees of freedom of a system in a multij Iscale treatment, Journal oj Physics A, 40, F281-F288J M. Praprotmk, K. Kremer, L. Delle SIte, 2007b, AdaptIve molecular resolutIon VIa] Fi contmuous change of the phase space dImensIOnahty, PhYSical ReView E, 75J 1117701 IS.G. Samko, A.A. Kilbas, 0.1. Marichev, 1993, Integrals and Derivatives of Fracs rional Order and Applications, Nauka i Tehnika, Minsk, 1987, in Russianj Find FractIOnal Integrals and Derivatives Theory and AppllcatlOns, Gordon and! IBreach, New York, 1993J ~E. Tarasov, 2004, Fractional generalization of Liouville equations, Chaos, 14j 1123-127 IV.B. Tarasov, 2005a, FractIOnal systems and tractIOnal Bogohubov hIerarchy equa1 ItIOns, Physical Review E, 71, 011102J ~E. Tarasov, 2005b, Fractional Liouville and BBGKI equations, Journal ofPhysics.j IConferenceSeries 7, 17-33J IV.B. Tarasov, 2006, Transport equatIOns trom LIOUVIlle equatIOns for tractIOnal SYS1 Items, International Journal oj Modern Physics B, 20, 341-3531 IY.E. Tarasov, 2007, Fokker-Planck equation for tractIOnal systems, Internationall IJournalofModern Physics B, 21, 955-967 j p.E. Uhlenbeck, G.W. Ford, 1963, Lectures in Statistical Mechanics, Americanl IMathematIcal SOCIety, ProvIdenceJ

IPart III ~ractional

Dynamics and ~ong-Range Interaction§

~hapter~

[Fractional Dynamics o~ Media with Long-Range Interaction

18.1 Introduction pynamics with long-range mteractlOn has been the subject of contmumg mteres~ 1m different areas of SCience. The long-range mteractlOns have been studied m diS1 ~rete systems as weII as m their contmuous analogues. Models of classical spm~ Iwith long-range interactions were studied in Refs. (Dyson, 1969a,b, 1971; Joycej ~969; Frohlich et aI., 1978; Nakano et aI., 1994a,b,c, 1995; Sousa, 2005). An in1 ltimte I -dimenslOnal Ismg model with long-range mteractIons was considered byl pyson (Dyson, 1969a,b, 1971). The d-dimenslOnal classical Heisenberg model withl Ilong-range mteractlOn was descnbed m Refs. (Joyce, 1969; Frohhch et aI., 1978)J landtheir quantum generahzatIon can be found m Refs. (Nakano et aI., 1994a,b,c).1 IKmks m the Frenkel-Kontorova model with long-range mterpartIcle mteractlOnsl Iwere studied m Ref. (Braun et aI., 1990). Sohtons m a l-dimenslOnal lattice Withl ~he long-range Lennard-Jones-type mteractlOn were considered m Ref. (IshimonJ ~982). The properties of time penodic spatIaIIy locahzed solutlOns (breathers) onl ~hscrete chams in the presence of algebrmcaIIy decaymg interactions were consid1 ~red in Refs. (Flach, 1998; Gorbach and Flach, 2005). Energy and decay proper-I ~ies of discrete breathers m systems with long-range mteractlOns were also studied! 1m the framework of the Klem-Gordon equatlOn (Braun and Kivshar, 1998; Flach] ~998; Baesens and MacKay, 1999; Braun and Kivshar, 2004), and discrete nonhn-I ~ar Schrodmger equatlOns(Gaididei et aI., 1997, 1995; Mmgaleev et aI., 1998, 2000j Rasmussen et aI., 1998). SynchromzatlOn of chaotic systems with long-range mterj lactionswas discussed in (Tessone et aI., 2006). Nonequilibrium phase transitions inl ~he thermodynamiC hmit for long-range systems were considered m (Bachelard e~ laI., 2008). Statistical mechamcs and dynamics of solvable models with long-rang~ ~nteractlOns were discussed m (Barre et aI., 2005; Campa et aI., 2009).1 [I'hedynamics descnbed by the equatlOns with fractlOnal space denvatIves can bel ~haracterized by the solutions that have power-like tails. Similar features were obj Iserved in the lattice models with power-like long-range interactions (Pokrovsky and! IVirosztek, 1983; Flach, 1998; Gorbach and Flach, 2005; Altimov et aI., 1998; Altij V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

1154

8 Fractional Dynamics of Media with Long-Range Interaction!

mov and Korolev, 1998; Alfimov et al., 2004). As it was shown in Refs. (Laskin andl IZaslavsky, 2006; Tarasov, 2006a,b; Tarasov and Zaslavsk)', 2006a,b), the equatIOn~ Iwith fractional derivatives can be directly connected to chain and lattice modelsl IWith long-range mteractions. Fractional dynamiCs of systems with long-range spacel Imteraction and temporal memory was also conSidered m (Zaslavsky et al., 2007j [Tarasov and Zaslavsky, 2007a,b; Korabel and Zaslavsky, 2007; Van Den Berg et alJ 12009). ~n this chapter, we consider chains and lattices with long-range interactions, andl ~ontmuous hmits of these discrete systems. The map of discrete models mto contm-I luous mediUm models is defined. A wide class of long-range mteractions that givel ~he fractional equations m the contmuous hmit is discussed. There is a connectionl I(Tarasov, 2006a,b) between the dynamiCs of system of particles With long-rang~ linteractions and the fractional continuous medium equations by using the transj ~orm operation. We conSider the lattice of coupled nonhnear OSCillators. We mak~ ~he transform to the contmuous hmit and denve the fractional equation, which dej Iscnbes the dynamiCs of the OSCillatory mediUm. We show how the contmuous hmi~ ~or the systems of OSCillators With long-range mteractIOn can be descnbed by thel ~orrespondmg fractIOnal equatIOn. The cham equatIOns of motion With long-rang~ ImteractIOn are mapped into the contmuum equation With the Riesz fractIOnal denva-I bYe. ~-~----~-~----~~-~~-~~~-~~-~-~ IUsually we assume for long-range mteraction that each cham particle acts on alII ~ham particles. There are systems where this assumptIOn cannot be used (Tarasov J 12008). We conSider a cham such that nth particle is mteracted only With kth particlesl Iwith k = n±a(m), wherea(m) EN andm = 1,2,3, .... As an example of such caseJ r(m) will be described by exponential type functions a(m) = b", where b > 1 and! 'r E N. In this case, the long-range interaction will be called fractal. We considerl ~he cham of coupled hnear OSCillators With the fractal long-range mteraction. Inl ~he contmuous hmit, the cham is an OSCillatory mediUm With the fractal dispersIOili ~aw. This law is represented by the Weierstrass functIOns whose graphs have non-I Imteger box-countmg dimenSIOn, l.e., these graphs are fractals. We prove (TarasovJ 12008) that the chams With long-range mteractIOn can demonstrate fractal propertie~ k1escnbed by fractal functIOnsJ ~n SectIOns 8.2-8.3, the equatIOns of motion for the system of particles With long-I Irange mteractIOn are conSidered. In SectIOn 8.4, the transform operatIOn that mapsl ~he discrete equatIOns mto contmuous mediUm equatIOn is defined. In SectIOn 8.5J ~he Founer senes transform of the equatIOns of systems With long-range mteracj ~ion is realized. In Sections 8.6, we consider a wide class of long-range interactionsl ~hat can give the fractIOnal equatIOns m the contmuous hmi!. In SectIOns 8.7-8.8,1 Iwe give a bnef reView of fractIOnal spatial denvatives to fix notatIOn and provid~ la convement reference. In SectIOn 8.9, the contmuous hmits of discrete equatIOn~ lare discussed. In SectIOn 8.10, the Simple example of nearest-neighbor mteractIOi1j liS conSidered to demonstrate the apphcatIOn of the transform operatIOn to the wellj Iknown case. In Section 8.11, the long-range alpha-interactions with positive integerl Ipowers are conSidered. In SectIOn 8.12, the long-range alpha-mteractIOns With frac1 ~IOnal powers and the correspondent contmuous mediUm equations are discussed. Inl

18.2 Equations of lattice vibrations and dispersion law

1551

ISection 8.13, the fractional reaction-diffusion equations are considered. In Sectionl 18.14, the nonlinear long-range alpha-interactions for the discrete systems are used tq klerivethe Burgers, Korteweg-de Vries and Boussinesq equations and their fractiona~ generalizations in the continuous limit. In Section 8.15, the fractional3-dimensionall ~attice with long-range interaction is considered. In Section 8.16, the appearance ofj ~he fractional derivatives from dispersion law is suggested. In Sections 8.17-8.18J [ractallong-range interaction and fractal dispersion law are considered. In Sectionl 18.19, the Griinwald-Letnikov-Riesz long-range interaction is suggested. The conj pusiOn IS gIven m SectIon 8.20J

18.2 Equations of lattice vibrations and dispersion la\\j [I'he crystallIne state of substances IS dIfferent from others states (gaseous, IIqUIdj lamorphous) m that the partIcles are m an ordered and symmetrIcal arrangementl ~aIled the crystal lattIce. The lattIce IS characterIzed by space perIOdIcIty. In ani lunbounded crystal we can define three non-coplanar vectors at, az, a3, such tha~ ~hsplacement of the crystal by the length of any of these vectors brIngs It back tq Iltself. The vectors ai, I - 1,2,3, are the shortest vectors by whIch a crystal can bel ~hsplaced and be brought back mto Itself. As a result, all spatIal lattIce pomts canl Ibe defined by the vector 0 = (n t ,nz, n3), where ni are integer. If we choose the coj prdmate orIgm at one of the sites, then the position vector of an arbItrary lattIce sItel Iwith 0 = (nt,nZ,n3) is writtenl BI

[(0)

= rn =

Lniai'

(8.n!

~n

a lattIce the SItesare numbered m the same way as the partIcles, so that the vectorl IS at the same tIme "number vector" of a correspondmg partIcle. It IS easy to seel ~hat the umt ceIl volume IS equal tq ~

k\lthough the mam translatiOn perIods are chosen arbItrarIly, the umt ceIl volum~ IstIlI remams the same for any chOIceof the vectors a., I = 1,2,3.1 IWe assume that the eqUIlIbrIum posItiOns of partIcles comclde WIth the lattIcel Isites r(o). A lattice site coordinate r(o) differs from the coordinate of the corre-I Ispondmg partIcle, when partIcles are dIsplaced relatIve to theIr eqUIlIbrIum pOSlj ~iOns. 'Ib define the coordmates of a partIcle, It IS necessary to mdlcate ItS dlsplacej fuent WIth respect to ItS eqUIlIbrIum posItiOn. We denote the dIsplacement of a parj ~ic1e with vector 0 from its equilibrium position by u(o,t). The existence of thel ~rystal state means that over a WIde temperature range the relatIve dlsplacementsl pf partIcles are smaIl compared wIth the lattIce constant a, whIch IS the least valuel pf fundamental translation vectors aj. Note that in a cubic lattice aj = az = a3 = aJ

1156

8 Fractional Dynamics of Media with Long-Range Interaction!

[herefore we consider crystal lattice vibrations with the case of small harmonicl bscillations· lu(n,t)1 « lall [1'0 descrIbe physical processes m crystalhne state, the crystal lattice constructe~ IWith vectors ai, 1= 1,2,3, m real space is associated With some perIodic structurel ~alled the reciprocal lattice (Kittel, 2004). The reciprocal lattice is constructed froml ~he vectors bi, i = 1,2,3, that are related to ai bYI

Iwhere ;Si; is the Kronecker delta. The vectors b i can be simply expressed throughl ~he vectors ad

Iwhere ek1m is a unit antisymmetric tensor. The parallelepiped constructed from bil liS called the umt cell of a reciprocal lattice. The umt cell volume m the reciproca~ ~attice is equal to the inverse value of the umt cell volume Vo of the regular latticej

[I'he space where the reciprocal lattice eXists is called reciprocal space. By analogY] ~o the translation vectors r = r(n) of the regular lattice, we can define translationl Ivectors m reciprocal space:1

~ = g(m)

=

Emibd ~

Iwhere m = (ml,m2,m3), and mi are integers. The vector g is called a reciproca~ ~attjce vector [t is easy to derIve equations for small lattice vibratiOns tf the forces m crystall larepotential. In thIS case, we can conSIder the potentIal energy of a crystal, whos~ Ipartlcles are dIsplaced from theIr eqmhbrIum pOSItiOns and express It through thel ~isplacement u( n, t). In general, the crystal energy is dependent on the coordinate~ pf the atomiC nuclei and electrons (Kittel, 2004). At every given moment the elec1 ~romc state can be deSCrIbed by a functiOn dependent on the pOSitiOns of the nucleil ~xcludmg the electromc coordmates from the crystal energy IS the essence of thel Iso-called adIabatIc approXImatiOn, WhICh IS justified both m studymg harmomc Vlj IbratiOns and m the case of small anharmomcity. The potential energy U of a crystall liS a functiOn only of the coordmates of atomiC nuclei. Therefore we can expand thel Ipotential energy U in powers of u(n,t) and consider the first nonvanishing expan-I Ision terms. Assuming the crystal to be in equilibrium at u(n,t) = 0, we representl I(Klttel, 2004) the potentIal energy m the form:1 1

=

Uo+

~ L,hl(n,n')uk(n,t)ul(n,t), n.n'

(8.2)

18.2 Equations of lattice vibrations and dispersion law

1571

Iwhere Ua = const and the summation is over all crystal sites. The italics k, I arel ~he coordinate indices. We assume the summation over doubly repeated coordinat~ lindices from 1 to 3. Since we consider that the crystal at o(n, t) = is in equilibriuml Istate, the matrix hi (n, n') should be defined positively. In particular, the following Imequahties must be satisfied!

°

f/kk(n,n) ;;? 0,

k = 1,2,3j

[he positively defined matrix with elements lkl (n, n') is called the matrix of (poten-I ~ial) force constants or dynamical matrix of a crysta1.1 [I'he equatIOns of motion of partIcles sulijected to the potential forces arel

a u (n,t)

au

at 2

auk n,t

IwhereM IS the mass of particle. Osmg a general expreSSIOn for the potentIal energY] 1(8.2), we obtaml

r"lu a2ukat(n,t ) 2

= -

~ ( ') I( , ) ';thl n,n u n,t .

OCt is easy see one important property of the coefficients lkl (n, n'). Assume the crystall ~o be displaced as a whole: uk(n,t) = uk = const. Then the internal crystal statel ~annot be changed m case of absence of external forces. As a result, Equations (8.3)1 ~ (8.4)1 n1

In'

[I'hese reqUIrements follow from the conservation of total momentum m the crystalJ k\nother reqUIrement IS that the conservatIOn law for total angular momentum mus~ Ibe automatically obeyed m the absence of external forces. We assume that rotatIOili pf the crystal IS deSCrIbed by the aXIal vector tt; Thenl

(8.5)1 Iwhere ek1m is a unit antisymmetric tensor and xm(n) are coordinates of r(n). Thi~ IrotatIOn causes no change m the mternal lattIce state If external forces are absentJ ISubstitutIOn of (8.5) mto equations of motion g1Ve~ ~klm.Q1 Llik(n,n')~(n') =

01

OJ

ISince the vector .Qk is arbitrary, we havel

Lhl(n,n')~(n') = L1km(n,n')xl(n')·1 In'

n1

[I'hese condItions should be satisfied for any partIcle m the lattIce, i.e., for any vectorl ~. Let us discuss the bulk properties of a crystal assuming the crystal lattice to bel lunbounded. For an unbounded homogeneous lattice, due to its homogeneity thel

1158 ~atrix lkl(n,n

8 Fractional Dynamics of Media with Long-Range Interaction!

f) has the forml

Iwhere elements of lkl(n,n') are satisfied by the conditionsj

(8.6)1 nJ

In' ~n

a simple lattice each particle is an inversion center. Then we havel

Rsing condition (8.6), we can represent equations of motion (8.3) in the formj

a u (n,t) at 2

~ ( f) [l( f )] =-J...lkln,n u n,t ) -u l( n,t. ,

(8.7)

[I'hese equatIOns of motIon illustrate the mvariance of a crystal structure wIth respec~ ~o Its dIsplacement as a whole.1 [TheseequatIOns of motIon are equatIOns for 3-dImensIOnal dIsplacement vectorsl ~n thIS chapter, we shaIl use the sImplest model to deSCribethe vIbratIOn of a crystall IlattIce, where all partIcles are dIsplaced m one dIrectIOn. Then the dIsplacement ofj IpartIcle from ItS eqUIlIbrium posItIon IS determmed by a scalar rather than a vectorJ [I'hIS model aIlows us to deSCribe the mam propertIes of a vIbratmg crystal usmgl ISImple equatIOns. ThIS model IS used to IIlustrate the calculatIOn scheme and thel Ildeology of mathematIcal methodsl ~et us conSIder the statIOnary crystal vIbratIOns for whIch the dIsplacements o~ lallparticles are time dependent only by the factor exp{ -Lo»}, i.e.]

for such vIbratIOns, equations of motion (8.7) gIV~ ~w2uk(n) = L1kl(n - nf)u' (n").

(8.8)1

iiJ

[I'he plane wave solutIOn of these equatIOns has the formj

(8.9)1 [I'he vector k IS analogous to a wave vector of crystal vIbratIOns and IS caIled thel ijuasI-wave vector. We conSIder k as a free parameter that determmes the solutIOnl ISubstItutmg (8.9) mto (8.8), we obtam the equatIOnsl ~w2uk(n) = Lhl(n - nf)u l exp{ik[r(n W

f ) -

r(n)]}.

(8.10)1

15~

18.2 Equations of lattice vibrations and dispersion law

[hese equations are linear with respect to the displacement uk. Using the definition! 1(8.1), we havel t3I

[(n') - r(n) = E[ni - n:] ai = r(n - n'),1

"

k'\s a result, we obtaml

(8.11)1 [&l(k) = EJkl(n)exp{ -ikr(n)}j II]

[I'he vectors n or r refer to the real space of SItes or the coordmates m a crystalJ [The vectors k refer to the reciprocal space. The value hl(n) is the force dynamica~ ~atrix in the site representation, and Jkl (k) is the same matrix in k representationj [The condition for Eqs. (8.11) to be compatible has the formj (8.12)1 [I'hlS relation IS called the charactenstic equation for elgenfrequencles and ItS soluj ~lon relates the frequency of pOSSIble crystal vIbrations to a quaSI-wave vector k.1 [I'he wave vector dependence of frequency IS called the dIspersIon law and the charj lactenstic equatiOn IS referred to as a dIspersiOn equatiOn. Solvmg the dlspersiOriI ~quation, we obtain the dispersion law m = m(k) for crystallattice vibrations. w~ rote that the matrix lkl(k) is realJ Vkl(k) = [hl(n)cos(kr(n))·1 lIIJ

IOsmg the force matnx property (8.4), we obtaml Vkl(k) = EJkl (n) [cos(kr(n)) n

-In

V\s a result, the plane wave vIbratiOn (8.9) can be traveltng m a crystal if ItS fre1 ijuency m IS connected wIth the quaSI-wave vector k by the dIspersiOn law (8.12)1 [n a scalar model mstead of the system of (8. I I) there IS only one equatiOn, and thel k11spersiOn law can be wntten expltcltI)j fVZ(k)

=M

1 [J(n)

exp{ -ikr(n)}

=M

1 [J(n)[cos(kr(n))

III

- 111

III

from thIS equatiOn, we can see that the dIspersiOn law determmes the frequenc5J fu(k) as a periodic function of the quasi-wave vector k with a periodg of a reciproca~ ~

~(k+g) = m(k)l

[This proves to be the basic distinction between the dispersion law of crystal vibra-I ~iOns and that of contmuous mediUm vIbratiOns. The difference between the quas11 Iwave vector k and the ordmary wave vector IS also observed here. Only values of kI

1160

8 Fractional Dynamics of Media with Long-Range Interaction!

~ying

inside one unit cell of a reciprocal lattice correspond to physically nonequiva-j states of a crystalj [Returning to the dispersion law as a solution of the dispersion equation (8.12), wei ~ake into account that it is a cubic algebraic equation with respect to 002. The rootsl pf Eq. (8.12) determme the three branches for crystal lattIce VIbratIons specIfied! Iby the dispersion law: 00 = oo;(k), i = 1,2,3, where i is the number of a branch ofj Ivibrations IWe note that the characteristic equation (8.12) only determines the squared fre-j guency 002 • As a result, the i-branch dispersion law connects each value of the vectorl ~ with two frequencies: 00 = ±oo;(k). Hence the spectrum of squared frequencies ofj la vIbratmg crystal seems to be doubly degenerate. However as follows from (8.12)1 ~he dispersion law is invariant relative to the change in sign of the quasi-wave vec-j ~or: oo2(k) = 002( -k). Therefore the wave with a quasi-wave vector k and frequenc~ kO = -loo(k)1 describes the same states of crystal vibrations as the wave with vecj ~or -k and frequency 00 = -Ioo( -k)l. As a result, in order to describe independentl ~rystal states It suffices to conSIder the frequency for one SIgn that corresponds t9 lall pOSSIble k vectors mSIde a smgle umt cell of the recIprocal lattIce. ThIS allow~ Ius to conSIder VIbratIOns WIth positive frequencIes only.1 ~ent

18.3 Equations of motion for interacting particles ~et us conSIder a I-dImensIOnal lattIce system of mteractmg partIcles that are de1 IscrIbed by the equatIOns of motIonj

Iwhere un(t) are displacements from the equilibrium. The terms F(u n) characteriz~ Ian interaction of the particles with the external on-site force. The operator t~ i§ ~efined by i±e<j

IL~uk(t)

=

L

J(n,m) um(t),

Im=-oq

m"cnl Iwhere we use the summatIOn condItIOn over repeated mdexesJ

OO:xample 1J OCf the function J(n, m) has the formj

fJ(n,m) then

=

On+l,m - On,m,1

(8.14)1

18.3 Equations of motion for interacting particles

16~

[The operator illS the forward fimte dIfference of first orderJ

OO:xample 2J ~

Iwe obtain Ak LnUk(t)

I

=

Un+l (t ) - 2u n(t ) +Un- l (t ) =,1 2un(t )lI

Iwherethe operator ,12 is the finite difference of second orderj

OO:xample 3j IWe can consider the long-range interaction that is given bYI

Iwhere a IS a pOSItIve real number. In thIScase, we have nonlocal couplIng gIven byl ~he power-law functIon. Constant a IS a phySIcal relevant parameter. Some mtegerl Ivalues of a correspond to the well-known phySIcal SItuatIons: Coulomb potentIall ~orresponds to a = 0, dIpole-dIpole mteractIOn corresponds to a = 2, and the lImI~ ~ -+ 00 IS for the case of nearest-neIghbor mteractIOnJ

IRemark IJ for the mteractIon term of Eqs. (8.13) the translatIon mvanance condItIon has thel ~

t±::e<J

[: J(n,m) = 0

(8.15)1

m=-oq

m"ctl [or all n. If (8.15) cannot be satisfied, we must define the operator (8.14) byl

IL~

=

L

J(n,m) [8~ - 8~],

(8.16)1

1111=-001

~

Iwhere 8~ is Kronecker's delta. For this operator the interaction term is translationl Imvanant. Note that the nomnvanant terms can lead to the dIvergences m the contm-I luaus limit

IRemark2J [n general, we can conSIder a nonlmear long-range mteractIOn defined by the equaj Itiill:iS

Iwith operator (8.16). For example, we consider f(u) = uZ and f(u) = u - gu Z tha~ give the Burgers, Korteweg-de Vries and Boussinesq equations and their fractiona~ generalIzatIOns m the contmuous lImltJ

1162

8 Fractional Dynamics of Media with Long-Range Interaction!

18.4 Transform operation for discrete modeI§ [n this section, we define the operation that transforms the equations of motion fo~ IU n (t) of discrete model into continuous medium equation for a scalar field u(x, t) j ~n order to obtain a continuous medium equation for discrete equation for un(t)j Iwe assume that un(t) are Fourier coefficients of some function u(k,t). We define thel lfield u(k,t) on I-K!2,K!21 by the equationj ~(k,t) =

IE'J

L

un(t) e- ikxn = »>L\ {un(t)},

(8.17)1

Iwhere X n = nL1x, Llx = 2n!K is distance between oscillators. The inverse Fourierl Isenes transform IS defined b5J

n(t) =

1

-

K

jK/2 dk u(k,t) eikxn = »>il{u(k,t)}.

(8.18)

-K 2

~quatlOns (8.17) and (8.18) are the baSIS for the Founer transform, whIch IS obtame~ Iby transformmg from dIscrete vanable to a contmuous one m the hmlt Llx ----+ 01 I(K ----+ 00). The Founer transform can be denved from (8.17) and (8.18) m the hml~ las Llx ----+ O. We replace the discrete functionj

Iwith continuous u(x, t) while

lettin~

[I'hen change the sum to an mtegral, and Eq. (8.17) and (8.18) becomg +00

dx e-ikxu(x,t)

=

»>{u(x,t)},

(8.19)

(8.20)1 IWe assume

tha~

~(k,t) =

2'u(k,t)J

Iwhere 2' denotes the passage to the limit Llx ----+ 0 (K ----+ 00). We note that u(k,t) lis a Fourier transform of the field u(x,t), and u(k,t) is a Fourier series transforml pf un(t), where we can use un(t) = (2n!K)u(nfh,t). The function u(k,t) can bel ~erived from u(k,t) in the limit Llx ----+ 01 k\s a result, we define the map from a discrete model into a continuous one bYI ~he followmg transform operatIOn (Tarasov, 2006a,b)1

pefinition 8.1. Transform operatIOn T IS a combmatlOnl

16~

18.5 Fourier series transform of equations of motion

pf the followmg operations:1 10) The Founer senes transform:1

(8.21)1 1(2) The passage to the hmlt Llx ---+ OJ

bSf:

u(k,t)

---+

2{u(k,t)} = u(k,t).

(8.22)1

---+

§-l{u(k,t)}

(8.23)1

1(3) The mverse Founer transformj

W- 1 :

u(k,t)

=

u(x,t).

[he operation t = §-12 §L\ is called a transform operation, since it allow§ Ius to realize transforms of discrete models of interacting particles into continuou~ rIedmm models. To prove the transformatIOn, we consider an apphcatlOn of thel pperation t to L~Uk(t) and F(un(t)) of Eqs. (8.13)J

18.5 Fourier series transform of equations of motion! OC=et us consider a discrete system of mfimte numbers of particles With mterpartlckj Imteraction. The followmg theorem descnbes the Founer transform of the mteractioilj term,

[Theorem 8.1. Let J(n,m) be such that the conditionsj

V(n,m) =J(n-m) =J(m-n),

IE IJ(n)1

2

<

00

(8.24)1 (8.25)1

~

'bold. Then the Fourierseries transform !# L\ maps the equation~ i±e<j

IL~uk(t)

=

E

J(n,m)[un(t) - um(t)],

(8.26)1

Im=-oq

m"cnl

Iwhere Un

= Un (t)

is a position of the nth particle, into the equatio~

(8.27)1 ~(k,t) = §,1 {un(t)}j

Va(kLlx)

=

§L\{J(n)}j

1164

8 Fractional Dynamics of Media with Long-Range Interaction!

IProof. To denve the Founer senes transform of the mteraction term (8.26), we muI1 ~iply (8.26) by exp{ -iknfix}, and summing over n from -00 to +00. Then! f±-oo

IL

e-iknf1xL~Uk=

In=-oo

+00

+001

L

L

e-iknl1xJ(n,m) [Un-Um]1

n=-oo m=-oq

rn"ctl f±-oo

1=

L

+00

+00

(8.28)1

In=-oo m=-oo

n=-oom=-oq

rn"cn

m"cnl

Rsing the conditions (8.24) and (8.25), we introduce the notationsj (8.29)1 11l=-OOI

~

~(k,t) =

L

e iknLlxun(t).

(8.30)1

In=-oq

lOsing J( -n) = J(n ), the function (8.29) can be represented b~ 1+00

+O<j

Va(kfix) = LJ(n)(e-iknl1x+eiknl1x) =2 LJ(n)cos(kfix).

(8.31)1

from Eq. (8.31), we can see that Ja(kfix) is a periodic functionl

Iwhere m IS an mteger. Usmg (8.30) and (8.29), the first term on the nght-hand sIdel pf (8.28) gIve§ f±-oo

+00

+00

+09

!Here we use (8.24), (8.25), andJ(m' +n,n) =J(m'), andl 09

f±-oo

Va(O)

L

= ij

[Using J(m,n' +m) form: l-l-co

= J(n'), +00

IL L

In--oo m--oo

fuin

J(n)

= 2

LJ(n). n

00

(8.33)1

II

the second term on the right-hand side of (8.28) has thel +00

e lknLlxJ(n,m)um= L

m--oo

+09

Um L n--OOI

e iknLlxJ(n, m)1

18.5 Fourier series transform of equations of motion

165]

(8.34) n'- 001

!Ji=-oo

r'"c~ ~quatlOns

(8.32) and (8.34) gIve the expresslOnj (8.35)1

q

Iwhere fa (k!1x) is defined by Eq. (8.29). [I'he followmg theorem descrIbes the FOUrIer transform of Eqs. (8.13)j

[Theorem 8.2. The Fourier series transform .# LI maps the equations ofmotionj +00 d un(t) dt 2 = g m~ooJ(n,m)[un(t) - um(t)] +F(un(t)),

(8.36)

m"cil

Iwhere J(n,m) satisfies the conditions (8.24) and (8.25), Un is the position of the nthl Ipartlcle, and F IS an externaL on-site force, Into the equatlOn:1 (8.37)

~(k,t)

= §LI{Un(t)}J

Va(k!1x)

= §LI {J(n)}J

I(1nd .# LI is an operator notation for the Fourier series transformJ IProof To derive the equation for the field u(k,t), we multiply Eqs. (8.36) byl ~xp{ -iknLlx}, and summing over n from -00 to +00. Thenl

+00

1+00

pg L L v

ooffl

L

n

00

§§]

FfC'j

~(k,t) =

L

e-iknLlXun(t)j

==""l

~he

+001

e-ikni1xJ(n,m) [un-u m] +

left-hand SIde of (8.38) has the formj

[Thesecond term of the right-hand side of (8.38)

i~

e-ikni1xF(un). (8.38)1 ®

1166

8 Fractional Dynamics of Media with Long-Range Interaction! IE'J

IE

e-ikmixF(Un)

= »>LI

{F(un)}j

=="'l

[The Fourier series transform S>LI maps the interaction term (8.26) into expressioIlj 1(8.27). As a result, we obtam Eg. (8.38) m the form:1 (8.39) Iwhere »>LI {F (un)} is an operator notation for the Fourier series transform of F (un).1 D

18.6 Alpha-interaction of particIe~ [TheFourier series transform S>LI of the interaction term is defined by (8.27), when.j

lJa(k)= [, e-1knJ(n) =2 [,J(n)cos(kn), H-

00

~

land u(k,t) =

»>LI

(8.40)1

n-U

{un(t)}.lfthe function fa(k) is given, then J(n) can be defined byl

IWe can define a specIal class of mterpartIcle interactions by the followmg defimtIOnl

lI>efinition 8.2. The interaction terml

~Uk(t)= [, J(n-m)[un(t)-um(t)],

(8.42)1

W=-oq

~

Iwhere conditions (8.24) and (8.25) are satisfied, is called a-interaction, or long-I Irange alpha-mteractIOn, If the functIOn (8.40) satIsfies the condItIOnj

Iwhere a

> 0 and 0 < IA a I < 00.1

K=ondItIOn (8.43) means

k\s a result, we havel

tha~

18.6 Alpha-interaction of particles

[or k

-+

0,

1671

wher~

[Using Eq. (8.41), we can obtain the functions J(n) that describe an a-interaction]

OO:xample 1J [Using (8.41) with the function:1

Iwe obtain

J(n) =A a

(8.44)

Iwhere LdJ.1, v,z) is the Lommel function (Luke, 1969). As a result, the interactioij ~erm (8.42) with J(n - m) of the form (8.44) defines the a-interaction with a > OJ

OO:xample 2J [The interaction that IS defined byl

~an be considered as an a-interaction. Using (see Section 5.4.2.12 of Ref. (Prud-I r.Ikov et al., 1986)]

Iwe obtam

tTherefore the function J (n) = n

2

defines a-interaction with a

= 1J

OO:xample 3J IWe can conSIder the long-range interaction withl

~(n) = (~~t. IUsmg (Ref. (Prudmkov et al., 1986), SectlOn 5.4.2.121

Ikl ~ n,

1168

8 Fractional Dynamics of Media with Long-Range Interaction!

Iwe obtain +00 (-l)n 1 2 n a(k) = 2 ~ ~cos(kn) = 2k - 6'

A

[Then we havel

~s

a result, we have the a-interaction with

a = 2~

[Example 4j [f we consider the functionl

~hen

~

(_1)n (n) = a2-n2 '

Eq. (8.40) glve§

A nI j J(k) = asin(na) cos(ak) -""";'i

I

for k ----> 0, we obtaml

llim fa(k) -fa(o) = k--->O

k2

.na . 2sm(na)

V\s a result, the functiOn (8.47) defines the a-mteractiOn WIth a

-

21

[Example 5j for non-mteger and odd numbers s,1 (8.49)1 lIS an rz-mteracuon wlthl

s, 2,

for 0

< s < 2 (s i

1), we hav~

for s - 1, EquatiOn (8.4l) glve§

for non-mteger s > 2J

0 < s < 2, s > 2.

16~

18.6 Alpha-interaction of particles

Iwhere ~(z) is the Riemann zeta-function] ~t

[Example 6j can be directly verified that the functionj

~

(n) =

(- I)n

qf3 +n)qf3 -

(8.50)1

n)'

Iwhere a = 213 - 2> -1, defines the a-interaction. Using the series (Ref. (Prudnikov ~t a1., 1986), Section 5.4.8.12)

L_ r(s+ 1 +n)r(s+ - I )" I-n) cos(nk 00

(

2s 1 -

L 2 [ q2s+ 1) sin

2s

(k)"2 - 2r

1

2(s+

I

1)'

Iwhere s > -1/2 and 0 < k < 2n, we ge~

[I'he bmIt k ----t 0 gIvesl llim fa(k) -fa(O) = k--->O

Ikl a

1

qa+l)"

(8.51)1

function (8.50) with 13 > 1/2 defines the a-interaction with a = 213 - 2 > -1 j

[Example 7~ !For V(n) = l/n!J

l~ ~ = Iwhere Ikl

<

00.

k eCOS

cos(sin k),

(8.52)1

The passage to the limit k ----t 0 givesl

rI'hen we have the rz-mteractron wIth a - 1] [Examples of functions J (n) for a-interactions can be summarized in the table.1

1170

8 Fractional Dynamics of Media with Long-Range Interaction!

len)

a

Aa

n- 2

1

-n

len) = (n!)-I

1

-4e

(-1) nn- 2

2

1/2

Inl vr-

(p

> L, P =j= 5,4, ... )

(-1)" [a 2 - n2tl

Inl-(J3+I)

(_I)n,,~

(0<[3 <2, [3 =1= 1)

(_I)n"I/2

-13- - (J3)lnl~

1/2

L1([3+ 1/2, l/2,nn)

(-1 )nr- I ([3+ n)r- I ([3 - n)

-s(a

L

1)

2

(naI2) sin -I (na)

[3

r( -[3)cos(n[3 12)

[3-1

1

2[3-2

r- I (2[3-1)

18.7 Fractional spatial derivatlves 'Biemann- Liouville fractional derivative~ [Let

la, bl

be a finite interval on the real axis R The Riemann-Liouville fractiona~ and xD~ of order a > 0 are defined (KUbas et al., 2006) byl

~erivatives aDr;

x>a,1 x
17~

18.7 Fractional spatial derivatives

bD~f(x) = xDgf(x) = f(x)j bD~f(x)

= D~f(x)l

~t can be directly verified that the Riemann-Lioville fractional differentiation ofj ~he power functions (x - a)13 and (b - x)13 yields powerfunctionsl

I

Da(x-a)f3

Lx

=

Da(b - x)f3 = x

I

b

r(f3+l) (x-a)f3-al qa+f3+l) , r(f3+l) (b_x)13- a1 qa+f3+l) 1

Iwhere f3 > -1 and a > O. In particular, if f3 = 0 and a > 0, then the Riemann] OC:louvllle tractIOnal derIvatIves of a constant C are not equal to zeroj

~D~C =

l+ r(a 1) (x-a)-a,1

IxDgc=

qa + l )(b -

1

X)- a j

IOn the other hand, for k = 1,2, ... , [aJ + 1, we havq bD~(x-a)a-k=OJ

kDg(b _x)a-k

=

0.1

[The equatIOili lIs valId If, and only IfJ ;"1J

If(x) = [.Ck(x-a)a-kj ~

Iwhere n = [aJ + 1 and Ci, k = 1, ... .n, are arbitrary real constants. The equationj

lIs valId If, and only If)

'" If(x) = [.Ck(b-x)a-kj ~

Iwhere n = Ia I+ 1 and Ci, k = 1, ..., n, are arbitrary real constants.1 [Let a> 0, and n = lal + 1. If f(x) E AC"la,bl, then the Riemann-Liouville fracj ~ional derivatives exist almost everywhere on [a,b], and can be represented in thel form:

8 Fractional Dynamics of Media with Long-Range Interaction!

1172

r

a n-l (D~f)(a) k-a 1 D~f(z) (aDxf)(x) = ~ r(k_a+l)(x-a) + r(n-a)Ja dZ(x_z)a-n+l'

l

Daf)(x) = n~ (-1/ (Dxf) (b) (b-x/-a+ 1 d D~f(z) . r(k-a+l) r(n-a) x z(z_x)a-n+l x b

'=

ICaputo fractional

b

derivative~

[he Caputo fractional derivatives ~ DC; and ; Dg can be defined for functions bej ~onging to the space ACn[a,b] of absolutely continuous functions. Let a> 0 and le~ be given by n = lal + 1 for a It N, and n = a for a EN. If f(x) E ACnla,bl, thenl ~he Caputo fractional derivatives exist almost everywhere on la,bl. If a It N, theij

r

Iwhere n =

lal + 1. If a

=

n E N, theij

t ;D~ f) (x) = t~D~f)(x) OCf a

D~f(x)J

= (-1)nD~f(x)~

It Nand n = Ia I + 1, then Caputo fractional derivatives coincide with thel

~Jemann-LiOuvl1le

fractiOnal denvatIves m the foIIowmg cases. We hav~

IiI

[f(a) = (D;f)(a) = ... = (D~-l f)(a) = oj landthe equalItyj lIS valId, If

If(b) = (D~f)(b) = ... = (D~

1 f)(b)

= oj

~t can be dIrectly venfied that the Caputo tractiOnal dIfferentIatiOn of the [unctions (x - a)f3 and (b - x)f3 yields power function§ C Da(x

a x I

- a)f3 =

r(f3+1) (x-a)f3- a qa+l3+ 1) , l

r(f3+1) (b-x)f3- a rIcDa(b-x)f3= r(a+f3+1) , l

b

powe~

17~

18.7 Fractional spatial derivatives

Iwhere f3 > -1 and a > O. In particular, if f3 = 0 and a ~ional derivatives of a constant C are equal to zero:1

> 0, then the Caputo fracj

for k = 0, 1,2, ... ,n -1, we havt:j

[he Mittag-Leffler function E a lA. (x - a)a] is invariant with respect to the Caputol ~erivatives ~ D~, i.e.]

Ibutit is not the case for the Caputo derivative ~

[Liouville fractional

Dg j

derivativ~

Let us define the Liouville fractional derivative on the whole real axis R The left~ Isided Liouville fractional derivative D~ of order a > 0 is defined (Kilbas et aLl 12006) by

r

rD~f)(x) = D~(I;

U)(x)

n 1 d r(n-a)dxn

t, (x-z)a-n+l f(z)dz j -00

[The right-sided Liouville fractional derivative DIJ, of order a ~t a1., 2006) byl rD~f)(x)

= =

I

(-ltD~(I~

r

~-lt

> 0 is defined

U)(x) n

d n - a) dx n

1+

00

x

f(z)dz 1 (z - x)a-n+tJ

~ere D~ = dn7d~ is the usual derivative of order n, where n = [aJ rIeans the integer part of a. In the case of a = n E N, we obtaull

rD~f)(x) = D~f(x)J

IIf a = 0, thenl ~f f(x)

tD~f)(x)

(Kilba~

= (D~f)(x) = f(x)J

ELI OR) and f3 > a > 1, thenl (D%/lff)(x) = f(x) ,I

+ 1 and [aJ

1174

8 Fractional Dynamics of Media with Long-Range Interaction!

(D~I~f)(x)

=

(I~-a f)(x)j

[ffractional derivatives (D%f)(x) and (D%+k f)(x) exist, thenl

[I'he FourIer transforms of the LiOuvIlle derIvatIves of order a > 0, are defined byl ~he relations'

~=fik)a =

Ik1aexp{

=fsgn(x)~} j

land!# denotes the Fourier transform operator]

18.8 Riesz fractional derivatives and integral§ ~et

us conSider Riesz fractiOnal derIvatIves and fractiOnal mtegrals. The opera-I of fractiOnal integration and fractiOnal differentIatiOn m the n-dimensiOnall !Euclidean space JRn can be considered as fractional powers of the Laplace opera-I ~or. For a > 0 and "sufficiently good" functions f(x), x E JR, the Riesz fractiona~ ~ifferentiation is defined in terms of the Fourier transform!# b5J ~iOns

(8.53)1 [The Riesz tractiOnal integration is defined 5)1

(8.54)1 [I'he Riesz fractiOnal mtegratiOn can be reahzed m the form of the Riesz potentiall (refined as the FOUrIer convolutiOn of the forml

(8.55)1 Iwhere the function Ka (x) is the Riesz kernel. If ~he function Ka(x) is defined byl

[f a

i n,n+ 2,n+4, ..., thenl

[The constant Yn (a) has the formj

a > 0, and a #- n, n + 2, n + 4, ... ,1

18.8 Riesz fractional derivatives and integrals

n(a) =

1751

2a nn/2r(a/2)/r(n-a)/2, (-1 )(n-a)/22 a- 1n n/2 r( a/2) r(1 + (a - n)/2),

a-l-n+2k, a = n + 2k.

nEN K8.56)1

!ObvIOusly,the Founer transform of the Riesz fractional mtegration is given byl

[his formula is true for functions f(x) belonging to Lizorkin's space. The Lizorkinj Ispaces of test functIOns on jRn is a hnear space of all complex-valued mfimtelyl ~ifferentiab1e functions f(x) whose derivatives vanish at the origin:1

W=

{f(x) : f(x) E S(lRn ), (D~f)(O)

= 0, Inl E

NL

(8.57)1

Iwhere S(lRn ) is the Schwartz test-function space. The Lizorkin space is invariantl Iwith respect to the Riesz fractional integration. Moreover, if f(x) belongs to thel Lizorkin space, thenl

~~ f(x)I~f(x)

=

I~+f3 f(x)J

Iwhere a > 0, and f3 > 0.1 for a > 0, the Riesz fractional derivative D~ = ~orm of the hyper-smgular mtegral byl

-aula Ixl u can be defined in thel

Iwhere m > a, and (.1;' fHz) is a finite difference of order m of a function f(x) withl la vector step z E jRn and centered at the pomtx E jRnj

.1;'f)(z)

=

m'

~(-1/ k!(m~k)!f(x-kz)

[The constant dn ( m, a) is defined byl

1N0tethat the hyper-singular integral D~ f(x) does not depend on the choice of m ~ rr; Fi'C[f"--f"'(x~)--'b-e-'lo-n-g-s--ct-o--ct'-he-sp-a-c-e-o--'f~''''s-u'''ffi~c~ie-n--ct1'-y-g-o-o-d ....".--cf"--u-n-ctc.-io-n-s-,--'th-e-n-t;'-h-e--'F~o-u~ri'----'erl ~ransform

S> of the Riesz fractional derivative is given byl

1176

8 Fractional Dynamics of Media with Long-Range Interaction!

[his equation is valid for the Lizorkin space (Samko et a1., 1993) and the spacel K:(lR n ) of infinitely differentiable functions on lRn with compact supportj [he Riesz fractional derivative yields an operator inverse to the Riesz fractiona~ Imtegrationfor a specIal space of functions. The formula:1 P~I~f(x) = f(x),

°

a>

(8.58)1

Iholds for "sufficiently good" functions f(x). In particular, Equation (8.58) holdsl [or f(x) belonging to the Lizorkin space. Moreover, this property is also valid fo~ ~he Riesz fractional integration in the frame of Lp-spaces: f(x) E Lp(lRn ) for 1 ~ Ip < n/a. Here the Riesz fractional derivative D~ is understood to be conditionall~ ~onvergent m the sense tha~ a ~a (8.59)1 x = I'tm D xe' £-----+0

'

Iwhere the limit is taken in the norm of the space Lp(lR n ) , and the operator D~e i~ ~efined by

t, = dn(m,a) ~I>e Izla+n(L1;'f)(z)dz, Iwhere m > a, and (L1;" fHz) is a finite difference of order m of a function f(x) withl la vector step z E lRn and centered at the point x E lRn . As a result, the following Ipropertyholds. If 0< a < nand f(x) E Lp(lR n ) for 1 ~ P < n/a, thenl P~ I~f(x)

= f(x),

a>

OJ

Iwhere D~ is understood in the sense of (8.59), with the limit being taken in the norml pfthe space Lp(lRY. This result was proved in (Samko et al., 1993) (see Theoreml ~

IWe note that the RIesz denvative can be represented a~

r~u(x,t) = 2cos(~a/2) (D~u(x,t) +D<:u(x,t)), Iwhere a #- 0, 1,3,5... , and ~Ives defined 5)1

51~

are left- and right-sided Liouville fractional deriva-I

1

d

m

~u(x,t) = r(m - a) dxm a _u(x,t)

dm r(m- a) dxm (_1)m

=

(8.60)1

JX -00

1

00

x

U

t d

(x - c;')a-m+I' u(c;,t)dc; (c; _x)a-m+I'

(8.61)

Iwhere m - 1 < a < m. We note that the RIesz tractIOnal denvatives appear in thel ~ontmuous lImIt of lattIce models wIth long-range mteractIOns.1

177]

18.9 Continuous limits of discrete equations

18.9 Continuous limits of discrete equation~ [I'he transform operatton t allows us to obtam contmuous medIUm equattons hoiTI] ~iscrete models of particles. We consider an application of the operation T to t~Uk(t j land F(un(t)) of Eqs. (8.13). Let us give the basic theorem regarding the forcesl

IF(un(t)).

[heorem 8.3. The transform operation t maps the function F(un(t)) into thefuncj Ition F(u(x,t)), i.e.j [' F(un(t)) = F(u(x,t)), (8.62)1 !where u(x,t)

=

Tun(t),

if the function F satisfies 2F(un) = F(2u n).1

IProof. The Founer senes transform leads t9

IWe note that

1§L\F(un) -I F(§L\u n) = F(u(k,t))J [Thepassage to the limit Llx ----; 0 give§

rI'hen we havel

Iwhere we use :£'<%L\ I§-l: ~s

= <%:£'. The inverse Fourier transform give~

§{F(u(x,t))} ----; §-I{§{F(u(x,t))}}

=

F(u(x,t))l

q

a result, we obtain (8.62).

~f the mterparttcle mteractIOn that kX-mteractIOn, then the functIOnl

IS

descnbed by (8.16), and (8.24), (8.25), IS ani

Va(k) = [, e 1knJ(n) =2 [,J(n)cos(kn) N

00

~

(8.63)1

n-U

Isattsfies the condltIOIJI

(8.64)1 [or k ----; 0, where

a > 0, 0 < IAal < 00, and!

1178

8 Fractional Dynamics of Media with Long-Range Interaction!

[n the continuous limit the equations of motion for particles with a-interaction giv~ ~he fractional equations for continuous mediumj

rrheorem 8.4. The transform operation t maps the discrete equations of motio~ (8.65)

IWlth non-Integer a-interactIOn Into the tractIOnal continuous medIUm equatlOn:1 ;j2u(x,t) ()Ix at 2 =GaAaaxau(x,t)+F(u(x,t)), Iwhere aa / a Ixl a

=-

D~

(8.66)

is the Riesz.fractional derivative (Kilbas et a1., 2006), and"

(8.67)1 liS

afimte parameter.1

IProof. The Fourier series transform §.1 of Eq. (8.65) gives (8.37). We will be in1 ~erested in the limit Llx ----; 0. Then Eq. (8.37) can be written a§

(8.68)

~ere we use (8.64), and c; IS a fimte parameter that IS defined by (8.67). Note IRa satIsfies the condItIonl

tha~

[The expression for g-a,.1 (k) can be considered as a Fourier transform of the operatorl 1(8.16). Note that g ----; 00 for the limit Llx ----; 0, if G a is a finite parameter.1 [n the limit Llx ----; 0, Equation (8.68) givesl a

u(~,t)

= Ga

!7a (k ) u(k,t) +§{F(u(x,t))},

~(k,t)

(8.69)

= ~u(k,t)l

[The inverse Fourier transform of (8.69) has the form! a 2u(x,t) at 2 =Ga3"a(x)u(x,t)+F(u(x,t)),

(8.70)1

18.9 Continuous limits of discrete equations

(8.71)1 ~ere, we use the connection between the Riesz fractional derivative and its Fouried OCransform (Samko et aI., 1993; Kdbas et aI., 2006)l

ISubstItutIOn of (8.71) mto (8.70) gIVesthe contmuous medIUm equatIOn (8.66).

q

[Examples ofthe interaction terms J(n) that give the operators (8.71) are summaj Irized in the following tablej

g-a(x)

J(n) (_I)n"a+1

(_I)n"I/2

~ - (a+I)lnla+I/2LI(a+3/2, 1/2,nn)

-aa/alxl a

(-i) nn- 2

-(1/2) D;

n?

-tx D;

Inl-(a+I)

(O
-2r( -a)cos(na/2) aa /alxl a

Inl-(s+l)

(s>2, s#3,4,···)

((s-I)D;

(-ltr-I(1 + a/2+n)r- 1 (1 + a/2-n)

(-I)" [a 2 - n2

(n!)-I

tl

(a> -1)

-r-I(a + 1) aa /alxl a -(an/2) sin- I (na) D;

4eiD;

1180

8 Fractional Dynamics of Media with Long-Range Interaction!

18.10 Linear nearest-neighbor interactioij [..:et us consider an apphcatton of the transform operatton 1(8.14) of the formj

t

to the mteractton teriTIj

[This term describes the nearest-neighbor interaction. In this case, we havel

l!(n,m) = On+l,m - 2on,m + on-l,m,1 Iwhere On,m is the Kronecker symbolj [..:et us give the well-known theorem regardmg the nearest-neighbor mteractton.1

[Theorem 8.5. The transform operation f maps the equations ofmotio~ (8.72)

linto the continuous medium equationl (8.73)1 Iwhere G2 = g(Llx? is a finite parameter.1 IProof. To derive the equation for the field u(k,t), we multiply Eq. (8.72) byl ~xp{ -iknLlx}, and summing over n from -00 to +00. Thenl +00

I

d

+00

I

Un = g e
-

[fhe first term on the fight-hand Side of (8.74) 1+00

2Un + Un-I]+

I+00

e-iknt.xp(Un)

n=-oo I~

+09

II

e-iknt.xJ(n,m)um=

In=-oo

I

e-iknt.x[Un+I-2un+un_tl!

n=-OOI

+00

FI

+00

e-iknt.xUn+1 - 2

~=-oo

I

n=-oo

e-iknt.xUn +

+09

I

e-iknt.xUn_~

n=-~

lOsing the definition of u(k,t), we obtainl ~

I

111=-001

e-iknt.xJ(n, m )Um = eikt.xu(k,t) - 2u(k,t) + e-ikt.xu(k, t

)1

18~

18.11 Linear integer long-range alpha-interaction

t= [e ikL1x +e- ikL1x ~s

2]u(k,t)

= 2(cos(Mx)

-l)u(k,t)j

a result, we havd cc

L.

e-iknL1xJ(n,m)Um

ISubstitution of (8.75) into (8.74)

= -4sin2 Mx 2 u(k,t).

(8.75)

give~

(8.76)

for L1x ---+ 0, the asymptotic behavior of the sine is sin ( ~) Ibe represented b51

rv

~. Then (8.77) canl

lOsing the finite parameter G2 = g(L1x?, the transition to the limit L1x ---+ 0 in Eql 1(8.76)gives (8.78)1 Iwhere we use 0

<

IG21

< 00. The inverse Fourier transform §

1

of (8.78) has thel

[Qi]ll

V\s a result, we obtam the contmuous medIUm equation (8.73).

q

18.11 Linear integer long-range alpha-interactionl ~et

us consider the cham with hnear long-range interaction that is defined by thel of motIOi1j

~quatIOns

(8.79)

IwhereJ(n,m) =J(ln-ml), andl (8.80)1 Iwith positive integer number f3 J

1182

8 Fractional Dynamics of Media with Long-Range Interaction!

[Theorem 8.6. The power-law interaction (8.80) for the odd number fl is a-interaction,

IWith a = 1 for fl = 1, and a = 2 for fl = 3,5,7.... For odd number fl, the transforml operation maps the equations ofmotion (8.79) with the interaction (8.80) into th~ "(;ontinuous medium equation with derivatives offirst order for f3 = l]

t

d u(x,t) _ .

1--=:-'-::2,...--'- -IGI

dU(X,t) (()) - - - + F u x,t ,

(8.81)

I(1nd the second orderfor other odd [3 ([3 = 2m - 1, m = 2,3,4, ... )l

(8.82)

I =

(8.83)

ngL1x,

fire the finite parameters.1 IProof From (8.37), we get the equation for u(k,t) in the forml d u(k,t)

A

1------::-'-::-2---'-- =

A

-g [fa (kLlx) - fa(O)] u(k, t)

Ifa (kLlx) = L

+ §,d F (un(t) n,

e-iknLlxlnl-(l+{3)·

(8.84)

(8.85)1

11=-001

~

[The functIOn (8.85) can be represented b51 +001 .. +001 a(kLlx) = ~ _ _ (e-lknLlx +elknLlx) = 2 ~ - - cos (knL1x). l n +/3 n l +/3

A

'=

'=

(8.86)

[Then we can use (Ref. (Prudnikov et al., 1986) Section. 5.4.2.12 and Section]

15.4.2.7) the relahon~ oo

L=1

cos(nk) n 2m

= (_1)m- (2n) m

B2m

2(2m)!

(~)

2n

'

0

~

k

~

2n

Iwhere m = 1,2,3, ..., and B2m(Z) are the Bernoulli polynomials (Bateman and Erdej Ilyl, 1953). These polynomIals are defined byl I1J

IBn(k)

=

L C~Bs~-s,1 ~

Iwhere B s are the Bernoulli numbers (Bateman and Erdelyi, 1953) froml

18~

18.11 Linear integer long-range alpha-interaction

[z] < 2nj for

f3 = 1, we us~

["hen we havel

[I'he bmlt k ---+ 0 glvesl

~im fu(k) -fu(O) --+0

V\s a result, we have the a-interaction with Iwe have A

u(k)

(

=

=

k

a=

-n.1 1. For

_1)m2m ( k ) (2m)! (2n) B2m 2n '

f3 = 2m 0:( k

1 (m

= 2,3, ... )J

s; 2n

k'\s a result, we obtaml

[he transition to the limit Llx ---+ 0 in Eq. (8.84) with {3

=

1 give~

d u(k,t) dt 2 =Gl ku(k,t)+§{F(u(x,t))},

(8.87)

Iwhere Gl = ngLlx is a finite parameter. The inverse Fourier transform of (8.87) lead§ ~o contmuous medIUm equatIOn (8.81) WIth coordmate derIvatIve of first orderJ [he limit Llx ---+ 0 in Eq. (8.84) with {3 = 2m - 1 (m = 2,3, ...) givesl (8.88)

h

r

2 =

(_1)m 1(2n)2m 2 ~ 4(2m-2)! B2m-2g(Llx)J

lIS a fillIte parameter. The mverse FOUrIer transform of (8.88) leads to partIal dlfter-I ~ntial equation (8.82) of second order. q

1184

8 Fractional Dynamics of Media with Long-Range Interaction!

lRemark for even numbers f3, and f3 = 0, the interaction (8.80) is not an a-interaction. Fofj 1f3 = 0, we have (Ref. (Prudnikov et al., 1986), Section. 5.4.2.9) the relationj

[Then the bmlt Ax ---+ 0 glVe§

Va(Mx) ~ -In(Mx) for even numbers

---+

ooJ

m

since the expression has the logarithmic poles]

18.12 Linear fractional long-range alpha-interactionl ~n

thIS section, we consIder the alpha-mteractIOn that IS defined by the functIOnl (8.89)1

Iwhere 13 is a positive non-integer number. In this case, the correspondent continuou~ ~quatl0ns can have tractIonal and mteger spatIal denvatlves dependmg on a sIgn ofj

rJ)=2. [Theorem 8.7. The power-law alpha-interaction (8.89) with non-integer 13 is a~ linteraction with a = 13 for 0 < 13 < 2, and a = 2 for 13 > 2. For 0 < 13 < 2 (13 -=I- 1) ,I Ithe transjorm operation t maps discrete equations (8.79) with the interaction (8.89)1 linto the continuous medium equation with Rieszjractional derivatives oj order a j a aa atZu(x,t)-GaAaalxlau(x,t)=F(u(x,t)),

0
a-=l-l.

Wor 13 > 2 (13 -=I- 3,4,5, ... ), the continuous medium equation has the l(ienvatlve of second ordea

a

a

atZu(x,t)+Gas(a-l)alxlzu(x,t)=F(u(x,t)),

a>2,

(8.90)

coordinat~

a-=l-3,4, ...

rs-:m

Pa = glL1xl liS a finite parameter.1

min { a,z},

(8.92)1

18.12 Linear fractional long-range alpha-interaction

1851

IProof From Eq. (8.37), we obtain the equation for u(k,t) in the form:1

;Pu(k,t)

+ g [fa(kLlx) A

1------=:--'-;;-2---'--

A

fa (0)] u(k,t) - '%,dF (un(t))} = 0,

(8.93)

(8.94)1 for tractional pOSItIve a, the functIOn (8.94) can be represented (LaskIn and! IZaslavsky, 2006; Tarasov and Zaslavsky, 2006a; Tarasov, 2006b) byl

Iwhere Liy(z) is a polylogarithm function (Lewin, 1981). Using the series represen-I ~atIOn of the polyloganthm (ErdelYI et aL, 1981)j (8.96)1 Iwhere Izi

< 2n, and Yi=- 1,2,3 ..., we obtainl

Iwhere a i=- 0, 1,2,3 ..., S(z) is the Riemann zeta-function,

IkLlxl < 2n, andi

Ea =2r(-a) cos (na) 2 . from (8.97), we

hav~

(8.98)1

fJa(O) = 2~(1 + a)·1

Aa(kLlx) - fa (0) = Aa ILlxl a

cc

Ikl a+ 2 L ..::....o...---,---,----'-(Llx)2n( _k2)n,

(8.99)

n=!

Iwhere a i=- 0, 1,2,3... , and IkLlxl < 2n1 ISubstItutIOn of (8.99) Into Eg. (8.93) gIve§

a ~;~,t) +gAalLlxla Iklau(k,t +2g ~ s( a ~~)~ 2n)(Llx?n(-k2)n u(k,t) - '%,dF (un(t))} = O. (8.100 ~n

the bmIt Ax

---+

0, EquatIOn (8.100) can be wntten In the sImple foririi

1186

a

8 Fractional Dynamics of Media with Long-Range Interaction!

u(~,t) +Ga Y a,L1(k) u(k,t)-S?L1{F(un(t))} =0,

Iwhere we use the fimte parameter (8.92),

ayfO,1,2, ... , (8.101

an~

[he expression for :o/a,L1 (k) can be considered as a Fourier transform of the interacj ~ion operator (8.14). From (8.92), we see that g ----+ 00 for the limit Llx ----+ 0, and finitel Iva1ue of Ga. [The transItIOn to the bmlt Ax ----+ a In Eg. (8.101) glVe§

a

u(~,t) +GaYa(k)u(k,t)-S?{F(u(x,t))} =0,

ayfO,1,2, ... ,

0< a < 2, a yf 1, a> 2, a yf 3,4, ....

(8.103

(8.104

[he Inverse Founer transfonn of (8.103) lsi

a u(x,t) at 2 +GaY'"a(x)u(x,t)-F(u(x,t))=O ayfO,1,2, ...,

0 2, a yf 3,4, .... IHere, we have used the connectIOn between the Riesz fractIOnal denvatIve and founer transform (Samko et aI., 1993)j

It~

k\s a result, we obtaIn the contInUOUS medIUm eguatIOn§

a u(x,t) aa at 2 -GaAa~u(x,t)=F(u(x,t)),

0
a u(x,t) a at 2 +Ga,(a-1)~u(x,t)=F(u(x,t)),

a>2,

ayf1,

(8.105

ayf3,4, ... (8.106)

[This ends of the proof.

q

18.13 Fractional reaction-diffusion equation

187]

18.13 Fractional reaction-diffusion equationl [..:et us consIder the equatIons of first order wIth respect to tIme denvatIvfj

(8.107~ Iwhere un(t) is displacement of nth particle from the equilibrium. The terms R(u n) ~haracterize an interaction of the particles with the external on-site force. Equation~ 1(8.107) wIth posItIve non-mteger number a descnbe cham wIth the power-law long-I Irange mterpartIcle mteractIonJ

rrheorem 8.8. The transform operation T maps Eqs. (8.107) with 0 < a < 21 I( a I- I), into the continuous medium equation with the Riesz fractional derivativel ~~ oforder a.j

ua~,t -GaD~u(x,t)=R(u(x,t)), IWhere

A a = 2 Tt.

0
-a) cos (n2a) ,

aI-I,

(8.108

(8.109~ (8.1lOj

lIS afimte parameter.1 !proof. Multiplying Eq. (8.107) by exp{ -iknLlx}, and summing over n from 1+00 , we obtaml

a

+00

l

L

e-iknL1x--.!:!..!:..+g at

=-00

+00

+00

L L

n=-oo m=-oo

m"cnl

-00

tq

- ikrusx

e

In - ml

a+!

t±::<Xj

FL

(8.111~

e-iknL1XR(un).

111=-001

~(k,t) =

L

e iknL1x un(t) 1

10=-oq

Iwe have

[Thenthe Fourier series transform !# L1 maps the equationsl

8 Fractional Dynamics of Media with Long-Range Interaction!

1188

It~Uk(t)

IE'J

=

L

J(n,m)[un(t) - um(t)],1

=="'l

m#nl linto the equationl

Iwhere we use Eqs. (8.26), (8.27), andj ~(k,t)

= §,1{u n(t )}J

Va(kLlx)

=

§,1{J(n)}j

[The Fourier series transform §,1 maps the interaction term (8.26) into expression! 1(8.27). As a result, we obtaIn Eq. (8.111) In the formj

(8.112

(8.113~

~ere §,1 {R(un)}

is an operator notation for the Fourier series transform of R(un)l for fractIOnal positrve a, the functIOn (8.113) can be represented byl

(8.114~ Iwhere a i- 0, 1,2,3 ..., '(z) is the Riemann zeta-function, pf (8.114) Into (8.112) glve~

IkLlxl < 2n. Substitutiorf

~ -gA a lL1xla Ikl a U(k,tl at\-2n (L1x)2n(-k2)n u(k,t) 2n !

= §,1 {R (un(t ))}.

[n the limit L1x ----; 0, Equation (8.115) can be written in the formj

Iwhere we use the finite parameter (8.110), andj

(8.115

18.13 Fractional reaction-diffusion equation

18~

[he expression for ga,L! (k) can be considered as a Fourier transform ofthe interacj ~ion operator. From (8.110), we see that g ----; 00 for the limit Lix ----; 0, and finite valu~ ~

[The transItIOn to the lImIt Ax ----; 0 m Eq. (8.116) glVe§

du(k t) - - ' - - Galklau(k,t) = §{R(u(x,t))},

(8.117

Iwhere 0 < a < 2, and aiL Using the inverse Fourier transform §-1 of (8.117),1 Iwe obtain the continuous medium equation (8.108) with the Riesz derivative D~ of prder a. This ends of the proof. q ~quation

(8.108) is a fractional reaction-diffusion equation. For R(u) = 0, Equa-I (8.108) is a fractional kinetic equation that describes the fractional superdif-I ~usIOn (Smchev and Zas1avsky, 1997; Uchmkm, 2003a,b; Gorenflo and MamardI) [997; Mamardl et a1., 2001)l fractIOnal reactIon-dIffusIOn equatIons are mathematIcal models that descnbg Ihow the concentratIOn of one or more substances dlstnbuted m space changes underl ~he mfluence of two processes: local chemIcal reactIOns m whIch the substances arel ~onverted mto each other, and nonloca1 superdiffusIOn, whIch causes the substancesl ~o spread out m space. FractIOnal reactIOn-dIffusIOn equatIOns can be naturally ap1 IplIedm chemIstry. These equatIons also descnbe tractIonal dynamIcal processes ml IbIOlogy and phYSICS. MathematIcally, tractIOnal reactIon-dIffusIOn equatIons havg ~he form of semI-lInear parabolIc tractIonal dIfferentIal equatIOns] [The most sImple fractIOnal reactIOn-dIffusIOn equatIOn concermng the concen-I ~ratIOn u of a smgle substance m one spatIal dImensIOn. It can also be referre~ ~o as the fractIOnal generalIzatIOn of the Kolmogorov-PetrovskY-PIskounov equa1 ~ion (Kolmogorov et al., 1937). If the reaction term vanishes (R(u) = 0), thenl ~he equatIOn represents a pure superdIffusIOn process. The correspondmg equatIOilj ~s the tractIOnal dIffusIOn-wave equatIOn (MamardI, 1996; Gorenflo et a1., 2000j Kgrawal, 2002). For R(u) = u(l - u), we have a fractional generalization of thel fIsher's equatIOn, also known as the Flsher-Kolmogorov equatIOn, that was ongI1 r.ally used to descnbe the spreadmg of bIOlogIcal populatIOns (FIsher, 1937). SUbj Istitution of R(u) = u(l - uZ) into (8.108), we obtain a fractional generalization o~ ~he Newell- WhItehead-Segel equatIOn that IS used to descnbe RayleIgh-Benard conj IvectIOn (Newell and WhItehead, 1969; Segel, 1969). A tractIOnal generalIzatIOn o~ ~he Zeldovich equation is obtained by using R(u) = u (1 - u)(u - a) and 0 < a < 1J IWe note that the Zeldovlch equatIOn anses m combustIOn theory (Zeldovlch et al.J ~938). Equation with a = 2 and R(u) = uZ - u3 is also called the Zeldovich equa~ ~IOn (GIldmg and Kersner, 2004). FractIOnal reactIOn-dIffusIOn equatIOns WIth CUbICI InonlIneanty and Brusselator model were consIdered m (Gafiychuk, 2009)J ~t IS easy to obtam tractIOnal two-component reactIOn-dIffusIOn equatIOns, whlchl lallow us to have a much larger range of possIble phenomena than theIr one1 klimensional equations. An important idea, which was first proposed by Alan Turin~ ~ion

1190

8 Fractional Dynamics of Media with Long-Range Interaction!

I(TurIng, 1952), IS that a state that IS stable m the local system should become unj Istable in the presence of diffusion. This idea seems unintuitive at first glance a§ kliffusion is commonly associated with a stabilizing effect. It is important to studYI 1mstabIlIty m the presence of superddfusIOn.1

18.14 Nonlinear long-range alpha-interactioIlj OC'et us conSIder the dIscrete equatIOns wIth nonlInear long-range alpha-mteractIOnl ~hat IS deSCrIbed by the mteractIOn terml t±::<Xj

~~f(Uk)

=

L

Ja(n,m)[f(u n) - f(u m)],

(8.118~

m=-OOI

m"cnl Iwhere f(u) is a nonlinear function of Un = un(t), and Ja(n,m) = Ja(n - m) definesl ~he a-interaction. If a is integer (a = 1,2,3,4), then the interaction with f( u) = u2 land f(u) = u - gu 2 give the Burgers, Korteweg-de Vries and Boussinesq equation~ lin the continuous limit. If we use the fractional a in Eq. (8.125), we can obtain thel ~ractIOnal generalIzatIOn of these equatIOns.1 1

[Theorem 8.9. The transform operation maps the equations ofmotionI (8.119

Iwhere F is an external on-site force, and Ja(n) defines the a-interaction, into th~ rontmuous medium equatlOnl

(8.120 Iwhere G« = glLlxlU is a finite parameter.1

IProof. The FOUrIer serIes transform of the mteractIOn term (8.118) can be presente~ as

+09

FL L

e iknL\xJ(n,m) [f(u n) - f(um)]1

In--oo m--oq

~

FL

In=-oo

+00 m=-oo

111'#

+00

+09

n=-oom=-oq

mini

19~

18.14 Nonlinear long-range alpha-interaction

for the first term on the right-hand side of (8.121 ):1 1+00

+001

[.

[. e-iknL1xJ(n,m)f(Unj

In=-oo m--oq

rn"ctl Efoo

F [.

+001

e-iknL1xf(u n) [. J(m')

= §L1 {f(u n)}

Ja(O),

(8.122~

m'--od

In-- oo

Im'"c~

Iwhere we use J(m' + n, n) = J(m')J for the second term on the nght-hand SIdeof (8.121)j +00

+09

In=-oo

m=-oq

IE [.

e-ikni1xJ(n,m)f(um)1

rn"cnl +00

F [.

+09

f(u m) [. e-iknL1xJ(n,mj

m=-oo

n=-oq ~"cml

+00

+00

f(Um)e-ikmL1x

e- ikn'L1x J(n')

= §L1

{f(u n)} Ja(Mx)

Iwhere we use J(m,n' +m) = J(n')J k\s a result, we obtaIn the equatIonj

d Ct(k,t) dt 2

A

=

A

g[Ja(O) -Ja(Mx)]§L1 {f(u n)} +§L1 {F(u n)},

Iwhere Ct(k,t) = §L1 {un(t)}, and Ja(Mx) = §L1 {J(n)}j for the hmIt Llx ----+ 0, EquatIOn (8.123) can be wntten

Iwhere we use finite parameter Ga

(8.123

a~

= glL1xl a, and!

aere, the functIOn R a satIsfies the condItIOnj

[n the limit L1x ----+ 0, we getl

d

2 u(k,t)

- G« :Ya(k) § {f(u(x,t))} -

§

{F (u(x,t))}

=

0,

(8.124

1192

8 Fractional Dynamics of Media with Long-Range Interaction!

[I'he mverse Founer transform of (8.124) gIve§

a

-z u(x,t) - Ga g-a(x) j(u(x,t)) - F (u(x,t))

=

0

Iwhere g-a(x) is an operatorl

q

k\s a result, we obtain the continuous medium equation (8.120). lRemark k\s an example of Ja(n), we can use the function§

(8.125~

(-It Ja(n) = r(l +a/2+n)r(1 +a/2-n)'

I

Iwhich defines the a-interaction] OC=et us conSIder examples of quadratIc-nonlInear long-range mteractIons (Tarasov J 12006a,b). !Example 1~ [The contmuous lImIt of the cham equatIOnsj

a l

(t) u at

Iwhere Ji(n) (i bon:

= gl

+00

+00

Im"cn

m"ctl

= 1,2) define

the ai-interactions, gives the fractional Burgers equaj 1

at u(x,t)

I

m~ooJI (n,m)[u~ - u~] + s:m~oo h(n,m)[un - um]1

+ Gal u(x,t) alxl a ) u(x,t) -

2

Ga2alxl a2 u(x,t) = O.

(8.126

IWe note that a speCIalcase of thIS equatIOn was suggested m (HIler et al., 1998). Fofj kxI = 1 and az = 2, Equation (8.126) gives the Burgers equation (Burgers, 1974)1 ~hat IS a nonlmear partIal dIfferentIal equatIOn of second orderj

19~

18.14 Nonlinear long-range alpha-interaction

[t is used in fluid dynamics as a simplified model for turbulence, boundary Ibehavior, shock wave formation, and mass transport.1

laye~

OO:xample 2J [Thecontinuous limit of the system of equations:1

aun(t) ---at=gl

L co

2

JI(n,m)[un -

2 U m]+g3

Im"cn

L

h(n,m)[un-u m ]

m"ctl

Iwhere Ji(n) (i = 1,3) define the ai-interactions with al = I and a3 = 3, givesl IKorteweg-de YrIes (KdY) equation:1

first formulated as a part of an analysis of shallow-water waves in canals, it ha~ Isubsequently been found to be mvolved m a wide range of phySICS phenomena,1 ~speclany those exhlbltmg shock waves, travellIng waves and solItons. Some theo1 Iretical phySICS phenomena m the quantum theory are explamed by means of KdYI Imodels. It IS used m flUId dynamiCs, aerodynamics, and contmuum mechanics as ij rIodel for shock wave formatIOn, solItons, boundary layer behaVIOr, turbulence, and! mass transport~ OCf we use non-integer ai-interactions for Ji(n), then we obtainl 1

3

at u(x,t) - Gal u(x,t) alxl a1u(x,t)

+ Ga3alxla3u(x,t) = 0

[This equation is a fractional generalization of KdV equation (Momani, 2005; Miski-j nis, 2005). OO:xample 3J [rhe contmuous lImit of the equatlOnsl

If(u) =u-guZ,1 land Ji(n) define the ai-interactions, give§

a aa2 at 2 u(x,t) - Ga2alxl a2u(x,t)

aa2

aa4

+ gGa2alxla2u2(x,t) + Ga4alxla4u(x,t) = 0

(8.127) rrhls equation IS the tractIOnal Boussmesq equation that takes into account a nonlo-I ~al interaction of medium particles. If a2 = 2 and a4 = 4, Equation (8.127) gives thel

1194

8 Fractional Dynamics of Media with Long-Range Interaction!

[Boussinesq equation that is a nonlinear partial differential equation of fourth

orde~

[This equation was formulated as a part of an analysis of long waves in shallowj Iwater. It was subsequently apphed to problems m the percolation of water m porou~ Isubsufface strata. It also crops up m the analysIs of many other phySIcal processes]

18.15 Fractional 3-dimensional lattice eguatioij [I'he generahzation for the 3-dlmenslOnal case can be easy reahzed. Let us conslderl ~he 3-dimensionallattice that is described by the equations of motion:1 (8.128

p'(n,m) =J(n-m) =J(m-n)l [To obtain the continuous equations, we assume that Un (t) are Fourier coefficients ofj ~he function u(k,t), where k = (kl,k2,k3), such tha~

~(k,t)

= L Un(t) e- 1kr n = §L\{un(tn] lIIJ

[Thevector r(n) is defined byl

Iwhere ai are the translatlOnal vectors of the lattice. The contmuous hmlt IS definedl Iby lad ----+ o. Multiplying (8.128) by exp{ -ikrn }, and summing over n, we obtain the equatioij [or u(k,t) in the forml (8.129 Iwhere §L\{F(u n)} is an operator notation for the Fourier series transform of F(un)j Ii.i.lli.I Ifa(ka) = e 1krn J(n)J

L lIIJ

18.16 Fractional derivatives from dispersion law

1951

[n 3-dimensionallattice, the a-interaction with a = (at, al, a3), is defined as ani linteraction that satisfies the conditionSl i = 1,2,3,

Iwhere0

< IA a;I <

00.

(8.130

Equation (8.130) means tha~ 13

Va(k) -Ja(O)

31

[Aa;lk;ja;

=

,

I

+ [Ra;(k),1 i

II

[n the continuous limit (lad ----+ 0), the a-interaction in the 3-dimensional latticel gives the continuous medium equations with the fractional derivatives JUijJlxlUij Ii - 1, 2, 3, m the form:1

aSu(r,t) __ '" A aaiu(r,t) (()) atS - g 6 a; alXil a; +F u r,t Iwhere s - 1, 2. ThiS equation descnbes multIfractlOnal properties of Imedium

contmuou~

18.16 Fractional derivatives from dispersion lawl ~et

us conSider a wave propagatIOn m a medIUm and obtam the nonlInear parabolIcl (Leontovlch, 1944; LlghthlIl, 1978; Kadomtsev, 1988; Sagdeev et al.J

~quatlon

If98ET. [The wave vector k can be represented in the formj ~

- ko + tc - ko + /(11 + /(1..,

(8.13q

Iwhere ko is the unperturbed wave vector and subscripts (II, -.l) are taken respectivel~ ~o the directIOn of ko. Let us conSider the disperSIOn lawj

[I'he disperSIOn law IS symmetnc Ifl ~(k) =

rrhen we

ro(k)l

represen~

fu(k) = ro(lkl)

= ro(ko+ Ilkl-kol)·1

1196

8 Fractional Dynamics of Media with Long-Range Interaction!

Ik -

for the case 11\"1 =

kol «ko = Ikol, we obtainl

(8.132~ v~ = ~n

m) ak (rP 2

.

k=ko

(8.133~

this case, we ge~ (8.134

[I'hen substttutmg (8.134) mto (8.132), we obtaml

(8.135~ Iwhere ~ = m(ko). Using the inverse Fourier transform for expressions (8.133) and! 1(8.135), we obtain!

.au

at

-

Iwhere u = u(r,t) hanstorm'

. au

=~U-IV

g

vg

vg

- - - L 1 j u - - L 1 llu + F(u) 2k 2 '

ax

= u(t,x,y,z), and x is along ko. Here

(8.136

we use the inverse Fourierl

(8.137~

(8.138~ (8.139~ ~quatIOn (8.136)

IS the nonlInear parabolIc equatIOn (LeontovIch, 1944; LIghthdlJ Kadomtsev, 1988; Sagdeev et aI., 1988). The change of vanab1es froi11l I(t ,x,y,z) to (t ,x - vgt,y,z) givesl ~978;

(8.140~ Iwhich IS also known as the nonlmear Schrodmger equatIOnJ IWave propagatIOn m the lattIce WIth long-range mteractIOn can be Iby rewriting the dispersion law (8.135) in the fonnj

generalIze~

(8.14q

18.16 Fractional derivatives from dispersion law

1971

Iwith new finite constants G a , and Ga. Using (8.137) and the connection betweenl Riesz fractional derivative and its Fourier transform (Samko et aI., 1993 )1

(-L1.d a / 2 ~ (1C~ )a/2 j r-L1II)f372 ~ (1CTI) f3 721 Iwe obtam from (8.141) the equatIOnj (8.142 Iwhere u = u(t,x,y,z). Let us change the variable from x to ~ ~

Iwe

= x- vgt in Eq. (8.142)J

obtain (8.143

[I'hIS IS the tractIOnal nonlInear parabolIc equatIonJ IRemark lJ

for Ga

= 0 and F(u) = blulzu, Equation (8.143) is the fractional Ginzburg-Landaul

~quation

(Weitzner and Zaslavsky, 2003; Tarasov and Zaslavsky, 2005; Milovano\j land Rasmussen, 2005; Tarasov, 2006c)j IRemark 2J

[The first and second terms of (8.143) on the nght-hand SIde are related to wavel Ipropagatlon m oscIIIatory medIUm WIth long-range mteractlon of partIcles. The terIll] Iwith F(u) on the right-hand side of Eqs. (8.142) and (8.143) correspond to the wavel ImteractIOn due to the nonlmear propertIes of the medIa. Thus, EquatIOn (8.143) canl Ibe used to descnbe non-local processes of self-focusmgJ IRemark 3J W'Jote that we can conSIder I -dImenSIOnal SImplIficatIOns of Eq. (8.143). For thel Ivariables (t,x), we obtainl

(8.144~ Iwhere u = u(t, ~), ~

= x - vgt. For the variables (r.z), Equation (8.143) give~

(8.145~

1198

8 Fractional Dynamics of Media with Long-Range Interaction!

Iwhere u = u(t,z)j

18.17 Fractal long-range interactionl [Usually we assume for long-range interaction (LRI) that each chain particle act§ pn all chain particles. There are systems where this assumption cannot be used! I(Tarasov, 2008). In general, the cham cannot be considered as a straight hne. Fo~ ~xample, the hnear polymers can be represented as some compact oliiects. It is weIll Iknown that "tertiary structure" of protems refers to the overall foldmg of the entir~ Ipolypeptide chain into a specific 3D shape (van Holde, 1998; Protein Data Bankj 12010; Kolinski and Skolnick, 2004). The tertiary structure of enzymes is often comj Ipact, globular shaped (van Holde, 1998; Protem Data Bank, 2010). In thiS case, wei ~an conSider that the cham particle is mteracted Withparticles of a ball Withradms R.I [I'hen only some subset An of cham particles act on nth particle. We suppose that nthl Iparticle is interacted only with kth particles with k = n ±a(m), where a(m) EN and! rI - 1,2,3, .... We can conSider fractal compactified hnear polymers (chams), suchl ~hat these "compact objects" satisfy the power-law N(R) rv Rd , where 2 < d < 3 an~ IN(R) is a number of chain particles inside the sphere with radius R. As an exam-I Iple of such case, a(m) will be described by exponential type functions a(m) = bmj Iwhere b > 1 and bE f':J. In thiS case, the LRI Will be called fractal mteractionl ~et us conSider a bnef reView of fimte difference operators to fix notations and! Iprovide convement references. A forward difference is defined byl

I4hu(x,t) = u(x+h,t) - u(x,t)J Dependmg on the apphcation, the spacmg h may be vanable or held constant. Wi.j ~an consider h = a(m)ho. The forward difference can be considered as a differenc~ pperator, which maps the function u(x,t) to LlhU(X,t). This operator can be repre-I Isented as

Iwhere Th is the shift operator With step h, defined bYI

rnu(x,t)

= u(x+h,t),

(8.146]

land I is an identity operator. Fimte differences of higher orders can be defined ml IrecurslVe manner as

k\nother pOSSible and eqUivalent defimtlOn i§

[The difference operator Llh is hnear and satisfies Leibmz rule. Applymg Taylor'sl ~heorem for u(x + h, t) with respect to h, we obtain the equationj

18.17 Fractal long-range interaction

t

(x+h,t)

19S1 k

co

= (;

hk! D~u(x,t),

(8.147~

IwhereD~ denotes the derivative operator, mapping u(x, t) to its derivative du(x, t)/ dxl ~quatIOn (8.147) allows us to represent the operator (8.146) m the formj

(8.148~ k\s a result, the forward difference operator has the formj

iLih = exp{hD;} -I = exp i{ -ihD;} -I.

(8.149j

formally mvertmg the exponentIal, we havt:j

[I'hIS formula holds m the sense that both operators gIve the same result when applIed! ~o a polynomIal. Even for analytIc functIOns, the senes on the nght IS not guaranteedl ~o converge. It may be an asymptotIc senes. However It can be used to obtam morel laccurate approXImatIons for the denvatIve. For mstance, usmg the first two terms ofj ~he series yields the second-order approximation to D~u(x, t ).1 OC:et us conSIder a I-dImensIOnal system of mteractmg partIcles, whose dlsplacej Iments from the equilibrium are un(t), where n E Z. We assume that the system i§ ~escnbed by the equations of motIOnj

(8.150~ (8.151 landh is the distance between the particles. Herea(m) andJ(n,m) are some function~ pf integer number m. The function a(m) takes integer values at integers m. The casel IJ(n,m) = J( In- ml) was considered in (Tarasov, 2006a). The right-hand side ofthel ~quatIOn descnbes an interaction of the partIcles m the system. In thIS section, wei consider the casel IJ(n,m) = b(m)l Irhen the operator L~ has the form]

(8.152

1200

Iwhere

8 Fractional Dynamics of Media with Long-Range Interaction!

0:

is Kronecker's delta]

IWe Illustrate thIS cham equatIOn (8.150) wIth well-known example (MaslovJ [973). In the case of nearest-neIghbor mteractIOnJ

kz(m)

=

1,

b(m) = OmO.

(8.153~

ISubstItutmg (8.153) mto Eq. (8.150), we havel (8.154 OC'et us gIve the basIc theorem regardmg the nearest-neIghbor mteractIonJ

[Theorem 8.10. Equation (8.154) is equivalent to the equatiorj (8.155 Iwhere u(x,t) is a smooth/unction such that u(nh,t) = un(t)l IProof Using a smooth function u(x,t) such thatl ~(nh,t) = un(t), ~quatIOn (8.154)

(8.156~

can be represented a~ (8.157

[I'hiS IS the dIfferentIal-dIfference equatIOn. EquatIon (8.148) for the shIft operatorl 1(8.146) gives ~xp i{ -ihD;}u(x,t) = u(x+h,t)j [Then we can rewnte Eq. (8.157) m the forml (8.158 Rsing the Euler's formula, we obtain Eq. (8.155). Equation (8.155) is a pseudo-I q khtterentIal equatIOn.

IRemark lJ [The properties of Eq. (8.155) were conSIdered m (Maslov, 1973)1 lRemark 2J IWe note that for h ---+ 0, we

hav~

20~

18.17 Fractal long-range interaction

land Eq. (8.155) g1Ve~

a u(x,t) _

f---=-'-::2-----'--

C

Diu x,t ) -- 0

2 2 (

[I'hIS equatIOn IS a wave equatIOn] ~f a(m) in Eg. (8.160) is not a constant function, then we have the long-rang~ linteraction of the chain particles. Note that the function a(m) should be integer-I Ivalued. For example, a(m) = m, a(m) = 2m and a(m) = m3m. The se~

IAn={n±a(m): mEN}c~ klescribes the numbers of particles that act on the nth particle] ~.

12.

If a(m) = m, where mEN, then An is a set of all integer numbers Z for all n, i.e.j IAn = Z. In thIS case, the nth partIcle mteracts WIth all cham partIclesJ If a(m) = 2m, where mEN, then An is a subset of Z, i.e., An C Z. In this case, thel bth partIcle mteracts only wIth cham partIcles WIth numbers n ± 2, n ± 4, n ± 8J

n±16 .... 13. If a(m) = m3m, where mEN, then An C Z. The nth particle interacts only withl ~ham

partIcles WIth numbers n ± 3, n ± 18, n ± 81, n ± 324 ..

J

IWe suppose that a(m) is exponential type function such that a(m) = am, wherel ~

> 1 and a E l"L ThIS functIOn defines the fractal long-range mteraction suggested!

1m (Tarasov, 2008)J ~efinition 8.3. The mteractIOn term (8.151) IS called fractal long-range mteractIOili lifthe functions a(m) and b(m) have the formj

(8.159~

Iwhere a> 1 and a E N, and 0

< b < 1, b E lRj

~s a special case, we can consider b = ad

2J

IRemark lJ

[The power-law a(m) = am, where a E N, and a > 1, can be realized for compactl Istructure of lmear polymer molecules. For example, a hnear polymer molecule I~ r.ot a straIght hne. Osually thIS molecule can be consIdered as a compact oliject. I~ lIS well-known that tertIary structure of protems refers to the overall foldmg of thel ~ntIre polypeptIde cham mto a specIfic 3D shape (van Holde, 1998; Protem Datij !Bank,2010; KohnskI and Skolmck, 2004). The tertIary structure of enzymes IS oftenl la compact, globular shape (van Holde, 1998; Protem Data Bank, 2010). In thIScasel Iwe can consIder that the cham partIcle IS mteracted WIth partIcles mSIde a spher~ IWIth radIUs R. Then only some subsets of cham partIcles act on nth partIcle. We asj Isumethat nth particle is interacted only with kth particles with k = n ±a(m), wher~ r(m) E Nand m = 1,2,3, .... The polymer can be a mass fractal object (Newkom~ ~t al., 2006). For fractal compactIfied hnear polymer chams, we have the power-Iawl

1202

8 Fractional Dynamics of Media with Long-Range Interaction!

Rd , where 2 < d < 3 and N(R) is the number of chain particles in the balll Iwith radius R. Then we suppose that a(m) is exponential type function such tha~ r(m) = am, where a> I and a E N. This function defines the fractal long-range inj

IN(R)

rv

~eraction

IRemark 2J lOne of the oldest fractal functions IS WeIerstrass functIOn (WeIerstrass, 1895):1 (8.160~

Imtroduced as an example of everywhere contmuous nowhere differentiable func-I ~ion by Karl Weierstrass around 1872. Maximum range of parameters for which thel labove sum has fractal propertIes was found by Godfrey Harold Hardy (Hardy, 1916)1 ~n 1916, who showed thatl P
(8.161~ functions whose graphs have non-mteger box-countmg dImenSIOn are called fractall functions IWe can consider Eq. (8.150) with a fractallong-range interaction. Let us give thel Ibasic theorem regardmg the fractal mteractIOnJ

[Theorem 8.11. Equations of motion (8.150) with fractal long-range interactionJ !which is defined by (8.151) and (8.159), are equivalent to the continuous mediumJ, ~quatlOn:

2

at u(x,t) +

4c

h2

+00

.

~oo b(m) sin

2( - -iha(m) 1) 2 -Dx u(x,t) = 0,

(8.162

Iwhere u(x,t) is a smooth function such that u(nh,t) = un(t)l IProof Using a smooth function u(x,t) such that u(nh,t) = un(t), we obtain thel khfferentIal-difference equatIOn:1 a u(x,t) at 2

=

c ~ h2 m~oo b(m)[u(x+a(m)h,t) - 2u(x,t) +u(x-a(m)h,t)].

(8.163

IUsmg Eq. (8.148) for the shIft operator (8.146), we obtaml ~xp i{ -ia(m)hD;}u(x,t)

[Then we can rewnte Eq. (8.163) m the form:1

= u(x+a(m)h,t),

(8.164j

20~

18.18 Fractal dispersion law

d U(X,t) c ~ .. 1 . . 1 dt 2 = h2 ~oo b(m) [exp l{ -za(m)Dx} - 2 +exp I {za(m)Dx}]u(x,t). (8.165) IUsmg the Euler's formula, EquatIOn(8.165) can be represented as pseudo-dIfferentIa~ ~quatIOn (8.162). q lRemark for a(m) = 1 and b(m) = omO, this equation gives Eq. (8.155) that describes particlesl [or the case of nearest-neighbor interactionj

18.18 Fractal dispersion lawl [t is mterestmg to study a connection between the dynamiCs of cham with fractall Ilong-range mteractIons (FLRI) and the contmuous medmm equations With fractall ~ispersIOn law. Here, we conSider the cham of coupled particles With FLRI. Th~ ~ransformatIOn to the contmuous field gives the contmuous equatIOn, which de1 Iscnbes the dynamiCs of the oscillatory medmm. In (Tarasov, 2008), we showed howl ~he oscillations of chams WithFLRI are descnbed by the fractal disperSIOn law. Thi§ ~aw is represented by the Weierstrass functions (Weierstrass, 1895) whose graphsl Ihave non-mteger box-countmg dimenSIOn, l.e., these graphs are fractals. Fractal§ lare good models of phenomena and objects m vanous areas of SCience (Mandelbrot,1 ~983). Note that fractals m quantum theory were recently conSidered m (Kroger) 12000; Berry, 1996; WOJCik et aL, 2000). We prove that the chams With long-rang~ ImteractIoncan demonstrate fractal properties descnbed by fractal functIonsJ [.:et us define the pseudo-dIfferential operatorj

(8.166~ Rsing this operator, Equation (8.162) takes the form:1

(8.167~ IWe can conSider an eigenvalue equation for the nonhnear differential operatorl 1(8.166):

I2'P(x,k)

=

It(k)P(x,k)l

rI'hefunctIOn

W(x,k) = Aexp{ikx}

(8.168]

lis an eigenfunction of the operator (8.166). The eigenvalue It (k) of the operator i~

IA(k)

=

2

m~

00

b(m) sin

2

(~k) j

1204

8 Fractional Dynamics of Media with Long-Range Interaction!

[Using the well-known formula:1 ~sin2(a/2)

= l-cosaj

Iwe obtain ~(k) =

E

b(m) [1 - cos(ha(m)k)].

(8.169~

~

[f mteractIon (8.152) IS the fractal long-range mteractIOn with the parameter§

la(m) = am,

b(m) = a(d 2)m,

a E N,

a> 1,

(8.170~

OChen Eg. (8.169) has the formj

IA(k) = C(hk)j IwhereC(z) is the cosine Weierstrass-Mandelbrot function (Mandelbrot, 1983; Feder1 If98ET,

lC(z)

I±<>aJ

=

L

a(d-2)m [1 - cos(a(d-2)mz)]·1

m=-oq

[Thebox-counting dimension of the graph of this function is d. The operator (8.166) Fan be called the Weierstrass-Mandelbrot operator. The spectral graph (k, C(hk)) o~ ~hls operator IS a fractal set with dimensIOn d.1 ISubstItutIOn of the functIOn (8.168) mto Eq. (8.167) g1Ve~

k

+ ¥:-C(hk) = OJ

[fhls equation descnbes the dispersIOn law for the cham with the fractallong-rang~ linteractions (8.152) and (8.170). As a result, the graph (k, w( k)) is a fractal] IWe note that the group veloclt)j IVgrou p

=~

[or the plane waves cannot be find, since C(z) is the nowhere differentiable function] IWe can consider a generalization of conditions (8.170) in the form (8.159). Notel ~hat (8.159) with b = ad 2 gives (8.170). For the parameters (8.159), we obtain thel

Ipseudo-dlfferentIal eguatIOi1j

a u(x,t) 4c . at 2 + ---;:;'2 sm

2( -2Dx ih 1) 8c ~ u(x,t) + ---;:;'2 ~ b

m

.

sm

2( --2iha" 1) Dx u(x,t) = O. (8.171)

[fhls equation can be represented in the forml

18.18 Fractal dispersion law

2051

(8.172 Iwhere A is the pseudo-differential operato~

k\ = L bm cos( -ihamD;) , m=]

land

~2

[ =

(8.173~

4b 1 h2(1-b)]

~guatIOn (8.172) describes the oscIllations m the case of the fractal long-rang~ Imteraction.The left-hand side of Eg. (8.172) m the hmit h -+ 0 is the Klem-Gordonl pperator. The right-hand side of Eg. (8.172) describes a nonlocal part of the mteracj

tion,

Let us consider an eigenvalue equation for the pseudo-differential operator] 1(8.173): [The eigenfunctions (8.168) is an eigenfunction ofthe operator (8.173)j

[The eigenvalue A.J\ (k) is the Weierstrass function W(hk!n) j

PCJ\(k)

=

W(hk!n)J

Iwhere W(x) is defined by Eq. (8.160). The box-counting dimension of the graph o~ ~he Weierstrass function W(x) is (8.161). The spectral graph (k,W(hk!n)) of thi~ pperator is a fractal set with dimensIOnl

[Therefore operator (8.173) can be called the Weierstrass operatorl k\s a result, we prove that the chams with long-range mteractIOn can demonstrat~ OCractal properties. We conSider chams with long-range mteractIOns such that eachl rth particle is interacted only with chain particles with the numbers n±a(m), wher~ &l = 1,2,3, .... The exponential type functions a(m) = bm , where b > 1 is integerJ lare used to define a fractal long-range mteractIOn. The equatIOns of cham oscilla"1 ~IOns are characterized by disperSIOn laws that are represented by Weierstrass and! IWeierstrass-Mandelbrot (fractal) functIOns. The suggested chams with long-rang~ ImteractIOns can be conSidered as a simple model for hnear polymers that are comj Ipact, fractal globular shapel

1206

8 Fractional Dynamics of Media with Long-Range Interaction!

18.19 Griinwald-Letnikov-Riesz long-range interactionl '(Jriinwald-Letnikov fractional derivative [The Grtinwald-Letnikov fractional derivatives are based on a generalization of thel lusual differentiation of a function f(x) of integer order n of the form~

Iwhere

:1;: and VI: are forward and backward fimte dIfferences of order n of a funct!oq

If(x) with a step h and centered at the point x. The nth-order forward and ~ifferences

backwar~

are respectively given bYI

ILlh'f(X)

=

E(-l/G)f(x+(n-k)h),

(8.l75~

rhf(X) = 1(-I/G)f(X-kh). [The dIfference of a tractIOnal order a

> 0 IS defined by the

(8.174~

mfimte senesj

(8.176~ Iwhere the bmomlal coeffiCIents are

for h > 0, the dIfference (8.176) IS called left-sIded tractIOnal dIfference, and folj '(z < 0 it is called a right-sided fractional difference. We note that the series in (8.176) ~onverges absolutely and uniformly for every bounded function f(x) and a> 01 for the fractIOnal dIfference, the semIgroup propertyj

lis valid for any bounded function f(x) and a > 0, f3 > 0.1 rrhe Founer transform of the tractIOnal dIfference IS given byj IY; {V~ f(x)}(k)

[or any function f(x) E £1 (lR.)l

= (1 - exp{ikh} )U Y; {j(x)}(k)J

18.19 Grunwald-Letnikov-Riesz long-range interaction

2071

~quatIOns

(8.174) and (8.175) are used to define the Grtinwald-Letmkov fracj derivatives by replacing n E N in by a > O. The value hn is replaced by haJ Iwhile the finite difference V;: is replaced by the difference V~ of a fractional orderl ~ional

IX:

n-[I'' h-e-rle-'f"-t--a-n--'d'--r~lg--'h--'t--s~ld'-e--'d"G""ru~'~'n-w-a-rld'---Y-L-e'-tn~lk'-o-v-d-re-r~lv-a"tl-ve-s-o---'f"--o-r--'d-er---=aC-C>c-TO--a-r-e-d'---;ej lfined by:

a f(x) fLD x+

= lim h-tO

Vff(x) ha

j

Irespectively. We note that these derivatives coincide with the Marchaud fractiona~ ~erivativesof order a> 0 for f(x) E Lp(!~), 1 ~ p < 00 (see Theorem 20.4 in (Samkol ~t al., 1993)). The properties of the Grtinwald-Letnikov fractional derivatives arel Irepresented m SectIon 20 of the book (Samko et al., 1993)J IWecan define a fracttonal denvattve of order a > 0 b5J

(8.177 rrhlS denvattve comclde wIth the Riesz fractIOnal denvatlve of order a > 01

[I'herefore the fractIOnal denvattve (8.177) IScalled (Samko et aL, 1993) the Grtinwald1 ILetmkov-Rlesz denvattve of order a > OJ

IChain with Griinwald-Letnikov-Riesz interaction OC:et us consIder a system of mteractmg partIcles, whose dIsplacements from thel ~quilibrium are un(t), where n E Z. We assume that the system is described by thel ~quatIOns of motIOili

(8.179~ (8.180 IWe consider the functionl

~ (n,m)

=

b(m)

=

qm-a;

r(m+ 1 .

(8.181~

1208

8 Fractional Dynamics of Media with Long-Range Interaction!

[his type of long-range interaction will be called the Griinwald-Letnikov-Riesz inj ~eraction. Let us give the main theorem regarding this interaction.1

rrheorem 8.12. In the limit h ----+ 0 equations (8.179), (8.180) with (8.181) give thq continuous medium equations ij2u(x,t) () a ( ) 1---:::-'-;:2-----'--+A a GLRDxux,t =0, Where

A(a)

(8.182

I

=

2r(1- a)cOs(a 1r ),

I

'rnd u(x,t) is a smooth function such that u(nh,t) IProof. We define a smooth function u(x,t) such ~(nh,t) =

=

2 un(t).1

tha~

un(t)l

[I'hen Eq. (8.182) can be represented a§

a u(x,t) at 2

L b(m) h1a [u(x+mh)(t) +u(x-mh)(t)].

+00 =

(8.183

m=O

lOSIng the left-sIded and rIght-sIded tractIOnal dIfferences, we obtaull

(8.184 lOSIng the Grtinwald-Letmkov-RIesz derIvatIve (8.178), EquatIon (8.184) can bel IrewrItten In the form (8.182). q

18.20 Conclusioril pIscrete system of long-range InteractIng oscIllators serve as a model for numer-I pUS applIcatIOns In phySICS, chemIstry, bIOlogy, etc. Long-range InteractIOns arel IImportant type of InteractIOns for complex medIa. We conSIder long-range alphaj IInteractIOn. A remarkable feature of suggested a-InteractIOns IS the eXIstence of ij ~ransform operatIOn that replaces the set of coupled IndIVIdual oscIllator equatIOn~ Iby the contInUOUS medIUm equatIOn wIth the space derIvatIve of non-Integer or1 ~er a. ThIs transform operatIOn allows us to conSIder dIfferent models by applyIng methods of tractIOnal calculus. The method of fractIOnal calculus can be a poweffu~ method for the analysIs of dIfferent lattIce systems.1 IWe note that a fractional derivative can be result from a fractional difference las InteractIOn term, Just as nth order dIfferences lead to nth derIvatIves. It followsl ~rom the representation of the RIesz tractIOnal derIvatIve by Griinwald-Letmkovl

References

209

hactIOnal denvatIve (KIlbas et aL, 2006; Samko et aL, 1993). We assume that thel ~ransform operator can be used for improvement of different scheme of simulationsl [or equations with fractional derivatives.1 IWe consider the interactions with symmetric function J(n - m) = J(m - n). Thel ~ontmuous lImit for thiS type of mteractIon gives the Riesz fractIonal denvatIvesJ IWe can assume that an asymmetric interaction term (J(n - m) =I- J(ln - min lead§ ~o other forms of the fractional derivative that use the Feller potentials (see Sec-j tion.Iz.I of (Samko et al., 1993)) instead of Riesz.1 IWe also prove that the chams with long-range mteractIon can demonstrate fracj ~al properties. Models of chams with long-range mteractIOns such that each nthl Iparticle is interacted only with chain particles with the numbers n ± a(m), wher~ = 1,2,3, ... are suggested. The exponentialfunctions a(m) = b m with integer b > 11 lare used to define a special form of long-range interaction that demonstrates fractall IpropertIes. The equatIOns of cham oscIllatIOns are charactenzed by disperSIOn law§ ~hat are represented by fractal functIOns. These functIOns are everywhere contmu-I pus nowhere dlflerentIable functIons. We assume that the suggested chams modell ~an be conSidered as a Simple model for hnear polymers that are compact, fractall globular shapel IWe note that self-SimIlar functIOns and hnear operators can be used (Michehtschl ~t aL, 2009) to deduce a self-SimIlar form of the Laplacian operator and of thel WAlembertian wave operator. The self-simIlanty as a symmetry property reqmresl ~he mtroductIOn of long-range mterpartide mteractIOns. In Ret. (Michehtsch et aLJ 12009), authors obtamed a self-SimIlar hnear wave operator descnbmg the dynamic~ pf a quasi-contmuous hnear cham of mfimte length with a spatIally self-SimIlar diS1 ~nbutIOn of nonlocal mterpartIde mteractIons. The self-simIlanty of the long-rangi.j ImteractIons results m a disperSiOn law with the Weierstrass-Mandelbrot functiOnJ Iwhich exhibits fractal features.1

rn

lReferencesl P.P. Agrawal, 2002, SolutiOn for a fractiOnal dlflusiOn-wave equation defined in ~ Ibounded domam, Nonlinear Dynamics, 29, 145-155J rrt: Alfimov, Y.M. Eleonsky, L.M. Lerman, 1998, Sohtary wave solutiOns of non-I Ilocalsine-Gordon equations, Chaos, 8, 257-27lj p.L. Alfimov, Y.G. Korolev, 1998, On multIkmk states descnbed by the nonloca~ Isme-Gordon equation, Physics Letters A, 246, 429-435J p. Alfimov, T. Pierantozzi, L. Vazquez, 2004, Numencal study of a fractiOna~ Isme-Gordon equatiOn, m: FractionaL dijjerentiation and its applications, A. Lei IMehaute, lA. 'I'enreiro Machado, L.c. 'I'ngeassou, 1 SabatIer (Eds.), Proceed-I lings ofthe IFAC-FDA Workshop, Bordeaux, France, July 2004, 153-162J R Bachelard, C. Chandre, D. FanellI, X. Leoncmi, S. Ruffo, 2008, Abundance ofj Iregular orbits and noneqmlIbnum phase tranSitIons m the thermodynamiC IImi~ Ifor long-range systems, Physics Review Letters, 101, 260603.1

1210

8 Fractional Dynamics of Media with Long-Range Interaction!

~.

Baesens, R.S. MacKay, 1999, Algebraic localisation of linear response inl Inetworks with algebraically decaying interaction, and application to discret~ Ibreathers m dIpole-dIpole systems, Helvetica Phjsica Acta, 72, 23-32J [. Barre, E Bouchet, T. Dauxois, S. Ruffo, 2005, Large deviation techniques applied! Ito systems with long-range interactions, Journal of Statistical Physics, 119, 677-1 l1ll. H. Bateman, A. ErdelYI, 1953, Higher Transcendental Functions, YoU, McGrawj !Hill, New York~ [fL. Yan Den Berg, D. Fanelh, X. LeoncmI, 2009, StatIOnary states and fractlOnall Idynamics in systems with long range interactions, E-print: arXiv:0912.3060j M.Y. Berry, 1996, Quantum fractals in boxes, Journal ofPhysics A, 29, 6617-6629.1 [p. Biler, T. Funaki, W.A. Woyczynski, 1998, Fractal Burger equation, Journal of! IDijjerential Equations, 148, 9-461 p.M. Braun, Y.S. KIvshar, 1998, Nonhnear dynamICS of the Frenkel-Kontorov~ Imodel, Physics Reports, 306, 2-108j p.M. Braun, Y.S. KIvshar, 2004, The Frenkel-kontorova Model: Concepts, Methods) landApplications, Sprmger, Behn, New YorkJ p.M. Braun, Y.S. KIvshar, LL Zelenskaya, 1990, Kmks m the Frenkel-Kontorov~ Imodel WIth long-range mterpartIcle mteractions, Physical Review B, 41, 7118-1 I7T3K ~.

Burgers, 1974, The Nonlmear DIffusIOnEquatIOn, ReIdel, Dordrecht, Amsterdam] V\. Campa, T. DauxOls, S. Ruffo, 2009, Statistical mechamcs and dynamICS of SOlV1 ~ble models WIth long-range mteractions, Physics Reports, 480, 57-159.1 fJ. Dyson, 1969a, EXIstence of a phase-transItion m a one-dImenSIOnal Ismg ferro-I Imagnet, Communication in Mathematical Physics, 12,91-107 j f.J. Dyson, 1969b, Non-existence of spontaneous magnetization in a onej klimensional Ising ferromagnet, Communication in Mathematical Physics, 12j 212-215 f.J. Dyson, 1971, An Ising ferromagnet WIth discontmuous long-range order, Com1 munication in Mathematical Physics, 21, 269-2831 V\. ErdelyI, W. Magnus, E Oberhettmger, EG. TrICOmI, 1981, HIgher Transcenden-I ~al FunctIOns, YoU, KrIeger, Melbboume, FlOrIda, New York] ~. Feder, 1988, Fractals, Plenum Press, New York, London1 ~.A. FIsher, 1937, The wave of advance of advantageous genes, Ann Eugen London) 17, 355-3691 IS. Flach, 1998, Breathers on lattIces WIth long-range mteractIOn, PhySIcal Revlefij IE, 58, R4116-R4119J IS. Flach, c.R. WIlhs, 1998, DIscrete breathers, Physics Reports, 295,181-264.1 ~. Frohhch, R. Israel, E.H. LIeb, B. SImon, 1978, Phase tranSItIOns and reflec1 ItIOn posinvity L General theory and long-range lattIce model, Communicationl lin Mathematical Physics, 62, 1-341 IV. Gafiychuk, B. Datsko, V. Meleshko, D. Blackmore, 2009, AnalySIS of the soluj Itions of coupled nonlinear fractional reaction-diffusion equations, Chaos, Solij ~ons and Fractals, 41, 1095-11041

IR eferences

2111

IYu.B. Gaididei, S.E Mingaleev, P.L. Christiansen, K.G. Rasmussen, 1997, Effectsl pf nonlocal dispersive interactions on self-trapping excitations, Physical Reviettj IE, 55, 6141-6150l [u. Gaididei, N. Flytzanis, A. Neuper, EG. Mertens, 1995, Effect of nonlocal interj Fictions on solIton dynamIcs m anharmomc lattIces, PhySical ReView Letters, 75J ~~40-22431

IB.H. Gilding, R. Kersner, ~004, Travelling Waves in Nonlinear Diffusion Convectionl IReaction, Birkhauser Verlag, Base1.1 k\.Y. Gorbach, S. Flach, ~005, Compactlike discrete breathers in systems with non-I llinear and nonlocal dispersive terms, Physical Review E, 72, 056607 j K Gorenflo, E MamardI, 1997, Fractional calculus: mtegral and dIfferentIal equaj Itions of fractional order, in Fractals and Fractional Calculus in Continuum Me~ '[hanics, A. Carpmten, E Mamardl (Eds.), Spnnger, New York, 223-276; and! IE-print: arxiv:0805.3823j K Gorenflo, Y. Luchko, E MamardI, 2000, Wnght functions as scale-mvanant soj IlutIOns of the dIffUSIOn-wave equation, Journal of ComputatIOnal and Applzedl IMathematics, 118, 175-19U p.H. Hardy, 1916, WeIerstrass's non-dIfferentiable functIOn, Transactions oj th~ IAmerican Mathematical Society, 17, 301- 325 J IK.E. van Holde, 1998, Principles of PhySical BIOchemistry, Prentice Hall, New Jerj ~

IY. {shImon, 1982, SolItons m a one-dImenSIOnalLennard-Jones lattIce, Progress oJI ITheoretical Physics, 68, 402-410.1 Joyce, 1969, Absence of ferromagnetism or antIferromagnetism m the ISOtropICI IHeisenberg model with long-range interactions, Journal of Physics C, 2, 1531-1 11533. !B.B. Kadomtsev, 1988, Collective Phenomena In Plasmas, Pergamon, New YorkJ ISection 2.5.5 and Section 3.4.4. Translated from RUSSIan: 2nd ed., NaukaJ IMoscow,1988.1 k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj klOnal Dijjerentzal EquatIOns, ElseVIer,AmsterdamJ ~. KIttel, 2004, IntroductIOn to Solzd State PhySICS, 8th ed., WIley, New YorkJ V\. KolInskI, J. Skolmck, 2004, Reduced models of proteins and theIr applIcatIOns) IPolymer, 45, 511-524J V\.I. Kolmogorov, I. Petrovsky, N. PIscounov, 1937, Etude de l'equatIOn de la dIffu-1 ISIOn avec crOIssance de la quantIte de matiere et son applIcatIOn a un problem~ Ibiologique, Moscow University Bulletin, Ser. International Sect. A, 1, 1-25j ~. Korabel, G.M. Zaslavsky, Y.E. Tarasov, 2007, Coupled OSCIllators WIth power1 Ilaw mteractIOn and theIr fractIOnal dynamICS analogues, Communications in Non1 ~inear Science and Numerical Simulation, 12, 1405-1417.1 IN. KorabeI, G.M. Zaslavsky, 2007, TranSItIOn to chaos m dIscrete nonlInearl ISchrOdinger equation with long-range interaction, Physica A, 378, 223-237 j ~. Kroger, 2000, Fractal geometry in quantum mechanics, field theory and spinl Isystems, Physics Reports, 323, 81-181J

rrs:

1212

8 Fractional Dynamics of Media with Long-Range Interaction!

W'l. Laskm, G.M. Zaslavsky, 2006, NonlInear fractIOnal dynamIcs on a lattIce withl Ilong-range mteractions, Phjsica A, 368, 38-54J M.A. LeontovIch, 1944, Izvestya Akademii Nauk SSSR. Seria Fizika, 8, 16J OC:. Lewm, 1981, PoLYlogarzthms and Associated FunctIOns, North-Holland, Newl IYillK: [. Lighthill, 1978, Waves in Fluid, Cambridge University Press, Cambridge. Sectionl ll.l IV. Luke, 1969, The Special Functions and Their Approximations, AcademIc PressJ INew York. Vol.!. Chapter 6j f. Mainardi, 1996, Fractional relaxation-oscillation and fractional diffusion-wavd Iphenomena, Chaos, Solitons and Fractals, 7, 1461-1477.1 f. Mainardi, Yu. Luchko, G. Pagnini, 2001, The fundamental solution of the spacej ItIme fractional dIffusIOn equatIOn, Fractional Calculus and Applied Analysis, 4J 1153-192

lB. Mandelbrot, 1983, The Fractal Geometry of Nature, Freeman, New YorkJ IY.P. Maslov, 1973, Operator method, Nauka, Moscow. In RUSSIan. Sect. 1.8j [T.M. MichelItsch, G.A. Maugm, EC.G.A. Nicolleau, A.E Nowakowski, S. Derogar,1 ~009, DIspersIOn relatIOns and wave operators m self-SImIlar quasicontmuou~ IlInearchams, Physical Review E, 80, 011135J k\.Y. Milovanov, J.J. Rasmussen, 2005, Fractional generalization of the Ginzburgj ILandau equatIOn: an unconventional approach to cntIcal phenomena m compleX] ImedIa, Physics Letters A, 337, 75-80.1 IS.E Mmgaleev, Y.B. GaIdideI, EG. Mertens, 1998, SolItons m anharmomc chamsi IWIth power-law long-range mteractions, Physical Review E, 58, 3833-3842j IS.E Mmgaleev, v.B. GaIdideI, EG. Mertens, 2000, SolItons m anharmomc chamsi Iwith ultra-long-range interatomic interactions, Physical Review E, 61, R1044-1 IR1D4T. f. MiskIms, 2005, Weakly nonlocal supersymmetnc KdV hIerarchy, Nonlinea~ IAnalysis: Modelling and Control, 10, 343-348J IS. Momani, 2005, An explicit and numerical solutions of the fractional KdV equaj Ition, Mathematics and Computers in Simulation, 70, 110-1118.1 WI. Nakano, M. TakahashI, 1994a, Quantum HeIsenberg cham WIth long-range ferj lromagnetic mteractIOns at low temperatures, Journal oj the Physical Society ojl 'flapan, 63, 926-933; and E-pnnt: cond-maU9311034J WI. Nakano, M. TakahashI, 1994b, Quantum HeIsenberg model WIth long-range fer1 lromagnetic interactions, Physical Review B, 50, 10331-10334j WI. Nakano, M. TakahashI, 1994c, Quantum HeIsenberg ferromagnets WIth long-I Irange mteractions, Journal oj the Physical Society oj Japan, 63, 4256-4257j WI. Nakano, M. TakahashI, 1995, Magnetic propertIes of quantum HeIsenberg ferro-I Imagnets WIth long-range mteractions, Physical Review B, 52, 6606-661Oj k\.e Newell, J.A. Whitehead, 1969, Finite bandwidth, finite amplitude convection,1 IJournalofFluid Mechanics, 38, 279-303j p.R. Newkome, P. Wang, eN. Moorefield, TJ. Cho, P.P. Mohapatra, S. Li, S.H.I IHwang, O. Lukoyanova, L. Echegoyen, J.A. Palagallo, Y. Iancu, S.W. RIa, 2006J

IR eferences

213

INanoassembly of a fractal polymer: A molecular "Sierpinski hexagonal gasket"j IScience, 312, 1782-1785l ~.L. Pokrovsky, A. Virosztek, 1983, Long-range interactions in commensurate-j lincommensurate phase transition, Journal ofPhysics C, 16, 4513-4525j k\.P. Prudnikov, Yu.A. Brychkov, 0.1. Marichev, 1986, Integrals and Series, Vol.! .j IElementary Functions, Gordon and Breach, New York~ [he RCSB Protein Data Bank (PDB), 2010j Ihttp://www.rcsb.org/pdb/home/home.dq IK.O. Rasmussen, P.L. Chnstiansen, M. Johansson, Yu.B. GaIdidei, S.P. Mmgaleev,1 11998, Locahzed excitations m discrete nonhnear Schroedmger systems: Effectsl pf nonlocal dispersive interactions and noise, Physica D, 113, 134-151j [R.Z. Sagdeev, D.A. Usikov, G.M. Zaslavsky, 1988, Nonlinear Physics. From th~ IPendulum to Turbulence and Chaos, Harwood Academic, New York] k\.I. SaIchev, G.M. ZaslavskY, 1997, FractiOnal kmetic equatiOns: solutiOns and apj Iplications, Chaos, 7, 753-764j IS.G. Samko, AA Kil5as, 0.1. Manchev, 1993, Integrals and Derivatives of Fracj rional Order and Applications, Nauka i Tehmka, Mmsk, 1987, m Russianj !ind Fractional Integrals and Derivatives Theory and Applications, Gordon and! IBreach, New York, 1993l [.-.A. Segel, 1969, Distant side-walls cause slow amplitude modulation of cellulafj convection, Journal of Fluid Mechanics, 38, 203-224j ~.R. Sousa, 2005, Phase diagram m the quantum XY model with long-range mter1 !ictiOns, European Physical Journal B, 43, 93-96J IVB. Tarasov, 2006a, Contmuous hmit of discrete systems with long-range mterac1 Ition, Journal ofPhysics A, 39, 14895-14910.1 ~E. Tarasov, 2006b, Map of discrete system into continuous, Journal ofMathemat-I [cal Physics, 47, 092901j IVB. Tarasov, 2006c, PSi-senes solutiOn of tractiOnal Gmzburg-Landau equatiOn) Vournal oj Physics A, 39, 8395-8407J ~E. Tarasov, 2008, Chains with fractal dispersion law, Journal of Physics A, 41j 035101. ~E. Tarasov, G.M. Zaslavsky, 2005, Fractional Ginzburg-Landau equation for frac-j Ital media, Physica A, 354, 249-261 J IVB. Tarasov, G.M. Zaslavsky, 2006a, FractiOnal dynamiCs of coupled oscillatorsl IWith long-range mteraction, Chaos, 16, 023110.1 ~E. Tarasov, G.M. Zaslavsky, 2006b, Fractional dynamics of systems with long-I Irange mteractiOn, CommunicatIOns In Nonlinear SCience and Numerical Simula-I rion, 11, 885-898l IV.E. Tarasov, G.M. Zaslavsky, 2007a, FractiOnal dynamiCs of systems with long-I Irangespace mteraction and temporal memory, PhysicaA, 383, 291-308J IVB. Tarasov, G.M. ZaslavskY, 20075, ConservatiOn laws and Hamiltoman's equaj ItiOns for systems with long-range mteractiOn and memory, CommunicatIOns I~ Wonlinear Science and Numerical Simulation 13, 1860-1878l

1214

8 Fractional Dynamics of Media with Long-Range Interaction!

rJ. Tessone, M. CencmI, A. TorcmI, 2006, SynchromzatIOn of extended chaotIcl Isystems with long-range interactions: An analogy to Levy-flight spreading of epij §emIcs, Physical Review Letters, 97, 224101J k\.M. Turing, 1952, The chemical basis of morphogenesis, The Philosophical Transj lactions ofthe Royal Society B (London), 237,37-68.1 ~.v. Uchaikin, 20ma, Self-similar anomalous diffusion and Levy-stable lawsJ IPhysics-Uspekhi, 46, 821-849J ~.v. Uchaikin, 20mb, Anomalous diffusion and fractional stable distributions, Jours Inal ofExperimental and Theoretical Physics, 97, 81O-825j f. Weierstrass, 1895, Uber kontinuierliche funktionen eines reellen arguments, diel Ifur kemen wert des letzteren emen bestImmten dIfferentIal quotIenten besItzenJ lIn Mathematische Wake II, 71-74. Mayer-Muller, Berlin.1 [H. Weitzner, G.M. Zaslavsky, 2003, Some applications of fractional derivativesj ICommunications in Nonlinear Science and Numerical Simulation, 8, 273-281 j p. WojCIk, I. BIalymckI-BIrula, K. ZyczkowskI, 2000, TIme evolutIOn of quantuml fractals, Physical Review Letters, 85, 5022-5026j p.M. Zas1avsky, M. Edelman, V.E. Tarasov, 2007, DynamICS of the cham of os=j ~I1lators wIth long-range mteraction: from synchromzatIOn to chaos, Chaos, 17J 043124 OCB. Zeldovich, D.A. Frank-Kamenetsky, 1938,Acta Physicochimica URSS, 9,341-1 1350.

~hapter~

~ractional

Ginzburg-Landau Equation

9.1 Introduction romplex Gmzburg-Landau equatIOn (Aranson and Kramer, 2002) IS one of thel rIOst-studled equatIOns m phYSICS. ThIs equatIOn descnbes a lot of phenomena m1 ~Iudmg nonlInear waves, second-order phase transItIOns, and superconductIvIty. W~ Inote that the Gmzburg-Landau equatIon can be used to descnbe the evolutIon ofj lamplItudes of unstable modes for any process exhIbItIng a Hopf bIfurcatIon. Thg ~quatlon can be consIdered as a general normal form for a large class of bIfurca-1 ~IOns and nonlInear wave phenomena m contmuous medIa systems. The compleX! pmzburg-Landau equation IS used to descnbe synchromzatIOn and coIIectIve OSCI!1 Ilatlon m complex medIal ~eginning with the papers of Winfree (Winfree, 1967) and Kuramoto (Kuj Iramoto, 1975, 1984), studies of synchronization in populations of coupled oscillaj ~ors become an actIve field of research m bIOlogy, chemIstry and phYSICS (Strogatzj 12000). SynchromzatIOn and coIIectIve oscIllatIOn are the fundamental phenomenal 1m phYSICS, chemIstry and bIOlogy (Blekhman, 1988; Plkovsky et aI., 2001), werel lactlvely studIed recently (see, for example, (Boccalettl et aI., 2002; Afrmmovlch e~ laI., 2006; Boccaletti, 2008)). An oscillatory medium is an extended system, when~ ~ach partIcle (element) performs self-sustamed osclIIatIOns. A weII-known phys-I Ilcal and chemIcal example IS the osclIIatory Belousov-Zhabotmsky reactIOn (Be1 Ilousov, 1951, 1959; Zhabotmsky, 1964a,b; Kuramoto, 1984). OscIllatIOns m chem1 Ilcal reactIOns are accompamed by a color vanatIOn of the medIUm (ZhabotmskyJ ~974; Garel, 1983; Field and Burger, 1985). Complex Ginzburg-Landau equationj I(Aranson and Kramer, 2002) IS a canomcal model for osclIIatory systems wIth loj ~al couplIng near Hopf bIfurcatIOn. Recently, Tanaka and Kuramoto (Tanaka and! IKuramoto, 2003) showed how, m the vlclmty of the bIfurcatIOn, the descnptIOn o~ Ian array of nonlocaIIy coupled osclIIators can be reduced to the complex Gmzburgj [.-andau equation. In Ref. (Casagrande and Mikhailov, 2005), a model of systeml pf diffusively coupled oscillators with limit cycles was described by the complex] pmzburg-Landau equation wIth nonlocal interaction, Nonlocal couplIng was con1 V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

1216

9 Fractional Ginzburg-Landau Equation

Isidered in Refs. (Shima and Kuramoto, 2004; Kuramoto and Battogtokh, 2002j rasagrande and Mlkhadov, 2005).1 fractional Ginzburg-Landau equation was suggested in (Weitzner and Zaslavskyj 12003; Tarasov, 2005; Milovanov and Rasmussen, 2005). We consider a model of ~oupled oscillators with long-range alpha-interaction (Tarasov, 2006a,b). A remark-I lable feature of this interaction is the existence of a transform operator (Tarasov j 12006a,b) that replaces the set of coupled particle equatIOns of motion mto the conj ~inuous medium equation with fractional space derivative of non-integer order a.1 [fhls lImit helps to consider dIfferent models and related phenomena by applymg ~ifferent tools of fractional calculus (KUbas et al., 2006; Samko et al., 1993). In thel Imodelof coupled oscillators, we show how their synchronIZatIOncan appear as a rej Isult of bifurcation, and how the corresponding solutions depend on the order a. Thel Ipresence of fractional derivative leads also to the occurrence of localized structuresj fartlcular solutIOns for tractIOnal time-dependent complex Gmzburg-Landau equaj ~Ion are derIved. These solutIOns are mterpreted as synchronIzed states and localIzed! Istructures of the oscillatory medium (Tarasov and Zaslavsky, 2006a,b)j ~n SectIOns 9.2-9.3, some particular solutIOns are derIved with a constant wavel rumber for the fractIOnal Gmzburg-Landau equatIOn. These solutIOns are mter1 Ipretedas synchronIZatIOn m the oscillatory medIUm. In SectIOns9.4, we derIve SOlU1 ~Ions of the tractional Gmzburg-Landau equatIOn near a lImit cycle. These solutIOnsl laremterpreted as coherent structures m the oscillatory medIUm with long-range mj ~eractIOn. Fmally, a short conclUSIOn IS given m SectIOn 9.5.1

~.2

Particular solution of fractional Ginzburg-Landau eguatioij

[The discrete model of particles with lInear nearest-neighbor mteractIOn can bel ~ransform mto contmuous medIUm model (Tarasov, 2006a,b). Complex Gmzburg1 OC=andau equation (Aranson and Kramer, 2002) IS canOnIcal model for contmuou~ pscdlatory medIUm with local couplIng near Hopf bIfurcatIOn. We can conSider ij [-dimensIOnal lattice of weakly coupled nonlInear OSCillators that are deSCrIbed bYI ~he equations

Iwherewe assume that all OSCillators have the same parameters. Usmg the parameter~ g(Llx) 2 and C2 = gc(Llx) 2, a transition to the continuous medium (PikovskYI ~t aL, 2001) glve~

h=

~lp(x,t) = (1 +ia)P(x,t) - (1 +ib)IP(x,t)1 2P(x,t)+g(1 +ic)D;P(x,t), (9.2)1 IwhlchIS a complex time-dependent Gmzburg-Landau equatIOn (Aranson and Kramer) 12002). The Simplest coherent structures for thiS equation are plane-wave solutIOnsl I(Pikovsky et al., 2001)j

~.2

2171

Particular solution of fractional Ginzburg-Landau equation

W(x,t)

=

R(K) exp{iKx- im(K)t+ 80},

(9.3)1

land 80 IS an arbitrary constant phase. These solutIOns eXist fo~

(9.5)1 ISolutIon (9.3) can be mterpreted as a synchromzed state (PlkovskY et aL, 2001).1 IWe can conSider the equations of motion for nonlInear OSCillators With long-rang~ kx-interaction (Tarasov and Zaslavsky, 2006a,b) with 1 < a < 2. The corresponding ~quatIOn m the contmuous lImit (Tarasov, 2006a,b) I~

Dip(x,t) = (1 + ia)P(x,t) - (1 + ib) IP(x,t)12P(x,t) + g(1 + ic)-a-P(x,t), a x

Iwhere JaZJlxl a = -D~ is the Riesz fractional derivative (Kilbas et aL, 2006), dej lfined by (9.7)1 ~quatIOn

(9.6) IS a fractIOnal generalIzatIOn of complex time-dependent Gmzburg"1 equatIOn (9.2). ThiS equatIOn can be derIved as a contmuous lImit of dlscretel Isystem for nonlInear OSCillators With long-range mteractIOn (Tarasov, 2006a,b).1 Let us conSidered time evolutIOn and "time-structures" as solutIOns of fractIonaE ~quation (9.6). Particularly, synchronization process will be an example of the soluj ~Ion that converged to a time-coherent structure] ~andau

[Theorem 9.1. Equation (9.6) with the initial conditionl

W(x,O) = Roexp{ i80+ iKx}J Vias the particularsolutionl W(x,t)

=

R(K,t) exp{i8(K,t) + iKx}J

IR(t) = Ro(1- gIKla)I!2(R5 + (1 - glKl a - R5)e 2(1 gIKIU)t) 1/2, 18(t) =

-~ In [(1- gIKla)-1 (R5 + (1 -

glKl a - R5)e-

kOa(K) = (b - a) + (c - b)gIKla,

(9.8)1

2at)] - ma(K)t + 80, (9.9)1

1- glKl a > O.

(9.10)1

IProof. We seek a particular solutIOn of Eq. (9.6) m the forml

W(x,t) =A(K,t) exp{iKx}.

(9.11)1

9 Fractional Ginzburg-Landau Equation

1218 ~guatIOn

(9.11) represents a partIcular solutIOn of (9.6) wIth a fixed wave numberl

IK.

ISubstttutmg (9.11) mto (9.6), and usmgj

(9.12)1 Iwe obtain

ID!A(K,t)

= (1 + ia)A -

(1 + ib)IAI 2A - g(1 + ic)IKlaA,

(9.13)1

Iwhere A(K,t) is a complex-valued field. Using the representationl

IA(K,t) = R(K,t)exp{i8(K,t)}, ~guatlon

(9.14)1

(9.13) glveij

(9.15)1 (9.16)1 IWe note that the hmlt cycle here IS a cIrcle wIth the

radm~

ISolution of Egs. (9.15) and (9.16) with arbitrary initial condition§ ~(K,O)

= Ro,

8(K,0)

= 80

(9.17)1

q

Ihasthe form (9.8) and (9.9).

[ThIS solutIOn can be mterpreted as a coherent structure m nonhnear oscIllatoryl fuedmm wIth long-range mteractIOn.1 IRemark lJ IS easy to prove that the hmlt cycle of Eqs. (9.15) and (9.16) IS a cIrcle wIth thel IiJ.i.ill.llS (9.18)1 ~t

[f we consIder the case

~hen

Eqs. (9.8) and (9.9)

glv~

~(t) = Ro,

8(t) = -OJa(K)t + 80.

(9.19)1

ISolution (9.19) means that on the limit cycle (9.18) the angle variable 8 rotates withl la constant velocity OJa(K). The plane-wave solution i~

(9.20)1

~.2

Particular solution of fractional Ginzburg-Landau equation

21S1

[I'hIS solutIon can be mterpreted as synchromzed state of the medIUmJ IRemark 2J

for the case, Ian addItIonal phase ShIft occurs due to the term, whIch IS proportIOnal to b m (9.9)J [There is a single generalized phase variable. To define this phase, we rewrite (9.16)1

as

ID! lnR = (1 - glKn - R2 , kJie = (a - cglKl a ) - bR2 •

(9.21)1 (9.22)1

ISubstitution of R from (9.21) into (9.22) give~ 2

(9.23)1 V\s a result, the generalIzed phase (PIkovsky et al., 2001) IS defined byl

W(R, e)

=

e-

blnR.

(9.24)1

from (9.23), we obtaml [This equation means that generalized phase cp(R, e) rotates uniformly with constan~ Ivelocity. For glKl a = (b - a)/(b - c) < 1, we have the lines of constant generalize~ Iphase. On (R, e) plane these lines are logarithmic spirals e - b lnR = const. It i§ ~asy to see that the decrease of a corresponds to the increase of K. For b = 0, wei Ihavestraight lines cp = instead of spirals]

e

IRemark 3J

from the case (9.10), the group veloCItYI

lIS equal to

b,g = a(c-b)gIKl a

1J

[The phase velOCIty I~

~s a result, the long-range interaction decreasing as Ixl-(a+l) with 1 < a < 2lead§ ~o increase the group and phase velocities for small wave numbers (K ----+ O)j

9 Fractional Ginzburg-Landau Equation

1220

~.3

Stability of plane-wave solutionl

ISolution of (9.20) can be represented asl

IX = R(K,t)cos(8(K,t) +Kx),

Y

= R(K,t)sin(8(K,t) +Kx),

(9.25)1

Iwhere X = X(K,t) = Re'P(x,t) and Y = Y(K,t) = Im'P(x,t), and R(K,t) and! 18(K,t) are defined by (9.8) and (9.9). For the plane-wave§

o(x,t)

(l-gIKn l 2cos(Kx-wa(K)t+8o) ,

=

(9.26)

1N0t all of the plane-waves are stable. To obtain the stability condition, consider thel Ivariation of (9.13) near solution (9.26)1

Iwhere oX and oY are small variations of X and Y, andl

~ll = 1 - glKl a - 2Xo(Xo - bYo) - (X6 + Y6)J Al2

= -a + gclKla - 2Yo(Xo - bYo) + b(X6 + Y
A21 = a - gclKla - 2Xo(Yo + bXo)- b(X~ + Y~),

kb = 1- glKl a -

(9.28)

2Yo(Yo+bXo) - (X6+ Y6)j

[TheconditIOns of asymptotic stabilIty for (9.27) arel ~II

+A 22 < 0,

A llA22 -A]2A 21 < 0.

(9.29)1

from Eqs. (9.26) and (9.28), we getl

(9.30)1 land the first condition of (9.29) 1(9.29) gives

IS

valId. SubstitutIOn of Eqs. (9.26) and (9.28) mtol

IAllA22- A12A2]

F (b(l- gIKI

U

) -

(a - gcIKI U))(3b(1- gIKI U )

[hen the second conditIOn of (9.29) has the forml

I(V - l)(V - 3) < 0,1 Iwhere

-

(a - gcIKI U ) ) . (9.31)1

~.4

22~

Forced fractional equation

~s

a result, we obtainl (9.32)1

Il.e., the plane-wave solutIon (9.20) IS stable If parameters a, b, c and g satIsfy (9.32)J ~ondition (9.32) defines the region of parameters for plane waves where the synj chronization exists I

~.4

Forced fractional equatiors

[Letus consider fractional equation (9.13) forced by a constant E forcel

ID}A = (1 + ia)A -IAlzA - g(1 + ic)IKlaA - iE,

ImE

=0

(9.33)1

Iwhere A = A (K, t). Here we put for simplicity b = 0, and K is a fixed wave numberl k\s a result, we have the so-called forced Isochronous case (b - 0) (PIkovskY et aLJ 12001). Usmg

fX = X(K,t) = ReA(x,t), Y = Y(K,t) = lmA(x,t)J Iwe obtam a system of the real

equatIOn~

(9.34)1 (9.35)1 ~umerIcal solutIOn of Eqs. (9.34) and (9.35) was performed (Tarasov and Zaslavsky J 12006a,b) wIth parameters a - I, g - I, c -70, E - 0.9, K - 0.1, for a wIthml linterval a E (1,2). For <Xo < a < 2, where <Xo ~ 1.51..., the only stable solution is ij Istable fixed pomt. ThIS IS regIOn of peflect synchromzatIOn (phase lockmg), when~ ~he synchronous OSCIllatIOns have a constant amplItude and a constant phase shIf~ Iwith respect to the external force. For a < <Xo the global attractor for (9.35) is aI ~ImIt cycle. Here the motIon of the forced system IS quasIperIodIc. For a - 2 therel lIS a stable node. When a decreases, the stable mode transfers mto a stable focus. A~ ~he tranSItIOn pomt It loses stabIlIty, and a stable lImIt cycle appears. As a result, wei Ihave that the decrease of order a from 2 to I leads to the loss of synchromzatIOnl I(Tarasov and Zaslavsky, 2006a,b; Zaslavsky et al., 2007)j IWe can characterIze the medIUm by a smgle generalIzed phase varIable (9.24)J ~quatIOn (9.24) can be rewritten m the forml

(9.36)1

9 Fractional Ginzburg-Landau Equation

1222

Iwhere X and Yare defined by (9.25).1 for E = D, the phase rotates uniformlYI

Iwhere wa(K) is given by (9.10) with b = D, and can be considered as a frequenc~ bI natural oscillations]

for E

i

D, Equations (9.35) and (9.36)

giv~

IDicp = -wa(K) -Ecoscpj [his equation has an integral of motion. If w2

OCf w2

> E 2 , the integral isl

< E 2 , then we havd

[I'hese expreSSIOns help to obtam the solutIOn m form (9.14) for forced case (9.33)1 Ikeepmg the same notations as m (9.14)j [Using the variables R = R(K,t) and 8 = 8(K,t), we getl

1 a Ecos8 Dt 8 = (a-cgIKI ) - - R - '

(9.37)1

1

~umencal solutIOn of (9.37) was performed m (Tarasov and Zaslavsky, 2006a,b~ IWIth the same parameters as for Eq. (9.35), i.e., a - 1, g - 1, c - 70, E - 0.9J IK = D.1, and a within interval a E (1,2)J

9.5 Conclusionl fractIOnal Gmzburg-Landau equatIOn can be used to descnbe synchromzatIOn and! ~ollective OSCIllatIOn m complex medIa WIth long-range mterparticle mteractIOnsl land nonlocal propertIes. The fractIOnal spatIal denvatives m equatIOns are hnked t9 Inonlocal properties of dynamICS of medIa. MedIa of mteractmg objects are a bench-I Imark for numerous apphcatIOns m phYSICS, chemIstry, and bIOlogy. All conSIdered! Imodels can be related to the partIcles WIth long-range alpha-mteractIOn. A remark-I lable feature of thIS mteractIOn IS the eXIstence of a transform operator (Tarasov J 12DD6a,b) that replaces the set of coupled particle equations of motion into the conj ~muous medIUm equatIOn WIth tractIOnal space denvative of non-mteger order a.1 [I'hiS hmlt helps to conSIder dIfferent models and related phenomena by applymgj

IR eferences

223

khfferent tools of fractIonal calculus (KIlbas et aI., 2006; Samko et aI., 1993). AI ~lOntrivial example of general property of fractional linear equation is its solutionl Iwith a power-wise decay along the space coordinate. Note that fractional equationj ~an be obtamed by usmg a generahzatIon of Kac mtegral (Tarasov and ZaslavskyJ 12008). The contmuous medmm equatIOns with fractIonal denvatIves demonstratt:j ~ffect of synchromzatIOn (Tarasov and Zaslavsky, 2006a,b; Zaslavsky et aI., 2007)J Ibreathers (Flach and Willis, 1998; Flach, 1998; Korabel et aI., 2007), fractional kij retics (Zaslavsky, 2002), and othersj

Referencesl ~S.

Afraimovich, E. Ugalde, J. Urias, 2006, Fractal Dimensions for Poincare Rej 'f;urrences, ElseVier, Amsterdaml [.S. Aranson, L. Kramer, 2002, The word of the complex Gmzburg-Landau equaj ItIOn, Reviews oj Modern Physics, 74, 99-143; and E-pnnt: cond-maU01061151 Baesens, R.S. MacKay, 1999, AlgebraiC 10cahsatIOn of hnear response ml Inetworks with algebraically decaymg interaction, and apphcatIOn to discretel Ibreathers in dipole-dipole systems, Helvetica Physica Acta, 72, 23-32j [3.P. Belousov, 1951, A periodic reaction and its mechanism, in Autowave Processesl lin Systems with Dijjusion Gorky State University, Gorkyj !B.P. Belousov, 1959, A penodic reactIOn and its mechamsm, m Collection oj Shortl IPapers on Radiation Medicine, Medgiz, Moscowl p. Blekhman, 1988, SYnchromzatwn In SCIenceand Technology, Amencan SOCietYI pfMechanical Engineers, 255p.; Translated from Russian: Nauka, Moscow, 1981 j IS. BoccalettI, 2008, The SYnchromzed DynamICs of Complex Systems, ElseVier, Amj Isterdan IS. Boccaletti, J. Kurths, G. OSipOV, D.L. Valladares, C.S. Zhou, 2002, The synchro-I Inization of chaotic systems, Physics Reports, 366, 1-101j ~ Casagrande, A.S. Mikhailov, 2005, Birhythmicity, synchronization, and turbu-I Ilence in an oscillatory system with nonlocal inertial coupling, Physica D, 205j 1154-169; and E-pnnt: nhn.PS70502015.1 ~.J. Field, M. Burger (Eds.), 1985, Oscillations and Traveling Waves in Chemica~ ISystems, WIley, New Yorkl IS. Flach, 1998, Breathers on lattices with long-range mteractIOn, PhySIcal Revlefij IE, 58, R4116-R4119J IS. Flach, c.R. WIlhs, 1998, Discrete breathers, Physics Reports, 295,181-264.1 p. Garel, O. Garel, 1983, Oscillations in Chemical Reactions, Spnnger, Berhnj V\.A. KIlbas, H.M. Snvastava, J.J. Trujillo, 2006, Theory and Applications oj Frac1 kwnal Dijjerentwl EquatIOns, ElseVier, AmsterdamJ IN. Korabel, G.M. ZaslavskY, Y.E. 'I'arasov, 2007, Coupled OSCillators With powerj Ilaw interaction and their fractional dynamics analogues, Communications in Nonj ~inear Science and Numerical Simulation, 12, 1405-1417.1

r.

1224

9 Fractional Ginzburg-Landau Equation

IY. Kuramoto, 1975, Self-entramment of a populatIOn of coupled non-lInear osj ~illators, in International Symposium on Mathematical Problems in Theoretica~ IPhysics, H. Araki (Ed.), Springer, Berlin, 420-422j IY. Kuramoto, 1984, Chemical OscillatIOns, Waves, and Turbulence, Spnnger] IHffl.i.i:i: IY. Kuramoto, D. Battogtokh, 2002, Coexistence of coherence and incoherence inl Inonlocal coupled phase oscillators, Nonlinear Phenomena in Complex SystemsJ S, 380-3851 k\.Y. Milovanov, J.J. Rasmussen, 2005, Fractional generalization of the Ginzburgj ILandau equatIOn: an unconventional approach to cntical phenomena m compleX] Imedia, Physics Letters A, 337, 75-80.1 k\. Pikovsky, M. Rosenblum, J. Kurths, 2001, Synchronization. A Universal Concepti lin Nonlinear Sciences, Cambridge University Press, Cambridge.1 IS.G. Samko, A.A. Kil5as, 0.1. Manchev, 1993, Integrals and Derzvatlves of Fracj klOnal Order and ApplicatIOns, Nauka i Tehmka, Mmsk, 1987, m Russianj ~nd FractIOnal Integrals and Derivatives Theory and ApplzcatlOns, Gordon and! IBreach, New York, 19931 IS. Shima, Y. Kuramoto, 2004, Rotatmg spiral waves Withphase-randomized core ml Inonlocally coupled oscillators, Physical Review E, 69, 0362131 IS.H. Strogatz, 2000, From Kuramoto to Crawford: explonng the onset of synchro-I Inization in populations of coupled oscillators, Physica D, 143, l-20j p. Tanaka, Y. Kuramoto, 2003, Complex Gmzburg-Landau equation With nonloca~ ~ouplIng, Physical Review E, 68, 026219J IVB. Tarasov, G.M. Zaslavsky, 2005, FractIOnal Gmzburg-Landau equation for frac1 Ital media, Physica A, 354, 249-261J ~E. Tarasov, 2006a, Continuous limit of discrete systems with long-range interacj Ition, Journal ofPhysics A, 39, 14895-14910.1 IVB. Tarasov, 2006b, Map of discrete system mto contmuous, Journal oj Mathemat-I lical Physics, 47, 092901J ~E. Tarasov, 2006c, Psi-series solution of fractional Ginzburg-Landau equation] IJournal ofPhysics A, 39, 8395-8407 j ~E. Tarasov, G.M. Zaslavsky, 2006a, Fractional dynamics of coupled oscillatorsl IWith long-range mteraction, Chaos, 16, 023110.1 IV.E. Tarasov, G.M. Zaslavsky, 20065, FractIOnal dynamiCs of systems With long-I Irangeinteraction, Communications in Nonlinear Science and Numerical Simula-I tion. 11, 885-898J IVB. Tarasov, G.M. ZaslavskY,2008, FractIOnalgeneralIzatIOnof Kac mtegral, Comj Imunications in Nonlinear Science and Numerical Simulation, 13, 248-258J ~. Weitzner, G.M. Zaslavsky, 2003, Some applIcatIOns of fractIOnal denvatives] ICommunications in Nonlinear Science and Numerical Simulation, 8, 273-28U k\.T. Winfree, 1967, Biological rhythms and the behavior of populations of couple~ pscillators, Journal of Theoretical Biology, 16, l5-42j p.M. Zaslavsky, 2002, Chaos, fractional kinetics, and anomalous transport, Physicsl IReports, 371, 461-580J

References

2251

p.M. Zaslavsky, M. Edelman, Y.E. Tarasov, 2007, Dynamics of the chain of os-j ~dlators WIth long-range InteractIon: from synchronIZatIOn to chaos, Chaos, 17J 043124 k\.M. Zhabotinsky, 1964a, Periodic liquid phase reactions, Proc. Acad. Sci. USSRj 1157, 392-395j k\.M. ZhabotInsk)', 1964b, PenodIc processes of malOnIC aCId OXIdatIon In a lIqUId! Iphase, Biofizika, 9, 306-311j k\.M. Zhabotinsky, 1974, Concentration Oscillations, Mir, Moscowj

~hapter

lQ

~si-Series

Approach to Fractional Equations

110.1 Introductionl fSi-senes approach to the questiOn of mtegrabihty is not concerned with the displayl pf exphcit functiOns. In this approach the eXistence of Laurent senes for each de1 Ipendent vanables is considered. In general, the senes may not be summable to ani ~XphCit form, but does represent an analytic function. The essential feature of thi§ OC:aurent senes is that it is an expansiOn about a particular type of movable smgularj lity, l.e., a pole. The eXistence of these Laurent senes is mtImately connected withl ~he smgulanty analysis of differential equatiOns (Ince, 1927). Begmmng with thel IpiOneenngcontnbutiOns by Pamleve (Pamleve, 1973), studies of these properties o~ ~onlinear differential equations become an active field of research (Bureau, 1964j ~osgrove and Scoufis, 1993; Tabor, 1989; Roy-Chowdhury, 2000)J IWe note that the connectiOn of smgular behaViOr and the solutiOn of partial dIfj ~erentIal equatiOns by the method of the mverse scattenng transform was noticed byl V\blowitz, Ramam, and Segur m (Ablowitz et aI., 1978, 1980a,b), who developedl Ian algonthm, caIIed the ARS algonthm, to test whether the solutiOn of an ordmai)1 khfferentIal equatiOn was expressible m terms of a Laurent expansiOn. If this was thel ~ase, the ordmary dIfferential equatiOn was smd to pass the Pamleve test and wa§ ~onJectured to be mtegrable. Under more preCise conditiOns Conte (Conte, 1993)1 Ishowed that the equatiOn is mtegrable. PSi-senes solutiOns of dIfferential equatiOn~ Iwere considered m (Tabor, 1989; Tabor and Weiss, 1981; Bountis et aI., 1982; Changl ~t aI., 1982). PSi-senes for nonlmear dIfferential equatiOn contam a lot of mforma-I ~iOn about the solutiOns of this equatiOn. We prove that solutiOns of the fractiOna~ ~quation can be derived (Tarasov, 2006) by using psi-series method with fractiona~ Ipowers. The leadmg-order behaViOrs of solutiOns about an arbitrary smgulanty, a~ IweII as their resonance structures, have been obtamed. It was proved that fractiOna~ pmsburg-Landau equatiOns of order a Withpolynomial nonlmeanty of order s hav~ ~he non-integer power-like behavior of order a 7(I - s) near the singularitYl [n Section 10.2, the singular behavior of the fractional equation is considered. Inl ISectiOn 10.3, we discuss the powers of senes terms that have arbitrary coefficientsl V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

1228

10 Psi-Series Approach to Fractional Equations

~hat are

called the resonances or Kovalevskaya exponents. In Section 10.4, we deriv~ psi-series and recurrence relations for I-dimensional fractional equation withl Irational order a (1 < a < 2). An example of differential equation with derivative~ pf order a = 372 is suggested. In Section 10.5, the next to singular behaviour fo~ larbItrary (ratIOnal or matIonal) order IS dIscussed. FInally, a short conclUSIOn I§ gIVenm SectIOn 1O.6J ~he

110.2 Singular behavior of fractional equationl [Let us note the basic idea allowing to generalize psi-series approach to the fractiona~ klifferential equations. For a wide class of fractional equations, we can use (Tarasov j 12006) the series

IlTf(X) -T

1

(x _ xo)m/2n

~ ak (_ )k/2n t:o x Xo ,

(10·1)1

Iwherek, m, n are the integer numbers. For the order a = min, the action of fractiona~ Herivative

~an

be represented as the change ak ---+ ak-2m of the number of term ak m Eq. (10.1)1 allows us to denve (Tarasov, 2006) the psi-series for the tractIOnal dIfferentIa~ ~quation of order a = min. For the fractional equation the leading-order singula~ IbehavIOur IS defined by power that IS equal to the half of denvatIve orderJ OC:et us consIder the tractIonal equatIOnl ~t

~D~IJI(x) +cD~IJI(x) +alJl(x) +bIJl3(x) = 0,

(10.2)1

Iwhere lJI(x) is a real-valued function, and D~ is the Riemann-Liouville fractiona~ BenvatIve of orBer 1 < a < 2J ~quatIOn (10.2) can be consIdered as a tractIOnal I-BImensIOnal Gmzburg"1 OC:andau equatIOn for a real-valued field. Note that the Gmzburg-Landau equatIOi1j IWIth fractIOnal denvatIves was suggested m Ref. (WeItzner and ZaslavskY, 2003)1 land It was conSIdered m (MI1ovanov and Rasmussen, 2005; Tarasov, 2005, 2006j rrarasov and Zaslavsky, 2006).1 IWe detect pOSSIble smgular behaVIOr m the solutIOn of a dIfferentIal equatIOn byl means of the leadmg-order analySIS. To determme the leadmg-order behaVIOr, wei ISet

W(x)

=

f(x-xoY,

(10.3)1

IwhereXo IS an arbItrary constant (the locatIOnof the smgulanty). Then we substItutel 1(10.3) mto tractIOnal dIfferentIal equation (10.2) and look for two or more dommantl terms. The detection of which terms are dominant is identical to the determinationl pf which terms in an equation are self-similarj

22~

110.3 Resonance terms of fractional equation

ISubstituting (10.3) into Eg. (10.2), and using the relationl

[f 1 <

a < 2, then p - a < p - 1. For the dommant termsJ (10.5)

k\s a result, we obtaml ~-a=3p,

(10.6)1

(10.7)1 ~guation

~f

(10.6) give§

I < a < 2, then -I < p < -1/2. Therefore the leading-order singular behaviorl

liS found:

b

al2

I"' (x) = f(x-xo)-,

f2

gr(1 - a/2) = 01 1

+ br(l- 3a/2)

land the singularity is a pole of order a/2. Evidently our psi-series starts at (x-=J ko)-aI2. The resonance conditions and psi-series are considered in the next sectionsj V\s a result, we get that tractiOnal ddferential equatiOns of order a with polyn01 rIial nonlinearity of order s have the noninteger power behavior of order a / ( I - s) r.ear the smgulantyJ

110.3 Resonance terms of fractional eguatioij ~et us

considerresonance terms offractional equation (10.2). The powers of (x - xo) have arbitrary coefficients are called the resonances or Kovalevskaya's expo-I r.ents. In order to find resonance, we use (Tabor, 1989) the subStitutiOnl ~hat

lP(x) = f(x - xo)P + l(x - xoy+r, land find the values r. In Eg. (10.8) parameters p and

f are defined bYI

(10.8)1

1230

10 Psi-Series Approach to Fractional Equations

gr(l- a/2) br(1 - 3a/2)'

ISubstItutIonof Eq. (10.8) mto (10.2) gIve§

T'tp + r + 1) 1 gl ---:---'-------'-(x - xo)P+r-a + cpl(x - xo)P+r- +al(x - xo)P+ T p+r+l-a ftbl 3(x -xo?p+3r + 3bzZf(x _xo)2 p+3r + 3blf2(x _xo)p+3r = O.

(10.10~

IOsmg Eq. (10.9), and consIderIng the lInear wIth respect to I terms of (10.10), wei IJ.1J.iYe

1r(1 + r - a/2)r(1 - 3a/2) - 3r(1 - a/2)r(1

+r -

(10.1 q

3a/2) = O.

~quatIOn

(10.11) aIIows us to derIve the values of r. We note that Eq. (10.11) can bel derIved by usmg the recurrence relatIOns. In the general case, the values o~ Ir can be lffatIOnal or complex numbers. The solutIOn of fractIOnal equatIOn (10.2)1 IWIth I < a ~ 2 has two arbItrary parameters. Therefore we must have two values ofj Ir that are the SolutIons of Eq. (10.11). It IS mterestmg to note that (10. I I) gIves twg Irealvalues of r only fo~ la > aol ~hrectly

Iwhere ao

ao

~ 1.300588. The order is a universal value that does not depends onl Ivalues of parameters g, a, b, c of tractIonal equatIOn (10.2)j [I'he SolutIons of Eq. (10. I I) correspond to the pomts of mtersectIOns of the corre-I

Ispondmg functIOn wIth the hOrIzontal aXIS. As a result, the nature of the [s summarIzed (Tarasov, 2006) as foIIowsj

resonance~

For a such that 1 < a < ao, the values of r are complex or r < -aj21 I_ For a such that lXo < a < 2, we have two the real values of r. Note that foIj P'O < a < 1.9999999995, the values r satIsfy the mequalIty r < 6.4261

I-

110.4 Psi-series for fractional equation of rational orrlelj ~et

us conSIder tractIOnal equation (10.2), wIth the ratIOnal order a such

m

~=-,

l
r!l

[Usingthe standard procedure (Tabor, 1989), we

tha~ (10.12~

substitut~

(lO.13~

23~

110.4 Psi-series for fractional equation of rational order

gr(l- a/2) br(1- 3a/2)

I

Imto tractIOnal equatIOn (10.2). The coefficIent ao IS a real number for two casesj

I- g/b? 0 and 1 < a < 4/3j

I-

g/b::::; 0 and 4/3 < a < 2j

for the rational order a lin the form

= min, we suggest (Tarasov, 2006) to use the cp-functionl (10.14~

~n

this case, Equation (10.13) give§ (10.15

(10.16~

13k = -k .

I

2n

k\s a result, the action of fractional derivative of order a = m/ n can be represente~ las the change of the number of term ak ----t ak-2ml (10.17 ~t

allows us to derIve the generalIzed pSI-SerIeS solutIOns of fractIOnal equatIOlJI

I{m2).

Let us consIder the serIes

L adx-xo)f3k

Ip(x) =

sr:

(10.18~

lk==O

Rsing (10.12) and (10.16), Equation (10.18) can be represented asl

P(x)

I

=

f adx-xo)2il.

ISubstItutmg (10.19) into Eqs. (10.2), we obtaIlJI

lHere we use the notationl

k m

(10.19~

1232

10 Psi-Series Approach to Fractional Equations

IWe can compute ak (k = 1,2, ...) through the equation of coefficients of hkel Ipowerof (x - xo) to zero in (10.20)j

L +a

I

~

L

k 3m

ak-2m (X- Xo) 2il

Ik-2m

ak-2m+2n------,---

+ 3baO2~ L ak (X-

~

XO) !"J

k-Ol

(10.2d Ik-Ol-I

lOsing

i-II

a6 = p, we ge~ 2n ~ akBk(n,m) + cak-2m+2n k-3m+ 2n

l

Ik=Tll

ft aak- 2m+ 3blak + b L L ak-i- jaia j = 0,

(10.22~

f=l j=~

Iwhere k - 0, 1,2, ... , and ak - 0 for k < O. Recurrence relation 00.22) can be rewntj [en as

2

f =-

gr(l- a12) br(l - 3a12)

gr( T)

(10.24

= - br(2n-3m)'

ISubstitutmg (10.24) mto (10.23), we obtaml kg (Bk(n,m) - 3Bo(n,m)) = - ca k- 2m+2n

k-3m+2n - i 2 -aak-2m- b ak-i-jaiaj n i=ij=l

[fhe resonances for the k satisfy the conditiOnl

~n

thiS case, the coefficient ak can be arbitrary.1 k'\s a result, we obtam the nonresonance term~

LL

23~

110.5 Next to singular behavior

Gk

=

-A(k, m,n) ( CGk-Zm+Zn

k-3m+2n

k-I

i

+ GGk-Zm + b L L Gk-i- jGiGj

)

i=1 '=1

2n

(10.26 [hese terms describe the coefficients ofthe series (10.l9)J ~xample.

[n (Tarasov, 2006), we consider fractional equation (10.2) with derivative of orderl ~ = 3/2. In this case, n = 2, m = 3, and the coefficients Gk are defined byl

_ (CGk-Z¥ +GGk-6+bL7~lL~=I Gk-i-jGiGj) T(¥)T(f) Gk-(I)) . g (T (HI) ----:1 T (5) =:T - 3T (k-:I-5)T:I ~n

(10.27

(Tarasov, 2006), we obtairj

Ip(x)

=

fGdx-xo)~

lk==O

1= Go (x-xo)-3/4 +Gz (x-xO)-1/4 +G4(X-xO)I/4 +G6 (x-xO?/4 +...

(1O.28~

~quatlon

(10.28) gIves a power pSI-serIes that represents the solutIOn of the fracj equation of order a = 1.5. The coefficients in Eq. (10.28) are defined by EqJ 1(10.27). For example, G - -b - C - g - 1 glve~ ~ional

kJo

~

0.961553,

~4 ~

0.001685,

Gz

~

G6

-0.238224j

~

0.3872131

~quatlOn

(10.28) wIth these coeffiCIentsrepresents a solutIOnof fractIOnal equatlOi1j Iby psi-series for a = 3/21

110.5 Next to singular behaviolj ~n general, we can conSIder not only the ratIOnal a. Let us assume that the order OJ lIS an arbItrary (ratIOnal or matlOnal). Instead of Imposmg a serIes commencmg a~ ~he power mdlcated by the smgularIty found by the leadmg-order analYSIS, we canl k1etermme the next to smgular behaVIOr by wrItmgl

(10.29] Iwhere

1234

10 Psi-Series Approach to Fractional Equations

gr(l- a/2) br(1 - 3a/2)' IWe can always write 'l'(x) in the form (10.29). In order to make the process usefulj Iwe require (Tarasov, 2006) that the first term be a leading-order term, i.e.J

(10.31~ ISubstItutmg (10.29) mto (10.2), and usmg (lOA), we havel

r(p+l) a 1 (x-xoY- +cpf(x-xoY- +af(x-xo) p+l-a

gf r

ftbf3(X-XO?p + gD~lJf(x) +cD~lJf(x) +alJf(x) +blJf3(x) (10.32~ ft3bf2(x - xofplJf(x) + 3bf(x - xo)P lJf2 (x) = O. Equations (10.32) and (10.30) givel ~D~lJf(x)

+ cD; lJf(x) + alJf(x) + blJf3 (x) + 3bp(x - xo)2plJf(x)1

ft3bf(x-xoYlJf2(x)cpf(x-xoY

1 +af(x-xoY =

01

~ultiplying this equation on (x-xo) 3p, and using condition (10.31), we get thel

without nonlinear terms for (x - xo) --+ OJ k'\s a result, we obtaml

~quation

IWith conditIon (10.31) for the solutiOns. EquatIon (10.33) is a hnear mhomogeneou§ OCractIonal equatIon. The solutIon of thiS equatIon allows us to find the solutIon ofj OCractIonal equatIon (1O.2)J ~et us consider Eq. (10.33) with c - 0, and the boundary conditiOn~

rfhen the solutiOn isl 121

IlJf(x)

=

L lJfk(X-XO)a

kEa,a+l_k[-a(x-xo)al

~

af

lX (x-xo - y)a-1Ea,a[-a(x-xo - y)a]y -xo)-a/2dy.

(10.34

~ere E a ,f3 [z] is a Mitag-Leffler function (Miller, 1993; Gorenflo et al., 1998; Goren-I Iflo, 2002; Kilbas et aI., 2006) that is defined byl

2351

110 6 Conclusion

[Let us consider the integral representation for the Mittag-Leffler functionl

(10.35~ Iwhere Ha denotes the Hankel path, a loop, which starts from -00 along the lowe~ Iside of the negative real axis, encircles the circular disc IgI:'( Izl!/a in the positivel klirection, and ends at -00 along the upper side of the negative real axis. By thel Ireplacement ~a -+ ~ equation (10.35) transforms into (Podlubny, 1999; Gorenflo]

I2OU2J I

Ea

Z =-.-

,/3 [ ]

2Jrla

1 y(a,o)

e~l a ~(!-/3)/a

~-

Z

d~,

I

< a < 2,

(10.36

Iwhere Jral2 < 8 < min{ Jr, Jra}. The contour y(a, 8) consists of two rays S-o ==i I{ arg(~) = -8, I~ I;? a} and S±o = {arg(~) = +8, I~ I;? a}, and a circular arc Co @ {IgI = I, - 8 :'( arc( g) :'( 8}. Let us denote the region on the left from y(a, 8) a§ K' (a, 8). Then (Gorenflo, 2002)j (10.37 land 8 :'( Iarg(z) I :'( Jr. In our case, z = -a(x-xo)U, arg(z) = Jr. As a result, wei larrIve at the asymptotic of the solutiOn, whIch exhIbIts power-lIke taIls for x -+ 00.1 [I'hese power-lIke taIls are the most Important effect of the non-Integer denvatIve Inl ~he tractIonal equatiOns]

[0.6 Conclusioril IWe prove that the pSI-senes approach (Tabor, 1989) can be generalIzed to the fracj ~iOnal dIfferential equatiOns (Tarasov, 2006). The suggested pSI-senes approach canl Ibe used for a WIde class of tractiOnal nonlInear equatiOns. The leadIng-order behav-I liOrs of solutIons about an arbItrary sIngulanty, as wen as theIr resonance structures] ~an be denved for fractiOnal equatiOns by the suggested generalIzatiOn of pSI-senes.1 IWe prove that fractiOnal dIfferentIal equatiOns of order a WIth a polynomIal nonlIn-1 ~ar term of order s have the non-integer power-like behavior of order al( 1- s) nea~ ~he sIngulantYl ~n general, the IntegrabIlIty of tractiOnal nonlInear equatiOns IS a very Importantl landInterestIng oliject for researches. There are a lot of problems that are connecte~ IWIth speCIfic propertIes of the fractiOnal calculus. For example, we must denve ij generalIzatiOn of the LIe algebra for the vector fields that are defined by fractiOna~ ~envatIves. For thIS generalIzatiOn, the JacobI Identity cannot be satisfied, and wei Ihave a non-LIe algebra. The defimtiOn of such "tractiOnal" LIe algebra IS an openl guestion and cannot be realized by a simple way. In order to formulate the fractiona~

1236

10 Psi-Series Approach to Fractional Equations

[generalization of a Lie algebra for derivatives of non-integer order, we can use thel Irepresentation of fractional derivatives as infinite series of derivatives of integerl kJrders (KIlbas et aI., 2006; Samko et aI., 1993). For example, the Riemann-LIOuvIll~ ~ractIOnal denvatIve can be represented as an Infimte senes of denvatIves of Integerl prders. In thIScase, the possIble realIzatIOns of the generalIzatIOn are connected withl OChe special algebraIc structures for Infimte jets (VInogradov et aI., 1986; VInogrado\1 land Krasil'shchik, 1997; Vinogradov and Vinogradov, 2002)j

Referencesl

IM.J. Ablowitz, A. Ramani, H. Segur,

1978, Nonlinear evolution equations and ordij Inary differential equations of Painleve type, Letters to Nuovo Cimento, 23, 333-1 I33I MJ. AblowItz, A. Ramam, H. Segur, 1980a, A connectIOn between nonlInear evoj IlutIOn equatIOns and ordInary dIfferential equations of P type I, Journal oj Math1 ~matical Physics, 21, 715-721J M.J. AblowItz, A. Ramam, H. Segur, 1980b, A connectIOn between nonlInear ev01 IlutIOn equations and ordInary ddferentIal equatIOns of P type II, Journal of Mathj emaucal Physics, 21, 1006-1015~ [T. BountIs, H. Segur, E VIvaldI, 1982, Integrable HamIltoman systems and thel IPaInleveproperty, Physical Review A, 25, 1257-1264J fJ. Bureau, 1964, DIfferential equations wIth fixed cntIcal POInts,Annali di Matem1 'atica Pura e Applicata, 116,1-116.1 lYE Chang, M. Tabor, J. WeIss, 1982, Analytic structure of the Henon-HeIles HamIlj Itoman In Integrable and non Integrable regIme, Journal of Mathematical PhyslcsJ 123, 531-538J ~. Conte, 1993, SIngu1antIes of dIfferential equatIOns and IntegrabIlIty, In Introduc-I klOn to Methods of Complex AnaLYSIS and Geometry for ClassIcal Mechamcs andl !Nonlmear Waves, D. Benest, C. Froeschle, (Eds.), EdItions FrontIeres, Gd-sur1 !Yvette. r.M. Cosgrove, G. Scoufis, 1993, PaInleve claSSIficatIOn of a class of ddferentIa~ ~quatIOns of the second order and second degree, Studies in Applied MathematicsJ ~8, 25-87. K Gorenflo, A.A. KIlbas, S. V. RogoSIn, 1998, On the generalIzed Mittag-Lefflerl Itypefunctions, Integral Transforms and Special Functions, 7, 215-224j ~. Gorenflo, J. Loutchko, IY. Luchko, 2002, ComputatIOn of the MIttag-Leffler func-I ItIOn and ItSdenvatIve, Fractional Calculus and Applied Analysis, 5, 491-518.1 V\.A. KIlbas, H.M. Snvastava, JJ. TruJIllo, 2006, Theory and Applications oj Frac1 klOnal Dijjerentzal EquatIOns, ElseVIer, AmsterdamJ IE.L. Ince, 1927, Ordmary DIfferential EquatIOns, Longmans-Green, LondonJ IK.S. Miller, 1993, The Mittag-Leffler and related functions, Integral Transformsl land special Functions, 1, 41-49J

IR eferences

2371

k\.V Miiovanov, J.J. Rasmussen, 2005, FractIOnal generabzatIOn of the Gmzburgj ILandau equation: an unconventional approach to critical phenomena in complex] ImedIa, Physics Letters A, 337, 75-80.1 f. Pamleve, 1973, Lecons sur la Theone AnaLYtlque des EquatIOns DijjerentlellesJ IHermann, Pans, 1897; Repnnted m: Oeuvres de Paul Pamleve, VoLl, Centre Naj Itional de la Recherche Scientifique, Paris.1 ~. Podlubny, 1999, Fractional Differential Equations, Academic Press, New Yorkj k\.K. Roy-Chowdhury, 2000, Painleve Analysis and Its Applications, CRC PressJ IBoca RatonJ IS.G. Samko, A.A. KIlbas, 0.1. Manchev, 1993, Integrals and Denvatlves of Fracj klOnal Order and AppllcatlOns, Nauka I Tehmka, Mmsk, 1987, m Russlanj !Ind Fractional Integrals and Derivatives Theory and Applications, Gordon and! Breach, New York, 1993j M. Tabor, 1989, Chaos and Integrabillty In Nonlinear DynamiCs, WIley, New YorkJ M. Tabor, J. WeISS, 1981, AnalytIC structure of the Lorenz system, PhySical Revlelij lA, 24, 2157-2167 j IVE. Tarasov, 2006, PSI-senes solutIOn of fractIOnal Gmzburg-Landau equatIOn) Vournal oj Physics A, 39, 8395-8407J IVE. Tarasov, G.M. Zaslavsky, 2005, FractIOnal Gmzburg-Landau equation tor frac1 Ital media, Physica A, 354, 249-261j ~E. Tarasov, G.M. Zaslavsky, 2006, Fractional dynamics of coupled oscillators withl Ilong-range mteraction, Chaos, 16, 023110j k\.M. Vmogradov, LS. KrasIl'shchIk, 1997, AlgebraIC aspects of dIfferentIal caIcu1 Ilus, (collectIOn of papers), Acta Applicandae Mathematicae, 49, SpeCIal Issue 3j k\.M. Vinogradov, LS. Krasil'shchik, v.v. Lychagin, 1986, Introduction to the Gej pmetry of Nonlinear Dijjerentwl EquatIOns, Nauka, Moscow. In RUSSIan] k\.M. Vmogradov, M.M. Vmogradov, 2002, Graded multIple analogs of LIe algej Ibras, Acta Applicandae Mathematicae, 72, 183-197J ~. Weitzner, G.M. Zaslavsky, 2003, Some appbcatIOns of fractIOnal denvatIves) ICommunications in Nonlinear Science and Numerical Simulation, 8, 273-281 j

IPart 1111

IFractional Spatial Dynamics

~hapter

1]

[Fractional Vector Calculusl

111.1 Introductionl [The calculus of denvatIves and mtegrals of non-mteger order go back to LeIbmzJ ~IOuvIIIe, GrUnwald, Letmkov and RIemann. The fractIOnal calculus has a long hIS1 ~ory trom 1695, when the denvatIve of order a - 0.5 was descnbed by LeIbm~ I(Oldham and Spanier, 1974; Samko et aI., 1993; Ross, 1975). The history of fracj ~IOnal vector calculus (PVC) IS not so long. It has only 10 years and can be reduce~ ~o the papers (Ben Adda, 1997, 1998a,b, 2001 ; Engheta, 1998; VelIev and EnghetaJ 12004; Ivakhnychenko and VelIev, 2004; NaqvI and Abbas, 2004; NaqvI et aI., 2006j ~ussam and NaqvI, 2006; Hussam et aI., 2006; Meerschaert et aI., 2006; Yongl ~t aI., 2003; Kazbekov, 2005) and (Tarasov, 2005a,b,d,e, 2006a,b,c, 2007, 2008)~ [I'here are some fundamental problems of conSIstent formulatIOns of PVC that canl Ibe solved by usmg a tractIOnal generalIzatIon of the fundamental theorem of caIj ~ulus (Tarasov, 2008). We define the tractIOnal dIfferentIal and mtegral vector 0P1 ~ratIOns. The tractIOnal Green's, Stokes' and Gauss' theorems are formulated. Thel Iproofs of these theorems are realIzed for SImplest regIOns. A tractIOnal generalIza-1 bon of extenor dIfferentIal calculus of dIfferentIal forms IS dIscussed. A consIsten~ fVC can be used in fractional statistical mechanics (Tarasov, 2006c, 2007), frac1 ~IOnal electrodynamICs (Engheta, 1998; VelIev and Engheta, 2004; Ivakhnychenkol land VelIev, 2004; NaqvI and Abbas, 2004; NaqvI et aI., 2006; Hussam and NaqvIJ 12006; Hussam et aI., 2006; Tarasov, 2005d, 2006a,b, 2005e) and tractIOnal hydro1 klynamics (Meerschaert et aI., 2006; Tarasov, 2005c). Practional vector calculus i§ Ivery Important to descnbe processes m complex medIa (Carpmten and MamardIJ If997). ~n

SectIOn I 1.2, the problems of conSIstent tractIOnal generalIzatIOn of vectorl are descnbed. FractIOnal generalIzatIOn of the fundamental theorems o~ k;alculus are considered in Section 11.3. In Sections 11.4-11.5, the differential andl lintegral vector operations are defined. In Sections 11.6-11.8, the fractional Green'sj IStokes' and Gauss' theorems are formulated. The proofs of these theorems are realj IIzed for SImplest regions, PmaIIy, a short conclUSIOn IS given m SectIOn I 1.9.1 ~alculus

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

1242

11 Fractional Vector Calculus

111.2 Generalization of vector calcUlus IVector calculus is a theory With differentiation and mtegration of vector fields. Thel Ivector calculus studies various differential and integral operators defined on scala~ pr vector fields. There are several important theorems related to these operatorsj Iwhich generalize the fundamental theorem of calculus to higher dimensions] for Cartesian coordmates, the gradient, divergence and curl operators can bel ~efined by gradf(x) = esDU(x) , (11.1)1 ~iv F(x) = D~Fs(x),

(11.2)1

IcurIF = eIClmnD~Fn,

(11.3)1

Iwhere D~ are derivatives of first order with respect to x s , s = 1,2,3. Here es , s =I L 2, 3, are orthogonal unit vectors, and Fs(x) are components of the vector fieldl [F(x) = Fs(x)es. In Eq. (11.3), clmn is Levi-Civita symbol, which is 1 if (i,j,k) is ani ~ven permutation of (1,2,3), (-1) if it is an odd permutation, and 0 if any index i§ Irepeated. [t seems that it is pOSSible to define a generahzation of grad, div, curl by usmg ij ~ractional derivative D~ instead of the derivative D1, where D~ are fractional (LH puvl1le, Riemann-LiOuvl1le, Caputo, etc.) denvatives (Oldham and Spamer, 1974j ISamko et aL, 1993; Miller and Ross, 1993; Podlubny, 1999; Kl1bas et aL, 2006) o~ prder a With respect to xs , s - 1,2,3. ThiS pomt of View was used m Refs. (Benl k\dda, 1997, 1998a,b, 2001; Engheta, 1998; Vehev and Engheta, 2004; Ivakhnyj ~henko and Veliev, 2004; Naqvi and Abbas, 2004; Naqvi et aL, 2006; Hussain and! ~aqvi, 2006; Hussam et aL, 2006; Meerschaert et aL, 2006; Yong et aL, 2003j IKazbekov, 2005) and (Tarasov, 2005a,b,d,e, 2006a,b,c, 2007). At such approachl ~here is a big arbitranness m defimtiOn of vector operators. Unfortunately it is no~ Ipossible to reach conSistent construction of a tractional vector calculus. The maml Iproblemof formulation of tractiOnal vector calculus appears, when we try to generj lahze not only differential vector operators, but also the mtegral theorems (Tarasov J 12008). In general, the tractiOnal vector calculus must mcIude generahzatiOns of thel ~tfferential operatiOns (gradient, divergence, curl), the mtegral operatiOns (flux, cir"1 ~ulatiOn), and the theorems of Gauss, Stokes and Green. In thiS chapter, the fracj ~iOnal differential operators Will be defined such that fractiOnal generahzatiOns ofj Imtegral theorems (Green's, Stokes', Gauss') can be reahzedJ ~et us descnbe a mam problem that appears when the curl operator and mtegrall ~ormulas are generahzed on a tractiOnal case. For simphficatiOn, we use a rectangu-I br domain on]Rz and the Cartesian coordinates. We consider the Green's formula inl ~artesian coordinates for functions F; = Fx(x,y) and Fy = Fy(x,y), which are define~ ~or all (x,y) in the region W.I [Theorem 11.1. (Green's Theorem for a Rectangle) ILet FAx,y) and Fy(x,y) be continuously differentiable real-valued functions in al I(:lomam that mcludes the rectangl~

243

111 2 Generalization of vector calculus

IW := {(x,y):

a

~

x

~

b, c

~

y

~

d}.

~

d]

(11.4)1

ILet the boundary ofW be the closed curve aw. Thea

IProof Let W be the rectangular domainl

IW := {(x,y) :

a

~

x

~

b, c

~

y

IWith the vertexes m the pomt§

IA(a,c),

B(a,d),

C(b,d),

D(b,c)·1

[he sides AB, BC, CD, DA form a boundary aW of W. Thenl I

r (Fxdx+Fydy) her Fxdx+ ~r Fxdx+ hBr Fydy+ ~r FydJ1 =

~w

~

lb

Fx(x,d)dx+

tl b

l

a

Fx(x,c)dx +

dx[Fx(x,d)-Fx(x,c)]+

jd Fy(a,y)dy+ l Fy(b,Y)d~ c

fd dy[Fy(a,y)-Fy(b,y)].

(11.6)1

[I'he mam step of proof of Green's formula is to use the Newton-Leibmz formulij

11

b

dxDxf(x)

= f(b) - f(a).

[Thefunction f(x) in (11.7) is absolutely continuous on 1(11.6) can be represented asl

la, bl. As a result, expressioij

[ThiS ends the proof.

q

lRemark [To denve a fractlOnal generahzatlOn of Green's formula (11.5), we should have generahzatlOn of the Newton-Leibmz formula (I 1.7) m the forml

~

bIt: aD~ f(x) = f(b) -

f(a),

(11.8)1

Iwhere some integral and derivative of non-integer order are used. This generaliza-I ~ion exists for specified fractional integrals and derivatives, and does not exist fofj

1244

11 Fractional Vector Calculus

larbitrary taken type of fractional derivativesj [Let us consider the left-sided Riemann-Liouville fractional integral and deriva-I The left-sided Riemann-Liouville fractional integral for x E la, bl is defined!

~ive.

IQY

I

1

l

x

dx' f(x') ( : f(x) := r( a) a (x _ x')I-a'

a> O.

[The left-sided Riemann-Liouville fractional derivative for x E la, bl and n - 1 < a IS defined byl

r


(11.10 ~n order to generalize the vector calculus for fractional case, we must take intol laccount the semlgroup property of tractIonal Integrals and denvatIvesJ

[Theorem 11.2. Let f(x) E Lp(a,b) and

a,f3 > 0, where Lp(a,b)

(1 < p < 00) is al

~et of those Lebesgue measurable functions on [a,bJfor whic~

[hen the semlgroup property for the left-Sided Riemann-LIOuville fractIOnal mtej Igrals

(11.11~

lis satisfied at almost every point x E [a,b]. If a + f3 > 1, then relation (11.11) hold~ fit any point of [a,b] (see Lemma 2.1 and Lemma 2.3 in (Kilbas et a1., 2006))1 IProof This statement was proved in (Samko et a1., 1993) (see Theorem 2.5 of Sec-j ~lOn 2.7 In (Samko et aL, 1993». q

[Theorem 11.3. Let f(x) E Lda,b) and aI': a f(x) E ACmla,bl. Then thefollowingl IrelatlOn IS satlsfiedl

Iwhere n - 1 <

a

~ n, m - 1 <

f3

~ m, and

a + f3 < n1

IProof This theorem was proved in (Kilbas et a1., 2006) (see Property 2.4 in (Kilba§ ~t al., 2006)). q

IRemark. [n general, the semigroup propertYI

(11.12~

2451

111 2 Generalization of vector calculus

lis not satisfied for fractional derivatives. For some special cases, Equation (l1.12~ ~an be used (see Theorem 2.5 m (Samko et aL, 1993)). For example, propert)] 10 1.12) is satisfied for the function§

li.e., Equation (11.12) is valid for f(x) ifthere exists a function g(x) E Lda,b) suchl that [The semigroup property for fractional derivatives is also valid if a = 0, b = 00 and! If(x) is infinitely differentiable (generalized) function on 10,00) (see Section 1.4.5 ofj I(Gelfand and Shilov, 1964) and Section 8.3 of (Samko et al., 1993»~ for fractional integral (11.10) and derivative (11.10), we have the following state-j Im.ent.

[Theorem 11.4. Let f(x) be a Lebesgue measurable function on la,blfor whic~

pnd alb a f(x) has absolutely continuous derivatives up to order (n -1) on [a,b]J rrhen the relatIOn (11.13

Iwhere D~- j

= d n - j / d~- j are integer derivatives, and n - 1 < a < n, holds almostl

~verywhere on

IProof.

la,blJ

ThIS theorem was proved m (Samko et aL, 1993) (see Theorem 2.4 m SectIOnl

q

12.6 of (Samko et aL, 1993». ~xample.

for 0

< a < 1, Equation (11.13) gIve~ ex

ex f (X ) = f () b - (b-a)ex-

alb aDx

ra

1

alb -

ex f (X ) ,

(11.14

PbVIOusly that Eqs. (11.14) and (11.13) cannot be conSIdered as a reahzatIon

o~

lQl.ID. IRemark lJ fropertIes (11.13) and (11.14) are connected WIth the definItIOn of the Rlemann1 ~IOuvIIIe fractIonal denvatIve, where the mteger-order denvatIve acts on the frac1 ~ional integralj (11.15] IaD~ = D~ al~ a, n - 1 < a < n.

1246

11 Fractional Vector Calculus

[his definition gives that the left-hand side of (11.14)

a

n

a

i~

n- a

D x-ax Ila x Q IaDx Q I x, I

(11.16~

Iwherethe integer derivative D~ is located between the fractional integrals. Since thel pperations D~ and aIr; are not commutativ~

Iwe get the additional terms, which cannot give the right-hand side of (11.8). Thi§ ~lOncommutativity can be represented as a nonequivalence of Riemann-Liouvill~ land Caputo denvatIves (Podlubny, 1999; KIlbas et al., 2006)J

ca()_ a()_n~ (x-a)i-a(i)() aDx f X - aDx f X r (j _ a + 1) Dxf a,

'=0

n- 1 < a < n

Iwherethe lett-SIded Caputo fractIOnal denvatIve IS defined by the equation (compar~ Iwith (11.15)) [The noncommutativity of D~ and aI~ in (11.16) does not allow us to use semigrou~ Iproperty (11.11) for the fractIOnal Integrals. As a result, we cannot formulate a con1 Ivement RIemann-LIOuvIlle fractIOnal analog of the Newton-LeIbmz formulal IRemark 2J ~n

order to have a tractIOnal generalIzatIOn of the Newton-LeIbmz formula of thel [orm (11.8), we must replace the left-sided Riemann-Liouville derivative aD~ inl ~q. (11.8), wherel Iby the left-sided Caputo derivative ~ D~, such that the left-hand side of (11.8) i~

[fhen we can use the semIgroup property (11.11), and]

bI~ ;D~f(x) [n partIcular, If n = 1 and 0

Ii: ~D~ f(x)

=

=

aI~ aI; aD~f(x)

=

aI;D~f(x)l

< a < 1, thei1j all

D~f(x) =

l dxD~f(x) b

=

f(b) - f(a)

[fhIS equatIon can be conSIdered as a convement tractIonal analog of the Newton1 ILeibniz formula I

V\s a result, to generalIze Gauss', Green's and Stokes' formulas for tractIOna~ we can use the equation WIth the RIemann-LIOuvIlle Integral and the Caputol

~ase,

2471

111 3 Fundamental theorems of fractional calculus

Herivative·

bIt: ~D~ f(x)

=

f(b) - f(a).

(1l.17j

[I'hIS equatIOn can be conSIdered as a tractIonal analog of the Newton-LeIbmz forj Ii:ll.iil.J.i:

111.3 Fundamental theorems of fractional calculus ["he first fundamental theorem of calculus states that the differentiation and inted 19ration are inverse operations: if a continuous function is first integrateH and thenl klifferentiated, the original function is obtaine~

k); aI;f(x) =

f(x).

(11.1sj

[I'he second fundamental theorem of calculus statesl bIlD~f(x) =f(b) - f(a).

(11.19j

[ntegral theorems of vector calculus (Stokes', Green's, Gauss' theorems) can bel ~onsIdered as generalIzatIOns of fundamental theorems of calculus.1 ~f we use the RIemann-LIOUVIlle mtegrals and derIVatIves (Samko et al., 1993j IKilbas et al., 2006), we cannot generalize (11.19) for fractional case, sincel

bIf: aD~f(x) -I f(b) -

f(a)J

[n thIS case, we have Eq. (11.13). The tractIonal generalIzatIon of the theoremsl [or finite interval la,bl can be realized (Tarasov, 200S). We suggest the fractiona~ lanalogs of Eqs. (11.18) and (11.19) m the forml

GD~ aI<; f(x) bI~ ~D~ F(x)

=

= F(x)

f(x) , - F(a),

a> 0, 0 < a < 1,

(11.20j (11.21~

Iwhere aI:: is the Riemann-Liouville integral, and ~Dr; is the Caputo derivative.1 ~et us gIve the baSIC theorems regardmg a connection of two central operatIOn~ pf calculus, dIflerentIatIOn and mtegratIOn.1

[Theorem 11.5. (FIrst Fundamental Theorem of FractIonal Calculus] ILet f(x) be a real-valuedfunction defined on a closed interval [a,b]. Let F(x) be th~ Ifunction definedfor x in [a,b] byl IF(x) = aI~f(x), Iwhere

aI~

is the Riemann-Liouville fractional

integra~

(11.22j

1248

11 Fractional Vector Calculus

~D~F(x) = f(x)

(11.23j

Ifor X E (a, b), where ~D~ is the Caputofractional derivativ~

Iwhere n -1 < a < nl IProof The proof of this theorem can be realized by using the Lemma 2.21 of (Kil-j bas et al., 2006)j for real values of a > 0, the Caputo tractIOnal derIvative proVIdes operatioIlj IInverse to the RIemann-LIouvIlle IntegratIon trom the left (see Lemma 2.21 (KI1ba§ ~t al., 2006) )J (11.24j ~D~ aIr; f(x) = f(x), a> 0 ~or

f(x) E L=(a,b) or f(x) E C[a,bn [I'hIS ends the proof.

q

[Iheorem 11.6. (Second Fundamental Theorem of Fractional Calculus) ILet f(x) be a real-valuedfunction defined on a closed interval [a,b]. Let F(x) be al Ifunction such tha~ (11.2Sj If(x) = ~D~F(x)

Iforall x in la, bl, the~ PJ:'!(x)

(11.26~

= F(b) -F(a),

PI; equivaLentLY]

lalt: ~D~F(x) = F(b) -F(a),

(11.27~

0 < a < 1.

IProof. The proof of thIS theorem can be reahzed by USIng the Lemma 2.22 of (KI1"1 Ibaset al., 2006). If f(x) E ACnla,bl or f(x) E Cnla,bl, then (see Lemma 2.22 (Kilba§ ~t al., 2006))

ale; ~D~ f(x) =

f(x) -

n~

:!

(x -

a/(D~f)(a),

n- 1< a

~ n,

(11.28

Iwhere Cn [a, b] is a space of functions, which are n times continuously differentiabl~ ~m [a,b]. In particular, if 0 < a ~ 1 and f(x) E AC[a,b] or f(x) E C[a,b], thenl

laIr; ;D~ f(x) = f(x) -

f(a)·1

[I'hIS equatIOn can be conSIdered as a tractIOnal generahzatIOn of the Newton"1 lLeIbmzformula In the form (11.8). ThIS ends the proof. q [n these theorems the spaces LIla, bl and Aqa, bl are used]

111.3 Fundamental theorems of fractional calculus

249

I_ Here AC[a,b] is a space of functions F(x), which are absolutely continuous onl [a,b].1t is known thatAC[a,b] coincides with the space of primitives ofLebesgu~ Isummable functions and therefore an absolutely continuous function F(x) has ~ Isummable derivative D~(x)F(x) almost everywhere on [a,b]. If F(x) E AC[a,b]j ~hen the Caputo derivative (0 < a < I) exists almost everywhere on la,bl (seel [heorem 2.1 of (KIlbas et al., 2006».1 I- We denote Lp(a,b) the set of those Lebesgue measurable functions f on [a,b] fo~ IWl.J.i.ai

for f(x) E Lp(a,b), where p > I, the Riemann-Liouville fractional integration~ lare bounded in L p ( a, b) and the semigroup property (11.11) is satisfied.1 IRemark 1J rI'he fundamental theorems of fractiOnal calculus (FI'FC) use the Rlemann-LiOuvIlI~ Imtegration and the Caputo dIfferentiation. The mam property IS that the Caputol ~ractiOnal denvatlve provIdes us an operatiOn mverse to the Rlemann-LiOuvIIle frac1 ~iOnal mtegratiOn from the left. It should be noted that consIstent fractiOnal gener1 lahzatiOns of the FI'C, the dtfferenttal vector operatiOns and the mtegral theoremsl ~or other fractional mtegro-dtfferentiation such as Rlesz, Grtinwald-Letmkov, WeylJ W'Jlshlmoto are open problems.1 IRemark 2J lIn the theorems, we use 0 < a ~ 1. We obtam the fractiOnal Green's, Stokes' andl pauss' theorems also for 0 < a < 1. Equation (11.20) is satisfied for a E lR±. Th~ W'Jewton-Lelbmz formula (11.21) holds for 0 < a ~ 1. For a> I, we have (11.28)1 k\s a result, to generalize the Green's, Stokes' and Gauss' theorems for a E lR±, wei ~an use Eq. (I 1.28) m the forml

f(b) - f(a) = [n partIcular, If I

aI~ ~D~ f(x) + n~ ~(b - a/(D~f)(a),
~

n- I < a

~ n.

(11.29

2, then n - 2 andl

If(b) - f(a) = aIg ~D~ f(x)

+ (b -

a) (DU) (a)·1

lRemark3J lIn the fundamental theorems of fractiOnal calculus, we use the left-sIded fractiOna~ Imtegrals and denvatives. The Newton-Lelbmz formulas can be represented for thel Inght-slded RIemann-LiOuvIlle fractiOnal mtegrals and the nght-slded Caputo fracj bonal denvatlves m the forml

1250

11 Fractional Vector Calculus

[n particular, if 0

< a :(:

I, thenl

IxI!:~Dgf(x) for a

> 0, and f(x)

E

L=(a,b) or f(x)

(11.30~

= f(x) - f(b).

E

C[a,b], then!

I;Dg xl!:f(x) =

(11.31j

f(x).

k\s a result, tractIOnal generalIzatIOn of dIfferentIal operatIOns and Integral theoremsl ~an be defined for the right-sided integrals and derivatives as well as for the leftj Isidedones

~1.4

Fractional differential vector operators

[To define fractIOnal vector operatIOns, we Introduce the operators that correspond tq ~he fractIOnal dIfferentIatIOn and fractIOnal IntegratIOn.1 [I'he left-SIded RIemann-LIOuvIlle tractIonal Integral operator lsi

IfIxa[x'] := r(a) 1 r dx I, (x-x')l-a'

a> O.

(11.32~

[The operator (11.32) acts on real-valued functions f(x) E LIla, bl byl

r

a" 1 f( )dx aIx [x ]f(x) = r(a) Ja (x-x)l-a'

(11.33

IWe define the left-sided Caputo fractional differential operator on [a, b] in the forml

n-l < a < n,

(11.34

[The Caputo operator (11.34) acts on real-valued functions f(x) E ACla,bl b~

r

a['] ( ') 1 dx aDxxfx =r(n-a)Ja (x-x')l+a-n

C

r;n f(x )

dX,n'

n-l
(11.35

IWe note that the Caputo operator can be represented a~

~D~[x'] =a I~ a[x']D n[x'] , ~quatIOns

n -1 < a < nl

(11.20)and (11.21) of the FI'FC can be rewntten In the forml

~D~[x'] aI~[xl]f(x") = f(x),

bIf:[x] ~D~[x']f(x') = f(b) - f(a),

a> 0,

(11.36j

0 < a < 1.

(11.37~

25~

111.4 Fractional differential vector operators

[his form is more convenient than (11.20) and (11.21), since it allows us to takel linto account the variables of integration and the domain of the operatorsj [Let f(x) and F(x) be real-valued functions that have continuous derivatives up t9 prder (n - I) on W C 1R 3, such that the (n - I) derivatives are absolutely continuous,1 li.e., f,F E ACnlwl. We define a fractional generalization of nabla operator byl (11.38~

Iwhere c D~ [xm] is the Caputo fractional derivatives with respect to coordinates xm.1 for the parallelepipe~

Iwe have

a [] CDa[] x, [D WX=ab

cDaw [y]

= C Da [y], c d

[The fractlOnal dIfferentIal vector operators are defined such that fractlOnal gener1 lalIzatlOns of mtegral theorems (Green's, Stokes', Gauss') can be realIzed by usmgl ~he RIemann-LlOuvIlle mtegratlOnl for SImplIficatIon, we define tractIonal gradIent, dIvergence and curl operator~ 1m the CarteSIan coordmates.1 ~efinition 11.1. If f = f(x,y,z) is (n - 1) times continuously differentiable scala~ lfield such that the derivative D~, If is absolutely continuous, then its fractional graj ~hent IS defined byl

IGrad~f = cD~f = el cD~[xLlf(x,y,z)1

t= ej cD~[x]J(x,y,z) +e2 cD~[y]J(x,y,z)+e3 cD~[z]f(x,y,z). (11.39~ pefinition 11.2. If F(x,y,z) is (n - 1) times continuously differentiable vector fieldl Isuch that D~l 1F/ are absolutely continuous, then its fractional divergence is define~ las a value of the expresslOl1j Piv~F = (cD~,F) = cD~[xLlFI(X,y,Zj

1= c D~[x]Fx(x,y,z) +

c D~[y]Fy(x,y,z) + c D~[z]Fz(x,y,z).

(1l.40~

~efinition 11.3. If F(x,y,z) is (n - 1) times continuously differentiable vector field! Isuch that D~, IF/are absolutely continuous, then the fractional curl operator is dej lfined by

~url~F = [cD~,F] = elClmk cD~[Xm]Fk = ej (CD~[y]Fz - CD~[z]Fy)

l+e2 (CD~[z]Fx - CD~[x]Fz) +e3 (CD~[x]Fy - CD~[y]Fx), (ll.4q Iwhere Fk

= Fk(X,y,Z) E ACnlwl

(k = 1,2,3)J

1252

11 Fractional Vector Calculus

IWe note that the fractional differential operators are nonlocal. The fractional graj klient, divergence and curl depend on the region W j [Let us give the basic relations for the differential vector operators of fractiona~ Calculus ~.

Forthe scalar field f = f(x,y,x), we hav~

Piv~ Grad~ f =

L (c D~ [xtl)2 f.

CD~ [xtl C D~ [xtlf =

(l1.42~

i=]

[Using notation (l1.38)j

(l1.43~ [n the general caseJ (l1.44~

lit is obvious troml

n = a D2a x + a r-a[D x x' a r-a]DnJ x x

1

n

rDx' a In L x

"l >. o:xa Inx

a_

a In x

12. The second relation for the scalar field f

aD xn =

a _ CDa -L 0 1 a x / 1

a Dx

= f(x,y, z) isl

ICurl~ Grad~ f = el £lmn CD~ [xm] CD~ [xn]f = 0,

(l1.4Sj

Iwhere £lmn is Levi-Civita symbol, i.e. it is I if (i,j,k) is an even permutation o~ 1(1,2,3), (-I) if it is an odd permutation, and if any index is repeatedl 13. For the vector field F - emFm, It IS easy to prove the relatIOnl

°

Piv~ Curl~F(x,y,z) = cD~[Xk]£klm cD~[xm]Fm(X,y,zj

1= £klm CD~[Xk] fl.

cD~[xtlFm(x,y,Z) = 0,

(l1.46~

Iwhere we use antisymmetry of £klm WIthrespect to 1 and ml There exists a relatIon for the double curl operator In the forml

[Using the relationl (l1.48~

Iwe obtain

25~

111.5 Fractional integral vector operations

(11.49j lRemark ~n

the general casej ~ D~ [x'] (j(x')g(x'))

i- (; D~ [x']f(x'))g(x) + (; D~ [x']g(x') )f(x).

(11.50j

for example (see Theorem 15.1 from (Samko et al., 1993», if f(x) and g(x) arel lanalytic functions on la,bl, thenl laD~[x'](j(x')g(x')) =

L"" a(a,j)(aD~- j[x']f(x'))( D{g(x)),

E' (a,J) =

qa+l) r(j+l)r(a-j+l)

(11.5q

j

k'\s a result, we havel

iPiv~(jF) i-

prad~(jg)

(Grad~

f)g +

(Grad~ g)f,

(Grad~f,F)+ f Div~F.

(11.52~ (11.53~

[I'hese relatIOns state that we cannot use the Leibmz rule m a tractional generahza-I bon of the vector calculus]

111.5 Fractional integral vector operation~ OC=et us define tractional generahzations of circulation, flux and volume mtegrall I(Tarasov, 2008). We consider the vector fieldl

(11.54] (11.55] IWe define the followmg tractIOnal generahzatIOns of mtegral vector operatIOnsl ~efinition 11.4. A tractIOnal circulatIOn is a tractIOnal hne mtegral of the vectorl lfield }' along a hne L that is defined by!

Wl(F)

= (If,F) = If [x]Fx + If [y]Fy+ If [z]Fz,

(11.56]

1254

11 Fractional Vector Calculus

for a = 1, EquatIOn (11.56) gIve§

(11.57~ Definition 11.5. A fractIOnal flux of the vector field F across a suftace S IS a ~ional surface integral of the field, such thatl

fiJff(F)

= (I~,F) = Iff [y,z]Fx + Iff [z,x]Fy + Iff[x,y]Fz ,

frac~

(11.58~

for a-I, we getl

Iwhere dS - e [dydz + e2dzdx + e3dxdy 1 ~efinition 11.6. A fractIOnal volume mtegral IS a trIple fractIOnal mtegral wIthm Iregion W in ~3 of a scalar field f = f(x,y,z) E L[(ffi.3),1

Iv*U) =

I~[x,y,z]f(x,y,z) = I~[x]I~[y]I~[z]f(x,y,z).

~

(11.60~

for a = 1, Equation (11.60) give§

IVJ,{f) :=

JJl.

dV f(x,y,z) =

JJfw

dxdydzf(x,y,z).

(11.61~

[This is the usual volume integral for the function f(x,y, z)J

~ ~.6

Fractional Green '8 formula!

~et

us conSIder a tractIOnal generalIzatIOn of the Green's formula. It IS known tha~ Green's theorem gIves the relatIOnshIp between a lIne mtegral around a sImplel ~losed curve aw and a double integral over the plane region W bounded by aW.1 [The theorem statement IS the followmg.1 ~he

[Iheorem 11.7. (Green's Theoremj ILet aw be a positively oriented, piecewise smooth, simple closed curve in the planel rnd let Wbe a region bounded by aW, If Fx and Fy have continuous partial deriva1 Itives on an open region containing W, thenl

(11.62~

2551

111 6 Fractional Green's formula

IOsmgthe fractIOnal vector operators for

a = 1, EquatIOn (11.62) can be rewnttenl

lin the fann

k1w[x]Fx(x,y) +I1w[y]Fy(x,y)1 I& [x,y] (D1w [y]Fx(x,y) - D1w[x]Fy(x,y))j

F

k\ fractIOnal generahzatIOn of the Green's formula (11.62) IS represented by thel ~ollowmg

statement (Tarasov, 2008)J

[Theorem 11.8. (FractIOnal Green's Theorem for a Rectangle)1 ILet Fx(x,y) and Fy(x,y) be absolutely continuous (or continuously differentiable)1 Ireal-valuedfunctlOns In a domain that Includesthe rectanglel

IW := {(x,y):

a ~ x ~ b, c

ILet the boundary ofW be the closed curve dW.

~

y

~

d}.

(11.63~

The~

IISw[x]Fx(x,y) + ISw[y]Fy(x,y]

FI~[x,y](c D~w[y']Fx(x,y') Iwhere 0 < a

~

c D~W[XI]Fy(X',y)),

(11.64~

1.1

IProof. In order to prove fractIOnal equatIOn (11.64), we change the double fractIOna~ lintegral I~ [x,y] to the repeated fractional integral~ ~~[x,y] =I~[x] I~[y]J

land then employ the Fundamental Theorems of Fractional Calculus. Let W be thel Irectangular domam (11.63) WIth vertexes m the pomt~

IA(a,c),

B(a,d),

C(b,d),

D(b,c).1

[The SIdes AB, BC, CD, DA of the rectangular domam (11.63) form the boundar5j law of W. For the rectangular region W defined by a ~ x ~ b, and c ~ y ~ d, thel Irepeated mtegral I~ ~~ [x] I~ [y]

= aIt:[x] elf [y] 1

[To prove of the fractIOnal Green's formula, we reahze the followmg transformatIOn~

1=

aIg [x] FAx, d) - aIg [x] FAx, c) + clf[y]Fy(a,y)dy- elf [y]Fy(b,y] 1= aIg[x] [FAx, d) - Fx(x,c)] + eIf[y] [Fy(a,y) - Fy(b,y)]. (11.65~

[The main step of the proof of Green's formula is to use the fractional Newtonj II ,eibniz formulal

1256

11 Fractional Vector Calculus

Fx(x, d) -Fx(x,c)

= Jf[y]fD~[y']F(x,y'),

Fy(a,y) -Fy(b,y)

=

-ala[x]~D~[x']F(x',y).

(11.66

k\s a result, expression (11.65) can be represented a~

[This is the right-hand side ofEg. (11.64). This ends the proof.

q

IRemark 1J

[n thIS tractIonal Green's theorem, we use the rectangular regIOn W. If the regIOIlj ~an be approxImated by a set of rectangles, the tractIonal Green's formula can alsq Ibe proved. In this case, the boundary aw is represented by paths each consisting o~ IhorIzontal and vertIcal hne segments, lymg m Wl

Remark 2J [1'0 define the double mtegral and the theorem for nonrectangular regIOns R, we canl Fonsider the function f(x,y), that is defined in the rectangular region W such tha~ IRe Wand f(x,y) = ~(x,y), (x,y) E R, (11.67

,

(x,y) E W /R.

k\s a result, we define a tractIOnal double mtegral over the nonrectangular regIOn RJ ~hrough the fractIOnal double mtegral over the rectangular regIOnWj II~ [x,y] F(x,y)

= I~[x,y] f(x,y).1

IRemark 3J

[To define double mtegrals over nonrectangular regIOns,we can use a general metho~ ~o calculate them. For example, we can do thISfor speCIalregIOnscalled elementarYI Iregions. Let R be a set of all points (x,y) such tha~

[Thenthe double mtegrals for such regions can be calculated byl (11.68~

IUsmgthe

relatIOn~

(11.69~

2571

111 7 Fractional Stokes' formula

Iwhere a

> 0, f3 > 0, we can consider the example§

~. p](x) =O,y= P2(x) =x2,F(x,y) =x+y.1 12. Cl'dx} = 0, y = P2(x) = x, F(x,y) = xy.1 13. CI'](x) =x3 , CI'](x) =x2,F(x,y) =x+y·1

for other relations see Table 9.1 in (Samko et aI., 1993). To calculate the Caputol klerivatives, we can use this table and the equationl

D~[x']f(x') = aD~[x']f(x') - ~ q{~ ~a~ 1)'

n-l


~ n.

(11.70

[Note that the Mittag-Leffler function E u [(x' - a )U] is not changed by the Caputol Herivative [I'hIS equatIon IS a tractIonal analog of the well-known property of exponentIal func-I ~ion of the form exp{x - a} = exp{x - a}. Therefore the Mittag-Leffler functiol1 ~an be conSIdered as a fractIOnal analog of exponentIal functIOnl

D;

~ ~.7

Fractional Stokes' formula

~et us conSIder a tractIOnal generalIzatIOn of the Stokes' formula for a SImple sur"1 [ace W. We denote the boundary of W by aw. Let F be a smooth vector field defined! pn the SImple surface W. Then the Stokes' formula has the forml

I

r

~dW

(F,dL) =

r (curlF,dS).

~w

(11.71~

[I'he fIght-hand SIdeof thISequatIOn IS the surface Integral of curlF over W, whereasl ~he left-hand side is the line integral of F over the line aw.1

[Theorem 11.9. (Stokes' Theorem] ILet W be a twice continuously dijjerentiable simple suljace with positively orientedl ~order aw. ifF is a continuously differentiable vector field defined on neighbor-I "hood of the trace of W, thel1j (11.72j [I'hus the Stokes' theorem IS the assertIOn that the lIne Integral of a vector fieldl pver the boundary of the surface W IS the same as the Integral over the surface o~ ~he curl of F'. For CartesIan coordInates, Equation (1 1.71) gIve~

[W 1=

(Fxdx + Fydy + Fzdz)1

JL

(dydz[DyFz -DzFy] +dzdx[DzFx-DxFz] + dxdy[DxFy -DyFx]).(11.73~

1258

11 Fractional Vector Calculus

[he fractional analog ofthe Stokes' theorem can be represented in the followin~ [orm (Tarasov, 2008).1

[Theorem 11.10. (FractIOnal Stokes' Theorem] ILet F = F(x,y,z) be a vector field such thatl

Iwhere Fx, Fy, Fx are absolutely continuous (or continuously dijJerentwble) real1 Ivalued functions on JR3. Then the fractional generalization of Stokes' formulal 1(11.73) is represented by the equation, (11.74~ ~n

I-

Eq. (11.74) we use the following fractional integral operatorsj

In the left-hand side of (11.74), I~w is a fractional surface integral over L =

aWl

Isuch that

(11.75~

I-

[he integral (11.75) can be considered as a fractional line integra1.1 In the right-hand side of (11.74), I~ is a fractional surface integral over S =

Wi

Isuch that

~~

I-

= I~ = eJI~[y,z] +e2I~[z,x] +e3I~[x,y].

(11.76~

The tractIOnal curl operation i~

ICurl~ F

=

e[£[mn C D~ [Xm]F~

1= eJ ( CD~ [y]Fz -

CD~ [z]Fy)

+ e2 ( CD~ [z]Fx -

CD~ [x]Fz]

l+e3(CD~[x]Fy- CD~[y]Fx).

(11.77j

for a-I, EquatIOn (11.77) gives the well-known expressIOnl ICurlWF = curlF

= e[£[mnDxmF~ 'F-e-J'("D'yP"'z------rD'ZP"'y')--;+-e-2'(ti;P"'x------r;D·xP""'J (11.78~

for CarteSian coordmates, the left-hand Side of Eq. (11.74) mean~ (11.79]

for CarteSian coordmates, the nght-hand Side of Eq. (11.74) mean§ ~I~,Curl~F) = I~[y,z] (CD~[y]Fz - CD~[z]Fy)1 f+-I~[z,x] (CD~[z]Fx - CD~[x]Fz) f+-I~[x,y] (CD~[x]Fy - CD~[y]Fx) .

(11.80~

111.8 Fractional Gauss' formula

259

[his integral can be considered as a fractional surface integral. As a result, Equationl 10 1.74) has the forml

FI~ [y, z]( cD~ [y]Fz I+I~[z,x]

C D~ [z]Fy

J

(C D~[z]Fx - CD~[x]FzJ

I+I~[x,y]( CD~[x]Fy -

CD~[y]Fx) J

[I'he tractional Stokes' formula m thIS form can be easIly proved for a rectangularl IregionW.

111.8 Fractional Gauss' formulal [The Gauss' theorem, whIch IS also known as the Gauss-Ostrogradsky theorem, IS ~ ~heorem m vector calculus that can be stated as foIIows. Let W be a regIOn m spacel Iwith boundary aw. Then the volume integral of the divergence of vector field FI pver Wand the surface integral of F over the boundary aWare related byl

(11.81~

V\ tractIOnal generalIzatIOn of the Gauss' formula (I 1.8I) IS represented by the fol1 ~owmg

theorem (Tarasov, 2008).1

[Theorem 11.11. (FractIOnal Gauss' Theorem for a ParaIIelepIped~ ILet FAx,y,z), Fy(x,y), Fz(x,y,z) be continuously differentiable real-valuedfunction~ 1m a domam that mcludes the paralleleplpedl

W:={(x,y,z): a c x

c b,

c~y~d, g~z~h}.

(11.82~

IIfthe boundary ofW be a closed surface aw, thenl rI~w,F) =I&Div~F.

IProof. For CarteSIan coordmates, we have the vector field F = Fxel + Fye2 Ii.i.lliI

(11.83~

+ Fze3J (11.84]

(11.85~

1260

11 Fractional Vector Calculus

[f W is the parallelepipedl

~hen

the integrals (11.84) arel II~[x,y,z]

=

aI~[x]

clf[y] gI~[z]J

V~w[y,z]

= cIJ[y] gIr[z]J II~w[x,z] = aIg[x] gIr[z]J IIXw[x,y] = alt[x] clf[y]J k\s a result, we can reahze the follOWIng transformatlon§ [I~,F) =

IXw[y,z]Fx + IXw[z,x]Fy + IXw[x,y]FJ

EcIJ[y]gIr[z] {Fx(b,y,z) -Fx(a,y,z) I Falr[x]gIr[Z] {Fy(x,d,z) -Fy(X,c,z)]

1+ aIr [xlcIJ [y] {Fz(x,y,g) -Fz(x,y,h)] aIr [x] cIJ[y] gIr[z] ;D~[XI]Fx(X',y,Z ~D~[y']Fy(x,y',Z)

+ ;D~[ZI]FZ(X,y,Z')

po I~(cD{t,Fj F I{tDiv~FJ [ThIS ends the proof of the tractIOnal Gauss' formula for parallelepIped region.

q

[The tractIOnal Gauss' formula (I 1.83) can be generahzed to non-parallelepIpedl IregIOns. In order to define the trIple Integral and the theorem for non-parallelepIpedl Iregions R, we consider the vector field G(x,y,z), that is defined in the parallelepipedl IregIOn W, R C W, such thatl

(x,y,z ) = ~t

F(x,y,z), (x,y,z) E R, (x,y,z) E W JR. 0,

(11.86

allows us to define a tractIOnal trIple Integral over the non-parallelepIped regIOili over the parallelepIped regIOn W.I

IR, through the fractIOnal trIple Integral

111.9 Conclusion

2M

111.9 Conclusionl

IA consistent formulation of fractiOnal vector calculus (FVC) by usmg a fractiona~ generalization of the fundamental theorems of calculus is considered. We define thel [ractional differential and integral vector operations. The fractional Green's, Stokes 1 land Gauss' theorems are formulated. The proofs of these theorems are realized fo~ ISimplest regiOns (Tarasov, 2008)J fractiOnal vector calculus is very important to descnbe processes m compleX] ~edia and systems. Fractional dynamics of particles and field is realized in nj klimensional space. Therefore fractional vector integro-differentiation can be used! [or the systems with a long-range power-law interaction (Laskin and Zaslavskyj 12006; Tarasov and Zaslavsky, 2006a,b; Tarasov, 2006e,d). A consistent FVC canl be used in fractional statistical mechanics (Tarasov, 2006c, 2007), fractional elec1 ~rodynamics (Engheta, 1998; Vehev and Engheta, 2004; Ivakhnychenko and VehevJ 12004; Naqvi and Abbas, 2004; Naqvi et aI., 2006; Hussam and Naqvi, 2006; HUS1 Isam et aI., 2006; Tarasov, 200Sd, 2006a,b, 200Se) and fractiOnal hydrodynamic~ I(Meerschaert et aI., 2006; Tarasov, 200Sc)J OC:et us note some possible extenSiOns of the fractional vector calculus.1 I_ It is important to prove the suggested fractiOnal mtegral theorems for a generall form of domains and boundaries] I- It is mterestmg to generahze the formulatiOns of fractiOnal mtegral theorems fo~ ~

I_ We note that most of the analytic results are easily understood, m a more generall ~orm, usmg the dlflerential geometry of which vector calculus forms a subsetJ pnfortunately a conSistent generahzatiOn of ddferenttal geometry, which use~ ~ractiOnal denvatives and mtegrals, is not reahzed yet.I [n the fundamental theorems of fractiOnal calculus, we use the Riemann-LiOuvill~ ImtegratiOn and the Caputo dlflerentiatiOn. The mam property is that the Caputol OCractiOnal denvative proVides us an operatiOn mverse to the Riemann-LiOuville fracj ~iOnal mtegratiOn from the left. Note that fractiOnal generahzatiOns of the ddferentta~ Ivectoroperators and the mtegral theorems for the fractiOnal mtegro-ddferenttatiOnl pf Riesz, Griinwald-Letmkov, Weyl, Nishimoto are an open problemJ [I'here are the foIIowmg pOSSible apphcatiOns of the fractiOnal vector calculusJ I_ A fractiOnal nonlocal electrodynamiCs that is charactenzed by the power lawl Inon-locahty can be formulated by usmg the FVq I- Nonlocal properties in claSSical dynamiCs can be descnbed by the FVq I_ DynamiCs of fractiOnal gradient and Hamiltoman dynamical systems can be dej Iscnbed by the FVC. Note that fractiOnal generahzatiOns of gradient and Hamilj ~onian systems were discussed in (Tarasov, 2005a,b).1 I- Usmg the FYC, the theory of smooth dynamical systems and bdurcatiOns of vec1 OCor fields (Birkhoff, 1927; Pahs and Melo, 1982; Gilmor, 1981; GuckenheimeIj landHolmes, 2002) can be generahzed. Smgulanty and dynamiCs on discontmu-I pus vector fields (Luo, 2006, 2010) also can be considered by FVq

1262

11 Fractional Vector Calculus

I_ The continuum mechanics of fluids and solids with nonlocal properties (with aI ~lOnlocal interaction of medium particles) can be described by the FVC. We not~ ~hat models of continuous media with long-range interparticle interaction wen~ kliscussed in (Tarasov, 2006e,d)~ I_ Osmg the FVC, fractIOnal dIfferentIal equatIons for conservatIon of mass, moj ~entum, energy can be obtained for mechanics of continuous media. We not~ ~hat the fractional conservation of mass was suggested in (Wheatcraft and Meerj Ischaert, 2008) by using fractional divergence~ I_ Osmg the FVC, fractIOnal dIfferentIal equatIons for conservatIOn of probabIlItY! ~an be conSIdered in framework of the nonlocal statIstIcal mechamcs (Vlasov J ~ 978); see also (Vlasov, 1966; Campa et aI., 2009). We note that fractional Liouj IvIlle and BogolIubov equatlOns were dIscussed m (Tarasov, 2006c, 2007).1 [n phySICS, applIcatIons of usual vector operatIOns, whIch are descnbed byl of first order, can be connected WIth nearest-neIghbor mterpartIcIe mj ~eractions of medium particles. The fractional derivatives of fractional vector 0P1 ~ratIOns can be connected WIth long-range alpha-mteractlOns (Tarasov, 2006e,d) ml ~omplex medIa. For denvatIves of non-mteger order WIth respect to coordmates, wei Ihave the power-lIke taIls as the Important property of the solutIOns of the fractIona~ ~envatIves

~quatIOns.

lReferencesl f. Ben Adda, 1997, Geometnc mterpretatIon of the fractIOnal denvatIve, Journal of] IFractional Calculus, 11, 21-52l

f. Ben Adda, 1998a, Geometnc mterpretatIOn of the dIfferentIabIlIty and gradIent o~ Ireal order, Comptes Rendus de l'Academle des ScIences. Serzes I: MathematzcsJ 1326, 931-934. In Frenchl f. Ben Adda, 1998b, The dIfferentIabIlIty m the fractIOnalcalculus, Comptes Rendu~ Ide l'Academie des Sciences. Series I: Mathematics, 326, 787-79U f. Ben Adda, 2001, The dIfferentIabIlIty in the fractIOnal calculus, Nonlinear Anal1 lysis, 47, 5423-5428J p.D. BIrkhoff, 1927, DynamIcal Systems, Amencan MathematIcal SocIety.1 k\. Campa, 1'. DauxOls, S. Ruffo, 2009, StatIstIcal mechamcs and dynamICS of solvj ~ble models WIth long-range mteractions, Physics Reports, 480, 57-159.1 k\. Carpmten, F. Mamardl (Eds.), 1997, Fractals and Fractional Calculus in Con1 ~inuum Mechanics, Spnnger, New Yorkl W'J. Engheta, 1998, FractIOnal curl operator m electromagnetIcs, MIcrowave and opj tical Technology Letters, 17, 86-91j ~.M. Gelfand, G.B. ShiIov, 1964, Generalized Functions I: Properties and Operas ~ions, AcademIC Press, New York and London.1 ~. GIlmor, 1981, Catastrophe theory jor SCIentIsts and Engineers, WIley,New Yorkl ~. Guckenheimer, P. Holmes, 2002, Nonlinear Oscillations, Dynamical Systems andi IBifurcations of Vector Fields, Springer, Berlin.1

References

263

k\. Hussam, S. Ishfag, Q.A. NagvI, 2006, FractIOnal curl operator and fractIOna~ IwavegUIdes, Progress In Electromagnetics Research, 63, 319-335J k\. Hussam, Q.A. NagvI, 2006, FractIOnal curl operator m chlral medIUm and fracj ItIOnal non-symmetnc transmISSIOn hne, Progress In ElectromagnetIcs Research] S9,199-213j 1M.v. Ivakhnychenko, E.!. Veliev, 2004, Fractional curl operator in radiation prob-I Ilems, 10th International Conference on Mathematical Methods in Electromag-s Inetic Theory, Sept. 14-17, Ukraine, IEEE, 231-233j IK.K. Kazbekov, 2005, Fractional differential forms in Euclidean space, Vladikavkazl !Mathematical Journal, 7, 41-54. In Russian,1 Ihttp://www.vmj.ru/articlesI2005.L5.pdfj k\.A. KIlbas, H.M. Snvastava, J.J. TrujIllo, 2006, Theory and Applications of Fracj tional Differential Equations, Elsevier, Amsterdam] N. Laskin, G.M. Zaslavsky, 2006, Nonlinear fractional dynamics on a lattice withl Ilong-range interactions, Physica A, 368, 38-54j ~.c.J. Luo, 2006, Singularity And Dynamics on Discontinuous Vector Fields, Else-] IVIer, Amsterdam.1 k\.c.J. Luo, 2010, Discontinuous Dynamical Systems in Time-Varying Domains] ISprmger, Berhn.1 M.M. Meerschaert, 1. Mortensen, S.w. Wheatcraft, 2006, Fractional vector calculusl Ifor fractional advection-dispersion, Physica A, 367, 181-190; and New ZealancA 'Mathematics Colloquium, Massey UmversIty, Palmerston North, New Zealand) IDecember 2005, http:77www.stt.msu.edu/ mcubed7MathsColloq05.pd~ IK.S. MIller, B. Ross, 1993, An Introduction to the Fractional Calculus and Frac~ kIOnal Differentzal EquatIOns, WIley, New YorkJ IQ.A. NagvI, M. Abbas, 2004, Complex and hIgher order fractIOnal curl operator ml ~lectromagnetics, Optics Communications, 241, 349-355.1 IS.A. NaqvI, Q.A. NaqvI, A. Hussam, 2006, Modelhng of transmISSIOn through ~ ~hIral slab usmg fractIOnal curl operator, Optics Communications, 266, 404-406.1 IK.E. Oldham, 1. Spamer, 1974, The FractIOnal Calculus: Theory and ApplzcatIOn~ r.f DijjerentzatIOn and IntegratIOnto ArbItrary Order, AcademIC Press, New YorkJ ~. Pahs, W. De Melo, 1982, GeometrIc Theory of DynamIcal SYstems. An Introduc-I ~ion, Spnnger, New YorkJ ~. Podlubny, 1999, Fractional Dijjerential Equations, AcademIC Press, New YorkJ lB. Ross, 1975, A bnef hIstory and exposition of the fundamental theory of fractIOna~ calculus, Lecture Notes in Mathematics, 457, 1-36J IS.G. Samko, A.A. KIlbas, 0.1. Manchev, 1993, Integrals and DerIVatIves of Fracj kional Order and Applications, Nauka I Tehmka, Mmsk, 1987, m RussIanj ~nd Fractional Integrals and Derivatives Theory and Applications, Gordon and! IBreach, New York, 19931 ~E. Tarasov, 2005a, Fractional generalization of gradient systems, Letters in Mathj emaucal Physics, 73, 49-58j ~E. Tarasov, 2005b, Fractional generalization of gradient and Hamiltonian systemsJ "Journal of Physics A, 38, 5929-5943J

1264

11 Fractional Vector Calculus

IVB. Tarasov, 2005c, FractIOnal hydrodynamIC equatIOns tor fractal medIa, Annal~ 'Pi Physics, 318, 286-3071 IV.E. Tarasov, 2005d, ElectromagnetIc field of fractal dIstnbutIOn of charged partIj ~les, Physics ofPlasmas, 12, 082106j ~E. Tarasov, 2005e, Multipole moments of fractal distribution of charges, Modernl IPhysics Letters B, 19, 1107-1118J IV.E. Tarasov, 2006a, Magnetohydrodynamics of fractal medIa, Physics of PlasmasJ 113, 052107l ~E. Tarasov, 2006b, Electromagnetic fields on fractals, Modern Physics Letters AJ 121, l587-l600j WoE. Tarasov, 2006c, Fractional statistical mechanics, Chaos, 16, 033108j IVB. Tarasov, 200M, Contmuous lImIt of dIscrete systems WIth long-range mteracj ItIOn, Journal oj Physics A, 39, 14895-14910.1 ~E. Tarasov, 2006e, Map of discrete system into continuous, Journal ofMathemat-I [cal Physics, 47, 09290lj IVB. Tarasov, 2007, LIOuvIlle and BogolIubov equatIOns WIth fractIonal denvatIvesJ Modern Physics Letters B, 21, 237-248J IV.E. Tarasov, 2008, FractIOnal vector calculus and tractIOnal Maxwell's equatIOns) IAnnals oj Physics, 323, 2756-2778J ~E. Tarasov, G.M. Zaslavsky, 2006a, Fractional dynamics of coupled oscillatorsl Iwithlong-range interaction, Chaos, 16, 023110.1 IV.E. Tarasov, G.M. Zaslavsky, 2006b, FractIOnal dynamICS of systems WIth long-I Irange mteractIOn, Communications in Nonlinear Science and Numerical Simula-I ~ion, 11, 885-8981 ~.I. VelIev, N. Engheta, 2004, FractIonal curl operator m reflectIOn problems, lOthl IInternatlOnal Conference on Mathematical Methods In Electromagnetic TheoryJ ISept. 14-17, Ukrame, IEEE, 228-230J k'\.A. Vlasov, 1966, Statistical Distribution Functions, Nauka, Moscow. In RussIan1 k'\.A. Vlasov, 1978, Nonlocal Statistical Mechanics, Nauka, Moscow. In RussIanl IS. W. Wheatcraft, M.M. Meerschaert, 2008, FractIOnal conservatIOn of mass, Adj Ivances in Water Resources, 31,1377-1381.1 ~hen Yong, Yan Zhen-ya, Zhang Hong-qmg, ApplIcatIOns of tractIOnal extenor dItj IterentIal m three-dImensIOnal space, Applied Mathematics and Mechanics, 24J 256-260

~hapter 1~

IFractional Exterior Calculus and IFractional DifferentiallForms

112.1 Introductionl pIfferentIaI forms and extenor calculus are Important theones m mathematIcs. Exte1 Inor calculus have found WIde apphcatIOns m fields such as general relatIVIty, theoryl pf electromagnetIc fields, thermodynamICs, theory of elastICIty, dIfferentIal geom-I ~try, topology and nonhnear dIfferentIal equatIOns. DIfferentIal forms are the mos~ Inatural language for expressmg electromagnetIc and gauge fields mathematIcaIIyJ [I'hIS language IS mdependent of coordmates. Extenor calculus of dIfferentIal formsl gIve an alternatIve to vector calculus, whIch IS ultImately SImpler and more naturel IOsmg denvatIves and mtegrals of non-mteger order, we can generahze the eX1 ~enor calculus of differentIal forms to tractIonal order. A tractIOnal generahzatIOnl pf the differentIal forms was suggested m (CottnII-Shepherd and Naber, 200Ia,b] I(see also (Tarasov, 2008a». An apphcatIon of tractIOnal dIfferentIal fonns to dyj ramIcal systems was conSIdered m (Tarasov, 2005a,b, 2006c, 2007). We note tha~ ~ractIOnal mtegral theorems for extenor calculus were not dIscussed m these papersl ~n thISchapter an approach to formulate a tractIOnal extenor calculus and tractIOna~ khfferentIal forms are suggested] [n SectIOn 12.1, we proVIde a bnef reVIew of dIfferentIal forms of mteger orderl ~o fix notatIOns and convement references. In SectIOns 12.3-12.4, tractIOnal gen1 ~rahzatIOns of extenor denvatIves and differential forms are suggested. In SectIOnl ~ 2.5, the Hodge star operator IS conSIdered. In SectIOn 12.6, we define the tractIOna~ Ivector operatIOns through fractIOnal differentIal fonns. In SectIOn 12.7, the fracj ~IOnal MaxweII's equatIOns m terms of fractIOnal forms are dIscussed. In SectIOn§ [2.8-12.9, the Caputo tIme denvatIves m electrodynamIcs and fractIOnal nonloca~ ~axweII's equatIOns are conSIdered. In SectIOn 12.10, tractIOnal electromagnetIq Iwaves are descnbed. FmaIIy, a short conclUSIOn IS given m SectIOn 12.I 11

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

1266

12 Fractional Exterior Calculus and Fractional Differential Forms

112.2 Difl'erentiaI forms of integer orderl [n mathematICS,dIfferentIal forms are an approach to multI-varIable calculus that I§ lindependent of coordinates. A differential form of degree k, or differential k-formj pn a smooth manifold M is formally defined as a smooth section of the kth exteriorl Ipower of the cotangent bundle of M (Westenholz, 1978). In this chapter we considerl 1M = lR n . A differential O-fonn is by definition a smooth function on M. A differentia~ ~-form W = pi (X)dXi is an objects dual to a vector field F = pi(x)D;j on lR n . Her~ landlater we mean the sum on the repeated index i from 1 to n. The object dXi, wherel Ii = 1, ... , n, are the orthononnal dIfferentIal I-forms on jRn. The forms aXi are duall pbjects to the vector fields D;j' i = 1, ... , n, on lRn .1 pIfferentIal forms can be multIplIed together usmg an operatIOn 1\ called thel Iwedge product. There IS also a dIfferentIal operator a on dIfferentIal forms called! ~he exterior derivative. The wedge product of kI-form and k2-form is a (k I + k2)j [onn, and the exterior derivative of k-fonn is (k + 1)-form. In particular, the exteriorl k{erivative of O-fonn j(x), which is a function on M, is its differential dj(x), whichl lis a I-fonn on MJ

lI>efinition 12.1. A difterentiall-fonnl (12.l)i lis called an exact l-form in lR n if the vector field F = e.F'(x) can be represented asl Ip'(x) Iwhere V

= V(x)

= -D~Yl

is a continuously differentiable function]

~fthe differential fonn (12.1) is an exact I-form, then W = -dV, where V = V(x) lIS a O-form. The exterIor derIvatIve extends the concept of the dIfferentIal of ~ ~unctIOn, whIch IS a form of degree zero, to dIfferentIal forms of hIgher degre~ I(Westenholz, 1978; Dubrovm et aI., 1992)J

a

pefinition 12.2. An exterior derivative d of k-form Wk is (k+ 1)-form dWk such ~he

followmg properties are satIsfiedj 1(1) The exterior derivative is an lR-linear mappingj

Iwhere k and I are integers, and cr , C2 E lRj 1(2) For k-form Wk and i-form Wt, the extenor derIvatIve gIvesl

1(3) av IS the dIfferentIal of the smooth functIOn Vj

tha~

112.2 Differential forms of integer order

2671

1(4) For any smooth function V = V(x), d(dV) = OJ

k\ k-form on W C JR" can be represented in components byl

(12.2)1 [he exterior derivative of (12.2) i§

(12.3)1

Id(O;l"';k(x)

= D;s (O;l ...ik(x) dx..

(12.4)1

IWe note that d(d(O) = 0 for any k-forms

(0.1

Definition 12.3. A differential k-form

IS caIIed a closed k-form m IR" If d(O - OJ

(0

Let us consIder the closed I -forms] [Theorem 12.1. Let F = e;F'(x) be a smooth vector field on a subset W ofJR". Thenl Ithe exterior derivative d ofdifferential1-form ( 12.1) i~

IProof. The extenor denvatlve of differential I-form (12.1)

glve~

IWe can rewnte thIS equation a§

~ (0= -D 1 1 . 1 l' j 2 Xj F'dx·/\dx·+-D J ' 2 Xj F'dx·/\dx· J , ~hangmg

the mdex notatiOn of the second term, we getl

psmg dx; /\dx;

= -dx; /\dx;, we obtaml

q

[I'hlS ends of the proof. PbviOusly, that the condItion for

(0

to be closed lsi

D XjI F; - D xil F j

I

k\s a result, we have the following statementj

= O.

(12.6)1

1268

12 Fractional Exterior Calculus and Fractional Differential Forms

[Theorem 12.2. If a smooth vectorfield F = ei p i (x) satisfies Eqs. (12.6) on a subsetl IW of]Rn, then (12.1) is the closed differentiall-form.1 OCf V(x) is a potential function, theij I

dV

av dx.. = dXi

[I'he extenor denvatIve of I-form (12.7), gIvesl

[The implication from "exact" to "closed" is then a consequence ofthe symmetry ofj ~he second derivativesl 2 2 d V d V (12.8)1 dXidxj - dXjdxi' 1

~fthe

function V = V(x) is smooth function, then the derivatives commute, and Eql

1(12.8) holds. ~t

IS well-known that any exact I-form on W C jRn IS closed. In general, thel statement does not hold. For example, the dIfferentIal I-forml

~onverse

E

= - -2--2 Y dx

lIn the regIOn

x

w {(x,y) =

+y

+ -2--2 x dy,I x +y

E ]R2: (x,y)

-I- (0,0)]

lIS a closed I-form, and It IS not exactl IWe can state that a closed I-form on W IS exact only If W IS SImply connectedl V\ regIOn IS SImply connected If It IS path-connected and every path between twq IpoInts can be contInuously transformed Into every other. A regIOn where any twg IpOInts can be JOIned by a path IS called path-connectedJ [n the theory of dIfferentIal forms, the fundamental result IS the POIncare theorem] ~t states that for a contractIble open subset W of jRn, any smooth k-form ro defined! pn W that IS closed, IS also exact, for any Integer k > O. ThIS has content only whenl Vi: IS at most n. ThIS IS not true for an open annulus In the plane, for some I-form~ Wthat fall to extend smoothly to the whole dISk, so that some topologIcal condItIOi1j lIS necessary. A space W IS contractIble If the IdentIty map on W IS homotopIC to ij ~onstant map. Every contractIble space IS SImply connectedl [I'he concepts of closed and exact forms are defined by the equatIOn d ro - 0 fo~ la gIven ro to be a closed form, and ro - -dV for an exact form. It IS known that tq Ibe exact IS a suffiCIentcondItIOn to be closed. The questIOn of whether thIS IS also ij r.ecessary condItIOn IS a way of detectIng topologIcal InfOrmatIOnJ

[heorem 12.3. If a smooth vector field F

= eiP' (x)

satisfies the relation~

(12.9)1

269

112.3 Fractional exterior derivative

~m a contractible open subset W ofJRn, then (12.1) is the exact I-form, and there isl 'rfunction V = V(x) such tha~

r> = -D!Y(x)dx;.

(12.10~

IProof. Let us consIder the forms (12.1). The formula for the extenor denvatIv(j pf (12.1) is (12.5). Therefore the condition for Q) to be closed is (12.9). If F' ~ I-av/ax;, then the implication from "exact" to "closed" is a consequence of thel Ipermutability of the derivatives. For the smooth function V = V(x), the derivative~ ~ommute, and Eq. (12.9) holds. q IWe note that this statement is a corollary of the Poincare theoremj

~ 2.3

Fractional exterior derivativel

V\ fractIOnal generalIzatIOn of dIfferentIal was presented In (Ben Adda, 1998, 1997)j V\ tractIonal generalIzatIon of the dIfferentIal forms was suggested In (CottnIIj IShepherd and Naber, 2001a,b); see also (Tarasov, 2008a). The applIcatIon of tracj ~ional differential fonns to dynamical systems was considered in (Tarasov, 2005a,b,1 12006c, 2007). We note that tractIOnal Integral theorems are not dIscussed. Let u~ rote the suggested tractIOnal generalIzatIOns of the extenor denvatIvej ~.

In the papers (Ben Adda, 1998, 1997), the tractIOnal dIfferentIal for analytICfunc-I IS defined b)1

~IOns

12.

(12.12~

13. In the papers (Tarasov, 2006c, 2008a), a tractIOnal extenor denvatIve IS defined! ~hrough the Caputo fractIOnal denvatIves In the forlllj

(12.13~

V\ defimtIOn of tractIOnal dIfferentIal forms must be correlated WIth a possIblel generalIzatIOn of the tractIOnal IntegratIOn of dIfferentIal forms. To denve tractIOna~ lanalogs of dIfferentIal forms and ItS Integrals, we conSIder a SImplest case that IS ani

1270

12 Fractional Exterior Calculus and Fractional Differential Forms

~xact I-form on the interval la, b] of R In order to define an integration of fractionall klifferential forms, we use the Riemann-Liouville fractional integrals. Then a fracj ~ional exterior derivative must be defined through the Caputo fractional derivative.1 [1'0 deSCrIbe a fractIOnal exterIor derIvative, we use the fundamental theorem ofj ~ractIOnal calculus (Tarasov, 2008a) that is deSCrIbed by the equatioIlj

laI~[x']~Dt[xl]j(x")

[Equation (12.14) can be rewritten

= f(x) - f(a).

(12.14j

a~

0< a

< 1,

(12.15

Iwhere f(x) E Lila,bl or f(x) E ACla,bl. Using

Idx'

=

sgn(dx')ldx'l = sgn(dx') Idx'11-aldx'l a,

0


~ Ij

[EquatIOn (12.15) can be represented byl

(12.16 Iwhere 0 < a < 1. The expreSSIOn m the big brackets of (12.16) can be las a fractional differential of the function f(x). As a result, we hav~

~jt[x]ad~f(x)

= f(b) - f(a),

0<

a < 1,

considere~

(12.17~

Iwhere a fractIOnal mtegral operation for fractional exact I-forms is defined by thel pperator

(12.18 Note that The fractional exterior derivative of O-form is a fractional differential of the functionl

bd~ f(x) := sgn(dx') Idxl a ;D~[x']f(x'),

0 < a ~ 1.

(12.19j

[EquatIOn (12.17) can be conSidered as a fractIOnal generahzatIOn of the mtegral fofj khfferential I-form. As a result, the fractIOnal exterIor derIvative i~

(12.20~ Iwhere we use the notationl

(12.2q

112 3 Fractional exterior derivative

2711

[here is also second reason to define [dXi]a by (12.21 )j IRemark lJ

[.Jet us note that Idxdu can be represented by the fractional exterior derivative ofj I(Xi - ai)u. To obtain this relation, we use the equations (see Property 2.16 in (Kilba~ ~t al., 2006), page 95)J

~Da[x'](x'-al= r(f3+I) (x-a)f3- a j x nf3 + 1- a) Iwhere n - I

< a ~ n, f3 > n -

I, and!

GD~[x'](x'-a/=O

k=O,I, ...,n-IJ

[hen the fractional exterior derivative of (Xi - ai)a give§

k'\s a result, we obtaml

(12.22~ [Therefore the tractiOnal extenor denvatlve (12.20) can be represented byl

(12.23 [I'he same equatiOn we have for tractlonal ddferentlalJ IRemark 2J

IOsmg the suggested defimtiOns of tractiOnal mtegrals and dIfferentIal forms, It I~ IpossIble to define a fractiOnal mtegratiOn of fractiOnal n-form over the hypercub~ [0, I]n. Unfortunately, a generalization of this fractional integral, which uses thel Imapping p of the region W C JRn into 10, lin, has a problem. For the integer caseJ Iwe use the equatiOnl for the tractiOnal case, the cham rule for dIfferentlatiOn (the tractiOnal denvatlve o~ ~omposIte functiOns) IS more complIcated (see SectiOn 2.7.3 of (Podlubny, 1999))J k'\s a result, a conSIstent defimtiOn of tractiOnal integration of dIfferentlal form fo~ larbItrary mamfolds IS an open questiOnl [I'heextenor denvatlve IS a natural coordmate and metnc mdependent dIfferentla~ pperator actmg on forms. The extenor denvatIve can be generalIzed to non-mtegeIj prders. To have a fractional exterior derivative d a, we must define a fractional powefj pfdxi,l- I, ... ,n. Wenotetha~

1272

~f

12 Fractional Exterior Calculus and Fractional Differential Forms

dx; < 0, thenl (dx;)U

= emu Idx;luJ

order to the fractional differential d a f(x) be real for real-valued functions f(x) on JRn , we usd ~n

(12.24~

linstead of (dx;)u. For a = m, Equation (12.24) givesl

k\s a result, we define a fractional exterior derivative of O-form f(x) byl

(12.25~ Iwhere [dx;]a is defined by (12.24), and 6'D~ is Caputo fractional derivative withl Irespect to X;. To SimplIfy equatton the imttal pomt of fracttonal extenor denvattvg ~an be set to zero (a; - 0), and we can use the notatiOnsl ~U

=

od~J

[The Idx;l u, i = 1, .. ,n, are fractional orthonormal differential forms on lRn • The formsl [dx;]a are objects dual to the fractional vector fields 6'D~ on lR~J ~et us conSider some Simple examples of the fractiOnal extenor denvative of 01 ~orm f(x). For integer values of the order of derivative (a E N) the following result~ lareobtamed: 1 Id f(x) = D;/(x) dx;J

Wf(x) = D'if(x) (dx;)2,1

IWe consider the function f(x)

5D~xf3=O, Iwe have

~axf=o,

IUsmgthe relatiOnl

=~

on the half axis JR+. Thenl

!3=O,I, ...,m-l, m-l
(12.26~

273

112 3 Fractional exterior derivative

Iwhere f3

OCf f3

>m-

1, and m - 1 < a ~ m, we obtainl

= 0, 1, ... , m -

1, thenl

~aXf = 01 for specific values of the power {3 the following results are obtained. For {3 haze

= a,

wei

(12.27~ ~f f3

= 1 and 0 < a < 1, thenl

IdaXi = for {3

=

1 and

r(2 ~ a)x}-a[dXi]a j

a > 1, we hav~ klUXi

= 01

Isince relations (12.26) hold. We note that Eg. (12.27) give~

f]a dx, =

r( a

(12.28~

1

+ 1) da Xia .

~quation (12.28) represents the fractional power of differentials

[dxd U through thel

~ractional basic I-forms

dUxd V\s a result, EquatIOn (12.25) WIth x E lR+ can be represented asl

a>O.

(12.29

IWe note that fractIOnal exterIor derIVatIves of functIOns can be conSIdered as a frac~ ~IOnal generalIzatIOn of dIfferentIal 1-form.1 lRemark OC:et us define fractIOnal dIfferentIal I-form of order WIth 0

kOl,a(X)

= F;(x) [dx;]u.

< a < 1 b5J (12.30]

[fhen we can conSIder a fractIOnal mtegral of dIfferentIal I-form (12.30) on the rej gion 1W:={xin ffi.: a;~x;~b;}J IWe define the mtegral bYI

1274

12 Fractional Exterior Calculus and Fractional Differential Forms

Iwhere we use fractional integral operation (12.18). Using definition (12.18) of opj frator a)~[Xj], we getl

[The definition Idxlu

= sgn(dx) Idxl with 0 < a < 1, give~

AIr

[Uw[x] WI,U(X) ~s

=

j

dXjFj(x)

r((X) Ja (bj -Xj)l-U

a result, we obtainl

Iwhere a/~ [Xj] is the Riemann-Liouville fractional integrals (Kilbas et aI., 2006) onl [a j, b j] c R This equation defines the fractional integral of fractional differentia~ I-form.

112.4 Fractional difl'erential forms for EuclIdean space jRn, we can gIve generalIzatIOn of dIfferentIal forms m compo-I rents. FractIOnal O-forms are contmuously dIfferentIable functlOn~ ~,U = ~,l = f(x)l

k\ tractIonal dIfferentIal I -form of order a can be defined byl (12.3q [fhe fractIOnal extenor denvatlve of the I-form (12.31) glve§

(12.32~ Iwhere Xi ?: ai. We note that tractIOnal dIfferentIal I-form (12.31) can be wntten

a~

(12.33~ [n general, the mltIal pomts

ai

of fractIOnal extenor denvatlve are not equal to zeroJ

[theorem 12.4. Let F'(x) be smooth functions on a subset W oflRn . Then Itionalexterior derivative d U ofdifferential I-form (12.31) i~

thefrac~

(12.34

2751

112 4 Fractional differential forms

IProof The exterior derivative of differential I-form (12.1)

give~

IduOOI,u = ~iD~iF; [dxj]U A [dx;]u.

(12.35~

IWe can rewnte Eg. (12.35) asl U

~hanging

001 , U =

~ ~D~F; [dx)·]U A [dx;]U + ~ ~D~F; [dx)·]U A [dx;]U ] J J ]

the index notation of the second term, we

ge~

q

Iwe obtam (12.34). k\ tractIonal dIfferentIal 2-torm ot order a IS defined byl

k\s a result, we have that 2-torm (12.34) can be conSIdered as a tractIOnal dIfferentIa~ 12-torm at order a withl

k\ tractIOnal k-torm on W C

jRn

can be represented m components byl (12.36~

rrhe fractional exterior derivative d U of the form (12.36) i~ (12.37~

~UOO;I"';k(X)

=

;P~iOO;I"';k(X) [dxj]u.

(12.38~

[IhIS IS the tractIOnal extenor denvatIve WIth the InItIal pomt taken to be zero] IRemark IJ IWe note that the fractional exterior derivatives d U of order a of k-forms of order aJ lare (k + I )-forms of order a. The fractional exterior derivatives d 2u of order 2a ofj ~-forms of order a are not (k + I )-forms of order a. As a result, we havel

(12.39j !Letus consider the fractional exterior derivative d U of the O-form

n

1276

12 Fractional Exterior Calculus and Fractional Differential Forms

(l2.40j [I'he fractIOnal extenor denvatIve of (12.40) give~

[he fractional exterior derivative d2a of the O-form

f i§

IWe see that mequaltty (12.39) holdsJ IRemark 2J IWe consider the fractional exterior derivative d U and fractional differential formsl ~k,a for kEN, a > 0, and a E R Obviously, that a E N are not fractional valuesj pifferential k-forms of arbitrary positive order a i= 1 (including a E N) will bel called fractIonal. Therefore

Wf(x) =D;/(x) (dXi?1 lis called fractional We note

tha~

Iwhered 1 = d is the usual exterior derivative.1 IRemark 3J

general, we can define fractIOnal dIflerentIal forms of nomnteger order at, suchl ~hat fractIOnal extenor denvatIve of the order a2 maps these forms mto the dtfferen-I ~ial forms of order at + a2. ThiS general case was conSidered by Cottnll-Shepherdl land Naber m Refs. (Cottnll-Shepherd and Naber, 2001a,b)J ~n

IRemark 4J [The fractional exterior derivative dU is an 1R-linearmappingj

Iwhere k and I are integers, and Ct, C2 E IRj [Remark 5J for the fractional exterior derivative dU of the wedge product of fractional forms thel lrelation ~a ((f)k,a II (f)1,a) = d a (f)k,a II (f)1,a + (-1/ (f)k,a II d a (f)1,J

lis not satisfied in general.1 ~et

us give defimtIOns of closed and exact fractIOnal differentIal k-forms.1

2771

112 4 Fractional differential forms ~efinition

12.4. A fractIOnal dIfferentIal k-form

OJk,a

IS called a closed fractIOna~

Ifurmlf IA fractIOnal dIfferentIal k-form OJk,a IS called an exact fractIOnal form If the forml ~an be represented a§ kOk,a = dUOJk-I,aJ

Iwhere OJk-I,a is a fractional (k - I )-formj ~t

is easily shown that the differentiall-formj kOI,a = F'(x) [dx;]U

(l2.4q

lis an exact fractional I-form if the functions F' (xl, i = I, ... , n, can be represente~

as IF'(x) =

- ~D~V(x),

(l2.42~

Iwhere V = V(x) is a continuously differentiable function, and ~D~ is Caputol ~envatIve of order a. Usmg the fractIOnal extenor denvatIve, the exact fractIOna~ ~ -form can be represented byl

rI'herefore we have (12.42). Note that expreSSIOn (12.41) IS a fractIOnal generahza-I ~IOn of the dIfferentIal form (12.1).1 lRemark [n the general case, the fractIonal k-forml

can be closed when the dIfferentIal k-forml

Iwith the same OJi] ...dx), is not closed. For example, the fractionall-formi

Iwhere x,y E JR+ and 0


I, is closed. At the same time the differential l-fomj

lIS not closed. ObVIOusly that fractIOnal k-form OJk,a can be exact, when the dIffer-I ~ntIal k-form OJk,1 IS not exactl for fractional I-forms, we can formulate the following statementj

1278

12 Fractional Exterior Calculus and Fractional Differential Forms

[Theorem 12.5. If smooth functions F = eiFi (x) on a contractible open subset W 011 ~n

satisfY the relationsl

~D~F'(x) - ~D~p1(x) 1 , 1

1

=

0,

(12.43~

Ithen (12.41) is an exact fractional1-for~

(12.44~

lrol,a = - ~D~V(x), Iwhere V(x) is a continuous differentiable function and ~P~V(x) = -F'(x)l

IProof. ThISpropOSItIon IS a corollary of the tractIOnal generalIzatIon of the Pomcar~ ~heorem (Cottrill-Shepherd and Naber, 2001b). The Poincare theorem is shownl I(Cottrill-Shepherd and Naber, 2001 a,b) to be true for exterior fractional derivativej

D

IRemark. that we can generalIze the defimtIOn of fractIOnal extenor derIvatIve asl

~ote

(12.45~ Iwhere a = (aI, a2, ..., an), and consider the fractional differential1-forms:1 (12.46~

[n thIS case, we can derIve equatIOns WIth derIvatIves of dIfferent orders ai. ~xample, the tractIOnal I-form (12.46) wIll be closed Ifl

~D~f Fi(x) - ~P~iFj(x)

=

Fo~

oj

for SImplICIty, we suppose that all a; - al

iExample. Let us conSIder the fractIOnal dIfferentIal I-forml (12.47~

Iwherex,y E lR±, andl

+ a12(x)yf32 ,

(12.48~

= a21 (y)x f33 +a22(x)yf34.

(12.49j

IFx(x,y) = IFY(x,y)

all (y)x

f31

[The fractional exterior derivative d of (12.47) give~ U

(12.50~ Iwherethe initial points of fractional exterior derivative are set to zero (al = a2 = O)j [flh > m - 1, where m - 1 < a ~ m, thenl

112.5 Hodge star operator

27~

(12.51) ~f

f32 = /33, equation (12.51) can be rewritten in the form! (12.52

IWe note that the I-form (12.47) with (12.48) and (12.49) is closed (daWI,a If32,f33 E TO, 1, ... ,m-I}, where m-I < a ~ ml

= 0), ifj

lRemark

k\ generalIzatIOn of the extenor denvatlve for tractIOnal case can be defined by usj Img the RIemann-LIouvIlle tractIonal dIfferentIatIon. If the partIal denvatIves m thel ~efinition of the exterior derivative d = dx;D;j are aIIowed to be with Riemannj [Liouville derivative of a fractional order a, then a fractional exterior derivative canl Ibe represented by the equatlOnl (12.53~

Iwhere oD~ is the Riemann-Liouville fractional derivatives (Kilbas et aL, 2006j ISamkoet aL, 1993), and the mltIal pomt IS set to zero. Note that Rlemann-LlOuvI1lg Berivative of a constant C need not be zera

(12.54~ rt'he Riemann-Liouville fractional exterior derivative of order a of X1' with the initiall Ipomt taken to be zero and n = 2, IS gIven byl

(12.55~ k\s a result, we obtam the relatlOnl

[Ihls equatIOn represents fractIOnal dIfferentIal through the fractIOnal power of dIfj ferential

112.5 Hodge star operato~ [fhe Hodge star operator * IS a lInear operator mappmg k-forms on n-dlmenslOnall Ivector space with inner product V to (n - k)-forms. We can define the Hodge sta~ pperator on an oriented inner product vector space V as a linear operator on thel

1280

12 Fractional Exterior Calculus and Fractional Differential Forms

~xterior algebra of V, interchanging the subspaces of k-vectors and (n - k)-vectors,1 Iwhere n = dim V, for 0 < k < n. Let us give a definition of the Hodge star of k-j Ivectors

pefinition 12.5. Let el, ez, ... ,en be an onented orthonormal baSIS of n-dlmenslOnall Ivectorspace V. Then the following property defines the Hodge star operator * com-j ~

(12.56~

Iwhere {il, ... , ik, ik±I, ... ,in} is an even permutation of {I, 2, ... ,n }j IWe have n!/2 relations (12.56), where only usual lexicographical order has the forml

~he

m

are independent. The first one inl

!Example 1~

k\ well-known example of the Hodge star operator IS the case n = 3, when It can bel ~aken

as the correspondence between the vectors and the skew-symmetnc matncesl pf that sIze. ThIS can be used In fractIOnal extenor calculus, for example to createl ~he cross product from the wedge product of two fractIOnal forms. For Euchdearil Ispace ]R3, we hav~

F[dx]U

=

[dy]U II [dz]Uj

F[dy]U

=

[dz]U II [dx]Uj

F[dz]U = [dx]U II [dy]Uj Iwhere Idxl u, IdYlu and Idzl u are the fractional orthonormal differential I-forms onl 3 • The Hodge star operator in this case corresponds to the cross-product in thre~ dImenSIOns.

m

!Example 2J Knother example is n = 4 Minkowski space-time with metric signature ( +,-, -, landcoordinates (t,x,y,z). For basis of fractional I-forms, we hav~

F[dt]U = [dx]U II [dy]U II [dz]U,1 F[dx]U = [dt]U II [dy]U II [dz]U,1 [dy]U = [dt]U II [dz]U II [dx]U,1 1* [dz]U = [dt]U II [dx]U II [dy]uJ

~

for fractlOnal2-formsJ

)

28~

112.6 Vector operations by differential forms

F[dt]U /\ [dx]U

_[dy]U /\ [dz]uJ 1* [dt]lX /\ [dy]lX = [dX]lX /\ [dZ]lX j 1* [dt]lX /\ [dZ]lX = _[dX]lX /\ [dy]lX,1 1* [dx]U /\ [dy]U = [dt]U /\ [dz]U ,I F[dx]U /\ [dz]U = -[dt]U /\ [dy]uJ ~ [dy]lX /\ [dZ]lX = [dt]lX /\ [dX]lX j Iwhere we use

~2.6

COl23

=

= -1 J

Vector operations by differential

form~

[I'he combmatIOn of the * operator and the exterior derivatIve d generates the classlj ~al operators grad, dlV, and curl, m three dimenSions. ThiS fact can be used to defin~ ~ractIonal generahzatIons of the differentIal operators grad, dlV, and curlJ ~.

For O-form ~ = f(x,y,z), the fractional exterior derivative gives the fractiona~ grad operatorj

[ThiS equatIOn can be considered as a tractIOnal generahzatIOn of the gradient byl tractIonal dtfterentIal forms]

Iwherethe region WC]R3 is given by x ~ a, y ~ b, z ~ cl 12 The fractional exterior derivative of the fractional I-forms (l2.57~

IdlXWl,lX

= (~D~Fx - ~D~Fy)[dy]lX /\ [dx]~ tt(~D~Fx - ~D~Fz)[dz]lX /\ [dX]lX + (~D~Fy - ~D~Fz)[dz]lX /\ [dy]lXJ

[I'hls fractional 2-form m components IS the fractIOnal curl operator:1

~lXWl,lX

= (fD~Fz- ~D~Fy)[dy]lX /\ [dz]4

t+-(~D~ Fz - ~D~Fx)[dx]lX /\ [dZ]lX + (~D~Fy - fD~Fx)[dx]lX /\ [dy]lX. (l2.58~ V\pplymg the Hodge star operator * to the 2-form (12.57) glvesj

1282

12 Fractional Exterior Calculus and Fractional Differential Forms

FdaWl,a = (~D~Fz - ~D~Fy)[dx]al

H;D~Fz - ~D~Fx)[dy]a + (;D~Fy - ~D~Fx)[dz]a.

(12.59~

13. Using the Hodge star operator * to fractional I-form (12.57), we obtainl (12.60~

[The fractIOnal extenor denvatIve of fractIOnal2-form (12.60) gIve§ (12.6q IWe can representl

k\ppying the Hodge star operator * to fractional 3-form (12.61), we obtain thel ITIi::iiJ:ll a [Dap (12.62j *Wl,a=a[Dap x x+b[Dap y y+c z z-

Ed

k'\s a result, we

hav~

[ThIS equatIOn defines a fractIOnal generalIzatIOn of the dIvergence through frac1 ~ional

differential forms I

lOne advantage of this expression is that the identity (d a lall cases, sums up the equatIon§ ICurl~Grad~/(x,y,z) =

lDiv& Curl& F

?=

0, which is true inl

oj

= OJ

k\s a result, fractIOnal generalIzatIons of the vector dIfferentIal operatIons can bel k1efined by fractIOnal dIfferentIal formsJ

112.7 Fractional Maxwell's equations in terms of fractional form§ [t IS well-known that the Maxwell's equatIOns take on a partIcularly SImple formJ Iwhenexpressed m terms of the extenor denvatIve and the Hodge star operator. W~ ~an use the Hodge star operator and the fractIOnal extenor denvatIve to descnbel ~ractIOnal dIfferentIal Maxwell's equations. We note that LIOUVIlle was a pioneer ml klevelopment of fractional calculus to electrodynamics (Lutzen, 1985)j !Let x P be coordinates, which give a basis of fractional I-forms [dxP]U in everYI Ipomt of the open set, where the coordmates are defined. Usmg the baSIS of I-form~ IdxPl u, J1 = 0, 1,2,3, and cgs-Gaussian units, we define the following notionsJ ~.

rn

The antisymmetric field tensor Illv(x), corresponding to the fractional 2-for

112.7 Fractional Maxwell's equations in terms of fractional forms

28~

(12.63~ Iwhere fJlv(x) are formed from the electromagnetic fields E and B. For examplej IA2 = Ez/c, 12,3 = -Bz, .... The electric and magnetic fields can be describe~ Iby the fractional differential2-form F( a) in a 4-dimensional space-time. In thel [ractional electrodynamics a fractional generalization of the Faraday 2-form, o~ ~lectromagnetIc field strength, IS (12.63). In the language of dIfferentIal formsJ Iwe use cgs-Gaussian units, not SI units. We note that xJl, J1 = 0, 1,2,3, are dij mensionless variables. Note also that the form (12.63) is a special case of thel Furvature form on the U (1) principal fiber bundle (Husemoller, 1966) on whichl IbothelectromagnetIsm and general gauge theorIes may be descrIbed. The 2-forml ~ F( a), which is dual to the Faraday form, is also called Maxwell 2-formj 12 The current fractional 3-form is

(12.64~ Iwhere jJ.l (x) are the four components of the current density. The fractional3-forml IJ(a) can also be called the electric current fractional form. The current J( a) i~ k{efined as a fractional 3-form here. We note that J ( a) can be defined as a fracj ~IOnall-form, I.e., the Hodge star of (12.64). The 3-form verSIOn IS much mcerJ Isince J (a) is closed rather than co-closed. The Hodge star operator * is a linea~ ~ransformatIon from the space of 3-forms to the space of I-forms defined by thel rIetric in Minkowski space, and the fields are in natural units, where 174neo = 11

IRemark. IWe use fgv(x) instead of Fgv(x), and jJ.l(x) instead of JJ.l(x). Note that the vectorl landtensor components and the suggested forms have dIflerent phySIcal dImensIOnsj Rsing (12.63) and (12.64), the Maxwell's equations can be written very comj Ipactlyas (12.65~ (12.66~ ~quatIOn

(12.65) reduces the Maxwell's equations to the BIanchI IdentIty.1 IWe note that the fractional exterior derivative of Eq. (12.66) give~

Iwhere we use the property d a d a = 0. As a result, the fractionaI3-formJ( a) satisfie§ ~he fractIOnal contmUIty equatIOn.1 rrhe current 3-form can be mtegrated over a 3-dImensIOnal space-tIme regIOnj [rhe phySIcal mterpretatIOn of thIS mtegrails the charge m that regIOnIf It IS spacej IlIke, or the amount of charge that flows through a surface m a certam amount ofj ~ime if that region is a spacelike surface cross a timelike intervalj V\s a result, we have the followmg equatIOnsj

1284 ~.

12 Fractional Exterior Calculus and Fractional Differential Forms

The fractional Bianchi identitYI

12. The fractional source equationj

13. The tractIOnal contInmty equatioIlj

lRemark

k\s the exterior derivative is defined on any manifold, the differential form versionj pf the BIanchI Identity makes sense for any 4-dlmensIOnal manifold. The sourc~ ~quation IS defined tf the manifold IS onented and has a Lorentz metnc. Therefor~ ~he dtfferential form verSIOn of the Maxwell equations IS a convenient to formulat~ ~he Maxwell equations In general relatlvltyl

112.8 Caputo derivative in electrodynamic§ [The behavior of electric fields (E,D), magnetic fields (B, H), charge density p(t, r)J landcurrent density j (t, r) is described by the Maxwell's equation~ ~ivD(t,r) =

p(t,r),

= -JtB(t,r), ~ivB(t,r) = 0, IcurlH(t,r) = j(t, r) + JtD(t,r). IcurIE(t,r)

(12.67~ (12.68~ (12.69~ (12.70~

~ere

r = (x,y,z) is a point of the domain W. The densities p(t,r) andj(t, r) describ~ sources. We assume that the external sources of electromagnetIc field arel given. [The relatIOnsbetween electnc fields (E, D) for the mediUm can be reahzed by! ~xternal

p(t,r)

=

eo L~= e(r,r')E(t,r')dr',

(12.71~

Iwhere eo is the permittivity of free space. Homogeneity in space gives e(r,r') ~ le(r - r'). Equation (12.71) means that the displacement D is a convolution ofj ~he electnc field E at other space pOInts. A local case corresponds to the Dlraq ~elta-function permittivity e(r - r') = e o(r - r'). Then Eq. (12.71) gives D(t,r) ~

leoeE(t,r). V\nalogously, we have nonlocal equation for the magnetic fields (R, H).I

285]

112.9 Fractional nonlocal Maxwell's equations

[Letus demonstrate a possible way of appearance of the Caputo derivative in thel electrodynamics. If we havel

~Iassical

P(t,x)

=

L~= e(x-x')E(t,x')dx',

(12.72~

Iwhere x E JR, thelll co

(D~,e(x - x')) E(t,x')dx'

(D;e(x-x'))E(t,x')dx' = lOSing the integration by parts, we getl +=

~et us consider the kernel

e(x-x') D~E(t,x')dx'.

(12.73

e(x - x') of integral (12.73) in the interval (a,x) such tha~

e(x-x') =

q>(x-x'), 0,

a < x' < x, x' >x, x' < a,

(12.74

IWith the power-hke functiOnl

[ , (x-i) a r(x-x)= r(l-a)'

O
(12.75~

0< a < 1

(12.76j

[Then Eq. (12.73) gives the relatiOl1I

ID;D(t,x) = ~D~E(t,x), IWith the Caputo tractional denvativeJ

112.9 Fractional nonlocal Maxwell's eguation§ IWe can consider fractiOnal non local differential Maxwell's equatiOns (TarasovJ 12008a) in the forml (12.77j IDiv~ E(t,r) = 81 p(t,r), K=url:iE(t,r) = -dtB(t,r),

(12.78j

Piv~B(t,r) =0,

(12.79j (12.80j

Iwhere as, S

= 1,2,3,4, can be integer or fractional, and 81,82,83 are constantsJ

1286

12 Fractional Exterior Calculus and Fractional Differential Forms

fractional integral Maxwell's equations, which use integrals of non-integer orj klers, were suggested In (Tarasov, 200Sd, 2006a,b, 200Se) to descnbe fractal dIstnj butions of electric chargesj [n the general form, the fractional Integral Maxwell's equations (Tarasov, 2008a)1 ~an be represented In the foriTIj tI~k,E(t,r)) =gII~}p(t,r),

(I~~,E(t,r))

=

(12.8q

_~(I~2 ,B(t,r)),

[I%"B(t,r)) = 0,

(12.82 (12.83j (12.84

fractIOnal coordInate denvatives are connected WIth nonlocal properties of thel Imedla. For example, a power-law long-range InteractIOn In the 3-dimensIOnallatj ~ice in the continuous limit can give a fractional equation (Tarasov, 2006e,d). Th~ Isuggested fractIOnal dIfferential and Integral equatIOns can be used to descnbe ani ~lectromagnetic field of medIa that demonstrate fractIOnal nonlocal propertIes. Thel Isuggestedequations can be conSIdered as a speCialcase of nonlocal electrodynamIc~ I(Brandt, 1972; Foley and Devaney, 1975; Belleguie and Mukamel, 1994; Genchevj ~997; Mashhoon, 2003, 2004, 2005)j ~et us denve a conservatIOn law equatIOn for denSIty of electnc charge In thel IregIOn W from the fractIOnal nonlocal Maxwell's equations, The time denvative o~ ~q. (12.77) I~ (12.8Sj piv;V dtE(t,r) = gt dtp(t,r). ISubstitution of (12.80) into (12.85) givesl k3DivW (g2CUr1~B(t,r) - j(t,r))

= gi dtp(t,r).

IDiv;V CUr1~ B(t,r) = 0,

(12.86~ (12.87j

landwe have the lawl ~t dtP(t,r)

+ g3 Div;V j(t,r) =

O.

(12.88j

fractIOnal equatIOn (12.88) IS a dIfferential form of charge conservation law ~ractIOnal nonlocal electrodynamIcs] [f at - a4, we can define the fractIOnal Integral charactenstics such a~

IQw(t) = gtI&'l [x,y,z]p(t,x,y,z),

fo~

(12.89j

Iwhich can be called the total fractIOnal nonlocal electnc charge, and! (12.90~

2871

112 10 Fractional waves

lis a fractional nonlocal current Then the fractional nonlocal conservation law is

~Qw(t) +JJw(t) =

O.

[This integral equation describes the conservation of electric charge in the ~lectrodynamics forthe case at = a4j

~ 2.10

(12.9l~ nonloca~

Fractional wavesl

OC'et us denve wave equatIons for electnc and magnetIC fields m a regIOn W trom thel fractioual nonlocal Maxwell's equations withj = 0 and p = OJ [I'he tIme denvatlve of Eq. (12.78) lsi (12.92j ISubstltutIOn of (12.84) andj = 0 mto (12.92) gIvesl

~?B

= -g2g3CurlW

Curl~B(t,r).

IOsmg (11.49) and (12.79) for a2 - a3 - a4, we

(12.93~

ge~

(12.94j k'\s a result, we obtaml Iwhere v2 = g2g3. Equation (12.95) is a fractional wave equation for the magneticl lfield B. Analogously, EquatIOns (12.78) and (12.84) gIve the tractIOnal wave equaj bon for electnc field] (12.96~ [The solution B(t, r) of Eq. (12.95) is a linear combination of the solutionsl ~+(t,r) and B_(t,r) ofthe equationsl

(12.97j

(12.98~ k'\s a result, we get the tractIOnal extension of d'Alembert expression that was con1 Isidered in (Pierantozzi and Vazquez, 2005).1 for the boundary condltIOn§ lim B(t,r)

I.11--+ ~he

00

= 0,

B(t,O) = G(t),

(12.99~

general solution of Eqs. (12.97) and (12.98) is given (Kilbas et aI., 2006) b)j

1288

12 Fractional Exterior Calculus and Fractional Differential Forms

1 2n

Bm±(t,r) = -

(12.100

Iwhere Gm(ro) = g?[Gm(t)], and Ea,/3 [z] is the biparametric Mittag-Leffler functio~ I(Kilbas et al., 2006). Here B±m(t,r), and Gm(t) are components of B±(t,r) andJ IG(t). for I-dimensional case, Bx(x,y,z,t) = u(x,t), By = Bz = 0, and we can considerl ~he fractional partial differential equationj

IDzu(x,t) - vZoD;au(x,t)

=

0,

x E JR, x> 0, v> 0,

(12.I01~

Iwith the conditions

(12.102~ Iwhere k = 0 for 0 < a ,,;; 1/2, and k = 1 for 1/2 < a ,,;; 1. If 0 < 2a < 2 and v > OJ OChe system of Eqs. (12.101), and (12.102) ISsolvable (Theorem 6.3 of (Kdbas et aLJ 12006)), and the solution u(x,t) is given b)j

11: G~a(x-y,t)!k(y)dy, 00

(x,t)

=

~a(x,t) =

n-I

< a,,;; n,

"2 vl'- a (-a,k+ 1 - a, vltlx- a).

(12.103

(12.104

lfIere ( -a,k+ 1 - a, vltlx a) is the Wright function (Kilbas et al., 2006)J IWe note that the solutIOns of equations as (12.97) and (12.98) are based pnmaryl pn the use of Laplace transforms for equations with the Caputo ~Dc; derivativesl [This leaves certain problems (Kilbas et al., 2006) with the fractional derivative~ ~Dc; for a E JR.I [The tractIOnal denvatIves m equatIOns can be connected WIth a long-rang~ Ipower-law mteractIOn (Tarasov, 2006e,d). The nonlocal propertIes of electrodynam"1 IICS can be conSIdered (Tarasov, 2008b) as a result of dIpole-dIpole mteractIOns withl la fractIOnal power-law screenmg that IS connected WIth the mtegro-dIfferentIatIOnl pf non-mteger order. For non-mteger denvatIves, we have the power-lIke tails as thel IImportant property of the solutIOns of the tractIOnal equatIOnsl

[2.11 Conclusionl fractIOnal dIfferential forms are Important to descnbe tractIOnal dynamICS of com"1 IplexmedIa and systems. FractIOnal extenor calculus can be used in tractIOnal statIs"1 ~ical mechanics (Tarasov, 2006c, 2007), fractional classical electrodynamics (Enj gheta, 1998; Veliev, and Engheta; Ivakhnychenko and Veliev, 2004; Naqvi and! V\bbas, 2004; NaqvI et aL, 2006; Hussam and NaqvI, 2006; Hussam et aL, 2006j [Tarasov, 200Sd, 2006a,b, 200Se) and tractIOnal hydrodynamICs (Meerschaert et aLJ

References

289

12006; Tarasov, 2005c). The fractIOnal denvatIves III equatIOns can be connected wIthl Ilong-range power-law IllteractIOns (Tarasov, 2006e,d)J [The theory of differential forms is very important in mathematics and physicsl I(Westenholz, 1978; Flanders, 1989). The fractIOnal dIfferential forms can be I111 ~erestIllg to formulate a fractional generalIzatIon of dIfferential geometry, Illclud-I ling symplectic, Kahler, Riemann and affine-metric geometries. We assume that thel [ractional differential forms and fractional integral theorems for these forms can bel lused to descnbe claSSIcal dynamIcs (GodbIllon, 1969; Dobronravov, 1976; VIlasIJ 12001; Westenholz, 1978) and thermodynamIcs (Chen and Byung, 1993; Westenj Iholz, 1978). It IS Important to have fractional generalIzatIons of symplectIc geome-I ~ry and POIsson algebra.1

lReferencesl f. Ben

Adda, 1997, Geometnc mterpretation of the fractIOnal denvatIve, Journal ojl IFractional Calculus, 11, 21-521 f. Ben Adda, 1998, Geometnc IllterpretatIOn of the dIfferentiabIlIty and gradIent o~ Ireal order, Comptes Rendus de l'Academle des ScIences. Series I: MathematlcsJ 1326, 931-934. In Frenchl ~. Bellegme, S. Mukamel, 1994, Nonlocal electrodynamIcs of weakly confined eX1 ~Itons III semIconductor nanostructures, Journal oj Chemical Physics, 101, 9719-1

9TI5: ~.H.

Brandt, 1972, Non-local electrodynamIcs III a superconductor WIth spaclallyl Ivarying gap parameter, Physics Letters A, 39, 227-228.1 M. Chen, c.B. Byung, 1993, On the IlltegrabIlIty of dIfferential forms related tg InoneqmlIbnum entropy and meversIble thermodynamIcs, Journal oj Mathemati1 fal Physics, 34, 3012-3029J IK. Cottnll-Shepherd, M. Naber, 2001a, FractIOnal dIfferential forms, Journal of! !Mathematical Physics, 42,2203-2212; and E-print math-ph/0301013j IK. Cottnll-Shepherd, M. Naber, 2001b, Fractwnal DIfferential Forms II, E-pnntl Imath-ph/m01 016.1 IV. V. Dobronravov, 1976, Foundations oj Analytical Mechanics, VIshaya ShkolaJ Moscow In Russian I ~.A. DubroVIll, A.N. Fomenko, S.P. NOVIkov, 1992, Modern Geometry - MethodJ, land Appllcatwns, Part I, Spnnger, New York.1 ~. Engheta, 1998, FractIOnal curl operator III electromagnetIcs, Microwave and 0P1 kical Technology Letters, 17, 86-91 J IH. Flanders, 1989, Dijjerentialjorms with applications to the physical sciences, 2ndl ~d., Dover, New York) [.1'. Foley, A.I. Devaney, 1975, Electrodynamics of nonlocal media, Physical Reviettj IB, 12, 3104-3112j IZ.D. Genchev, 1997, GeneralIzed nonlocal electrodynamICs of dIstnbuted tunnell IJosephson Junctions, Superconductor Science and Technology, 10, 543-546J

1290 ~.

12 Fractional Exterior Calculus and Fractional Differential Forms

GodbIllon, 1969, Geometrie Dijjerentielle et Mecanique Analytique, HermannJ lI'aris.

f.A.

Griffiths, 1983, Exterior Differential Systems and the Calculus of Variationsj IBirkhauser, Bostonl p. Husemoller, 1966, Fibre Bundles, Mcgraw-Hill, New Yorkj k\. Hussam, S. Ishtaq, Q.A Naqvi, 2006, FractiOnal curl operator and fractiOna~ Iwavegmdes, Progress In Electromagnetics Research, 63, 319-335J k\. Hussam, Q.A Naqvi, 2006, FractiOnal curl operator m chiral medmm and fracj ItiOnal non-symmetnc transmiSSiOn hne, Progress In Electromagnetlcs ResearchJ S9,199-213j M. v. Ivakhnychenko, E.!. Vehev, 2004, Fractional curl operator m radiation prob-I Ilems, 10th International Conference on Mathematical Methods in Electromag-s Inetic Theory. Sept. 14-17, Ukraine, IEEE, 231-233j IK.K. Kazbekov, 2005, Fractional differential forms in Euclidean space, Vladikavkazl !Mathematical Journal, 7, 41-54. In Russian,1 Ihttp://www.vmj.ru/articles/20052-S.pd!i k\.A KIlbas, H.M. Snvastava, J.J. Trujillo, 2006, Theory and Applications oj Frac1 ~ional Dijjerential Equations, ElseVier, Amsterdam1 ~. Lutzen, 1985, LiOuVille's dtfferential calculus of arbitrary order and itSelectrody-I Inamical ongm, m Proc. 19th NordIc Congress Mathenzatlczans, IcelandiC Mathj ~matical Society, ReykjavikJ ~. Mashhoon, 2003, Vacuum electrodynamics of accelerated systems: Nonlocall IMaxwell's equations, Annalen der Physik (LeipZig), 12, 586-5981 ~. Mashhoon, 2004, Nonlocal electrodynamics of hnearly accelerated systems] IPhysical Review A, 70, 062103j ~. Mashhoon, 2005, Non10cal electrodynamics of rotating systems, Physical Reviettj lA, 72, 052105j M.M. Meerschaert, J. Mortensen, S. W. Wheatcraft, 2006, Fractional vector calculusl Itor fractiOnal advectiOn-disperSiOn, Physica A, 367, 181-190; and New Zealand! IMathematics Colloq mum, Massey Omversity, Palmerston North, New ZealandJ IDecember 2005 J Ihttp://www.stt.msu.edu/mcubed/MathsColloq05.pdfj IQ.A Naqvi, M. Abbas, 2004, Complex and higher order fractiOnal curl operator ml ~lectromagnetics, Optics Communications, 241, 349-355.1 IS.A Naqvi, Q.A Naqvi, A Hussam, 2006, Modelhng of transmission through ~ ~hira1 slab using fractional curl operator, Optics Communications, 266, 404-406.1 IK. Nishimoto, 1989, FractIOnal Calculus: IntegratIOns and DijjerentzatlOns of Arj '{Jitrary Order, University of New Haven Press, New HavenJ [T. Pierantozzi, L. Vazquez, 2005, An mterpolatiOn between the wave and diffusiOlll ~quatiOns through the fractiOnal evolutiOn equations Dirac hke, Journal oj Math1 emaucal Physics, 46, 113512j [. Podlubny, 1999, FractIOnal Dlfferentlal EquatIOns, Academic Press, New YorkJ IS.G. Samko, AA KIlbas, 0.1. Manchev, 1993, Integrals and Derlvatlves of Fracj klOnal Order and AppLzcatlOns, Nauka i Tehmka, Mmsk, 1987, m Russtanj

References

29~

~nd Fractional Integrals and Derivatives Theory and Applications, Gordon and! Breach, New York, 1993~ IVB. Tarasov, 2005a, FractIOnal generahzatIOn of gradIent systems, Letters in Mathj ~matical Physics, 73, 49-58j ~E. Tarasov, 2005b, Fractional generalization of gradient and Hamiltonian systemsJ flournal oj Physics A, 38, 5929-59431 IVB. Tarasov, 2005c, FractIOnal hydrodynamIC equatIOns for fractal medIa, Annal~ 'Pi Physics, 318, 286-3071 ~E. Tarasov, 2005d, Electromagnetic field of fractal distribution of charged partij ~les, Physics ofPlasmas, 12, 082106j ~E. Tarasov, 2005e, Mu1tipo1e moments of fractal distribution of charges, Modernl IPhysics Letters B, 19, 1107-1118J IV.E. Tarasov, 2006a, MagnetohydrodynamIcs of fractal medIa, Physics of PlasmasJ 113,0521071 ~E. Tarasov, 2006b, Electromagnetic fields on fractals, Modern Physics Letters AJ 121, 1587-16001 IV.E. Tarasov, 2006c, FractIonal statIstIcal mechamcs, Chaos, 16, 033108j IVB. Tarasov, 200M, Contmuous hmIt of dIscrete systems wIth long-range mterac1 ItIOn, Journal oj Physics A, 39, 14895-14910.1 ~E. Tarasov, 2006e, Map of discrete system into continuous, Journal ofMathemat-I [cal Physics, 47, 092901j IY.E. Tarasov, 2007, LIOuvIlle and Bogohubov equations wIth fractIOnal denvatIves) Modern Physics Letters B, 21, 237-248J IY.E. Tarasov, 2008a, FractIOnal vector calculus and fractIOnal Maxwell's equatIOns) IAnnals ofPhysics, 323, 2756-2778j IVB. Tarasov, 2008b, Umversal electromagnetIc waves m dlelectnc, Journal of] IPhysics A, 20, 1752231 ~.I. Vehev, N. Engheta, 2004, FractIOnal curl operator m reflectIOn problems, lOthl IInternational Conjerence on Mathematical Methods in Electromagnetic TheoryJ ISept. 14-17, Ukrame, IEEE, 228-230J p. ViiasI, 2001, HamIltOnian DynamICs, World SCIentIfic PUbhshmg, SmgaporeJ ~. von Westenholz, 1978, DIfferential Forms Tn MathematIcal PhySICS, Northj !Holland, Amsterdamj ~hen Yong, Yan Zhen-ya, Zhang Hong-qmg, 2003, ApphcatIOns of fractIOnal exte1 Inor dIfferentIal m three-dImensIOnal space, Applied Mathematics and Mechanics) 124, 256-2601

~hapter 1~

[Fractional Dynamical System~

113.1 Introductionl [The dynamIcal system IS a notIOnfor any fixed map, whIch descnbes the tIme depen-I ~ence of a posItIOn m ItS space of states. At any gIven tIme a dynamIcal system has ~ Istate gIven by a vector x, whIch can be represented by a pomt m an appropnate statel Ispace. InfimtesImal changes m the state of the system correspond to small changesl 1m the vectors. The evolutIon map of the dynamIcal system IS a fixed rule that dej Iscnbes what future states follow from the current state. The notIons of gradIent and! ~amlltoman systems anse m dynamIcal systems theory (HIrsh and Smale, 1974j pubrovm et aI., 1992; VIlasI, 2001). In thIS chapter, generalIzatIOns of gradIent and! ~amIltoman systems are suggested. We use dIfferentIal forms and extenor denva-I ~Ives of tractIOnal orders. Its allow us to define HamIltoman and gradIent dynam-I lical systems (Gilmor, 1981; Dubrovin et aI., 1992; Vilasi, 2001; Godbillon, 1969)1 pf non-mteger (tractIOnal) orders. In the general case, the tractIOnal HamIltomalli I(orgradIent) systems cannot be consIdered as Hamlltoman (gradIent) systems. Thel Isuggested class of tractIOnal gradIent and Hamlltoman systems IS WIder (Tarasov J 12005a,b)than the usual class of gradient and Hamiltonian dynamical systems. Th~ Isystems of gradIent and Hamlltoman type can be consIdered as a specIal case ofj ~ractIOnal gradIent and Hamlltoman systems.1 ~n SectIOn 13.2, we consIder a tractIOnal generalIzatIOn of gradIent systems. Inl ISectIOn 13.3, some examples of tractIOnal gradIent systems are suggested. The welH Iknown Lorenz and Rossler systems are consIdered as generalIzed gradIent systems.1 [n SectIOn 13.4, we consIder locally and globally HamIltoman systems. In SectIOnl p.5, fractional generalizations oflocally and globally Hamiltonian systems are sugj gested. Fmally, a short conclusIOn IS given m SectIOn 13.61

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

13 Fractional Dynamical Systems

1294

113.2 Fractional generalization of gradient system~ rJradient systeme [The gradient system is described by the equationj

~= Iwhere x

- grad

V(x)J

= Xiei E IFtn and V(x)

~oordinates, the

is a continuously differentiable function. In Cartesianl gradient is given bYI

~rad V(x) = eiD!y(x)j IWecan gIve the followmg defimtIon of gradIent systemsJ

pefinition 13.1. A dynamIcal system that IS descrIbed by the equatIOnsj

~ =F;(x), ~s

i= I, ... ,n

(13·1)1

called a gradIent (or globally potentIal) system m jRn If the dIfferentIall-form:1

fh =

F;(X)dxi

lisan exact form such that ro = -dV, where V ~unctIon (O-form).1

(13.2)1

= V(x) is a continuously differentiabl~

IWe can define locally potentIal systemsJ ~efinition

13.2. A dynamIcal system that IS desCrIbed by Eqs. (13.1) IS called system in IFt n if differential I-form (13.2) is closed.1

~

~ocally potential

[The globally potentIal system IS locally potentIal. In general, the converse state"1 ment does not hold. It IS well-known that the locally potentIal systemj

Idt linthe region W = {(x,y) E IFt z : (x,y) =I- (0,On is not gradient (globally potential)l OCf W C 1Ft is simply connected, then a locally potential system is globally potentialj fl'he concepts of locally and globally potentIal systems wIth the vector field }' ~ ~iF; are connected wIth the equatIOn d ro - 0 for the gIven form ro - F;dxi to be ~ posed form and the equation ro = -dV for an exact form. We can state that to bel globally potential (gradient) is a sufficient condition to be locally potential systems.1 [The question of whether this is also a necessary condition is a way of detectingl ~opologIcal properties of the dynamIcal systeml

113.2 Fractional generalization of gradient systems

2951

[Theorem 13.1. If a smooth vector field F = e;Fi(x) of system (13.1) satisfies th~ IrelationS • D;iFj = 0, i,l = 1, ... ,n (13.3)1

r;jFi -

pn a contractible open subset W

ofJR n , then dynamical system (13.1) is a gradienn

[SYstem such thatl

Iwhere Fi(x) = -D;y(x)j [I'hIS proposItIon IS a corollary of the Pomcare theorem that states that for a conj open subset W of JR n , any smooth I-form (13.2) defined on W, which i§ closed, is also exact] ~ractible

pefinition 13.3. A dynamical system that is described by Eqs. (13.1) is called generalIzed dIssIpatIve system, Ifl

aI

P(x) = E'" D;iFi(X) # oj I=]

land It IS caIIed a dISSIpatIve system,

I~

P(x) = E D;iFi(X) < OJ I=]

[The equations of motion for the gradIent system on a contractIble open

subse~

IW of JRn can be represented in the form (13.4). Therefore the gradient systems arel ~efined

by the potential functions V = V(x)l Ilf the exact differential I-formJ

ro = -dV = -dx;D;y(x1 lIS equal to zero, then we get the equatIOnsj

ID;y(x) =0,

i= 1, ... ,nJ

k\s a result, we obtaml

nx) -C=O,

(13.5)1

Iwhere C IS a constant. EquatIOn (13.5) defines the stationary states of the gradlentl ~ynamIcal system (13.4)1

!Fractional gradient systemsl [Let us consider a dynamical system that is described by Eqs. (13.1) on a subse~ IW of JRn. In general, we can consider a dynamical system that is described by thel

13 Fractional Dynamical Systems

1296

[ractional differential equations (Podlubny, 1999)j

(13.6)1 Iwhere fiDf is Caputo fractional derivative with respect to time. The fractional anaj Ilog of defimtlon of the gradient dynamical systems has the formJ

pefinition 13.4. A dynamical system that IS descnbed by the equatlOnsj

tfi= = F;(x),

(13.7)1

1

Iwhere i = 1, .. ,n, x = e;x;, and F = e;F;(x), is called a fractional gradient (or fracj ~Ional globally potential) system, If the tractional dIfferential 1-formj (13.8)1 lis an exact fractional form such that COI,a = -day, where Y = Y(x) is a continu-I pusly differentiable functIOn. Dynamical system (13.7) IS called a fractlOnallocallyl Ipotentlal system If the fractIOnal dIfferential I-form (13.8) IS a closed fractlOna~ I-form. IWe note that Eq. (13.8) IS a fractIOnal generalIzatIOn of (13.2). If 1(13.8) gives (13.2)J

a-

1, thenl

IRemark IJ IWe consider fractIOnal gradient systems of order a E lR+. ObVIOusly, that a E l"J I~ Inot fractional value. Fractional gradient systems of arbitrary positive order a #- 11 I(mcludmg a E l"J) will be called tractlOnall IRemark2J

V\ dynamical system that

IS descnbed by Eqs. (13.6) IS called a tractIOnal gradlentl I(or tractIOnal globally potential) system If tractIOnal dIfferential I-form (13.8) IS eX1 lact. Dynamical system (13.6) IS called a tractIOnal locally potential system If (13.8)1 liS a closed fractIOnal 1-form.1 OC:et us define the left-sided Caputo fractIOnal denvatIve b5J

(13.9)1 Iwhere x? a. If f(x) E ACm(lR) or f(x) E Cm(lR), thenl

~ere we use x ? a. In general, we can consider the Caputo derivative on the laxis R In this case, the parameter a is - 0 0 , andl

whol~

113.2 Fractional generalization of gradient systems

~f

2971

fractIOnal I-form (13.8) is exact, thenl

p>l,a = -daV = -[dx;]a~p~V~ rrherefore we have F;(x) = -~D~V(x)J [Theorem 13.2. If the force field F;(x) is potentialj

1F;(x) = - ~D~V(x),

(13.10~

Iwhere V (x) is a continuously differentiable function, then the conditionsj (13.1

q

IProof. SubstitutIOn of (13.10) mto (13.11) glveij

~p~ =oI~ a[x';]D~J Iwhere m - 1 <

a

~

m, we obtaml

rfhe relatIOn: lallows us to represent (13.12) m the form:1

[f V(x) is a continuously differentiable function, thenl

V\s a result, we obtam Eqs. (13.11). ~n

q

the general case, the converse statement does not holdl

[Theorem 13.3. If smooth functions F; = F;(x) on a contractible open subset W 011 ~n satisfY the relationsj

(13.13~

13 Fractional Dynamical Systems

1298

Ithen dynamical system (13.7) is afractional gradient system such thati

(13.14~ !where V(x) is a continuous differentiable function and ~P~V =

-F;j

IProof This proposition is a corollary of the Poincare theorem. Note that thel [poincare theorem is shown (Cottrill-Shepherd and Naber, 2001a,b) to be true fo~ ~he fractional exterior derivative n

pefinition 13.5. Dynamical system (13.7) is called a fractional generalized dissipaj ~Ive system, Ifj I1J

ra(X)

=

DivltF =

E ~D~F;(x) -1= oj 1=]

landit is called a fractional dissipative system, if]

f!2a(X)

=

DivltF =

E'" ~D~F;(x) < 0·1

I=l

IWe note that relatIOns (13. I I) are the fractIOnal generalIzatIOn of relatIOns (13.3)1

I

[Theorem 13.4. The stationary states of fractional gradient system (13.14) are dej fined by the equatlOnj (13.15 Iwhere tan:

Ck1 ...k"

are constants, and m

IS

the first whole number greater than or

equa~

IProof. In order to define the stationary states of a tractIOnal gradIent system, wei ~onsIder a solutIOn of the equatIOnj

~D~V(x)

=

(13.16j

O.

OC:et m be the first whole number greater than or equal to a. Then Eq. (13.21) has thel solution (Kilbas et a1., 2006; Samko et a1., 1993)1

IV(x)

=

E Ak(XI,". ,X;-l ,X;+l, .. · ,x,,) (x; -

a;)k,

(13.17~

1c=DJ

Iwhere A k are functions of the coordinates x; with i #- k. Using Eq. (13.17) for i ~, ... ,n, we obtam the solutIOnof the system of Eqs. (13. 16) m the form (13. 15).

~

q

iExample. for n = 2 such that x =

Xl ;;:

a and y =

X2 ;;:

b, we have a fractional gradient systeml

113.2 Fractional generalization of gradient systems ~hat

29S1

is described by the equations:1

~ = - ~D~V(x,y),

if = - fD~V(x,y).

(13.18~

[I'he statIOnary states for thIS system are defined byl

Iwhere Ckl are constants, and m is the first whole number greater than or equal to aj

IRemark lwe note that the Riemann-Liouville fractional derivative of a constant need not hel Izero. Therefore we see that constant C in the equation V(x) = C cannot define ij stationary state ofthe gradient system (13.14). It is easy to see thaq

~

a ( ) a PxY X = aPxp =

(Xi - ai) a

r( 1 _ a) C -I 0

j

[n order to define statIonary states of tractIOnal gradIent systems WIth Rlemannj [.:IouvIlle derIvatIve, we conSIder the equatIOnsj

~P~V(x) =

OJ

land solutIOn of these equatIOnsl

rrheorem 13.5. The stationary states of the fractional gradient system:1

(13.19~ Iwith the Riemann-Liouville derivatives aiD~, are defined by the equationj

(13.20 IHere Ckl ...kn are constants and m is the first whole number greater than or equal t~ IT:

IProof. The statIOnary states of a fractIOnal gradIent system are defined by the equaj Itiill:i:: bP~V(x)

= O.

(13.2q

[.:et m be the first whole number greater than or equal to a. The solutIOn (Samko e~ lal., 1993; Oldham and Spamer, 1974) of Eq. (13.21) has the form:1

IV(x)

= IXi-ail

a

!u=JJ

L ak(xI, ... ,Xi-I,Xi+l, ... ,Xn)(Xi- ail,

(13.22~

13 Fractional Dynamical Systems

1300

Iwhere ak are functions of the coordinates Xi wirh i iF k. Using Eq. (13.22) for i L... ,n, we obtain solution (13.20) ofthe system of Eqs. (13.21). [Example. consider n = 2 such that x = form: ~f we

Xl ~

0 and y =

X2 ~

q

0, then Eqs. (13.19) have thel

if = -oD~V(x,y).

~ = -oD~V(x,y),

=i

(l3.23~

[I'he statIonary states of thIS tractIonal gradIent system are defined by the equatIon:1

IV(x,y) _Ixyla-I

~

L L Ckl~yl = oj

1(=01=01

[The Ckl are constants and m is the first whole number greater than or equal to aj IRemark 1J [I'he RIemann-LIOuvIlle tractIonal denvatIve has some notable dIsadvantages Inl IphysIcal apphcatIOns such as the hyper-sIngular Improper Integral, where the orderl pf sIngulanty IS hIgher than the dImenSIOn, and nonzero of the fractIOnal denva-I ~Ive of constants, whIch would entaIl that dIsSIpatIon does not vanIsh for a systeml lIn eqUIhbnum. At the same tIme the definItIon of Caputo denvatIve IS of more rej IstnctIve that RIemann-LIOuvIlle denvatIve, In that reqUIres the absolute IntegrabIhty! pf the denvatIve of order m. The Caputo fractIOnal denvatIve first computes an or1 ~Inary denvatIve followed by a tractIOnal Integral to achIeve the deSIre order o~ ~ractIOnal denvatIve. The RIemann-LIOuvIlle tractIOnal denvatIve IS computed Inl ~he reverse order. The Caputo denvatIve can be represented through the Rlemannj LIOuvIlle denvatIveJ

CDaf(x) a

x

=

Daf(x) _ a x

m~ (x_a)k-a f(k) (a+). i...

r(k-a+l)

(13.24

[t IS observed that the second term In Eq. (13.24) regulanzes the Caputo fractIOna~ ~envatIve to aVOId the potentIally dIvergence from SIngular IntegratIOn at X - a+. I~ Ishould be noted that Caputo fractIOnal dIfferentIatIOn of a constant results In zero. Ifj ~he Caputo tractIOnal denvatIve IS used Instead of the RIemann-LIOuvIlle tractIOna~ ~envatIve then the statIOnary states of tractIOnal gradIent systems are the same a~ ~hose for the usual gradient systems (V(x) - C = 0). The Caputo formulation ofj OCractIOnal calculus can be more apphcable to gradIent systems than the Rlemannj LIOuvIlle formulatIOn] IRemark2J [n the general case, the fractIOnal gradIent systems cannot be conSIdered as gradI-1 ~nt systems. The class of fractIOnal gradIent systems IS a WIder class than the usuall pass of gradIent dynamIcal systems. The gradIent systems can be conSIdered a~ IspecIaI case of tractIOnal gradIent systems. Therefore It IS pOSSIble to generahz~ ~he application of catastrophe and bifurcation theory from gradient to a wider clas§ pf fractional gradient dynamical systems. Note that the order of fractional deriva-I

113.3 Examples of fractional gradient systems

30~

~ive

Fo~

can be considered as an additional parameter that can lead to bifurcation. fractional gradient system with the potentialj

~xample,

Ihas the stationary states that are defined by the equation:1

Ixl l - a(x3 +a'x+b') = 0,1 Iwhere bifurcatIOnis defined by the new values of the parameters:1

IWe can see that these parameters depend on aJ

IRemark 3J [I'he tractional generahzation of differential forms (Cottnll-Shepherd and NaberJ 1200la,b) leads us to the followmg open questions: Is there a tractional analog ofj ~he homology and cohomology theones? Is there a connection with the nonloca~ ~haracter of the fractIOnal denvative and the topological properties of the fractIOna~ ~hfferenttal forms? These mterestmg open questions require the additIOnal researchJ

~3.3

Examples of fractional gradient systems

[The class of tractIOnal gradient systems is wider than the usual class of gradientl klynamical systems. The gradient systems can be conSidered as a speCial case ofj OCractional gradient systems. Let us conSider some Simple examples of tractIOna~ gradient systems that cannot be conSidered as gradient systemsJ

!Example IJ IWe assume that the dynamical system is defined by the equatIOnsj

(13.25~ Iwhere the nght hand Sides have the formj

(13.26j [This system cannot be considered as a gradient dynamical system if a

Iwe get that

OJ -

Fxdx + Fydy is not a closed form, smcel

#- 0. Using

13 Fractional Dynamical Systems

1302

W']ote that the equation:1 pD~Fx - OD~Fy

= OJ

Iwith Riemann-Liouville fractional derivatives of order a is satisfied for the systeml 1(13.26) wIth x > 0 and y > 0, If a - k and the constant C ISdefined byl

rI'herefore thIS system can be consIdered as a tractIOnal gradIent system wIth thel IIInear potentIal functionj

Iv(x,y) = r(1- a)(ax+b)J Iwhere a = k

!Example 2~ [.-etus consider the dynamical system that is defined by Eq. (13.25) withl

Iwhere k

IFx =

an(n - 1)x" 2 + ck(k - 1)0 2yl,

(13.27j

fy =

bm(m - 1)ym-2 + cl(l- 1)1'/-2,

(13.28j

¥- 1 and I ¥- 1. It is easy to obtainl

landthe differential form ro = Fxdx + Fydy is not closed, i.e., d ro ¥- O. Therefore thi~ Isystem IS not a gradIent dynamIcal system. Usmg the condItIOnj

~or

the generalIzed dIfferentIal form:1

Iwe obtain dUrol,a = 0 for a = 2. As a result, we have that this system can be conj ISIdered as a generalIzed gradIent system wIth the potential functIOnj

~n the general case, the tractIOnal gradIent system cannot be consIdered as a gradlentl Isystem. The gradIent systems are a specIal case of tractIOnal gradIent systems suchl ~hat a = 1

30~

113.3 Examples of fractional gradient systems

OO:xample 3J [Let us prove that dynamical systems that are defined by the well-known Loren~ ~quatIOns (Lorenz, 1963; Sparrow, 1982) are fractIOnal gradIent systems. The Loren~ ~quations (Lorenz, 1963; Sparrow, 1982; Neimark and Landa, 1992) are defined byl

Idx dt--Fx,

dy -F dt -

y,

dz -F dt -

z,

(13.29~

Iwhere Fx, Fy and Fz have the formsj

IFx = cr(y-x),

Fy = (r-z)x-y,

Fz =xy-bz.

(13.30~

[I'he parameters cr, rand b can be equal to the valuesj

p- =

10,

b = 8/3,

r = 470/19::;::j 24.74 j

for these values the dynamical system has a strange attractor. The dynamical sys-j ~em, whIch IS defined by the Lorenz equatIOns, cannot be consIdered as a gradlentl ~ynamIcal system. The dIfferentIal 1-form ro = Fxdx + Fydy + Fzdz I~

kiro = -(z+ cr - r)dx /\ dy+ ydx /\dz+ 2xdy /\ dz. [t IS easy to see

(13.3Q

tha~

ID~Fx - D!Fy = z + cr - rJ ID1Fx - D!Fz = -y,1

[I'herefore (13.31) IS not a closed I-form. Note that the Lorenz system IS a dIssIpatIvg Isystem, sInce

for the Lorenz equations, condItIOns (13.11) can be satIsfied In the formj

(13.32~ V\s a result, the Lorenz system can be consIdered as a fractIOnal gradIent systeml IWIth the potentIal functIOnj (13.33 [rhe potentIal (13.33) umquely defines the system. USIng equatIOn (13.20), we Ob1 ~aIn the stationary states of the Lorenz system In the form of the equatIOn:1 (13.34]

13 Fractional Dynamical Systems

1304

Iwhere COO, cx, CY' Cz Cxy, Cxz, and CyZ' are the constants and a

= m = 2.1

OO:xampIe 4J [The Rossler system (Rossler, 1976; Neimark and Landa, 1992) is defined by Eqsj 1(13.29) wIth the forcesj

fx = -(y+z), ~t

Fz = 0.2+ (x-c)z.

Fy =x+0.2y,

(13.35~

is easy to see tha~

P~Fx - D~Fy =

-2j

ID;Fx-D;Fz = -l-zj ID1Fy - D~Fz

= oj

rI'herefore ()) - Fxdx + Fydy + Fzdz IS not a closed I-form. In general, the system is a generalized dissipative system]

Rossle~

rondItIOns (13.11) can be satIsfied m the form (13.32). As a result, the Rossler SYS1 ~em can be consIdered as a fractIonal gradIent system wIth a - 2 and the potentIall functIon:

(13.36~ [I'hIS potentIal umquely defines the Rossler system. The statIonary states of thel Rossler system are defined by Eq. (13.34), where the potential function is (13.36)j OO:xample 5j IWe can consIder the fractIOnal dIfferentIal equations (Podlubny, 1999) such thatl

Iwhere the forces are defined by (13.30) or (13.35). In this case, we have fractiona~ generahzatIOn of Lorenz and Rossler systems. These systems can be consIdered a~ ~ractIOnal-gradIentsystems of second order. We note that so-called fractIOnal umfied! Isystems (Deng and LI, 2008) also can be consIdered as fractIOnal-gradIent systems.1 IRemark. OC:et us note the mterestmg quahtatIve property of suffaces (13.34). The suffacesl pf the statIOnary states of the Lorenz and Rossler systems separate the three dI1 ImensIOnal Euchdean space into some number of areas. We have eIght areas for thel [.-orenz system, and four areas for the Rossler system (Tarasov, 2005b,a). These sepj larations have the interesting property for some values of parameters. All regions arel ~onnected with each other. Beginning movement from one of the areas, it is possibl~

113.4 Hamiltonian dynamical systems

3051

~o appear in any other area, not crossing a surface. Any two points from differentl lareascan be connected by a curve, which does not cross a surfacej

~3.4

Hamiltonian dynamical systems

[Hamiltonian dynamics is a reformulation of classical dynamics that was introducedl Iby WIlham Rowan HamIlton. The Hamlltoman approach (VIlasI, 2001) dIffers hoiTI] ~he LagrangIan approach m that mstead of expressmg second-order dIfferentIa~ ~quatIOns on n-dImenSIOnal coordmate space of dynamIcal system wIth n degree~ pf freedom, it expresses first-order equations on 2n-dimensional phase spacej !Letus consider the phase space ]R2n with the canonical coordinates (qI, ... , qn ,I IPI,... ,Pn). In general, dynamical system is described by the equationsj

~=

Gi (q,p),

([Pi

i

dt =F(q,p),

i=I, ... ,n.

(13.37~

for a closed system wIth potentIal mternal forces, we can descrIbe the motIon byl lusing Hamiltonian function H(q,p), which is the sum of the kinetic and potentiall ~nergy of the systemj i

= I, ... ,n.

(13.38

~n

general, we cannot descrIbe the motIon by usmg a umque functIOn. The defimtIOIlI pf HamIltoman systems can be reahzed m the foIIowmg form (Tarasov, 200Sc,b,1

rmr. ~efinition 13.6. Dynamical system (13.37) on the phase space ]R2n is called a lo~ ~aIIy

Hamlltoman system If] 173

= G'(q,P)dpi - F'(q,p)dqi

(13.39~

lis a closed I-form, df3 = OJ

pefinition 13.7. A dynamIcal system IS called a globally Hamlitoman system, I~ I-form (13.39) IS exact. A dynamIcal system IS caIIed a non-HamIltomanl Isystem if (13.39) is non-closed, i.e., df3 i OJ ~hfferentIaI

[n the canonical coordinates (q,p), the exterior derivative of the O-form H(q,p) be represented a~

~an

(13.40~ [Here and later we mean the sum on the repeated index i from I to nj

[Theorem 13.6. If the right-hand sides of Eqs. (13.37) satisfY the conditionsl

13 Fractional Dynamical Systems

1306 I G]-O ~IPi G' - D Pi ,

I p' D qi

DqIGJ +Dpl.P' = 0, I I

Dlp]-O qi -,

-

(13.4q

Ifor all (q,p) EWe JR2n, then the dynamical system (13.37) is a locally Hamiltonianl [SYstem In the regIOn W] IProof. Let us consIder the I-form (13.39). The extenor denvatIve of (13.39) IS wntj ~

Idf3 =d(G'dpi)-d(P'dqi)j [Then we obtainl

IOsmg the skew-symmetry of the wedge product /\, EquatIon (13.42) can be rewntj ~en

in the fonnj

[t is obvious that conditions (13.41) lead to the equation d{3

q

= O.

~quatlOns

(13.41) are called the Helmholtz condItIons (Helmholtz, 1886; Tarasov,1 2005c,b) for the phase space] [I'heglobally Hamlltoman system IS locally Hamlltoman. In general, the converst:j Istatementdoes not hold. If W C JR2n is simply connected, then a locally Hamiltoniaq Isystem IS globally Hamiitoman] ~997,

rtheorem 13.7. Dynamical system (13.37) on the phase space W C JR2n, is a glob-I rlly Hamiltonian system that is defined by the Hamiltonian H = H(q,p) iftheforml 1(13.39) is an exactjorm, 1{3 = dH,1 Iwhere H = H(q,p) is a continuous differentiable unique function on W, and W ~2n is simply connected.1

q

Iproof. Suppose that the form (13.39) is exact (/3 = dH) on W C JR2n, where H ~ IH(q,p) is a differentiable unique function on Wand We JR2n is simply connectedl

r:rn.en

- JH(q,P)d. JH(q,P)d. 1/3 - : lOPi P'+:loqi q,.

~quatlOns (13.39)

(13.43~

and (13.43) gIvel

Pi(q,p)= JH(q,p) :l

OPi

,

pi(

q,p

)

= _ JH(q,p) :l' oqi

(13.44~

113.5 Fractional generalization of Hamiltonian systems

3071

[f H = H(q,p) is a continuous differentiable function, then conditions (13.41) holdj land Eqs. (13.37) describe a globally Hamiltonian system. Substitution of (13.44~ linto (13.37) gives (13.38). As a result, the equations of motion are uniquely defined! Iby the Hamiltonian H = H (q, p ). t::j [£the exact differential I-form /3 is equal to zero (dH

= 0), then the equationl (13.45~

lH(q,p)-C=o,

Iwhere C IS a constant, defines the statIOnary states of globally Hamlltoman systeml 1(13.37).

113.5 Fractional generalization of Hamiltonian system§ fractIOnal generalIzatIOn of HamIltoman systems was suggested m (Tarasov, 200Sb)1 IWe can consider the fractional differentiall-forml (13.46~

Iwhere a > O. ThIS I-form IS a fractIOnal analogue of (13.39). Let us conSIder thel ~quatIOns of moholl] api

dt

=

i

F (q,p).

(13.47~

IWe can conSIder a fractIOnal derIVatIve WIth respect to tIme, such thatl

(13.48j [I'he fractIOnal generalIzatIOn of Hamlltoman systems can be defined by usmg fracj ~ional differential forms 1

~efinition 13.8. Dynamical systems (13.47) and (13.48) on the phase space ]R2n arel ~alled

fractIOnal locally HamIltoman systems, If (13.46) IS a closed fractIOnal forml

~Uf31,a

=

°

(13.49~

~or all (q,p) EWe ]R2n, where d a is the fractional exterior derivative.1 ~efinition 13.9. DynamIcal systems (13.47) and (13.48) are called fractIOnal glob-I lally Hamiltonian systems, if (13.46) is an exact fractional I-form. The system i§ ~alled a fractional non-Hamiltonian system if (13.46) is non-closed fractional formj Ii.e., dU/31,a i- 01

~n the canonical coordinates (q,p), the fractional exterior derivative d" for thel Iphase space ]R2n is defined b~

13 Fractional Dynamical Systems

1308

(13.50~ Iwhere we use the Caputo derIvatives of order a > O. For example, the fractiona~ ~xterior derivative of order a of l, with the initial points taken to be zero (a = b ~ 0), and n = 2, is given b)j

(13.5q [I'he Caputo derIvative gIvesl

lifk

>m-

1, and m - 1 <

a :::; m. If k = 1,2, ... ,m -

1, then]

rrhe fractional exterior derivative d U can be defined for the Riemann-Liouville derIvatives. In thIS case, we hav¢1

a k a r(k+ l)qk-a d q = [dq] r(k+l-a) I

a k [ ]a pq-a p = dq r(1- a)

a lp-a I r(1-a)1

+ [dp]

[jar(k+l)p-a r(k+ 1- a)

+ dp

Iwhere the mItial pomts taken to be zero (a - b - 0), and n - 2,1 ~et us consIder a fractIOnal generalIzatIOn of the Helmholtz condItIOns.1

rrheorem 13.8. If the right-hand sides oj Eqs. (13.47) satisfY the conditionsl (13.52~ ~DaGj+ C D a pi - 0 q'I b·I PtJ -, 1. ~

i

a pi D qi -

C D a pj - 0 Gi qi -,

(13.53~ (13.54~

Ithen dynamical systems (13.47) and (13.48) are fractIOnal locally HamiltOnian sysj reJiJX.

IProof. Let us consIder Eqs. (13.47) and (13.48). The correspondent fractIOnal dIfj ferential I -form iil ~l,a = G'(q,p) [dp;]U - P'(q,p) [dqi]u,

(13.55~

rrhe fractIOnal extenor derIvative of (13.55) IS grven by the relatIOnj

(13.56j ~quation

(13.56) can be represented a§

30~

113.5 Fractional generalization of Hamiltonian systems

IOsmgthe relation§ Idad = ;p~id[dqj]a + fp~P;[dpj]a j IdaF' = ~P~jFl[dqj]a + ~D~jFl[dpj]aJ Iwe obtain

IOsmgthe skew-symmetry of the wedge product, EquatIOn (13.57) can be rewnttenl lin the fom,. af31,a =

;p~Gj + fP~jF; [dq;]a!\ [dpj]

F~ (fp~pj - fp~p;) [dp;]a !\ [dPj]1 "2 (;P~jF; - ~D~Fj) [dq;]a!\

[dqj]a

~t is obvious that conditions (13.52), (13.53), (13.54) lead to the equation d a{31,a ~

p, i.e., {31,a is a closed fractional form.

q

IWe can define the Hamlltoman function for a specIal class of fractIOnal Hamlltoj Iman systems.1 rtheorem 13.9. Dynamical system (13.47) in region W of]R2n is a fractional glob-I l(111y Hamiltonian system with the Hamiltonian H = H(q,p), if(13.55) is an exactl I fractIOnal form such than {3I,a = daH, (13.58~ Iwhere H = H(q,p) is a continuous differentiable function on W, and We ]R2n isl connectedJ

~imply

IProof Suppose that the fractional form (13.55) is an exact fractional form, i.e.j {3I,a = daH. Using the fractional exterior derivative (13.50), we obtainl

(13.59~ ~quations

(13.55) and (13.59) givel

~i(q,p)

= ~D~iH,

rrhen Eqs. (13.47) can be wntten

Fi(q,p)

= -;P~Hj

a~

(13.60~

13 Fractional Dynamical Systems

1310

[hese equations describe the motion of fractional globally Hamiltonian systemsj D

[he fractional differential I-form !31,a for the fractional globally Hamiltonianl Isystem with Hamiltonian H can be written as !31,a = d U H. If the exact fractionall k1ifferentiall-form !31,a is equal to zero (daH = 0), then we can get the equation] ~hat defines the statIOnary states of the tractIOnal globally Hamiltoman system]

[Theorem 13.10. The stationary states ofthe fractional globally Hamiltonian systeml 1(13.60) are defined by the equation:1 m-l

~-1

IH(q,p)-

E

k]=O,l] =0

E

~

(13.6q

Ck] ...knlJ,..lnll(q;-a;/i(p;-b;)'i=O, ;=~

kn=O,ln= 0

Iwhere C k J...kn,l] ,...,In are constants and m IS the first whole number greater than orl Ifqual to a. OC=et us gIve the theorem for equatIons WIth the Rlemann-Lmoville denvatIves.

tractIona~

[Theorem 13.11. The stationary states of the fractional globally Hamiltonian sysj Item, whIch IS described by equatIOns WIth the Rlemann-Lluovllle derivatIves, ar~ fiefined by m

H(q,p) -ID(q; - a;)(p; - -

m1

m 1

nl

(13.62~

Iwhere Ck J...kn,l] ,...,In are constants and m IS the first whole number greater than orl Ifqual to a.

IProof.

ThIS propOSItIOn ISa corollary of propertIes of statIOnary states for gradIent systems.

fractIOna~

q

!Example 1~ OC=et us conSIder a tractIOnal dynamIcal system that IS defined by the equatIOnsj

0/

t =_cDaH 0 q ,

Iwhere q

> 0, p > 0, 0 < a

(13.63~

~ 1, and the HamiltonianH(q,p) has the formj

(13.64~ rrhe equations of motion arel

1136 Conclusion

3111

~

1

eft = mr(3 - a) p

2-a

,

dP dt

mill Z

= - r(3 - a)

q

2-a j

[These equations describe a non-Hamiltonian system that is a fractional Hamiltoj ~uan system. For a = 1, EquatIOns (13.63) WIth (13.64) defines the bnear harmomcl kJscIllator. USIngEq. (13.64), we obtaIn the follOWIng equatIOn for statIOnary state§

(13.65~ ~f

a = 1, then we get the equation H(q,p) = C, which describes the ellipsej

OO:xample 2J !Letus consider a dynamical system in phase space]Rz (n = 1) that is defined by thel ~quations with the Riemann-Liuoville derivativesj

(13.66~ Iwhere 0 < a ~ 1 and the Hamiltonian H(q,p) has the form (13.64). If a = 1, thenl ~qs. (13.66) describes the linear harmonic oscillator. If the exact fractional differ-I entIal1-form

lis equal to zero (dUH

= 0), then the equationl lH(q,p) -C1(q-a)(p-b)l a- 1 =

oj

Iwhere C IS a constant, defines the stationary states of the system (13.66). It Iwe get the usual stationary-state equation (13.45)j

a - 1J

[3.6 Conclusioril IUSIng the fractIOnal derIvatives and fractIOnal dIfferential forms, we conSIder fracj ~IOnal generabzatIOns of the notIOns of gradIent and HamIltoman systems. In genj ~ra1, the fractIOnal HamIltoman/gradient systems cannot be conSIdered as HamIl1 ~oman/gradient systems. The class of fractIOnal HamIltoman/gradient systems I~ IWIder than the usual class of HamIltoman/gradient dynamIcal systems. The HamIlj ~oman/gradient systems form a speCIal case of fractIOnal HamIltoman/gradient sysj ~ems. We note that the fractIOnal gradIent systems lead us to a pOSSIble extenSIOn ofj ~he theory of catastrophe and bIfurcatIOns (GIlmor, 1981). It IS pOSSIble to general-I IIze the theory for derIvatives of non-Integer order. The notIOn of fractIOnal gradientl Isystem allows us to formulate the theory of bIfurcatIOn of fractIOnal globally poten-I

1312

13 Fractional Dynamical Systems

~ial

vector fields. At this moment construction of the consistent theory of fractionall and bifurcations is not realised yetj IWe can assume that the ways of some chemical reactions with dissipation and! Isystems with determmistic chaos can be considered by the analysis of generahze~ Ipotential surfaces for fractional dynamical systems. Let us note the mterestmg prop-I ~rty of potential surfaces for systems with strange attractors. The surfaces of the sta-j ~ionary states of the Lorenz and Rossler equations separate the 3-dimensional Eu-j ~hdean space mto some number of areas (Tarasov, 2005b). We have eight areas fo~ ~he Lorenz equations and four areas for the Rossler equations. ThiS separation ha§ ~he interesting property: all regions are connected with each other (Tarasov, 2005b)j ~egmmng movement from one of the areas, it is possible to appear m any othe~ larea, not crossing a surface. Any two points from different areas can be connected! Iby a curve, which does not cross a surface.1 IOsmgthe notion of fractional gradient system, we can study a Wideclass of deterj Immistic dynamical systems with regular and strange attractors (Amschenko, 1990j W'Jeimark and Landa, 1992). Quantum analogs of fractional denvatives (TarasovJ 12008a), allow us to consider a generahzatiOn of the notion of fractiOnal Hamiltomaril Isystem. In thiS case, a Wide class of quantum non-Hamiltoman systems (TarasovJ 12008b) can be considered as fractiOnal Hamiltoman systems.1 ~atastrophe

Referencesl IY.S. Amschenko, 1990, Complex OSCIllatIOns tn SImple Systems, Nauka, Moscowj lIn Russian] IK. Cottnll-Shepherd, M. Naber, 2001a, Fractional differential forms, Journal of] lMathematical Physics, 42, 2203-2212; and E-pnnt: math-ph/0301013j IK. Cottnll-Shepherd, M. Naber, 20mb, FractiOnal dtfferential forms II, E-pnntj Imath-ph/03010 16.1 IW.H. Deng, c.P. Li, 2008, The evolutiOn of chaotic dynamics for fractiOnal umfie~ Isystem, Physics Letters A, 372, 401-407 j ~.A. Dubrovm, A.N. Fomenko, S.P. Novikov, 1992, Modern Geometry - Metho(Jj, landApplications, Part I, Spnnger, New York.1 ~. Gtlmor, 1981, Catastrophe theory jor Scientists and Engineers, Wiley, New Yorkj ISectiOn 14] ~. Godbillon, 1969, Geometrie Dif{erentielle et Mecanique Analytique Hennannj IE:.i.iJ.S: IH. Helmholtz, 1886, Journaljur die Reine und Angewandte Mathematik, 10, 137-1

rr:nn: M. Hirsh,

S. Smale, 1984, DIfferential EquatIOns, DynamIcal SYstems and Lmearl IAlgebra, AcademiC Press, New York.1 k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj kwnal Dijjerentlal EquatIOns ElseVier, AmsterdamJ

IR eferences ~.N.

313

Lorenz, 1963, DetermmIstIc nonpenodIc flow, Journal of the Atmospheric Scij 20, 130-14U IYu.I. Neimark, P.S. Landa, 1992, Stochastic and Chaotic Oscillations, Kluwer Aca-l demIc, Dordrecht and BostonJ IK.E. Oldham, 1. Spamer, 1974, The FractIOnal Calculus: Theory and Applzcatwn~ pi Differentiation and Integration to Arbitrary Order, Academic Press, New York] ~. Podlubny, 1999, Fractional Differential Equations, Academic Press, New Yorkj p.E. Rossler, 1976, An equatIOn for contmuous chaos, Physics Letters A, 57, 397-1 I39R: IS.G. Samko, A.A. KIlbas, 0.1. Manchev, 1993, Integrals and Denvatlves of Fracj kwnal Order and Applzcatwns, Nauka I Tehmka, Mmsk, 1987, m Russlanj !Ind Fractional Integrals and Derivatives Theory and Applications, Gordon and! Breach, New York, 1993j ~. Sparrow, 1982, The Lorenz EquatIOns: Bzjurcatwn, Chaos, and Strange Attracj kors, Spnnger, New York.1 IVE. Tarasov, 1997, Quantum dISSIpatIve systems: III. DefimtIon and algebraIC strucj Iture, Theoretical and Mathematical Physics, 110, 57-67.1 IVE. Tarasov, 2005a, FractIOnal generalIzatIOn of gradIent systems, Letters in Mathj ~matical Physics, 73, 49-58J ~E. Tarasov, 2005b, Fractional generalization of gradient and Hamiltonian systemsJ IJournalofPhysics A, 38, 5929-5943J IVE. Tarasov, 2005c, Phase-space metnc for non-HamIltoman systems, Journal oj1 IPhysics A, 38, 2145-2155J IY.E. Tarasov, 2008a, Weyl quantization of fractIOnal denvatIves, Journal oj Mathej matical Physics, 49(10), 102112.1 IV.E. Tarasov, 2008b, Quantum Mechanics of Non-HamiltOnian and DlSSlpatlve Sysj ~ems, ElseVIer, AmsterdamJ p. VIlasI, 2001, Hamiltonian Dynamics, World SCIentIfic PublIshmg, SmgaporeJ ~nces,

~hapter

141

[Fractional Calculus of Variations in

Dynamic~

114.1 Introductionl ~n mathematIcs and theoretIcal phYSICS, vanatIOnal (functIOnal) denvatIve IS a gen1 eralization of usual derivative that arises in the calculus of variations In a variationl 1mstead of dIflerentIatmg a functIOn WIth respect to a vanable, one dIfferentIates ~ ~unctIonal WIth respect to a functIOn. Usmg the tractIOnal calculus, we conSIder ij ~ractIOnal generalIzatIon of vanatIOnal (functIOnal) denvatIvesJ IWe define a tractIOnal generalIzatIon of an extenor denvatIve for vanatIOnal caIj ~ulus (Tarasov, 2006). The HamIlton and Lagrange approaches are conSIdered. Wfj ~enve the HamIlton and Euler-Lagrange equatIOns WIth denvatIves of non-mtege~ prder. FractIonal equatIons of motIon are obtamed by tractIonal vanatIon of Laj granglan and HamIltoman that have only mteger denvatIves. Usmg the vanatIon ofj OCractIOnal order, we conSIder a generalIzatIOn of stabIlIty notIon (Tarasov, 2007)J ~n SectIOns 14.2-14.3, we define the tractIOnal vanatIOns m HamIlton's approachl ~o descnbe the motIon of claSSIcal systems. The tractIOnal generalIzatIOn of statIOn1 lary actIOn pnncIple IS suggested. In SectIOns 14.4-14.5, we dISCUSS the tractIOna~ IvanatIOns m Lagrange's approach, and the fractIOnal generalIzatIOn of correspondj lingstationary action principle. In Sections 14.6-14.7, we consider the generalizationl pf actIOn pnncIple of non-Hamlltoman systems. The tractIOnal equatIOns of motIOili IWIth fnctIOn are presented. In SectIOn 14.8, we dISCUSS the stabIlIty WIth respect tq IvanatIOn of non-integer order. Fmally, a short conclUSIOn IS given in SectIOn 14.9.1

~4.2

Hamilton's equations and variations of integer orderi

!Let us consider Hamiltonian systems in the extended phase space M 2n + 1 = jR 1 d ~n x jRn of coordinates (t,q,p). The motion of systems is defined by stationaryl Istates of the action functionali

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

14 Fractional Calculus of Variations in Dynamics

1316

~[q,p] =

I

(p;Diq; -H(t,q,p))dt,

(14·1)1

Iwhere H tS a HamtItoman of the system, and both q and p are assumed to be mj klependent functions of time. In classtcal mechamcs, the trajectory of an oliject t§ klerivedby finding the path for which the action integral (14.1) is stationary (a minj limum or a saddle point).1 ~n Hamilton's approach the action functional (14.1) can be written asl

IS[q,p]

=

I

WH,

Iwhere the differential I-forml

'rJH = p;dq; -

Hdt,

(14.3)1

Iwhich is called the Poincare-Cattan l-fonn or the action l-fonn Here and later wei rIean the sum on the repeated mdex i from 1 to nl [I'he Pomcare-Cartan I-form looks Itke the mtegrand of the action or the Laj grangian. However it is a differential form on the extended phase space M 2n +\ ofj I(t,q,p), not a function. Once we integrate it over a curve C in M 2n +\ , we get thel ~

[he integration is taken from A to B in the extended phase space M 2n +\ J OC:et us mtegrate from A to B along two sltghtly dtfferent paths and take the dtfj ~erence to get a close loop mtegral. To evaluate thts mtegral we can use Stokes 1 ~heorem (Dubrovm et aI., 1992). In the language of dtfferenttal forms, the Stokesl ~heorem is

Iwhere W is an m-dimensional compact orientable manifold with boundary aw and! ~ is an (m - 1)-form. We note that W can be a submanifold of a larger space, so tha~ ~he Stokes' theorem actually tmpltes a whole set of relatiOns mcludmg the famtlta~ pauss and Stokes laws of ordmary vector caIculusl IUsmg Eq. (14.5) to the dtfference of actiOns computed along two netghbonngj Ipaths with (q, t) fixed at the endpoints, we obtainl

Iwhere L denotes the sufface area in the extended phase space bounded by the twq Ipathsfrom A to Bl [The principle of stationary action states that (5S = 0 for small variations abou~ ~he true path with (q, t) fixed at the end points. This will be true for arbitrary smal] Ivariations,if and only if dWH = 0 for the tangent vector along the extremal path.1

114.3 Fractional variations and Hamilton's equations

317]

IWe can consider the exterior derivative of Poincare-Cartan l-form, and derivd equations of motion from the condition dOJH = O. Using this condition, we ge~ ~he Hamilton's equations of motion. This condition is equivalent to the stationaryl laction principle 8Slq,pl = OJ ~he

[Theorem 14.1. The exterior derivative of the Poincare-Cartan I-form (14.3) is dej I

fined by the equatlOn.j

(14.7)1 IProof The exterior derivative of the form (14.3) can be calculated from the equaj tion: IdOJH = d(pidqi) - d(Hdt)

EDi Pidt I\dqi +D~;PidqjI\dqi +D1;PidPj I\dqil t=DiHdt I\dt - D~iHdqi I\dt - D1iHdpi I\dt.

(14.8)1

psmg dt 1\ dt = 0, dPi I\dt = -dt 1\ dPi, andl

q

Iwe obtam (14.7).

rrheorem 14.2. The trajectory oj a Hamiltonian system can be derived by findinifl Ithe pathjor which the Poincare-Cartan I-jorm OJH is closed, i.e.,1

(14.9)1 [I'hIS statement IS a statIOnary actIon prIncIple m HamIlton's approach. Usmg thel IprInCIple (14.9), we get the equatIOns of motIOn:1

(14.1O~ k'\s a result, we obtaml

W=DIH dt

Pi'

(14.1l~

IwhIch are the well-known HamIlton's equatIOnsl

~4.3

Fractional variations and Hamilton's equations

~n order to generalIze the action prIncIple tor fractIOnal case, we define a fractIOna~ [poincare-Cartan I-form. The fractional generalization of the I-form (14.3) can bel klefinedby

1318

14 Fractional Calculus of Variations in Dynamics

(14.12~

[I'he tractiOnal dIfferentIal I -form (f)H,a wIll be called ajractlOnal Pozncare-Cartanl ~ -form orfractional action I-form. We can consider the fractional exterior derivativ~ pfthe form (14.12), and use dU(f)H,a = Oto obtain the fractional equations ofmotionJ [Thereexists the following important statementj

[Theorem 14.3. The fractional exterior derivative ofthe fractional form (14.12)

i~

~a(f)H,a = (~Df Pi + ~p~H) [dtja 1\ [dq;] a - (fp~iPi[dqi]a - ~D~iH[dt]a) 1\ [dp;] a j [I4.T3]

IProof The fractional exterior derivative d U of the fractional I-form (14.12) give§ ~a(f)H,a =

da(pMqi]a) - da(H[dtja)

F dUPi /\ [dq;]U -

dUH /\ [dtjuJ

IOsmgthe followmg fractiOnal extenor denvatIves of O-formsj

Iwe obtam

Ida(f)H,a

=

(~Df Pi)[dt]a /\ [dqija + (;P~iPi)[dqj]a /\ [dq;]al

1+ (~P~jPi)[dpj]a

/\ [dq;]a - (~DfH)[dt]a /\ [dtjal

~ (~D~H)[dq;]a /\ [dtja - (fp~iH)[dp;]a /\ [dt]a.

(l4.14~

IOsmgthe property of the wedge productj

landthe propertIes of the Caputo denvatIvesj

Iwe rewnte Eg. (14.14) as (14.13).

q

IRemark. IWe note that the RIemann-LiOuvIlle tractiOnal denvatIve leads us to dependence o~ independent coordinates D~Pi = PiD~ 1 #- O. Therefore the fractional equations are ore com Icate or t e RIemann-LiOuvI e envatIves

114.4 Lagrange's equations and variations of integer order

31S1

rrheorem 14.4. The trajectory oj a Hamiltonian dynamical system can be derived b)j lfinding the path for which the fractional Poincare-Cartan I-form WH,a is fractionall closedform; i.e.j (14.15j ~a WH,a = O. [Here, we consider only fractional Hamiltonian systemsj [I'hIS theorem represents a tractIonal actIOn prIncIple m HamIlton's approach] Rsing (14.13) and (14.15) with the initial points taken to be zero, andj

Iwe obtain

(14.16~ Iwhere I

-

1,... ,n. These equatIons are the tractIOnal generahzatIon of HamIlton'sl

~quatIOns.

lEarthe fractional Poincare-Cattan I-forml

(14.17~ Iwhere k

~

m, kEN, m -1

< a < m, Equations (14.16)

ar~

CDa k cDaH o tPi=-O qj •

(14.18

W'Jote that we cannot use the rule of dIfferentIatmg a compOSItefunctIOn for tractIona~ ~erivative ~Df Therefore equations (14.18) with k i=- 1 are more complicated thanl ~q. (14.16).

p7.

~4.4

Lagrange's equations and variations of integer order]

[LetL(t,q, v) be a Lagrangian of dynamical system, where qi, i = 1, ... ,n, are coordi-I r.ates, and Vi, I - 1, .., n, are the velOCItIes. We conSIder the varIatIOnal problem fofj me action functIOnall

J

(14.19~

ID!qi = Vi,

(14.20j

~O[q, v] =

L(t,q, v)dt,

IWIth the addItIOnal condItIOnsl Iwhere both qi and Vi are assumed to be mdependent functIOns of tIme. In thIS casel Ipi, i = 1, ... , n, play the role of mdependent Lagrange multIphers. ObVIOusly, thel lindicated variational problem is equivalent to the problem on the extremum of thel laction,

1320

14 Fractional Calculus of Variations in Dynamics

(L(t,q, v) + p;(Diq; - v;))dt,

(14.21

Iwhere already all the variables q, v, p have to be varied. The corresponding Laj [grange's equations ar~

(14.22~ IWe can introduce the extended Hamiltonian in the space of variables (t,q,p, v) IH*(t,q,p, v) = P;V; -L(t,q, v).

a~

(14.23~

[The corresponding extended Poincare-Cattan I-form i§

(14.24] [I'he extenor denvatIve of thIS form IS descnbed by the followmg theoremJ [Theorem 14.5. The exterior derivative of the form (14.24) is defined by the equa-I Itton:

IProof. The extenor denvatIve of (14.24) I§ (14.26~ ~quatlOn

(14.26) can be represented a~ IdWH* = Di p;dt /\ dq; + D~jP;dqj /\ dq~ I+DbiP;dpj /\dq; +D~iP;dvj /\dq; - DiH*dt /\dtl

(14.27~ IOsmg at /\ at = 0, aq; /\ at = -at /\ aq;,

an~

D~p; =0, I ~quatlOn

(14.27)

gIve~

IdWH* = (Di P;+D~iH*)dt r.dq, - idq, -DbiH*dt) r.dp, -D~iH*dv;r.dt . (14.28~

from (14.23), we

ge~

32~

114.5 Fractional variations and Lagrange's equations

q

k'\s a result, we obtam (14.25).

[Theorem 14.6. The trajectory ofa Lagrangian dynamical system can be derived b)j lfinding the path for which the form (14.24) is closed, i.e.,1 (14.29) [This theorem is the stationary action principle in Lagrange's form. From Eqsj 1(14.25) and (14.29), we getl

(14.30~ [t IS easy to see that Eqs. (14.30) comcIdes wIth the Lagrange's equatIons (14.22) ~hat can be represented asl

(14.3q k'\s a result, we obtaml 1 j L=O, D q1 L - D! DD ,qt

I

i= 1, ... ,n.

I

(14.32~

[:\quatIOns (14.32) are the Euler-Lagrange equatIons for LagrangIan systemJ

114.5 Fractional variations and Lagrange's equation~ [Let L( t, q, v) be a Lagrangian of the system, and the extended Hamiltonian is defined!

!bY IH*(t,q,p, v)

= Pi!(Vi) -

L(t,q, v),

(14.33~

Iwhere f(vi) is a function of Vi. Let us defin~

(14.34] IwhIch IS a tractIOnal generalIzatIOn of the extended Pomcare-Caftan I-form (14.24)j

[theorem 14.7. The fractional exterior derivative of the fractional I-form ( 14.34) lis defined by

~aroH*a

= (5Dfpi -

~P~L) [dt]a A [dqi]~

1- ~P~iPi([dqda -

f(vi)[dt]a) A [dpd1

Hpi~D~f(vi) -~ D~L)[dvi]a A [dt]a.

IProof.

The tractIOnal extenor denvatIve of (14.34) gIve~

(14.35~

1322

14 Fractional Calculus of Variations in Dynamics

[Usingthe relationsj

Iwe obtain

~aWH*a = (~D~ p;)[dt]a 1\ [dq;]a + (~P~iP;)[dqj]a 1\ [dq;]al

1+ ( ~ Dpa p; ) [dpj ] a 1\ [dq;]a + (~D~p;)[dvj]a 1\ [dq;]dIJ I

J

J

J

H~D~H*)[dt]a 1\ [dt]a - (~P~H*)[dq;]a 1\ [dt]al

H~D~iH*)[dp;]a 1\ [dt]a - (gD~H*)[dv;]a 1\ [dt]a.

[Using [dt]a 1\ [dt]a = 0, [dq;]a 1\ [dt]a = -[dt]a 1\ [dq;]a,

(14.36~

an~

Iwe can rewrite (14.36) a~

~aWH*a = (~D~p;+ ~D~H*)[dt]a 1\ [dq;]al

H~P~;p;[dqj]a - ~D~iH*)[dt]a) 1\ [dp;]J (14.37j IOsmg the extended Hamiltoman (14.33) and propertIes of the Caputo denvatIves, we hav~

tractIona~

~D~iH* = ~D~i(Pjf(vj) - L) = f(vj)~D~iPj - ~D~iL(t,q, v) = f(v;)~p~iPd gD~H* ~s

=

gD~(pjf(vj) -L)

a result, we obtain (14.35).

= pjgD~f(vj) -

gD~Lj

q

rrheorem 14.8. The trajectory ojjractional Lagrangian systems can he derived b)j lfinding the path for which the fractional extended Poincare-Cartan I-form (14.34] liS closed form, I.e.] (14.38) [I'hIS theorem IS the tractIOnal action pnncIple in Lagrange's form (TarasovJ 12006). Using (14.35) and (14.38), we obtainl

32~

114.6 Helmholtz conditions and non-Lagrangian equations

Ip;~;D~f(v;) - ~;D~L =

OJ

~f f(v;) = vf, v; > c, = 0, then the relation:1

(14.39~ Iwhere f3

> m - I, and m - I < a

~

m, gives

(14.40~

(14.41~

(14.42~ Substituting (14.42) into (14.40), we obtairr

(14.43 [t is easy to see that Eq. (14.43) looks unusually even for /3 1/3 = a for the Hamiltonian (14.33), and the I-form (14.34)J ~f f(v;) = v?, v; > c; = 0, we can usel

=

1. Therefore we usel

k\s a result, we obtam the fractIOnal extended Lagrange's equattonsj

(14.44~ ISubstItutmg the thIrd equation from (14.44) into the first one, we obtaml

(14.45~ ~hat

ISthe fractIOnal Euler-Lagrange equations. For equations (14.32)1

a - 1, EquatIOns (14.45) are thel

~uler-Lagrange

114.6 Helmholtz conditions and non-Lagrangian equationsl [t is well-known that the Helmholtz conditions (Helmholtz, 1886; Fillipov et al.j ~992; Tarasov, 2008) are necessary and sufficient for equations to be the Eulerj ~agrange equations that can be derIved from stationary action prIncIple.1

[Theorem 14.9. The necessary and sufficient conditions fori

1324

14 Fractional Calculus of Variations in Dynamics

IE.( . ... ,q (N)) t t,q,q,

° ,

i=l, ... ,n

Ito be equatIOns that can be obtained from the statIOnary actIOn prinCiple

(14.46~ ar~

(14.47

dEi dq(m) h (k) wereqi

I

= k~ (-1) N

k(k) (d )k-m ( dEj ) m dt dq~k)'

m

= 1,...,N,

(14.48

k .. 1, ... ,nan~ = D tqi,I,J=

(14.49~ q

IProof. ThIs proposItIon was proved in (FI1hpov et al., 1992). ~xample.

[f we consIder the equatIons:1 IEi(t,q,q) =0, ~hen

i=l, .. ,nJ

condItIons (14.47) and (14.48) have the formj

(14.50~ _J-O ~ JE· - , :\. +:\. aqj o q, t

~quations

(14.50)

i,j

= 1, ... ,n.

giv~

i,j,K:= 1, ...

,nj

rthese conditions are satisfied for the linear dependence E, with respect to D}q 1 [ThIS theorem has the followmg corollanesl

rrheorem 14.10. The necessary and sufficient conditions to derive the equations:1 [Ei(t,q,i]) = Aij(t,q)i/j +Bi(t,q) = 0,

i = 1, ... ,n.

(14.52~

Ijrom the stationary action principle have the jorm.j IAij=-A ji,

(14.53]

114.6 Helmholtz conditions and non-Lagrangian equations

~ sx; __ dAk'' =0 l] +_]_+ dqk

dqi

dt

dB· dq;

dq;

I]f'] __'+_]

3251

,

dB· =0 dqi .

(14.54~ (14.55~

IProof. SubstitutIOn of Eg. (14.52) mto Egs. (14.50) and (14.51) gIVes (14.53),1 1(14.54), and (14.55)~ D

[Theorem 14.11. The necessary and sufficient conditions fori 'f£i(t,q,q,ij) = 0,

i = 1, ... ,nl

Ito be equations that can be derived from stationary action principle ar~

(14.58

Rsing Eg. (14.57), we can rewrite condition (14.58) in the more symmetric form:1

[.-etus give a definition of non-Lagrangian system (Tarasov, 2006)j pefinition 14.1. A dynamIcal system IS called non-LagrangIan system If the eguaj ~IOns of motion (14.46) cannot be represented m the form:1

Iwith some function L = L(t,q,q, ... ,q(N)), where q(k) = D~qj ~t

IS weB-known that the equations of second order cannot be represented a~

~n the general case, the Lagrange's equations have the additional term Qi(t,q,q)j Iwhich is a generalized non-potential force. This force cannot be represented a~

1326

14 Fractional Calculus of Variations in Dynamics

[or some function U Istem, 2002) ar~

= U(t,q,q). In general, the Euler-Lagrange equations (Goldj

~f we consider non-potential forces and non-Lagrangian systems, then the non-I Iholonomic vanational equation suggested by Sedov (Sedov and Tsypkm, 1989; Sej k!ov, 1968, 1965, 1997) should be used instead of stationary action principlej

114.7 Fractional variations and non- Hamiltonian system~ [n general, the phase space equations of motion cannot be represented m the formj

(l4.60~ Iwhere H = H (t, q, p) is a smooth function. The Hamilton's equations are written a§

(l4.61~ Iwhere H = H(t,q,p) is a Hamiltonian of the system. The functions G'(t,q,p) and!

IF' (t, q, p) describe the non-potential forces, which act on the system. For mechani-I Fal systems, we can considerG'(t,q,p) = O. If the functions G'(t,q,p) andF'(t,q,p) ~o not satisfy the Helmholtz conditiOns:1

(l4.62~ then (14.61) is a non-Hamiltonian system (Tarasov, 2008)j

[n general, the extenor denvative of the Pomcare-Cartan l-fonn is not equal t9 Izero (dWH i 0). This derivative is equal to differential 2-fonn e that is defined byl Inon-potentialforcesj ~ = F'(t,q,p)dt I\dqi - G'(t,q,p)dt I\dPi

(l4.63~

~or the non-Hamiltonian system (14.61). For example, the linear friction force F i ~ YPi gives 1

(14.64) [Theorem 14.12. The differential 2-form I

e of non-potential forces

is non-closecA

form. IProof If differential 2-fonn e is a closed fonn (de = 0) on a contractible openl Isubset W of ]R2n, then the fonn is the exact fonn such that a function h = h(t, q, P j ~XiStS,

and

e- dh. In this case, we have a new Pomcare-Cartan l-fonnl

114.7 Fractional variations and non-Hamiltonian systems

3271

such that dol = 0, and the system is Hamiltonian. [..:et us gIve a generalIzatIon of statIOnary actIon prInCIple for the systems wIthl Inon-potentIal forces]

rrheorem 14.13. The trajectory oj a non-Hamiltonian system can be derived by find-I ling the path for which the exterior derivative of the action I-form (14.3) is equal tg Ithe non-closed 2-form (14.63), i.e.J

(14.65] [I'hIS theorem IS the actIon prInCIple for non-Hamlltoman systems. EquatIon~ 1(14.7), (14.63) and (14.65) gIVethe equatIOns of motIon (14.61) for non-HamIltomanl ~

[..:et us define a tractIOnal generalIzatIOn of the form (14.63) byl (14.66~

[I'hIS form allows us to derIve tractIonal equatIons of motIon for non-HamIltomanl Isystems.

rrheorem 14.14. The trajectory oj a jractional system subjected by non-potentiall I

forces can be denved by finding the path for which the fractIOnal extenor denvat/vel 'Pf the fractional action I-form (14.12) is equal to non-closedfractionaI2-for~ 1(14.66), i.e., (14.67j [ThIS statement can be conSIdered as a tractIOnal action prInCIple for non"1 Hamiltonian systems (Tarasov, 2006). Using (14.13), (14.66) and (14.67), we

ge~

k'\s a result, we obtaml

(14.68~ (14.69~ [I'hese equatIOns can be conSIdered as a fractIOnal generalIzatIOn of equatIOns motion for non-Hamiltonian systems (Tarasov, 2006)j psmg the relatIOnj

Iwhere q; > a; = 0, and considering dUq; as a fractional differential of q; = q;(t):1

o~

1328

14 Fractional Calculus of Variations in Dynamics

Iwe can assume

tha~

[n thIS case, EquatIOns (14.68) gIVel

Daq.(t)=qI-apa-IcDaH+ t

Iwhere i

~4.8

I

I

I

0

Pi

1 T 2-

a d(tqp) ",

(14.70

= 1, ... , n. These equations are fractional differential equation for qi(t ).1

Fractional stabilitj]

fractional integrals and derivatives are used for stability problems (see, for example] I(Momani and Hadid, 2004; Hadid and Alshamani, 1986; Chen and Moore, 2002j IKhusainov, 2001; Matignon, 1996; Li et al., 2009». In this section, we formulatcj IstabIlIty WIthrespect to motIon changes at tractIonal changes of vanables. Note tha~ ~ynamIcal systems, whIch are unstable "Ill sense of Lyapunov", can be stable wIthl Irespect to fractIOnal vanatIOns.1 [.:et us conSIder a dynamIcal system that IS descnbed by the dIfferentIal equatIOnsj

~!xi=F;(x),

(14.71j

i=l, ... ,n,

Iwhere Xl, ... , x., are real vanabies that define the state of the systemJ IWe can consider variations OXi of the variables Xi. The unperturbed motion i~ Isatisfied to zero value of the variations, OXi = O. The variations OXi describe a~ [unction f(x) at arguments Xi change varies] [.:et us conSIder the case n = I. The first vanatlon descnbes a functIon change a~ ~he first power of argument change:1

~f(x)

= oXD~f(x).

(14.72~

[The second vanation descnbes a functIOn change at the second power of ~hange:

argumen~

(14.73~

[he variation on of integer order n is defined by the derivative of integer orderl ID~f(x), such thatl

Ion f(x) = (oxt D~f(x).1 IWe can define (Tarasov, 2006) a vanation of tractIOnal order m - I


~

m by thel

~quatIOn:

(14.74~

114.8 Fractional stability

32S1

land 5D~ is the Caputo fractional derivative (Samko et al., 1993; Kilbas et al., 2006)1 Iwith respect to x. The fractional variation of order a describes the function chang~ lat change of the fractional power of argument. The variation of fractional order i§ klefined by the denvattve of tracttonal order] OC--et us obtain equations for fractional variations 8aXi. We consider the fractiona~ Ivariation ofEq. (14.71) in the formj ~aDixi

= 8 aF;(x), i= 1, ... ,n.

[Usingthe definition of fractional variation (14.74), we

~aF;(x) = ~D~;F;(x)[8xj]a,

(14.75~

hav~

i,j= 1, ... .n.

(14.76~

from Eq. (14.76), and the property of variationj

(14.77~ Iwe

obtain

(14.78~ !Note that in the left hand side of Eq. (14.78), we have fractional variation of oUx;j landin the right hand side-the fractional power of variation lox; IU .1 OC:et us constder the tractIOnal vanatlon of the van able Xi. Usmg Eq. (14.74), wei bbtam (14.79~ for the Caputo tractIOnal denvattve, we

hav~

c Gj DaxjXi = 8ij C a;

I

o:Xi , Xi

(14.80~

Iwhere O;j is the Kronecker symbol. Substituting Eq. (14.80) into Eq. (14.79), wei ~an express the fractional power of variation [ox;]U through the fractional variatioi1l ~ f8Xj]a = (~D~xj) 18axj. (14.8q ~

J

J

ISubstItutIOn of (14.81) into (14.78) gIvesl

(14.82~ Wlere we mean the sum on the repeated mdex i from 1 to n. EquatIOns (14.82) arel ~quations for fractional variations. Let us denote the fractional variations oU Xi byl

(14.83~ V\s a result, we obtam the dIfterenttal equation for tractIOnal vanatIOns:1

(14.84~

1330

14 Fractional Calculus of Variations in Dynamics

[Using the matrix zt = (ZI, ...,Zn), andA a = IIAij(a)ll, we can rewrite Eq. (14.84] lin the matrix fonnl (14.86~ k?iZ(t) =AaZ(t). ~quation (14.86) ~o

is a linear differential equation. To define the stability with fractional variations, we consider the characteristic equationj pet(A a - AE) = 0

respec~

(14.87]

Iwith respect to A.. If the real part Re[A.kJ of all eigenvalues A.k for the matrix Ad lare negatIve, then the unperturbed motIon IS asymptotIcally stable WIth respect t9 [ractional variations. If the real part RelAkl of one of the eigenvalues Ak of the matrixl IA a is positive, then the unperturbed motion is unstable with respect to fractiona~ Ivariations V\ system is said to be stable with respect to fractional variations if for every e~ ~here is a value (50 such thatl (14.88] ~or all t > to, where x( a,t) describes a state of the system at t ? to. The dynamica~ Isystemis called asymptotically stable with respect to fractional variations x( t, a] liT (14.89~

oa

IWe note that the notIon of stabIlIty WIth respect to tractIOnal vanatIons (Tarasov J 12007) is wider than the usual Lyapunov or asymptotic stability (Malkin, 1959j pemIdovIch, 1967; Tchetaev, 1990). FractIOnal stabIlIty mcludes concept of "m1 ~eger" stabIlIty as a specIal case (a - 1). A dynamIcal system, whIch IS unstablel IWIth respect to first vanatIOn of states, can be stable WIth respect to tractIOnal van1 latIOn. Therefore tractIOnal denvatIves expand our pOSSIbIlIty to study propertIes ofj k1ynamlcal systemsJ V\s a result, the notIon of tractIOnal vanatIOns allows us to define a stabIlIty o~ Inon-mteger order. FractIOnal vanatIOnal denvatIves are suggested to descnbe thel IpropertIes of dynamIcal systems at tractIOnal perturbatIOns. We formulate stabIlItyl IWlth respect to motIon changes at fractIOnal changes of vanables. Note that dynam-I Ilcal systems, whIch are unstable "m sense of Lyapunov", can be stable WIth respec~ [0 fractIOnal vanatIOns.1

[4.9 Conclusioril rrhe tractIOnal extenor denvatIves can be used to conSIder a tractIOnal generalIza-1 ~IOn of vanatIOnal calculus (Tarasov, 2006). The HamIltOnIan and LagrangIan ap1 Iproaches WIth tractIOnal variations are conSIdered. HamIlton's and Lagrange's equa1

IR eferences

3311

~ions with fractional derivatives are derived from the stationary action principlesl I(Tarasov, 2006) by fractIOnal VarIatIOns. We prove that fractIOnal equatIOns can bel klerivedfrom actions, which have only integer derivatives. Derivatives of non-intege~ prder appear by the fractional vanation of LagrangIan and HamIltoman.1 k\pplIcatIOnof fractional vanatIOnal calculus can be connected WIth a generalIza-1 ~ion of variational problems. The gradient systems form a restricted class of ordinar)j klifferential equations. Equations for gradient systems can be defined by one func-I ~ion that is called potential. Therefore the study of these systems can be reduced tq Iresearch of potentIal. As a phySIcal example, the ways of some chemIcal reactIOnsl lare defined from the analysIs of potential energy suffaces (Levme and BernstemJ [974; FukUI, 1970, 1981; MIller et aI., 1980). The fractIOnal gradIent systems wer~ Isuggested m (Tarasov, 2005a,b). It was proved that gradIent systems are a specIal1 ~ase of such systems. A set of fractional gradient systems includes a wide class ofj Inon-gradlent systems. For example, the Lorenz and Rossler equatIOns can be conj Isidered as generalized gradient systems (Tarasov, 2005a,b). Therefore the study ofj ~he non-gradIent system, whIch are fractional gradIent systems, can be reduced tg Iresearch of potentIal.1 IOsmg the fractIOnal extenor calculus and the notion of fractIOnal vanatIOna~ ~envative, we can generalIze the extenor vanatIOnal calculus (Aldrovandl and! IKraenkel, 1988; Olver, 1986). We note that the fractional variational (functional) klerivatives can have wide applications in statistical mechanics (Bogoliubov, 1960j ~970, 1991; VasIlev, 1998), quantum field theory (BogolIubov, 1995; BogolIubo\j land ShIrkov, 1980; Ryder, 1985), and stochastic processes (Klyachkm, 1980). ~ rote that the generatmg functIOnal (for example, m quantum theory (BogolIubov and! IShirkov, 1980; Ryder, 1985» can be defined by the Mittag-Leffierfunctions (MiIIerj ~ 993; Gorenflo et aI., 2002; Kilbas et aI., 2006) instead of the exponential functionj [t IS connected WIth the fact that the MIttag-Leffler functIOn IS mvanant WIth respec~ ~o left-sided Caputo fractional derivative ~ DC; (see Lemma 2.23 in (Kilbas et aLl

~

[I'he fractIOnal vanatIOns can be used to define a fractIOnal generalIzatIOn of graj type equatIOns that have a WIde applIcatIOn m the theory of dIssIpatIve strucj ~ures (NIcolIs and Pngogme, 1977; Sagdeev et aI., 1988). The fractIOnal gradlentl ~ype equatIOns are generalIzatIOn of fractIOnal gradIent systems (Tarasov, 2005b~ ~rom ordmary dIfferentIal equatIOns mto partIal dIfferentIal equatIOns. ThIS general-I IIzatIOn can be realIzed by using de Donder-Weyl Hamlltoman and Pomcare-CartalJl n-form. ~Ient

lReferencesl

K

AldrovandI, R.A. Kiaenkel, 1988, On extenor vanatIOnal calculus, Journal of! IPhysics A, 21, 1329-1339~ ~.N. BogolIubov, 1960, Problems oj Dynamic Theory in Statistical Physics, Tech1 ImcalInformatIOn ServIce, Oak RIdge.1

1332

14 Fractional Calculus of Variations in Dynamics

[N.N. Bogoliubov, 1970, Method of functional derivatives in statistical mechanicsj lin Selected Works, Naukova Dumka, Kiev, In Russian, 197-209~ [N.N. Bogoliubov, 1991, Selected Works. Part II. Quantum and Classical Statistica~ !Mechanics, Gordon and Breach, New York) [N.N. Bogoliubov, 1995, Selected Works. Part IV. Quantum Field Theory, Gordonj and Breach, Amsterdam] [N.N. Bogoliubov, D.V. Shirkov, 1980, Introduction to the Theory of QuantizetA !Field, 3rd ed., Wiley, New York; and 4th ed., Nauka, Moscow, 1984. In Russianj IY.Q. Chen, K.L. Moore, 2002, AnalytIcal stabIlIty bound for a class of delaye~ Ifractiona1-order dynamic systems, Nonlinear Dynamics, 29, 191-200.1 !B.P. Demidovich, 1967, Lectures on the Mathematical Theory of Stability, NaukaJ !Moscow In Russian I !B.A. Dubrovin, A.N. Fomenko, S.P. Novikov, 1992, Modern Geometry - Method~ landApptzcatlOns, Part I, Spnnger, New York.1 IY.M. FI1lIpov, Y.M. Savchm, S.G. Shorohov, 1992, VariatIOnal Prmclples for Nonj 1P0tentwi Operators, Modern Problems of MathematICs, The Latest AchIeve-I Iments, VoI.40, Moscow, VINITI. In RussIanl IK. FukUI, 1970, A formulatIOn of the reaction coordmate, Journal oj Physical Chem1 listry, 74, 4161-4163J IK. FukUI, 1981, The path of chemIcal reactIons-the IRS approach, Accounts of! IChemicalResearch, 14, 363-368~ ~. Goldstem, 1950, Classical Mechanics, AddIson-Wesley, Cambndgej IH. Goldstem, c.P. Poole, J.L. Safko, 2002, Classical Mechanics, 3nd ed., AddIson-I IWesley, San FransIscoj R Gorenflo, J. Loutchko, Y. Luchko, 2002, ComputatIon of the MIttag-Lefflerfunc-1 Itionand its derivative, Fractional Calculus and Applied Analysis, 5, 491-518.1 IS.B. Hadid, J.G. A1shamani, 1986, Liapunov stability of differential equations ofj Inomnteger order, Arab Journal oj Mathematics, 7, 5-171 IH. Helmholtz, 1889, JournalJur die Reine und Angewandte Mathematik, 10, 137-1 1166. [r.D. Khusamov, 2001, StabIlIty analySIS of a lInear-fractIOnal delay system, Dijjerj ential Equations, 37, 1184-1188~ V\.A. KI1bas, H.M. Snvastava, J.J. TruJIllo, 2006, Theory and Applications oj Frac1 ~ional Dijjerential Equations, ElseVIer, Amsterdamj 1V.1. Klyachkm, 1980, Stochastic Equations and Waves in Randomly Inhomogeneousl !Medw, Nauka, Moscow. In RUSSIan) RD. Levme, J. Bernstem, 1974, Molecular ReactIOn DynamICs, Oxford UmversItYI IPress OxfofdJ IY. LI, Y.Q. Chen, I. Podlubny, 2009, MIttag-Leffler stabIlIty of tractIOnal order non-I IlIneardynamIC systems, Automatica, 45, 1965-1969.1 [.G. Malkin, 1959, Theory of Stability of Motion, United States Atomic Energ)j KO:ommIssIOn, WashmgtonJ p. Matignon, 1996, Stability result on fractional differential equations with app1icaj ItIOns to control processing, m IMACS - SMC Proceeding, Litle, France, 963-9681

IR eferences

333

IK.S. MIller, 1996, The MIttag-Leffler and related functIOns, Integral Transformsl land special Functions, 1, 41-49J IW.H. Miller, N.C. Hardy, J.E. Adams, 1980, Reaction path Hamiltonian for poly-I atomic molecules, Journal of Chemical Physics, 72, 99-112.1 IS. Momam, S.B. HadId, 2004, Lyapunov stabIhty solutIOns of fractIOnal mtegrodj IIflerentIal equatIOns, International Journal of Mathematical Sciences, 47,2503-1

rrsm:::

p. Nicolis,

I. Prigogine, 1977, Self-Organization in Nonequilibrium Systems: Froml IDlsslpatlve Structures to Order through FluctuatIOns, WIley, New YorkJ f.J. Olver, 1986, Application ofLie Groups to Differential Equations, Springer, Newl [ork. Chapter 4, and Section 5Aj [..:.H. Ryder, 1985, Quantum Field Theory, Cambndge UmversIty Press, CambndgeJ K=hapter 6. RZ. Sagdeev, D.A USIkov, G.M. ZaslavskY, 1988, Nonlmear PhySICS. From th~ IPendulum to Turbulence and Chaos, Harwood AcademIC, New York) IS.G. Samko, AA KIlbas, 0.1. Marlchev, 1993, Integrals and Derivatives of Fracj rional Order and Applications, Nauka I Tehmka, Mmsk, 1987, m Russlanj !ind Fractional Integrals and Derivatives Theory and Applications, Gordon and! IBreach, New York, 19931 [.-.1. Sedov, 1965, Mathematical methods for constructing new models of continuou~ media, Russian Mathematical Surveys, 20, 123-182.1 [..:.1. Sedov, 1968, Models of contmuous medIa WIth mternal degrees of freedom) flournal oj Applied Mathematics and Mechanics, 32, 803-819J [..:.1. Sedov, 1997, Mechanics oj Continuous Media, Volume 1, World SCIentIfic Pub1 Ihshmg, SmgaporeJ [..:.1. Sedov, AG. Tsypkm, 1989, Prmclples of the MIcrOSCOpIc Theory of GravttatwrlJ, land Electromagnetism, Nauka, Moscow, SectIOn 3.7, SectIon 3.8-3.12, SectIon 4J IV.B. Tarasov, 2005a, FractIOnal generahzatIOn of gradIent systems, Letters in Math1 ~matical Physics, 73, 49-58J ~E. Tarasov, 2005b, Fractional generalization of gradient and Hamiltonian systemsJ IJournalofPhysics A, 38, 5929-5943j ~E. Tarasov, 2006, Fractional variations for dynamical systems: Hamilton and Laj grange approaches, Journal oj Physics A, 39, 8409-8425J IY.E. Tarasov, 2007, FractIOnal denvatIve as fractIOnal power of denvatIve, Interna-I rional Journal oj Mathematics, 18, 281-299.1 IV.B. Tarasov, 2008, Quantum MechaniCS of Non-HamIltOnian and DISSIpatIve Sysj ~ems, ElseVIer,Amsterdam1 ~.G. Tchetaev, 1990, Stability oj Motion, 4th ed., Nauka, Moscow. In RussIan.1 k\.N. VasIlev, 1998, Functional Methods in Quantum Field Theory and StatisticaA IPhysics, Gordon and Breach; and Lemngrad State UmversIty, Lemngrad, 1976J 1m RussIan.1

~hapter

151

Fractional Statistical Mechanics

115.1 Introductionl IStatIStIcal mechanICS IS the applIcatIOn of probabIlIty theory to study the dynam-I ~cs of systems of arbItrary number of partIcles (GIbbs, 1960; BogolIubov, 1960j ~ogolyubov, 1970). Equations WIth denvatIves of non-mteger order have many ap1 Iplications in physical kinetics (see, for example, (Zaslavsky, 2002, 2005; Uchaikinj 12008) and (Zaslavsky, 1994; SaIchev and Zaslavsky, 1997; WeItzner and ZaslavskyJ 12001; Chechkm et aI., 2002; Saxena et aI., 2002; ZelenYI and MIlovanov, 2004j IZaslavsky and Edelman, 2004; NIgmatullIn, 2006; Tarasov and Zaslavsky, 2008j IRastovIc, 2008)). FractIOnal calculus IS used to descnbe anomalous dIffUSIOn, and] ~ransport theory (MontroIl and Shlesmger, 1984; Metzler and Klafter, 2000; Zaj Islavsky, 2002; OchaIkm, 2003a,b; Metzler and Klafter, 2004). ApplIcatIOn of fracj ~IOnal mtegratIOn and dIfferentIatIOn m statIstIcal mechanICS was also conSIdered] [n (Tarasov, 2006a, 2007a) and (Tarasov, 2004, 2005b,a, 2006b, 2007b). FractIOnall IkmetIc equatIons usuaIly appear from some phenomenologIcal models. We sugges~ ~ractIOnal generalIzatIOns of some baSIC equatIOns of statIstIcal mechanICS. To Ob1 ~am these equatIOns, the probabIlIty conservatIOn m a fractIOnal dIfferentIal volum~ ~lement of the phase space can be used (Tarasov, 2006a, 2007a). This element canl Ibe conSIdered as a smaIl part of the phase space set WIthnon-mteger-dImensIOn. W~ ~enve the LIouvIIle equatIOn WIthfractIOnal denvatIves WIth respect to coordmate~ landmomenta. The fractIOnal LIOuvIlle equatIOn (Tarasov, 2006a, 2007a) ISobtamed] OCrom the conservatIOn of probabIlIty to find a system m a fractIOnal volume elementJ [fhIS equatIOn IS used to denve fractIOnal Bogolyubov and fractIOnal kmetIc equaj ~IOns WIth fractIOnal denvatIves. StatIstIcal mechanICS of fractIOnal generalIzatIOnl pf the HamIltOnIan systems IS dIscussed. LIOuvIlle and Bogolyubov equatIOns wIthl fractional coofdinate and momenta derivatives are considered as a basis to derive OCractIOnal kmetIc equatIOns. The Vlasov equatIOn WIth denvatIves of non-mteger orj kler is obtained. The Fokker-Planck equation that has fractional phase space deriva-I ~ives is derived from fractional Bogolyubov equationj

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

1336

15 Fractional Statistical Mechanid

[n SectIOn 15.2, we obtam the LIOuvIlle equatIOn WIth fractIOnal denvatIves froIll] conservation of probability in a fractional volume element of phase space. Inl ISectIOn 15.3, the first Bogolyubov hIerarchy equatIOn WIth fractIOnal denvatIve~ lin phase space is derived. In Section 15.4, we consider the Vlasov equation withl [ractional derivatives in phase space. In Section 15.5, the Fokker-Planck equationj Iwith fractional derivatives with respect to coordinates and momenta is obtained fromj OCractIOnal Bogolyubov equatIOn. Fmally, a short conclUSIOn ISgIven m SectIOn 15.6j ~he

~S.2

Liouville equation with fractional derlvatives

lOneof the basic principles of statistical mechanics is the conservation of probabilitYI 1m the phase space (LIboff, 1998; Martynov, 1997). The LIOuvIlle equatIon IS ani ~xpressIOn of the pnnciple m a convement form for the analySIS. We denve thel [.:Iouville equatIon WIth fractIOnal denvatIves from the conservatIOn of probabIlIty! lin a fractional volume element] OCn the phase space ffi.2n with dimensionless coordinates (xl, ... ,x2n) = (ql,'" ,qn,1 IPI,... ,Pn), we consider a fractional differential volume elementj (15.1)1 ~ere,

d U is a fractional differentia1. The action of d U on a function j(x) is defined!

!bY ~Uj(x)

IWI

=

E ;;P~j(X)[dXk]U,

land ~D~ is the Caputo fractional derivative (Samko et a1., 1993) of order Irespect to xi, The Caputo denvatIve IS defined byl 1

U

Dxj(x)

=

t

j

n

(z)

r(n-a)Ja (x_z)a+I-n dz,

(15.2)1

a withl

(15.3)

Iwhere n - 1 < a < n, and j(n) (z) = D~j(z). We note that ;;p~ 1 = 0, and ;;P~XI [or kit. Using (15.2), we obtaii1j

= 01

(15.4)1 ~quatIOn

(15.4)

gIve~

[dXk]U = (;;p~ (Xk - ak)) for the Caputo fractional derivative (15.3), we

1

dU(xk - ak)'

hav~

337]

115.2 Liouville equation with fractional derivatives

(15.6) Iwhere f3 > m - 1, m - 1 <

a

~

m and Xk > as, Equations (15.5) and (15.6)

giv~

(15.7)1

for 0 < a ~ 1.1 for the usual phase space volume element dV, the conservation of probabIlIty I§ gIven by the equationj l-dV apy;x)

=

d(p(t,x) (u,dS)).

[The conservation of probability for the fractional volume element (15.1), can bel Irepresented b5J

(15.9)1 ~ere,

p (t, x) is a density of probability to find the dynamical system in daV, u ==i

~(t, x) is the velocity vector field in ]R2n, daS is a dimensionless surface element]

landthe brackets ( , ) is a scalar product of vectorsj In

12n

2il

(15.10~

Iwhere ek are the baSIC vectors of CartesIan coordmate system, and! ~aS k =

d aXl'" d a Xk-I d aXk+l'" d a X2n'

(15.1 q

[The functions Uk = Uk(t, x) define Xk components of u(t, x), which is a rate at whichl Iprobability density is transported through the area element daSk. In the usual casel I( a-I), the outflow of the probabIlIty m the Xk dIrectIOnI§

~(puk)dSk = (D;kPUk)dxkdSk = (D;kPUk)dV. [The fractional generalization (a

(15.12~

#- 1) of Eq. (15.12) i~

~a(puk)daSk

= (gD~PUk)[dx]adaSkJ

Rsing (15.11), (15.1) and (15.5), we getl

~a[puk]daSk = ~D~(pUk) (~D~Xk)

1= (~D~Xk) ISubstitutIOn of (15.13) mto (15.9)

gIve~

1

1

daXkdaS~

~D~(pUk)daV.

(15.13~

1338

15 Fractional Statistical Mechanid

(15.14 k'\s a result, we obtaml

(15.15~ Iwhere we lise the notationj

[I'hIS IS the LIOUVIlle equatIon WIth the derIvatIves of tractIonal order a. EquatIoIlj 1(15.15) describes the probability conservation for the fractional volume elemen~ 1(15.1) of the phase spaceJ for the dimensionless coordinates (ql,'" ,qn,PI,··· ,Pn), Equation (15.15) i§

p

~:'P + tD~k(P(t,q,P)Gk)+ tD~k(P(t,q,P)Fk) =0, k=1

(15.16

k=1

Iwhere Gk = Uk> and Fk = Uk+n (k = 1, ... ,n). The function p(t,q,p) describes a dis~ ~ribution of probability in the phase space. The functions Gk = Gk(t,q,P) are thel Fomponents of velocity field, and Fk = Fk(t,q,p) are the components of the forcel lfield. In general, these fields are non-potentIal. EquatIOn (15.16) IS LIOUVIlle equa1 ~IOn WIth tractIOnal derIvatIves WIth respect to phase space coordmates (Tarasov J 12006a, 2007a)~ pefinition 15.1. DynamIcal system that IS defined by the equatIOnsj

(15.17~ lis called a Hamiltonian system, if the right-hand sides of Eqs. (15.17) satisfy thel Helmholtz condItIOnsl

(15.18~ IWe note that a tractIOnal generalIzatIOn of Hamlltoman systems was suggested! [n Ref. (Tarasov, 2005c)J !Letus consider conditions (15.18) for a simply connected region of jR2n. A re~ gIOn IS SImply connected If every path between two pomts can be contmuousl5J ~ransformed mto every other, and any two pomts can be jomed by a path. In thI§ ~ase, we have the followmg theoreml rtheorem 15.1. If dynamical system (15.17) is Hamiltonian in the region W ojjR2n,1 I(:lnd this region simply connected, then the functions Gk and Fk can be representedl lin the form:

(15.19~

33~

115.2 Liouville equation with fractional derivatives

Ithat is uniquely defined by the Hamiltonian H

=

H(t,q,p)j

[n general, we have the mequahtyj (15.20~

[I Fk does not depend on Pk, and Gk does not depend on qb then Eq. (15.16) glVe§

for fractional generalization of Hamiltonian system (Tarasov, 2005c, 2006c), thel [unctions Gk and Fk can be represented a§ (15.22~

Iwhere H(q,p) is a generalized Hamiltonian function. For a = 1, we have the usuall Hamiltonian system (15.17) with (15.19). Substituting (15.22) into (15.21), we obj rtaln n

f-'--'-::--"--"-'-

+

L (D~kH(q,p)D~kP(t,q,p) - D~kH(q,p) D~kP(t,q,p)) = O. (15.23)

IWe can define the bracketsJ

KA,B}a

=

L (D~kAD~kB-D~kBD~kA).

(15.24~

~

~xpression

(15.24) can be represented in the form:1

I{A,B}a =

t (~D~kqk ~D~kPkrl (~D~kA ~D~kB- ~D~kB ~D~kA).

~

(15.25~

for a = 1, Equation (15.25) gives the Poisson bracketsj

I{A,Bh =

L (D~kA D~kB - D~kB D~kA) J

~

IWe note that

I{A,B}a = -{B,A}a,

{l,A}a =OJ

[n general, the Jacobi identity for (15.25) cannot be satisfied. Using Eqs. (15.25) land (15.23), we obtainl

IdP(t,q,p) at

+ {( P t,q,p ) ,H (q,p )} a=O.

(15.26~

1340

15 Fractional Statistical Mechanid

[Equation can be interpreted as Liouville equation for fractional generalization ofj HamIltoman systems (Tarasov, 2005c, 2006c). For a - 1, EquatIOn (15.26) gIvesl ~he usual Liouville equation for Hamiltonian systems in phase space.1

~S.3

Bogolyubov equation with fractional derlvatives

OC'et us conSIder a claSSIcal system WIth fixed number N of IdentIcal partIcles. Supj Iposethat kth partIcle IS deSCrIbed by the generalIzed coordmates qks and generalIze~ Imomenta Pks, where s - 1, ... ,m. In thIS case, we have the 2mN-dImensIOnal phas~ Ispace. The state of this system can be described by the distribution functionj

ij=(qI, ...,qN),

qk=(qkl, ...,qkm),1

P=(PI,· ..,PN),

Pk=(Pkl, ..·,Pkm)

lare the coordmates and momenta of the partIcles. The normalIzatIOn condItIOn I~ ~[1, ...,N]PN(q,p,t) = i,

Iwherei[l, ...,N] is the integration with respectto qI ,PI, ...,qN,PN over phase spacel [I'he mtegratIon can be WrItten byl

1l[1, ...,N] = i[1 ]i[2] .. ·l[NJl Iwhere l[k] is the integration with respect to qk, Pk such tha~

fractional Liouville equation (15.16) is represented b)j

(15.27 Iwhere G k IS a velOCIty of kth partIcle, Fk IS the force that acts on kth partIcle,

an~

(15.28~ (15.29~ pefinition 15.2. The one-partIcle reduced dIstrIbutIOn functIOn PI can be IQy

define~

34~

115.3 Bogo1yubov equation with fractional derivatives

(15.30~

Iwhere1[2, ... ,N] is an integration with respect to qz, ..., qN, pz, ..., PNJ !Obviously, that the function (15.30) satisfies the normalization conditionj

pefinition 15.3. The s-partIde reduced dIstrIbutIOn functIOn Ps can be defined byl IPs(q,p,t) = p( ql ,PI, ... , qs,Ps,t) = frs, ...,N]PN(q,P,t),

(15.3Ij

Iwhere f[s, ... ,N] is an integration with respect to qs, ..., qN, Ps, ..., PN.I [The Bogolyubov hIerarchy equatIOns (Bogolyubov, 1946; Boer and OhlenbeckJ [962; Bogohubov, 1960, 1991; Gurov, 1966; PetrIna et aL, 2002; Martynov, 1997)1 ~escrIbe the evolutIOn of the reduced dIstrIbutIOn functIons. These equatIons can bel ~erIved from the LIouvIlle equatIon. Let us derIve the Bogolyubov equatIons WIthl ~ractIOnal derIvatIves from the fractIOnal LIOuvIlle equatIOnj

[Theorem 15.2. Let F k be a force ofthe binary interactionsJ 1M

IFk = F%+ EFkl ,

(15.32~

~

Iwhere FA: = Fe(qk,Pk,t) is the external force, and Fkl = F(qk,Pk,ql,PI,t) are the inj Iternalforces. Fractional Liouville equation (15.27) gives the first Bogolyubov equa-I ItlOn of the formj (15.33

(15.34~

rnd pz is two-particle reduced distribution function] IProof. To obtam the first Bogolyubov equatIOn wIth fractIOnal derIVatIvesfrom Eqj 1(15.27), we conSIder the dIfferentIatIOn of (15.30) wIth respect to tImej

(15.35~ IOsmg (15.27) and (15.35), we getl

(15.36~ ~et us consider the integration f[qk] over qk for kth particle term of Eq. (15.36j [or k = 2,3, ... ,N. Using the fact that the coordinates and momenta are independentl

1342

15 Fractional Statistical Mechanid

Ivariables, we obtain! = 0,

(15.37

Iwhere 1a[qk] is a fractional integration with respect to variables qk. In Eq. (15.37),1 Iwe use that the distribution PN in the limit qk ----> ±oo is equal to zero. It follows fromj ~he normalization condition. If the limit is not equal to zero, then the integration overl Iphase space IS equal to mfimty. SImIlarly, we hav~

[Then all terms m Eq. (15.36) WIth k - 2, ... ,N are equal to zero. We have only thel OCerm WIth k = 1. Therefore Eq. (15.36) has the formj (15.38 ISmce the vanable ql IS mdependent of q2, ... , qN and P2, ... ,PN, the first term m Eql 105.38) can be wntten aij

[I'he force

F\ acts

on the first partIcle. For the bmary mteractIOnsJ IFI =Fr + [ F lk ,

(15.39~

~

Iwhere Fr = Fe(ql,PI,t) is the external force, and F lk = F(ql,PI,qk,Pk,t) are thel ~nternal forces. Usmg (15.39), the second term m (15.38) I~

IUsmg the defimtIOn of one-partIcle reduced dIstnbutIOn functIOn (15.30), we obtaml 0Vl

1[2,...,N]ng (FIPN) = ng (Frpt) + L ngJ[2, ... ,N](FlkPN). 1 1

1

(15.40~

~

IWe assume that the dIstnbutIOn functIOn PN IS mvanant under the permutatIOns IldentIcal partIcles (Bogolyubov and Bogolyubov, 1982):1

o~

[Then PN is a symmetric function, and all (N - 1) terms in Eq. (15.40) are identical:1

34~

115.4 Vlasov equation with fractional derivatives rJ1

IE 1[2, ... ,N] D~lS (FlkPN) = (N -1)1[2, ... ,N] D~l (Fl2PN). Rsing

(15.4q

1[2, ...,N] = 1[2]/[3, ...,N]' we rewrite the right-hand side of (15.41) in thel

Ifilli:l:i::

[[2, ...,N] D~l (F l2PN) = 1[2] D~l (F l2/ [3,...,N]PN) = D~J[2]Fl2Pz,

(15.42~

!where IPz = P(ql,PI,qZ,PZ,t) =

1[3, ...,N]PN(q,P,t)

lis a two-particle distribution function. Finally, we obtain Eq. (15.33).

(15.43j

q

IRemark 1J IWe note that the integral (15.34) describes a velocity of particle number change inl t!m-dimensional two-particle elementary phase volume. This change is caused bYI ~he mteractions between partlclesl Remark 2J (15.33) is a fractional generalization of the first Bogolyubov equation. Ifj Ki = 1, then we have the first Bogolyubov equatIon for non-Hamlltoman system§ I(Tarasov, 2005d). For Hamlltoman systems) ~quation

(15.44~ land Eq. (15.33) WIth a = 1 has the weB-known form (Bogolyubov, 1946; Boer and! Rh1enbeck, 1962; Bogoliubov, 1960, 1991; Gurov, 1966; Petrina et a1., 2002)j

115.4 Vlasov equation with fractional derivativesl OC:et us consIder the partIcles as statIstIcal Independent systems. Thenl (15.45~

ISubstItutlOn of Eq. (15.45) Into Eq. (15.34) g1Ve~

(15.46~ IwherepI =PI(ql,PI,t).1 Let us define the effectIve forcel

IUsing Fe!!, we can rewrite Eq. (15.46) in the formi

(15.47~

1344

15 Fractional Statistical Mechanid

ISubstItutmg Eq. (15.47) mto Eq. (15.33), we obtaml (15.48 [This is a closed equation for the one-particle distribution function with the externa~ [orce Fj and the effective force Fe!!. Equation (15.48) is a fractional generaliza-I ~ion of the Vlasov equation (Vlasov, 1938, 1968, 1945, 1961) that has phase spacel ~envatIves of non-mtegerorder. For a-I, we get the Vlasov equatIon for the non-I Hamiltonian systems (Tarasov, 2005d). For Hamiltonian systems (15.44), Equation] 105.48) WIth a - 1 has the usual form (Vlasov, 1938, 1968, 1945, 1961)l [..:et us conSIder a speCIal case of fractIOnal kmetIc equatIOn (15.33) such tha~ II(P2) = 0, G 1 = p/m = v, and Fe = eE, B = O. Then this equation has the formj

t£ + (V,D~PI) +e(E,D~pd

=

0,

(15.49~

Iwhere PI IS the one-partIcle denSIty of probabIlIty, andl !11J

~v,D~pd = L(vs,D~sPI)'

(15.50~

~

~f we take into account the magnetic field (B #- 0), then we must use the generalIzatIOn of LeIbmtz rulesj

fractiona~

(15.51 Iwhere s are mteger numbers. In thIS case, EquatIOn (15.49) has the addItIon term:1

(15.52 [..:et us conSIder the perturbatIOn (Ecker, 1972; Krall and Tnveiplece, 1973) of thel Histribution function in the form'l (15.53~

Iwhere PI is a homogeneous stationary density of probability that satisfies Eq] 1(15.49) for E = O. Substituting (15.53) into Eq. (15.49), we ge~

345]

115.5 Fokker-Planck equation with fractional derivatives

b2 + (v,D~opd +e(E,D~pd =

0.

(15.54~

Equation (15.54) is linear fractional kinetic equation for the first perturbation op] pf the distribution function. Solutions of fractional linear kinetic equations (15.54) Iwere considered in Ref. (Saichev and Zaslavsky, 1997). For E = 0, the function OP] lis described by the functionj

(15.55~ Iwhere Cs = vs(~D~qs)-I, and La [x] is the Levy stable probability density functionj I(Feller, 1971). For a - 2, we have the Gauss dlstnbutlOn. For I < a ~ 2, the func-I ~ion La [x] can be represented by expansions (15.80) and (15.81). The asymptotic ofj ~he solution, exhibits the power-like tails for x ----+ ooj

~s.s

Fokker-Planck equation with fractional derivatives

[The Fokker-Planck equatIOns wIth fractIOnal coordInate denvatIves have been sug1 gested In (ZaslavskY, 1994) to descnbe chaotIc dynamICs. It IS known that Fokker-I flanck equatIOn for phase space can be denved from the LIouvIlle equatIon (Islj Ihara, 1971; Resibois, and Leener; Forster, 1975). The Fokker-Planck equation withl hactlOnal denvatIves was obtaIned In (Tarasov, 2006a) from the fractIOnal LlOuj IvIlle equatIOn. USIng the generalIzed Kolmogorov-Feller equatIOn wIth long-rang~ IInteractlOn, the Fokker-Planck equatIOn wIth fractIOnal denvatlves wIth respect tq ~oordInates was denved In (Tarasov and Zaslavsky, 2008).1 OC:et us conSider a system of N IdentIcal partIcles and the Browman partIcle tha~ liS descnbed by the dlstnbutlOn functIon:1

rN+I

=

PN+I(q,P,Q,P,t)J

Iwhere q and p are the coordInates and momenta of the particles, andl

Q={Qs: s=l, ... ,m},

P={Ps: s=l, ... ,m~

lareBrowman partIcle coordInates and momenta. The normalIzatIOn condItIOn I~

1i[1, ...,N,N+1]PN+I

=

1.

(15.56j

[I'he dlstnbutlOn functIOn for the Browman partIcle IS defined by!

rB(Q,P,t)

=

i[l, ...,N]PN+I (q,p, Q,P,t).

(15.57j

[he LIOuvIlle equatIOn for PN+I I§

(15.58~

1346

15 Fractional Statistical Mechanid

IJY;Tij

ILNP = i

L (D~h (G~ P) + D~h (FskP)) j [l(;S]

IJY;Tij

ILBP

= i

E(DQs(gsp) + Df,(fsp))j [l(;S]

Here, LN and LB are LlOuvIlle operators Withtracttonal denvattves, andl

[he functions G~ and F,k are defined by the equations of motion for particle,1

qks

dt =

Gk( s

q,p ),

Pks =Fsk(q,p, Q) -----:it ,P,

k= 1, ... ,N.

(15.59

[I'he Hamilton equattons for the Browman parttclej

st; --;jf ~efine gs and


(15.60~

1s1

IWe conSider the boundary conditton m the formj

(15.6q

rN(q,p,Q,T)

=

exp{f3(§ -H(q,p,Q))}

(15.62~

lis the canonical Gibbs distribution with the Hamiltonian-I

V!(q,p,Q)

=

HN(q,P) +

E VB(qk,Q)·

(15.63~

It=]

Here, HN is a Hamiitoman of N-particle system, and VB is an energy of mteractlOi1j Ibetween k-particles and Browman particle. In the case,1

Iwe have the Hamiltonianl

~n general, we can consider G~ = G~(q,p), and g, = gs(Q,p)l

347]

115.5 Fokker-Planck equation with fractional derivatives

[t IS known that boundary condItion (15.61) can be reahzed (Zubarev and! lNovikov, 1972) by the infinitesimal source term in the Liouville equationj

t§i# - i(LN + LB)PN+I = -e(PN+I - PNPB).

(15.64~

[ntegrating (15.64) by i[l, ... ,N]' we obtainl (15.65 Iwhich is the Liouville equation for reduced distribution function of the Brownianl IpartIcle. [The formal solution ofEq. (15.64) has the formj

o N+l (t) = PB(t)PN -

~('r,LN,LB) =

d'r.£'( 'r,LN,LB) PB(t + 'r)PN,

(15.66

eere-ir(LN+LB)(:'r -i(LN+LB))1

ISubstItutmg(15.66) mto (15.65), we getl

~B + [,OQS(gsPB) + [,Of,PBl [I ,...,N](fsPN s=1

s=1

Irhe expression 1[1, ...,N]/sPN can be considered as average value of the force for canomcal GIbbs dIstnbutlOn (15.62) It IS equal to zero. Usmg the relatlOnj

Iwhere

Is.1

r}p) is a fractional potential force (Tarasov, 2005e,c):1 lAp) - _Oa U

Vs

-

Qs

B,

(15.68~

Iwe obtam

[t IS easIly shown by mtegratlOn that the termj

(15.69~

1348

15 Fractional Statistical Mechanid

[n Eq. (15.67) does not contnbute. Then Eq. (15.67) has the formj

(15.70 ~quation

(15.70) is a closed integro-differential equation for the distribution func-I bon PB. Note that Is can be represented as Is = Ap) + An), where lip) is a potentiall

dn

[orce (15.68), and ) is a non-potential force that acts on the Brownian parti~ FIe. For the equilibrium approximation, we have P rv (MkT)I/2, iLB rv M- 1/ 2 an~ liLN rv m- 1/ 2 . 1f M > > m, then the perturbation theory can be usedj [he approximation PB(t + -r) = PB(t) for Eq. (15.70), give§

t

P;t

m

m

s=1

s=1

+ ED{2s(gsPB) + EDf,(Mj3-1D~,,(Ass'PB(t))+BssIPsIPB(t)) =0 (15.71)

Iwhere PB(t) = PB(Q,P,t), andl (15.72

(15.73 V\s a result, we denve a fractIOnal generalIzatIOn of the Fokker-Planck equatIOili I(Zaslavsky, 1994; Tarasov, 2005f) for the phase space.1 ~f gs is a velocity (G s = PsiM) of Brownian particle and PB = PB(t,Q), then Eql 1(15.71) gives

(15.74~ Iwhere we mean the sum on the repeated Index s from I to m. EquatIOn (15.74) IS ~ OCractIOnal equatIOn In coordInate space. In thIS case, the functIOn PB IS descnbed a§

(ast )-I/aLa [Qs(ast )-I/aj,

(15.75~

Iwhere as = M 1Ps(D{2s Qs) 1, and La [x] is the Levy stable probability density func-I ~IOn (Feller, 1971)1 for a = 1, the function Lalxl gives the Cauchy distributionj

(15.76~ land (15.75) I§ (15.77

349

115.6 Conclusion

for a = 2, the function La [x] is the Gauss distributionj

(15.78~ land the function (15.75) i~ (15.79 for 1 <

a

~

2, the function La [xl can be represented as the expansionj

La[x] = _~ ~ (_ltr(l +nja) sin(nnj2)x". nx':: n! [I'he asymptotIc (x

----+

00, I

(15.80

< a < 2) IS gIven byl -----'-----'-- sin(nnj2)x- na.

(15.81

V\s a result, the asymptotIc of the solutIOn exhIbIts the power-lIke taIls for x ----+ 00.1 [These taIls are the Important property of solutIOns of equations wIth the non-mtege~ Herivatives

~ 5.6

Conclusionl

IWe HerIve the LIouvIlle, Bogolyubov, Vlasov anH Fokker-Planck equatIOns wIthl OCractIOnal HerIvatIves wIth respect to coorHmates anH momenta. To HerIve the fracj ~IOnal LIOuvIlle equatIon, we consIHer the conservatIOn of probabIlIty m the fracj ~IOnal dIfferentIal volume element. ThIS element IS HefineH by dIfferentIals of frac1 ~IOnal order. Usmg the fractIOnal LIOuvIlle equatIOn, we obtam a fractIOnal gener1 lalIzatIOn of the Bogolyubov hIerarchy equatIOns. These equatIOns descrIbe the ev01 IlutIOn of the reHuceH HensItIes of probabIlIty. FractIOnal Bogolyubov equatIOns arel lused to derive fractional kinetic equation (Tarasov, 2006a, 2007a). The fractiona~ fokker-Planck equatIOn, fractIOnal Vlasov anH lInear kmetIc equatIOns are obtame~ ~rom the suggesteH fractIOnal LIOuvIlle equatIons. The fractIOnal equatIons contaml ~erIvatIves of non-mteger orHer, anH the power-lIke taIls are the Important propert)1 pf the solutIOns of the equatIOns] IOsmgthe fractIOnal vector calculus (Tarasov, 2008), we can obtam the fractIOna~ klIfferentIal equatIOns for conservatIOn of probabIlIty such as the LIOuvIlle anH Boj golIubov equatIOns. We note that the fractIonal LIOuvIlle equatIon (Tarasov, 2006aJ 12007a) can be derIved by the method suggesteH in (Wheatcraft anH MeerschaertJ 12008).

1350

15 Fractional Statistical Mechanid

[t is known that fractional spatial derivatives describe nonlocal properties of thel klistributed system. Therefore the fractional statistical mechanics is connected withl OChe nonlocal statIstIcal mechanICS (Vlasov, 1978, 1966). At the same tIme fracj OCIOnal denvatIves are connected WIth long-range mterpartIcle mteractIOns. In Refs.1 I(Tarasov, 2006e,d), we proved that nonlocal alpha-interactions between particles ofj ~rystallattice give continuous medium equations with fractional derivatives with rej Ispectto coordinates. In Ref. (Vlasov, 1978), a nonlocal statistical model of crystall ~attice was suggested. We can conclude that the nonlocal and fractional statisticall ImechanIcs are dIrectly connected WIth statIstIcal dynamICS of systems WIth long-I Irange mteractIons (Campa et aI., 2009)J

lReferencesl ~.

de Boer, G.B. Uhlenbeck, (Eds.), 1962, Studies in Statistical Mechanics, North] !Holland, Amsterdaml ~.N. Bogolyubov, 1946, KmetIc equatIOns, ZhurnaL EksperimentaL'noi i Teoretich1 ~skoi Fiziki, 16, 691- 702. In RussIan; and JournaL oj Physics USSR, 10, 265- 2741 N.N. Bogoliubov, 1960, Problems of Dynamic Theory in Statistical Physics, Oak! IRidge, Technical Information Service; and OGIZ, Moscow, 1946. In Russianj ~.N. Bogolyubov, 1970, SeLected Works, VoI.2, Naukova Dumka, KIevJ ~.N. Bogohubov, 1991, SeLected Works. Part II. Quantum and ClassicaL Statistica~ IMechanics, Gordon and Breach, New Yorkl ~.N. Bogolyubov, N.N. Bogolyubov, Jr., 1982, IntroductIOn to Quantum Statlstlca~ !MechaniCS, World SCIentIfic Pubhshmg, Smgapore.1 k\. Campa, 1'. DauxOls, S. Ruffo, 2009, StatIstIcal mechanICS and dynamICS of solvj ~ble models WIth long-range mteractions, Physics Reports, 480, 57-159.1 V\.v. Chechkm, vvo. Gonchar, M. Szydlowsky, 2002, FractIOnal kmetIcs for relax1 ation and superdiffusion in magnetic field, Physics ofPlasmas, 9, 78-88j p. Ecker, 1972, Theory of Fully IOnized PLasmas, AcademIC Press, New York] IV. Feller, 1971, An mtroductwn to ProbabILIty Theory and Its AppLIcatIOns, VoI.2j IWIley, New YorkJ p. Forster, 1975, HydrodYnamics FLuctuations, Broken Symmetry, and CorreLationl lFunctions, Benjamin, London1 ~.W. Gibbs, 1960, Elementary Principles in Statistical Mechanics, Dover, Newl [ork, 1960; and Yale University Press, New Haven, 1902j IK.P. Gurov, 1966, Foundation oj Kinetic Theory. Method oj N.N. BogoLyubov,1 INauka, Moscow. In RussIanJ V\. ISIhara, 1971, StatisticaL Physics, AcademIC Press, New York. AppendIX IV, and! ISection7.5J k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj kwnaLDIfferentiaL EquatIOns, ElseVIer,AmsterdamJ ~.A. Krall, A.w. Tnvelplece, 1973, PrmclpLes oj PLasma PhYSICS, McGraw-HIlIJ INew YOrk

References

3511

[R.L. Liboff, 1998, Kinetic Theory: Classical, Quantum and Relativistic Descriptionj gnd ed., Wiley, New Yorkj p.A. Martynov, 1997, Classical Statistical Mechanics, Kluwer, Dordrecht] K Metzler, J. Klafter, 2000, The random walk's gmde to anomalous diffUSIOn: 3j Ifractional dynamics approach, Physics Reports, 339, 1-77 j [R. Metzler, J. Klafter, 2004, The restaurant at the end of the random walk: recenf f:!evelopments in the description of anomalous transport by fractional dynamicsj flournal oj Physics A, 37, RI61-R208.1 ~.w. MontroIl, M.E Shlesmger, 1984, The wondeftul world of random walks, Inl IStudles In Statistical Mechamcs, VoLlI, J. LebOWItz, B. Montroll (Eds.), North1 1H0Iland,Amsterdam, 1-121) KR. Nigmatulhn, 2006, FractIOnal kmetic equatIOns and umversal decouphng of ij Imemory functIOn m mesoscale regIOn, Phjsica A, 363, 282-298J p. Ya. Petnna, V.l. Gerasimenko, P.V. Mahshev, 2002, Mathematical FoundatIOn of! IClasslcal Statistical Mechamcs, 2nd ed., Taylor and FranCIS, London, New YorkJ 1338p.; and Naukova Dumka, Kiev, 1985. In Russianj p. Rastovlc, 2008, FractIOnal Fokker-Planck equatIOns and artificial neural net1 Iworks for stochaStiC control of tokamak, Journal oj Fusion Energy, 27,182-1871 f. Resl50ls, M. De Leener, 1977, Classical Kinetic Theory oj Fluids, Wiley, Newl IYork. SectIOn IXA.I k\.l. Saichev, G.M. ZaslavskY, 1997, FractIOnal kmetic equatIOns: solutIOns and apj IphcatIOns, Chaos, 7, 753-7641 IS.G. Samko, AA Kil5as, 0.1. Manchev, 1993, Integrals and Derivatives oj Frac1 rional Order and Applications, Nauka I Tehmka, Mmsk, 1987. m Russianj ~nd FractIOnal Integrals and Derivatives Theory and AppllcatlOns, Gordon and! IBreach, New York, 1993J KK. Saxena, AM. Mathai, H.J. Haubold, 2002, On fractional kmetic equatIOnsJ IAstrophysics and Space Science, 282, 281-287J IVB. Tarasov, 2004, FractIOnal generahzatIOn of LIOuville equations, Chaos, 14J

1123-127. ~E.

Tarasov, 2005a, Fractional systems and fractional Bogoliubov hierarchy equaj Itions, Physical Review E, 71, 011102j IV.E. Tarasov, 20055, FractIOnal LIOuville and BBGKI equations, Journal oj Physics.] IConjerence Series 7, 17-33J IVB.Tarasov, 2005c, FractIOnal generahzatIOn of gradIent and Hamiltoman systemsJ IJournalofPhysics A, 38, 5929-5943j ~E. Tarasov, 2005d, Stationary solutions of Liouville equations for non-I IHamiltoman systems, Annals oj Physics, 316, 393-4131 IVB. Tarasov, 2005e, FractIOnal generahzatIOn of gradIent systems, Letters in Math 1 ~matical Physics, 73, 49-58J ~E. Tarasov, 2005f, Fractional Fokker-Planck equation for fractal media, Chaos 15j 023102. ~E. Tarasov, 2006a, Fractional statistical mechanics, Chaos, 16, 033108j IVB. Tarasov, 20065, Transport equatIOns from LIOuville equatIOns for fractIOna~ Isystems, International Journal oj Modern Physics B, 20, 341-353.1

1352

15 Fractional Statistical Mechanid

IV.B. Tarasov, 2006c, FractIOnal vanatIOns for dynamIcal systems: HamIlton and Laj grange approaches, Journal of Physics A, 39, 8409-8425J IV.B. Tarasov, 200M, Contmuous lImIt of dIscrete systems WIth long-range mteracj Ition, Journal ofPhysics A, 39, 14895-14910.1 ~E. Tarasov, 2006e, Map of discrete system into continuous, Journal ofMathemat-I lical Physics, 47, 09290lJ ~.E. Tarasov, 2007a, Liouville and Bogo1iubov equations with fractional derivativesj Modern Physics Letters B, 21, 237-248J IV.E. Tarasov, 2007b, Fokker-Planck equatIOn for tractIOnal systems, InternatlOnall IJournal ofModern Physics B, 21, 955-967 j IV.E. Tarasov, 2008, Fractional vector calculus and tractional Maxwell's equatIOnsJ IAnnals oj Physics, 323, 2756-27781 ~.E. Tarasov, G.M. Zas1avsky, 2008, Fokker-P1anck equation with fractional coorj dinate derivatives, Physica A, 387, 6505-6512.1 IV. V. Ochmkm, 2003 a, Self-SImIlar anomalous dIffUSIOn and Levy-stable lawsJ IPhysics-Uspekhi, 46, 821-849~ IV.V. Uchmkm, 20mb, Anomalous dIffUSIOn and tractIOnal stable dIstnbutIOns, Jour~ Inal oj Experimental and Theoretical Physics, 97, 810-8251 IV.V. UchaIkm, 2008, Method oj Fractional Derivatives, ArtIshok, Ulyanovsk. Inl lRussian. k\.A. Vlasov, 1938, The VIbrational propertIes of an electron gas, Zhurnal Ekspenj Imental'noi i Teoreticheskoi Fiziki, 8, 29 I-3 I 8j k\.A. Vlasov, 1968, The VIbratIOnal properties of an electron gas, Soviet Physicsl IUspekhi, 10, 721-733J k\.A. Vlasov, 1945, On the kinetic theory of an assembly of particles with collectiv~ linteraction, Journal ofPhysics USSR, 9, 25-40.1 k\.A. Vlasov, 1961, Many-particle Theory and its Application to Plasma, Gordon! ~nd Breach, New York.1 k\.A. Vlasov 1966, Statistical Distribution Functions, Nauka, Moscow. In RussIan.1 A.A. Vlasov, 1978, Nonlocal Statistical MechaniCS, Nauka, Moscow. In RussIanJ R Weitzner, G.M. Zaslavsky, 2001, Directional fractional kinetics, Chaos, 11, 384-1 l396. IS.W. Wheatcraft, M.M. Meerschaert, 2008, FractIOnal conservation of mass, Ad~ Ivances in Water Resources, 31, 1377-1381.1 p.M. Zaslavsky, 1994, FractIOnal kmetic equation for Hamiltoman chaos, Physical ID, 76, 110-122j p.M. Zaslavsky, 2002, Chaos, fractIOnal kmetIcs, and anomalous transport, Physlcsl IReports, 371, 461-580J p.M. Zaslavsky, 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford Om1 Iversity Press, Oxford.1 p.M. Zaslavsky, M.A. Edelman, 2004, FractIOnal kmetIcs: from pseudochaotic dyj namics to Maxwell's demon, Physica D, 193, 128-147 ~ IL.M. Zelenyi, A.V. Milovanov, 2004, Fractal topology and strange kinetics: fromj IpercolatIOn theory to problems m cosmic electrodynamICs, Physics Uspekhi, 47J 1749-788.

References

353

[D.N. Zubarev, M.Yu. Novikov, 1972, Generalized formulation of the boundary conj !:lition for the Liouville equation and for BBGKY hierarchy, Theoretical and] [Mathematical Physics, 13, 1229-1238J

IPart IVI ~ractional

Temporal Dynamics

~hapter 1~

!Fractional Temporal Electrodynamics

116.1 Introductionl [The dielectnc susceptIbihty of a wide class of dielectnc matenals foIIows, overl ~xtended frequency ranges, a fractiOnal power-law frequency dependence that i~ ~aIIed the "umversal" response. The electromagnetic fields m such dielectnc medi~ ~an be descnbed by tractiOnal dIfferential equations (Tarasov, 2008a,b) with tImel ~envatIves of non-mteger order. An exact solutiOn of the tractiOnal equatiOns for ij ImagnetIc field is denved. We obtam equatiOns that descnbe "umversal" Cune-voIlj ISchweidler and Gauss' laws for such dielectnc matenals (Tarasov, 2008a). Thesel ~aws are represented by tractiOnal dIfferential equatiOns such that the electromag-I InetIc fields m the dielectnc matenals demonstrate "umversal" tractiOnal dampmgj I(Tarasov, 2008b). The typical features of "umversal" electromagnetic waves m dij ~Iectnc are common to a wide class of matenals, regardless of the type of physica~ Istructure, chemical composrtion, or of the nature of the polanzmg species, whethe~ ~ipoles, electrons or iOns.1 ~n SectiOn 16.2, the umversal response laws are discussed. In SectiOn 16.3, wei ~onsider a bnef reView of hnear electrodynamics of medmm to fix notatiOn and! Iprovide a convenient reference. In Section 16.4, the fractional equations for law§ pf umversal response are suggested. In SectiOn 16.5, we obtam the tractiOnal equa1 ~iOns of the Cune-von Schweidler law. In SectiOn 16.6, the tractional Gauss' law~ lor electnc field m dielectnc are discussed. In Sections 16.7-16.8, the tractIona~ ~lectromagnetIc wave equatiOns, which can be considered as umversal equatiOnsJ lare suggested. In Section 16.9, we discuss the fractional damping effects for magj ~etic field in dielectric. Finally, a short conclusion is given in Section l6.10j

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

16 Fractional Temporal Electrodynarnics

1358

116.2 Universal response law~ IA groWIng number of dlelectnc relaxatIon data show that the claSSIcalDebye behav-I [or (Debye, 1912; Debye and Huckel, 1923; Debye, 1945) IS hardly ever observe~ ~xpenmentally (Jonscher, 1983, 1996, 1999; Ramaknshnan and LakshmI, 1984). Inj Istead It has been denved (Jonscher, 1983, 1996, 1999; Ramaknshnan and LakshmIJ [984) that power-laws are a common feature of the dlelectnc response of most maj ~enals for wIde frequency ranges. The fact that dIfferent dlelectnc spectra can bel klescribed by the power-laws is confirmed in many measurements (Jonscher, 1983j [996, 1999) for a WIde class of vanous substances. The dlelectnc susceptIbIlIty ofj ~ost materials follows, over extended frequency ranges, a fractional power-law frej guency dependence, whIch IS called the law of unIversal response (Jonscher, 1983J ~ 996, 1999). This law is found both in dipolar materials beyond their loss-peak frej guency, and In matenals where the polanzatlon anses from movements of eIthe~ lionic or electronic hopping charge carriers. These power-law responses are mosf ~asIly dIsplayed In terms of the dlelectnc susceptIbIlIty:1

Ix(«i) =

X'(ro) - i X"(ro ~

las a functIon of frequency ro. It has been found (Jonscher, 1977, 1978; Ngm e~ lal., 1979) that the frequency dependence of the dlelectnc susceptIbIlIty follows ij ~ommon unIversal pattern for almost all kInds of matenals. The behavIO~

X"(ro)rvron

1,

O
ro»rop ,

(16·1)1

(16.2)1

Iwhere X' (0) is the static polarization and rop the loss-peak frequency, is observe~ pver many decades of frequency. Expressions (16.1) and (16.2) serve as the definij ~IOn of the unIversal response behavIOrl IAn Important consequence of the power laws IS that the ratIO of the ImagInary tq ~he real component of the susceptIbIlIty IS Independent of frequency. The frequenc)1 klependence given by Eq. (16.1) implies that the real and imaginary components ofj ~he complex susceptibility X(ro) = X'(ro) - i X"(ro) obey at high frequencies thel lrelation-

~ X'(ro) = cot (Jrn) 2 '

(16.3)1

[The expenmental behavIOr of Eq. (16.2) leads to a SImIlar frequencY-Independentl lrule for the low-frequency polanzatIOn decrementj 1

X"(ro) X'(O)-X'(ro)

= tan

(Jrm) 2

'

116.2 Universal response laws

35~

[his being a unique consequence of Kramers-Kronig relations and does not depen~ pn any particular physical processj [here are many models that describe and explain the universal response laws. Le~ Ius note some of them. The memory function approach and scahng relationshIps canl Ibe used (Ryabov and Feldman, 2002) as a basIs for the dynamIc model of symmetncl klielectric spectrum broadening. The correspondence between the relaxation timeJ ~he geometrical properties, the self-diffusion coefficient and the exponent of powerj ~aw wings was established in (Ryabov and Feldman, 2002). In Ref. (Ryabov an~ feldman, 2003), a memory function equatIOn and scahng relatIOnshIps were used! ~or the phySIcal mterpretatIOn of the Cole-Cole exponentj

Iwhere Xo IS a static susceptibIhty, and W p IS the frequency of the peak of the dlelec1 ~nc losses. The correspondence between the relaxatIOn time, the geometncal prop-I ~rties, the self-dIffUSIOn coeffiCIent and the Cole-Cole exponent was conSIdered ml I(Ryabov and Feldman, 2003). The fractIOnal power-laws frequently observed m dI1 ~Iectnc measurements can be mterpreted (Frennmg, 2002) m terms of regular smgu1 liarpomts of the underlymg rate equation for the process gIvmg nse to the dlelectnci Iresponse. The reVIew (Jonscher, 1999) presents a wlde-rangmg broad-brush PICturg pf dlelectnc relaxatIOn m sohds, makmg use of the eXIstence of a umversahty ofj ~helectnc response regardless of a wIde dIverSIty of matenals and structures, wIthl ~hpolar as weIl as charge-carner polanzatIOn.1 IWe note general equatIOns for the susceptibIlIty of dIsordered systems IS pro-I Iposed m (Bergman, 2000). In the tIme domam, the Kohlrausch-WI1hams-Watt~ Istretched exponentl lIS used (see also (Bergman, 2000; MI1ovanov et. ai, 2008, 2007)). In Ret. (Brouersl land Sotolongo-Costa, 2005), the authors derived a general two-power-law relaxj latIOn functIOn for heterogeneous matenals usmg the maXImum entropy pnnCIple fofj r.onextensIve systems. The asymptotIc power-law behaVIOrs comCIde WIth those ofj ~he Weron's generalIzed dlelectnc functIOn denved m the stochastIc theory from ani ~xtensIOn of the Levy central lImIt theorem. These results (Brouers and Sotolong01 rosta, 2005) agree WIth the Jonscher's umversahty pnncIple (Jonscher, 1983, 1996J If999).

[n Refs. (Nigmatullin, 1986; Nigmatullin and Ryabov, 1997; Novikov and Prij Ivalko, 2001; YI1maz et aI., 2006; NIgmatulhn et aI., 2007) fractIOnal calculus wa~ lundertaken m order to understand the nature of non-exponential relaxatIOn. In (NIg1 r.Iatulhn, 1986) the fractal geometry of an mhomogeneous medIUm was used ml ~he explanatIOn of the umversal response phenomenon. In Ret. (NIgmatuIlm and! [Ryabov, 1997), an equatIOn contammg operators of fractIOnal mtegratIOn and dIfj [erentiation was obtained and solved, which the relaxation function obeys in thi§ ~ase. In Ret. (NOVIkov and Pnvalko, 2001), the fractIOnal denvatives were used tq pbtam all three known patterns of anomalous, nonexponential dlelectnc relaxatIOlJI

16 Fractional Temporal Electrodynarnics

1360

pf an inhomogeneous medium in the time domain. It was assumed (Novikov and! IPrivalko, 2001) that the fractional derivative is related to the dimensionality of aI ~emporal fractal ensemble in a sense that the relaxation times are distributed overl la self-similar fractal system. In Ref. (Yilmaz et al., 2006), the model of dielectricl IrelaxatIOn based on the fractIOnal killetlcs contaillillg the complex power-law expo-I rents was used to receive a confirmation in description of the real part of the comj Iplex conductivity by the fitting function. The coincidence between the suggested! fuodel and measured data gIves (YIlmaz et al., 2006) the pOSSIbIlIty to suggest ij Imore relIable physIcal pIcture of the swellIng process that takes place ill neutrall land charged gels. The real process of dlelectnc relaxatIon dunng the polymenzaj ~Ion reactIOn can be descnbed (Nlgmatuilln et al., 2007) ill terms of the fractIona~ Ikinetic equations containing complex-power-law exponents. Based on the physica~ landgeometrical meanings of the fractional integral with complex exponents there i§ la pOSSIbIlIty (Nlgmatuilln et al., 2007) of developillg a model of dlelectnc relaxatIoIlj Ibased on the self-SImIlar fractal character of the averaged mIcroprocesses that take§ Iplace ill the mesoscale regIOnJ

116.3 Linear electrodynamics of mediuD1j ~et

us conSIder a bnef reVIewof lInear electrodynamIcs of medIUm (Jackson, 1998)1 fix notatIOn and proVIde a convement referenceJ [The behaVIOr of electnc fields eE, D), magnetIc fields (B, H), charge densltYI I(p(t,r)), and current density (j(t,r)) is described by the Maxwell's equationsj ~o

~ivD(t,r)

purlE(t,r) =

= p(t,r), -

~ivB(t,r)

dB(t,r)

at '

= 0,

(16.5)1 (16.6)1

(16.7)1

pUrlH(t,r) =j(t,r) + dDt,r). [The densities p (t, r) and j (t, r) describe an external source of field. We assume tha~ ~he external sources of electromagnetIc field are gIvenl [n lInear matenals the quantItIes of magnetIC fields (B, H) can be lillearly related:1

r(t,r)

=

~B(t,r),

(16.9)1

Iwhere /lIS the magnetIc permeabIlIty of the medIUm.1 [I'he relatIOn between electnc fields (E, and D) can be defined by the polanzatIOIlj density P(t,r) asl

lD(t,r)

=

foE(t,r) +P(t,r),

(16.l0~

36~

116.3 Linear electrodynamics of medium

Iwhere £0 is the electric permittivity of free space. The polarization density P(t, r) lis the vector field that expresses the density of permanent or induced electric dipol~ ~oments in a dielectric material. The vector P(t, r) is defined as the dipole momentl Iperumt volume] [n a homogeneous hnear and IsotropICdlelectnc medIUm, the polanzatIOn IS pro-I Iportional to the electric fieldj IP(t,r) = £oXE(t,r).

(16.11]

[n the case of a vacuum, X = O. The electric susceptibility X of a dielectric materiall lIS a measure of how easIly It polanzes In response to an electnc field.1 [n general, a matenal cannot be polanzed Instantaneously In response to an apj Iplied field. The quantities P(t, r) and E(t, r) are generalIy related byl

r(t,r)

=

£0 L~oo K(t,t')E(t',r)dt'.

(16.12~

!Homogeneity in time gives K(t,t ') = K(t - t'), and causality requires K(t) = 0 fo~ r < O. The function K(t) is calIed the dielectric response. Using the unit step func-I ~ion, we can write down K(t) = X(t)8( -t), and so the equation for P(t,r) as ~ functIon of time IS

Ip(t,r)

= £0

[00 X(t -t')E(t',r)dt',

(16.13~

Iwhere X (t) is the electric susceptibility of the medium. Equation (16.13) meansl ~hat the polanzatIOn IS a convolutIOn of the electnc field at prevIOUS tImes wIthl ~Ime-dependent susceptIbIhty. An Instantaneous response corresponds to DIrac deItij ~unction susceptibility X(t) = X 8(t). Then Eq. (16.13) gives P(t, r) = £oXE(t, r)l [t is convenient to take the Fourier transform and write relationship (16.13) as aI ~unctIOn of frequency. Then we are Interested In propertIes of the Founer transforml 1(§K)(w) of the kernel K(t), and the Fourier transform X(w) = (§X)(w) of thel ~lectric susceptibility X(t). We can writel

IX(W) = X'(w) - iX"(w) =

K(t)cos(wt)dt, ~quatIOns

l'"

X"(w) =

K(t)e-iW!dt,

o

K(t)sin(wt)dt.

(16.14~

(16.15

(16.15) give the foIIoWIng propertIes:1

Ix'(-w) =

X'(w),

X"(-w)

lX' (0) = L'" K(t )dt,

=

-X"(w)j

X" (0) = 0·1

Rsing the convolution theorem, the integral (16.13) disappears, and we

hav~

16 Fractional Temporal Electrodynarnics

1362

IP(oo,r) = £oX(oo)E(oo,r),

(16.16j

Iwhere X( (0) is defined by (16.14). Thefunctions P( oo,r), andE( 00, r) are the Fourierl han storm s' 1 1 P(oo,r) = (»>P) (oo,r) = 2n 1_= P(t,r)e-lW1dtj

r:

-

.

1

~ t(oo,r) = (»>E)(oo,r) =

lr+=. 2n 1-= E(t,r)e-lWldt. 1

[The susceptibility X( (0) is a function of the frequency of the applied field. Whenl ~he field is an arbitrary function of time t, the polarization is a convolution of thel fourier transform of X(00) with the :E( 00, r). This reflects the fact that the dipoles inl ~he materIal cannot respond mstantaneously to the applIed field (Slim and RuhadzeJ ~ 961; Kuzelev and Rukhadze, 2009)j

116.4 Fractional equations for laws of universal

respons~

OC:et us consIder the laws of unIversal response m a hIgh-frequency regIOn. For 00 ROJ}, the unIversal fractIonal power-law (16.1) can be represented m the form:1

Ix(oo)=Xa(ioo) u, IWIth some positrve constant Xa and

a

=

»1

(16.17~

O
1 - n. Usmg

Iwe get

IX'(oo)

=

Xa Ioolacos(sgn(oo)¥)j

k"(oo) = Xa Ioolasin(sgn(oo)¥) J ~s

a result, we obtain relation (16.3)j IUsmg Eq. (16.16), the polarIZatIOn densIty can be wntten

r(t,r)

= »>-1 (P(oo,r)) = £0»>-1

a~

(X(oo)E(oo,r)) ,

(16.18~

Iwhere P(oo,r) is a Fourier transform»> of P(t,r). Substitution of (16.17) intol 1(16.18) gives IP(t,r) = £oXa»> 1 ((ioo) aE(oo,r)). (16.19~

36~

116.4 Fractional equations for laws of universal response

~quatIOn (16.19) can be represented by mtegrals of non-mteger order a = 1 - nJ [The left-sided LIOuvIlle fractIOnal mtegral (Samko et aL, 1993; KIlbas et aL, 2006)1 lis defined by ex 1 f(t')dt'

(/+f)(t)

=

Jt

(16.20~

rya) _= (t-t,)I-ex'

I

[f we define the Fourier transform operator!# b5J

(16.21~ ~hen

the Fourier transform of integral (16.20) for f(t) E LI (lR), is given (see Thej prem 7.1 m (Samko et aL, 1993) and Theorem 2.15 m (KIlbas et aL, 2006)) by thel lrel atiOil"

1(§/~f)(W) =

dF(§f)(W)J

k\s a result, the fractional power-law (16.17) give§

r(t,r)

= €oXex(/~E)(t,r),

0
1.

(16.22j

[This equation means that polarization density P( t, r) for the high-frequency regioij liS proportIOnal to the LIOuvIlle fractIOnal mtegral of the electnc field (Tarasov J 12008a,b). OC:et us conSider the laws of umversal response m a low-frequency regIOn. Fo~ « wp , the universal fractional power-law (16.2) can be represented a~

r>

~(W)=x(O)-Xf3(iW)f3,

0<13<1

(16.23~

Iwith some positive constants X/3' X(O), and f3 = m. It is not hard to prove that EqJ 1(16.4) IS satIsfied. The law (16.23) can be represented by the left-Sided LIOuvIllel kractional derivative (Samko et aL, 1993; Kilbas et aL, 2006) that is denoted by D~.I [The differential operator D~ of order 13 is defined by the equationl

f3 _ k k-f3 _ 1 a (D+f)(t) -Dt(I+ f)(t) - r(k-f3) atk

Jt -c-co

f(t')dt' t-t' f3-k+I'

(16.24

Iwhere k - 1 < {3 < k. If we define the Fouriertransfonn operator § by Eq. (16.21),1 ~hen the Fourier transform § of fractional derivative (16.24) for f(t) E LI (lR) (seel [Theorem 7.1 in (Samko et al., 1993) and Theorem 2.15 in (Kilbas et al., 2006» i~ given by (§D~f)(w) = (iw)f3(§f)(w), 0 < 13 < 1.1 V\s a result, the fractIOnal power-law (16.23) gives the polanzatIOn densltyj

IP(t,r) = Eo§-l (X(w)E(w,r)) lin the form'

(16.25~

16 Fractional Temporal Electrodynarnics

1364

IP(t,r)

= eoX(O)E(t,r) - eoX/3 (D~E)(t,r),

0 < f3

< 1.

(16.26~

[his equation means that polarization density P(t, r) for the low-frequency region i§ klefinedby the Liouville fractional derivative of the electric field (Tarasov, 2008a,b )j ~quations (16.22) and (16.26) can be considered as the universal response law§ l(Jonscher, 1983, 1996, 1999) for time-domain. These equations allow us to derivcj ~ractIOnal equatIons for electnc and magnetIc fields (Tarasov, 2008a,b)J

116.5 Fractional equations of the Curie-von Schweidler laW [Let us consider the Curie-von Schweidler law (Curie, 1889a,b; von SchweidlerJ [907). USIng (16.22) and (16.26), the polanzatIOn current densIty:1

(16.27~

IJpol(t,r) = D!P(t,r) lIS descnbed by the tractIOnal equatIonsj

~pol(t,r) = eoXa (D~ aE)(t,r),

0 < a < 1,

~POI(t,r) = eoX(O)D!E(t,r) - eoX/3 (D~+f3E)(t,r),

0 < f3

(16.28j

< 1.

(16.29~

for constant electric field E(t, r), Equations (16.28) and (16.29) show that the timel klependence of the relaxation of the polarization current density (16.27) after thel Isuddenremoval of a polanZIngfield follows the power-laws, whIch are wIdely obj IservedIn practIce (Jonscher, 1983) and are known as the Cune-von Schweldler lawl I(Curie, 1889a,b; von Schweidler, 1907). For the changeable field E(t, r), Equation~ 1(16.28) and (16.29) can be consIdered as a generalIzatIOn of the well-known Cune1 Ivon Schweldler law. Let us consIder some examples of thIS generalIzatIOn]

!Example IJ IUsIng (16.28) and (16.29), we can denve the usual Cune-von Schweldler law. Thel Imost elementary of the applied field E( t, r) is the step function:1 ~(t,r) =

Eo(r)H(t -a),1

Iwhere H(t) is the Heaviside function, also called the unit step function. The Heav~ liside function H(t) is a discontinuous function, whose value is zero for negativ~ largument and one for positive argument. In this case, Equations (16.28) and (16.29) give land

365]

116.5 Fractional equations of the Curie-von Schweidler law

Iwhere t

> a and aDf is the fractional derivative:1 (16.30

land k - 1 <

a < k. Using the relation:1 t > a,

a>o,1

Iwe obtain the usual Curie-von Schweidler law in the formJ

E(t,r)

(t-a)a-I

eoXaEo(r)

=

ra (t-a)-

E(t,r)=-eoX{3Eo(r) Iwhere t

0 < a < 1,

,

-I

r(-f3) ,

0<13<1,

(16.31

(16.32

> a.

OO:xample 2J [Theexperimental applied field E(t, r) can be represented a~

OO:(t,r)

=

(16.33~

Eo(r) sin(Xt).

IOsmg the relatIon (Samko et al., 1993)j (16.34~ ~quatlOns

(16.28), (16.29) and (16.33) gIVel ~pol(t,r)

[or 0

= eoXaEo(r)A.. u sin (At + (1- a)n/2)

< a < 1, andl

~POI(t, r) = eoEo(rl(.i (0) A.. cos(A..t) - X{3A.. 1+{3sin (At + (1 + f3)n /2)) ~or 0

(16.35~

(16.36~

< f3 < 1. Equation (16.36) can be rewritten in the forml ~pol(t,r)

= eoA(f3)Eo(r) sin(Xt + cp(f3)),

(16.37~

Iwhere A([3) and p([3) are the amplitude and phase changes. These functions arel ~efined by the equatlOnsj

r(f3) =

Ja

2(f3)

+ b2 (f3 ), cp(f3) =

arctan

(:~~;) j

16 Fractional Temporal Electrodynarnics

1366

la(f3) = X(O)A - X[3A 1+{J cos(f3n/2)J Ib(f3) = X[3A 1+13 sin(f3n/2),1 land0 < f3

< 1.1

OO:xample 3J for the applied field:1

IE(t,r) = Eo(r) H(t - a) g(t)J

Iwith some function g(t), exact expressions for the polarization current densitYI IJpol(t,r) can be derived by using the list of fractional derivatives of the functiotj Ig(t) (see Tables 9.1-9.3 in (Samko et al., 1993)). For g(t) = (t-ay, where s > -lj Iwe have ----:-----'--------'----,--(t - a y-a,

s > -1, a > O.

(16.38

[The fractional derivative of g(t) = cos(X(t - a)) isl

t-a Dfcos(A(t-a)) = 2r(1- a) (I Fl(l, l-a,iA(t-a))+ IF1(1, I-a, -iA(t-a))), Iwhere IFt{a,b,c) is a hypergeometricfunction (Erdelyi, 1981). For g(t) = exp{ -Xt},l wetrse

(16.39~ Iwhere Ea,13[z] is the Mittag-Leffler function (Miller, 1993; Gorenflo et al., 19981 12002):

(16.40~ ~f

a = f3 = 1, then El,l [z] = exp{ z}, where nk + 1) = k! for positive integer kJ

V\s a result, fractIOnal relatIOns (16.28) and (16.29) can be consIdered as a gener1 lahzatIOn of the formulatIOn of the Cune-von Schweldler law (Tarasov, 2008a) froi11l la constant electric field into changeable fields E(t, r) J

[6.6 Fractional Gauss' laws for electric field [n electrodynamIcs, Gauss' laws are laws relatIng the dIstnbutIOn of electnc charg~ ~o the resultIng electnc field (Jackson, 1998). In Integral form, Gauss' law states tha~ ~he electnc flux through any closed sufface IS proportIOnal to the enclosed electnq ~harge. In dIfferentIal form, Gauss' law states that the dIvergence of electnc dIS1 placement field D(t,r) is equal to the charge density p(t,r). The universal response' ~aws are represented by fractional integral equation (16.22) for the high-frequencyl

1166 Fractional Gauss' laws for electric field

3671

IregIOn and fractIOnal dIfferentIal equatIOn (16.26) tor the low-frequency regIOn. Osj ling the Gauss' lawj k1ivD(t,r) = p(t,r)j land the equatIOnj p(t,r)

= foE(t,r} +P(t,r),1

§J divE(t,r) +divP(t,r)

=

p(t,r).

(16.4q

ISubstitution of (16.22) and (16.26) into (16.41) give§ ~'P(t,r) +foXa(I~'P)(t,r)

IfoX(O)'P(t,r) - foXf3(D~ 'P)(t,r)

= p(t,r), 0 < a < 1,

= p(t,r),

0

< f3 < 1,

(16.42~

(16.43~

Iwhere 'P(t,r) = divE(t,r). Using Lemma 2.20 of (Kilbas et aI., 2006) in the forml rD~I~'P)(t,r) = 'P(t,r)J

Iwe obtam the equatIon§

(D~'P)(t,r) + xa'P(t,r) = ~(D~p )(t,r), f3 X0 (D+'P)(t,r) - -'P(t,r)

X

1 = --p(t,r),

foX

0 < a < 1,

0 < f3

< 1.

(16.44

(16.45

[I'hese equatIons represent fractIonal dIfferentIal equatIons of Gauss' law tor thel ~lectric field E(t, r) 1 ~et us consider the integral form of the Gauss' law for the electric field E(t, r) .1 [Thetotal electnc charge of the region W of medIUm IS defined byl

[The electric field E = Iflux:

E(t,r) passing through a surface S = I
=

h(E,dS)

=

aw gives the electriq

fw diVEdvj

IWe assume that the regIOn W IS a fixed (statIOnary) regIOn of medIUm. Then the mtej gration of Eqs. (16.44) and (16.45) over the region W gives the fractional equationsj (D~
e

X0

1

(D~
X

foX

0 < a < 1, 0 < f3 < 1.

(16.46

(16.47

16 Fractional Temporal Electrodynarnics

1368

[hese equations represent the integral Gauss' laws for the electric field in dielectricl §edia (Tarasov, 2008a). Note that D% is the differential operator of order a that i§ kiefinedby Eq. (16.24). IfE(t,r) is defined byl IE(t,r) = Eo(t,r)H(t -a)j Iwhere H(t) is the Heaviside function, then the Liouville fractional derivative D~ ~ransforms into the Riemann-Liouville derivative aDf, which is defined by Eql 106.30). IWe can consider Eqs. (16.46) and (16.47) in the general formj IaD~U(t)-AU(t)=f(t),

O
(16.48j

Iwhere aDf is the Riemann-Liouville fractional derivative (16.30), and U(t) repj Iresents the values P(t,r) or cI>E(t). In this case, the parameter X is -Xa o~ Ix(O)!x13' Thefunction f(t) is defined as (1/ eo) aDf p (t, r) and (-1/ eoX13 )p(t, r)j pr (1/ eo) aDfQ(t) and (-1/ eoX13 )Q(t)l ~et us consider the Cauchy type problem (Kempfle and Schafer, 2000; Fuku1 raga and Shimizu, 2004; Heymans and Podlubny, 2006) for tractiOnal dtfferentIa~ ~quatiOn (16.48) with the Initial conditIonsj (16.49j Iwhere aD~ 1 = all a is the Riemann-Liouville fractional integral with 0 < a < 11 ~f f(t) is an integrable function on (a,b), then the problem (16.48), (16.49) has thel lumque solutiOn (Barrett, 1954) (see also Theorem 4.1 and Example 4.1 In (Kdba~ ~t aL, 2006)) grven b51

(t) = C (t - a)a-l Ea,a[A(t - a)a] +

1

(t - t,)a-l Ea,a[A(t - t,)a]f(t')dt'

(16.50) Iwhere Ea,alz] is the Mittag-Leffler function defined by (16.40). For f(t) = 0, wei bbtaIn (16.51j [To consider the asymptotic behaviOr of the solutiOns (16.50) and (16.51), we canl lusethe Integral representatiOn (Podlubny, 1999; Gorenflo et aL, 2002; MaInardi and! porenflo, 2000; KUbas et al., 2006) of the Mittag-Leffler function:1

lEa

,13

[z] =

~1 ~a-f3expg} d~. 2n

y(a)

~a_z

[Thepath of integration y(a) is a loop, which starts and ends at - 0 0 and encircles thel Fircular disk Is I ~ Izil/ a in the positive sense: larg(z) I ~ non y(a). The integran~ lin (16.52) has a branch point at S= O. The complex S-plane is cut along the negativ~ Irealaxis, and in the cut plane the integrand is single-valued: the principal branch ofj

36~

116.7 Universal fractional equation for electric field

~a is taken in the cut plane (KUbas et al., 2006). Then the asymptote of (16.52) ha§ OChe form (Gorenflo et al., 2002; Mamardl and Gorenflo, 2000; KIlbas et al., 2006)l

117 a ,j3[z]

r

Er(f3 I

00

=-

I

ak) Zk'

Izl---->

00,

(16.53~

land (5 ~ arg(z) ~ 7[, In our case, Z = X(t - a)a, arg(z) = 7[, As a result, we aITiv~ lat the asymptote of the solutIon, whIch exhIbIts power-lIke taIls. These power-lIktj ~ails are the most important effect of the non-integer derivative in the fractiona~ 1

1

~quations.

~6.7

Universal fractional equation for electric field

IOsmg the tIme-domam laws of dIelectrIc responses, we can obtam the wave equa1 ~ions with fractional time derivatives for electric field E(t, r) in dielectric medial [Thetime derivative of Maxwell equation (16.8) give~

d D(t,r) dt 2 IUsmg (16.9), we

1

1.

curlDtH(t,r) -DtJ(t,r).

=

ge~

d D(t,r) d2 t

=

I

1

1.

-curlDtB(t,r) -DtJ(t,r) f.1

from Eq. (16.6), we obtainl

d D(t,r) d2 t

I

= --curl curlE(t,r)

f.1

1.

-DtJ(t,r).

(16.54

IUsmg the relatIOnj

~url curlE = grad divE -

v2 E ,1

Iwe rewnte Eq. (16.54) asl

d D(t, d 2 r) t

. 2 ) +-I (graddIvE-VE

f.1

1. ( ) =-DtJt,r.

(16.55

ISubstItutIOn of (16.10) mto (16.55) gIvesl Eo

d E(t, 2 ( . E>- V2E ) d 2 r) +DtP t,r ) + -I ( grad div t

f.1

=

1. ( -DtJ t,r ) .

(16.56

for the region w » wp , the polarization density P(t, r) is related with E(t, r) byl (16.22). SubstItutmg (16.22) mto (16.56), we obtam the fractIOnal equation fo~ electric field' ~q.

16 Fractional Temporal Electrodynarnics

1370

12 a E(t,r) 2-a)( . a 2 + 2Xa( D+ E t,r ) + (grad dIVEV 2E ) v t v

1.() = -J.lDtJ t,r,

(16.57

Iwhere v2 = l/(EoJ.l) and 0 < a < 1. Note that divE -=I- 0 for p(t,r) = oj for the region (0« (Op, the fields P(t, r) and E(t, r) are connected by Eq. (16.26)j [Then Eq. (16.56) givesl (16.58 Iwhere 0 < {3 < 1, andl

~quations ~he

(16.57) and (16.58) describe the time evolution of the electric field inl dIelectrIc materIals (Tarasov, 2008b). These equatIons are tractIOnal ddferentIa~

~quatIOns.

116.8 Universal fractional equation for magnetic fieldl IOsmg the tIme-domam laws of dIelectrIc responses, we can obtam the wave equa1 ~ions with fractional time derivatives for magnetic field B( t, r) in dielectric media] ISubstItutIOn of (16.9) and (16.10) mto (16.8) gIvesl (16.59 [The time derivative ofthe Faraday's law (16.6)j

la 2 B(t , r ) _ at2

-

_

cur

I aE(t,r)

at

(16.60~

.

[Then we havel

-----,:-----,: -

~ j (t,r) . Eo

(16.61

ISubstituting(16.61) into (16.60), we obtainl

a B(t,r) a2 = t

a

1 1 1. --curl curlB(t,r) +-:;-cur1P(t,r) +-curlJ(t,r). £0

Eo at

£0

(16.62

[Using the re1ationj Icurl curlB = graddivB- V 2 B,

(16.63)

37~

116.8 Universal fractional equation for magnetic field

land div B = 0, we getl

a 2B(t,r) 1 2 1 a 1. 2 = - \ 7 B(t,r) + -:;-curlP(t,r) + -cur1J(t,r). t £0 £0 at £0

a

(16.64

for the case X(w) = X = const and homogeneous (or potential) current densitYI I(curlj(t,r) = 0), Equation (16.64) has the usual form:1

a 2B(t,r) 1 2 a a 2 - - \ 7 B(t,r) - X:;-curlE(t,r) = 0 t £0 at [Using Eq. (16.6) and f = £0(1 + X), Equation (16.64) gives the well-known wavel ~guation:

a B(t,r) _ _1 n2B( )= 0 v t, r . :l

at

2

f

for the power-laws frequency dependences of X( w), we obtain fractional differen-I ~Ial equatIOns.1 ~n the general case, we have X(w) =1= const depends on w. For the region W » wpj ~he polarization density P(t, r) is related to E(t, r) by Eq. (16.22), and we obtain thel ~ractIOnal eguatIOnj

a B(t,r) 1 2 Xa I a 1. a 2 = - \ 7 B(t,r) + -DtI+curlE(t,r) + -curlJ(t,r). t £of.1 £0 £0

(16.65

Rsing Eg. (16.6), we can represent (16.65) in the formj

a B(t,r) 1 2 Xa 2 a 1. a 2 = - \ 7 B(t,r) - -DtI+B(t,r) + -curlJ(t,r). t

fo

£0

fo

(16.66

[The experimental applied field B(t, r) can be represented a~ ~(t,r)

= B(t,r)H(t)j

Iwhere H (t) is the Heaviside function, and we use the same notation for the magneticl lfield. As a result, we obtam the tractIOnal equation tor magnetic field:1

1 a B(t, Xa ( 2-a) ) ) 2" a 2 r) +20Dt B (t,r ) -\72 B ( t,r =f.1curlJ• ( t,r, v t v

(16.67

Iwhere 0 < a < 1, v2 = 1/ (£of.1 ), and oD~ a is the Riemann-Liouville derivativg I(Kilbas et al., 2006) on [0,00) such tha~

t

2-a 1 a f(t')dt' (oD+ f)(t)=qa)at 2}0 (t-t'F-a' for the regIOn W

«

O
wr ' we use (16.26). As a result, we obtaIij

16 Fractional Temporal Electrodynarnics

1372

I 2" v{3

a B(t,r) a2 t

a{3

2+{3

2" (oD t

2

B)(t,r) - V' B(t,r)

v{3

•

= J.l curl jtr.r},

0 < f3

for the case of the homogeneous or potential current density (curlj(t,r) ~ions (16.67) and (16.68) givcj

a2B(t ,r)

1-~2-----'--+Xa


= 0), Equa-I

(2-a) 2 2 oDt B (t,r)-vV' B(t,r) =0,

(16.69

(16.70 ~quations (16.67) and (16.68) are fractional differential equations that describe thel rIagnetIc field In dielectnc media (Tarasov, 2009)1 ~quatiOns (16.57) and (16.58) for electnc fields and Eqs. (16.67) and (l6.68~ ~or magnetic fields are umversal equatiOns (Tarasov, 2008b) for waves In dielectncl Imatenals SIncethey are common to a wide class of matenals, regardless of the typg pf physical structure, chemical composition or of the nature of the polanzatIonJ

116.9 Fractional damping of magnetic field] ~edefimng

parameters, EquatiOns (16.67) and (16.68) can be represented In the gen1

era] form'

Iwhere B( a, r) = 0, and the curl of current density of free charges is considered as ani external source terml r(t,r) = J.l A2 curlj(t,r)l !Equation (16.71) gives Eq. (16.67) for a

= 2, I < f3 < 2, ane]

!Equation (16.68) can be derived from (16.71) with 2 <

a < 3, f3 = 2, ane]

V\n exact solutiOn of (16.71) can be represented In terms of the Wnght functiOn~ I(Kilbas et aI., 2006; Mainardi and Gorenflo, 2000). Using the 3-dimensional Fourierl ~ransform of Eq. (16.71) with respect to coordinates and Theorem 5.5 of (Kilbas e~

373

116 10 Conclusion

laL, 2006), we obtam (Tarasov, 2008b) the solutIOnl (16.72

IWl.1ere

G(-r,k)

=

2)' 1lI'l [ f (- k2A (s+ 1, 1) IAI-rCX-f3] -r cxs+cx s! (as+a,a-f3)

I.

(16.73

=

[TheWright function IIf'I is defined by the series:1 If'I [ (s+ 1, 1)

(as+f3,a)

Iz] _ -

qs+k+ 1)

l

~ qas+f3+ak) k!' 00

IWe note that the Wright functions can be represented as the integer-order derivative~ ID~Ecx,[3 [z] of the Mittag-Leffler function Ecx,[3[z] (see (KUbas et al., 2006; Mainardil land Gorenflo, 2000))j ~quation (16.72) describes the fractional field damping of magnetic field in thel ~iIelectnc medIa. An Important property of the evolutIOn descnbed by the fractIona~ ~quatlons for electromagnetIc waves IS that the solutIons have power-lIke taIls]

[6.10 Conclusionl IWe prove that the electromagnetIc fields and waves m a WIde class of dlelectncl Imatenals are descnbed by fractIOnal dIfferentIal equatIOns wIth denvatlves of non-I Imtegerorders wIth respect to tIme. The orders of these denvatIves are equal to 2 - al land 2 + f3, where the parameters 0 < a = 1 - n < 1 and 0 < f3 = m < 1 are defined! Iby exponents nand m of the umversal response laws for frequency dependence ofj ~he dlelectnc susceptIbIlIty. The typIcal features of the "umversal" electromagnetIq Iphenomenon and the suggested fractIOnal equatIOns (Tarasov, 2008a,b) are commolJl ~o a WIde class of matenals, regardless of the type of phYSIcal structure, chemlcall ~omposltIOn or of the nature of the polanzmg speCIes, whether dIpoles, electrons ofj lIOns. We assume (Tarasov, 2008b) that the lInk between the dlelectnc loss of IOW1 Iloss matenals and the phySIcal/structural features m the matenal can be connecte~ IWlth a fractIOnal screemng of dIpole-dIpole mteractIOnsl for small fractionality of a (or {3), it is possible to use £-expansion (Tarasovl land Zaslavsky, 2006) over the small parameter £ = a (or £ = 1 - f3). We note tha~ ~he suggested fractIOnal dIfferentIal equatIOns, whIch descnbe the electromagnetH:j lfieldm dlelectnc medIa wIth power-law response, can be solved numencaIIy. Then~ lare several numencal methods to solve fractIOnal equatIOns (Samko et aL, 1993j porenflo, 1997; Agrawal, 2002; TadJeran et aL, 2006). For example, the Grtinwald-I ~etmkov dIscretIzatIOn scheme (Samko et aL, 1993) can be used to compute frac1 ~ional equations for electromagnetic field in dielectricj

1374

16 Fractional Temporal Electrodynarnics

lReferencesl P.P. Agrawal, 2002, SolutIOn for a fractIOnal dIffusIOn-wave equation defined m 3j Ibounded domam, Nonlinear Dynamics, 29, 145-155J ~.H. Barrett, 1954, DIfferential equatIOns of non-mteger order, Canadian Journal otl IMathematics, 6, 529-54U K Bergman, 2000, General susceptibIhty functions for relaxations m disordere~ Isystems, Journal ofApplied Physics, 88, 1356-1365j f. Brouers, O. Sotolongo-Costa, 2005, RelaxatIOn m heterogeneous systems: A rarel ~vents dommated phenomenon, Phjsica A, 356, 359-374J [. Curie, 1889a, Recherches sur Ie pouvoir inducteur specifique et la conductibilit~ kles corps cristallises, Annales de Chimie et de Physique, 17, 385-434. In Frenchj ~. Cune, 18895, Recherches sur la conductiblhte des corps cnstalhses, Annales dlj IChimie et de Physique, 18, 203-269. In Frenchj IP. Debye, 1912, Some results of kinetic theory of isolators. Preliminary announce-] Iment, Physikalische Zeitschri]t, 13,97-100. In GermanJ !p. Debye, 1945, Polar Molecules, Dover, New YorkJ !p. Debye, E. Huckel, 1923, The theory of electrolytes I. The lowenng of the freezmg Ipoint and related occurrences, Physikalische Zeitschrift, 24, 185-206. In Germanj V\. Erdelyl, W. Magnus, E Oberhettmger, EG. Tncoml, 1981, Higher Transcenden-I ral Functions, YoU, Kneger, Melbboume, Flonda, New YorkJ p. Frennmg, 2002, Dlelectnc-response functIOn determmed by regular smgular1 Ipoint analysis, Physical Review B, 65, 245117 j M. Fukunaga, N. ShImIZU, 2004, Role of prehistones m the Imtial value problem~ pffractional viscoelastic equations, Nonlinear Dynamics, 38, 207-220.1 ~. Gorenflo, 1997, FractIOnal calculus: some numencal methods, m A. CarpmtenJ IE Mamardl (Eds.): Fractals and Fractional Calculus in Continuum Mechanics! ISpnnger, Wlen and New York, 277-290.1 K Gorenflo, A.A. Ktlbas, S. Y. Rogosm, 1998, On the generahzed Mittag-Leftlerl Itypefunctions, Integral Transforms and Special Functions, 7, 215-224j ~. Gorenflo, J. Loutchko, Y. Luchko, 2002, ComputatIOn of the Mittag-Leftler func-I ItIOn and ItSdenvative, Fractional Calculus and Applied Analysis, 5, 491-518.1 ~. Heymans, I. Podlubny, 2006, PhySIcal mterpretatIOn of mitial condItions for frac1 ItIOnal dIfferential equatIOns WIthRIemann-LIOuvIlle fractIOnal denvatives, Rheo-I [ogica Acta, 45, 765-771j ~.D. Jackson, 1998, Classical Electrodynamics, 3rd ed., WIley, New YorkJ V\.K. Jonscher, 1977, Omversal dlelectnc response, Nature, 267, 673-679.1 V\.K. Jonscher, 1978, Low-frequency dIspersIOn in camer-dommated dlelectncsJ IPhilosophical Magazine B, 38, 587-601 j A.K. Jonscher, 1983, DielectriC RelaxatIOn In Solids, Chelsea Dlelectncs Press! lLondon. A.K. Jonscher, 1996, Universal Relaxation Law, Chelsea Dlelectncs Press, Londonj V\.K. Jonscher, 1999, Dlelectnc relaxatIOn m sohds, Journal oj Physics D, 32, R571 IR7O:

References

3751

IS. Kempfle, I. Schafer, 2000, Fractional differential equations and initial conditionsj !Fractional Calculus and Applied Analysis, 3, 387-400.1 k\.A. KIlbas, H.M. Snvastava, J.J. TrujIllo, 2006, Theory and Applications of Fracj klOnal Dijjerentwl EquatIOns, ElseVIer,AmsterdamJ M. V. Kuzelev, A. A. Rukhadze, 2009, Methods of Waves Theory In DisperSive Mej Idia, World Scientific Publishing, Singapore.1 f. Mainardi, R. Gorenflo, 2000, On Mittag-Leffler-type functions in fractional evoj IlutIOn processes, Journal of Computational and Applied Mathematics, 118, 283-1 IZ!N: IK.S. MIller, 1993, The MIttag-Leffler and related functIons, Integral Transformsl land Special Functions, 1, 41-49j k\.Y. Milovanov, K Rypdal, J.J. Rasmussen, 2007, Stretched exponential relaxation] ~nd ac umversahty In dIsordered dlelectncs, Physical Review B, 76, 104201J k\. V. Miiovanov, J.J. Rasmussen, K. Rypdal, 2008, Stretched-exponentIal decayl IfunctIOns from a self-conSIstent model of dlelectnc relaxatIon, PhySICS Lettersl lA, 372 2148-2154j IKL. NgaI, A.K Jonscher, c.T. WhIte, 1979, OngIn of the umversal dlelectnc re1 IsponseIn condensed matter, Nature, 277, 185-189J ~.R. NIgmatulhn, 1986, The reahzatIon of the generahzed transfer equation In ~ Imedium with fractal geometry, Physica Status Solidi B, 133, 425-430.1 KR. NIgmatulhn, Ya.E. Ryabov, 1997, Cole-DaVIdson dlelectnc relaxatIon as ij Iself-SImIlarrelaxatIon process, Physics oj Solid State, 39, 87-90; and Fizika Tver1 Idogo Tela, 39, 101-105. In RussIanJ ~.R. NIgmatulhn, A.A. Arbuzov, F. Salehh, A. GIZ, I. Bayrak, H. CatalgIl-GIzJ gOO?, The first expenmental confirmatIon of the fractIonal kInetIcs contaInIng Ithe complex-power-Iaw exponents: Dlelectnc measurements of polymenzatIOnl Ireactions, Physica B, 388, 418-434j IY.Y. NOVIkov, Y.P. Pnvalko, 2001, Temporal fractal model for the anomalous dlelec1 Itnc relaxatIOn of Inhomogeneous medIa WIth chaotIC structure, Physical RevieMl IE, 64, 031504j [. Podlubny, 1999, FractIOnal Differential EquatIOns, AcademIC Press, New YorkJ v. Ramaknshnan, M.R. LakshmI, (Eds.), 1984, Non-Debye RelaxatIOn In Con1 Idensed Matter World SCIentIfic PubhshIng, SIngaporeJ IYa.E. Ryabov, Yu. Feldman, 2002, Novel approach to the analySIS of the non-Deby~ ~helectnc spectrum broademng, Physica A, 314, 370-378.1 IYa.E. Ryabov, Yu. Feldman, 2003, The relatIOnshIp between the scalIng parameterl ~nd relaxatIOn tIme for non-exponentIal relaxatIOn In dIsordered systems, Fracj tst; 11, 173-1831 IS.G. Samko, A.A. KIlbas, 0.1. Manchev, 1993, Integrals and Derivatives oj Frac1 kional Order and Applications, Nauka 1 Tehmka, MInsk, 1987. In RussIanj ~nd FractIOnal Integrals and Derivatives Iheory and AppllcatlOns, Gordon and! IBreach, New York, 1993J E. von Schweldler, 1907, Studlen ber anomahen 1m verhalten der dlelektnkal IC'StudIes on the anomalous behaVIOr of dlelectncs"), Annallen der Physlkl I(LeIpzIg), Senes 4, 24, 711-770. In GermanJ

rr.

1376

16 Fractional Temporal Electrodynarnics

IY.P. SIlm, A.A. Ruhadze, 1961, Electrodynamic Properties oj Plasma and Plasma1 [ike Media, Gosatomizdat, Moscow. In Russian] Tadjeran, M.M. Meerschaert, H.P. Schettler, 2006, A second-order accurate nuj ImerIcalapproxImatIon for the fractIonal dIffusIOnequatIon, Journal of Computa-I kional Physics, 213, 205-213.1 ~.E. Tarasov, 200ga, Fractional equations of Curie-von Schweidler and Gauss lawsJ flournal oj Physics A, 20, 1452121 ~.E. Tarasov, 200gb, Universal electromagnetic waves in dielectric, Journal of! IPhysics A, 20, 175223j IY.E. Tarasov, 2009, FractIonal mtegro-dIfferentIal equatIons for electromagnetIg Iwaves in dielectric media, Theoretical and Mathematical Physics, 158, 355-359j IV.E. Tarasov, G.M. Zaslavsky, 2006, DynamICS WIth low-level fractIOnalIty, Physical lA, 368, 399-4151 [. Yilmaz, A. Gelir, F. Salehli, R.R. Nigmatullin, A. A. Arbuzov, 2006, Dielectricj Istudy of neutral and charged hydro gels dUrIng the swellIng process, Journal of! IChemical Physics, 125,234705.1

r.

17] [Fractional Nonholonomic Dynamicsj ~hapter

117.1 Introductionl ~onholonomIc dynamIcs descnbes systems constramed by nOllIntegrable relatIOn-I IshIpS. The constramt, caned holonomIc constramt, depends only on the coordmatesJ ~t does not depend on the velocItIes. VelocIty dependent constramts such as could! Ibe holonomIc constramts If It can be mtegrated mto a sImple holonomIc constramt] k\ constramt that cannot be mtegrated mto a holonomIc constramt IS caned non-I holonomic (Tchetaev, 1962; Dobronravov, 1970; Rumiantsev, 1978, 2000, 1982)1 [The constramt, caned fractIOnal nonholonomIc constramt (Tarasov and ZaslavskyJ 12006a), depends on the denvatIves of non-mteger orders (Samko et aL, 1993; KI1ba~ ~t al., 2006). The fractional nonholonomic constraints are interpreted as constraintsl Iwith long-term memory (Tarasov and Zaslavsky, 2006a). Fractional derivatives al-j Ilow one to descnbe constramts wIth power-law long-term memory by usmg thel hactIOnal calculus (Samko et aL, 1993; KI1bas et aL, 2006).1 pynamIcs of systems wIth fillIte degrees of freedom, whose generalIzed coor1 ~mates are qb k - I, ..., n, are dIscussed. We assume that the systems are defined! Iby LagrangIan L - T - U, non-potentIal forces Qk, and nOllIntegrable relatIOnshIpsl Its - O. In thIS chapter, we conSIder two fonowmg specIal cases of fractIOnal non-I IholonomIc dynamIcsj 1(1) FractIOnal dynamIcal systems wIth nonholonomIc constramtsj

1(2) DynamIcal systems wIth fractIOnal nonholonomIc constramtsj

~n these cases, the forces are Qk = Qk(q,D}q,t). Here aqJta denotes a fractiona~ ~envatIve wIth respect to t 1

rI'he equatIOns of constramed motIon are denved usmg vanatIOnal pnncIple tha~ lIS caned the d'Alembert-Lagrange pnncIple. We conSIder fractIOnal dynamICS o~ V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

17 Fractional Nonho1onomic Dynamics

1378

ronholonomic systems that are described by Lagrangian and Hamiltonian withl [ractional temporal derivatives of generalized coordinates. Using the d' Alembert-I [Lagrange principle, we also obtain fractional equations of motion by using usuall [..:agranglan and Hamlltoman, whIch have only mteger-order derIVatIves, and a spej ~Ial type of constramts wIth memory. We dISCUSS the applIcabIlIty of the statIonaryl laction principle for fractional nonholonomic dynamicsj ~n Section 17.2, we provide a brief review of nonholonomic systems, fix notationsl land convenient references. In Section 17.3, the fractional temporal derivatives arel ~efined. In SectIOn 17.4, the nonholonomIc systems are consIdered as holonomIcJ OCn Sections 17.5-17.7, we define a fractional generalization of nonholonomic conj Istramts. Some examples are suggested. In SectIOns 17.8-17.9, we dISCUSS the applIj ~ability of the stationary action principle for fractional nonholonomic constraintsj finally, a short conclusion is given in Section 17.10.1

~7.2 ~n ~o

Nonholonomic dynamics

thIS sectIOn, a brIef reVIew of dynamIcs of nonholonomIc systems IS fix notatIOns and proVIde convement referencesJ

consIdere~

WAlembert-Lagrange principlel OCt IS known that the d'Alembert-Lagrange prInCIpleallows us to derIve equatIons ofj ImotIon WIth holonomIc and nonholonomIc constramts. For N-partIcle system It ha§ ~he form of the vanation equatIOn:1

r dt F) Dr· = 0 d(mvi) -

1

1

,

(17·1)1

Iwhere ri, 1 - 1, ... ,N, IS a radIUs-vector of Ith partIcle, Vi - ri IS a velOCIty, and F11 lIS a force that acts on ith partIcle, and the sum over repeated mdex i IS from 1 to N.I [To exclude holonomic constraints, the general coordinates qk, k = 1, ... , n, are use~ I(Pars, 1964). Here, n = 3N - m is a number of degrees of freedom, where m is aI r.umber of holonomIc constramts. Then ri IS a functIOn of generalIzed coordmate§ landthe nme varIable:1 h = ri(q,t). (17.2)1 [The general coordinates qk are functions of the time variable t and the initial dat~ ~hat are described by parameters a. We note that the variations Dl (t) can be define~ I(Dobronravov, 1976) byl

fq k(t ) -_

1<-

dl(t,a)d I da a,

117.2 Nonholonomic dynamics

37S1

Iwhere a are parameters that are determined by initial conditions. The variations land 8l are constrained by the relationshipl

or,1

~ r, = aqk art k 8q , Iwhere we use the sum over repeated index k is from 1 to n. Using (17.3), Equationj 1(17.1) gives

_

( I

d (mv t ) art _ F. art) 8 k = O. dt aqk ' aqk _ q

IWe can define (Pars, 1964) the generalIzed forcesj

k= 1, ... ,nj ~n general, the force Gk can be represented as a sum of the potential force

Gf =J

I-au / al, defined by the potential function U = U(q,t), and the non-potential forc~ Qk = Qk(q,q,t). We note that the force Qk is non-potential if at least one of thel condItIons: laQk _ aQ, =~ aqk

r:M:

J

Iwhere k,1- 1, ... , n, IS not satIsfied~ pIfferentIatIon of (17.2) wIth respect to tIme, gIves the velocItyj

IVt = drt(q,t) = art dl dt

~t

IS easy to see

aqk dt

+ art. at

tha~

Differentiation of (17.5) with respect to l givesl

~f

r, = r, (q, t) is continuous functions, thenl

k'\s a result, we havel

[dVi

W-

d

dt

(art) 0l .

IOsmgthe product rule for the time denvatIve, we obtaml

17 Fractional Nonho1onomic Dynamics

1380

(17.9) ISubstItutIOn of (17.6) and (17.8) mto (17.9) glVe§

IWe can rewnte thIS equatIon m the formj

~(mVi) ari d a (m ) Idt aqk = dt ail "2 Vi Vi -

a (m ) aqk "2 Vi Vi

I

•

k\s a result, EquatIOn (17.4) gIve§

(17.1O~ ~

mv2

N

~

r =2=~21 lis a kinetic energy. Using (17.5), the function T is represented by the equationj

(17.11~

A(q,t)

I

arar.

= ari ari

I

lOsing the Lagrange function L = L(q,q,t) = T - U, Equation (17.10) has the form:1

r

d

st.

st.

)

dt ail - aqk - Qk oq

k

= o.

[ThIS vanatIOnal equatIOn defines the d'Alembert-Lagrange pnncIple for ~oordmates. We note that the generalIzed potentIal forcesj

hp _ d au au i k - dt aqk - aqk' lalso give equation (17.12).1

k= 1, ... ,nl

(17.12~ generalIze~

38~

117.2 Nonholonomic dynamics

[Nonholonomic dynamical system~ [..:et us consider a mechamcal system with fimte degrees of freedom, whose generj lalized coordinates and velocities are l, qk, k = 1, ... , n. We assume that the sysj ~em is subjected to potential forces -aV /aqk, defined by the potential functiog IV = V(q,t), and to non-potential forces Qk = Qk(q,q,t). A nonholonomic systeml liS a system constramed by the nomntegrable relationshipsj

l!s(q,q,t) =0,

(17.13~

s=l, ... .r «:n.

~n general, Equations (17.13) are nonlinear with respect to i/ = dl / dt. We assume' ~hat constraints (17.13) are independent, i.e.j

[TheseequatlOns can be solved to some r dependent generahzed velocities. If the ve1 Ilocities il, k = 1, ... ,m, where m = n - r, are assumed independent, then Eq. (17.13j ~an be represented m the formj

°

IF (q,q,t .) =q·m+s -gs ( q,q·1 , ... ,q.m) ,t = , )S

(17.14~

Iwheres-1, ... ,rl for nonholonomic systems, we may assume that the constraints are ideal and thel Ivariations oqk, k = 1, ... , n, satisfy the Tchetaev's conditional

~ol =0,

(17.15~

s= 1, ... ,r.

for constraints (17.14) the conditions (17.15) ard S

dg 0 k 0qm+s = a' qk q,

1

s

= 1, ... ,rJ

~n order to descnbe nonholonomic dynamiCs, we must define variations. Thel Ivariations of generalized coordinates k = 1, ... , n, are defined by the relation o~ me ideal constramtsl (17.16j

ol,

Iwhere Rk are components of the constramt force vector R. Rk can be considere~ las a contnbutlOn of the reactlOn associated with the constramt to the generahze~ ~orce Qk. Because a reactlOn force does no work m a virtual movement that is conj ISistent With the correspondmg kmematical restnctlOn, we conclude that R must bel Iperpendicular to any oq that satisfies the constraint equations. Thus, if satisfYI ~onstraint equations, we have RkOl = 0. We now consider which condition must bel Ireahzed m order to satisfy constramt equatlOns. We can denve the usual equatlOn~ pf motion only under the condition (17.16). For nonholonomic systems a defimtlOllI

ol

17 Fractional Nonho1onomic Dynamics

1382

pf the vanatIOns was suggested by Tchetaev (Tchetaev, 1932, 1962). The vanatIOnsl ~l are defined byl s

=

1, ... ,r.

(17.17~

~quatlon (17.17) IS called the Tchetaev's condItIon (Dobronravov, 1970). Osmgj 1(17.16) and (17.17), we see that the functions x, are linear combinations of lsi

a all

~

(17.18~

Iwhere As, s = 1, ... , r, are Lagrange multipliers. The Tchetaev's definition of variaj ~Ions states that the actual constramed motIon should occur along a trajectory obj ~ained by normal projection of a force onto a constraint hypersurface. The constrain~ [orce Rk is minimum when Rk is chosen perpendicular to the constraint surface o~ Iparallel to the gradient alsiaqkj [n general, the nonholonomlc system IS sulijected to actIon of the generahze~ ~orce Qk and the constramt force Rk. Then the vanatIonal equatIon lsi

(17.19~ from (17.18), we obtaml

(17.20~ [To sImphfy out transformatIOns, we consIder the case of r - 1. Using Eq. (17 .17~ oql, I = 1, ... , n - 1, as independent variations. Then oqnl lIS not mdependent, and Eq. (17.17) gIvesl ~or r = 1, we can consider

ISuppose that X satisfies the equation:1

[Then the term WIth k = n m (17.20) IS equal to zero, and Eq. (17.20) has n - 11 ~erms WIth k-l, ...,n-l. In Eq. (17.20), the vanatIOns WIth k-l, ... ,n-l arel Imdependent, and the sum IS separated on n - 1 equatIOns. As a result, EquatIOi1j 1(17.20) IS eqUIvalent tq

k= 1, ... ,n.

(17.21~

38~

117.2 Nonholonomic dynamics

[Equations (17.17) and (17.21) form a system of (n + 1) equations with (n + 1) unj Iknowns A and l, where k = 1, ... , n. Solutions of these equations describe motion! las an evolution of system with nonlinear nonholonomic constraint (17.13)j [he canonical momenta pk can be defined b~

(17.22~ [Using L = T - U with U = U(q,t) and T from Eq. (17.11), we obtaiij

~k Iwhere we assume that

=

mAkl(q,t)i/ +mAk(q,t),

aU/ail =

k = 1, ... ,n.

(17.23~

O. Then Eqs. (17.21) have the form:1

[f

~hen

Eqs. (17.23) glVg

(17.24~

IWhereA1k(q,t) is defined byl

~quatiOns (17.24) allow us to represent equations of motion for phase space van1 ables land pk, k = 1, ... ,nj

lfVonholonomic system as a holonomic syste"" OC:et us prove that equatiOns of motion WIth nonholonomic constramt can be repre-I Isentedas equatiOns for a holonomic systemJ [1'0 sImplify our transformatiOns, we consIder the case of r = 1. Then we havg pn1y one nonho1onomic constraint. Using (17.24), we can defin~

If(p,q,t)

=

/(tj(q,p,t),q,t).

(17.25~

IWe assume that the constraint IS an mtegral of motion. Then the total time denvativ~ pf (17.25) is equal to zeroJ

~J(p,q,t) = [This equation give§

oj

1384

17 Fractional Nonho1onomic Dynamics

(17.26~ ISubstituting (17.21) into (17.26), we obtainl (17.27 ~quatlon

(17.27) gIve§ (17.28

[I'hen the Lagrange equatIons (17.21) have the formj

~t aiIst. Jqk st. = Qk+Rk(q,P,t),

(17.29~

fractIOnal generalIzatIOn of Eq. (17.29) has the formj (17.30 ~quatIOns

(17.29) and (17.30) descrIbe the motIon of a holonomIc system wIth ~ of freedom. For any trajectory of the system III the phase space, we havel If(q,p,t) = O. If the initial values qk(O) and qk(O) satisfy the constraint condition:1 ~egrees

If(q(O),q(O),to)

= OJ

~hen

the solutIOn of Eq. (17.29) descrIbes a motIon of the nonholonomIc systemJ OC:et us define a generalIzed force Ak - Qk + Rk, WhICh depends on generalIze~ Ivelocities qk, generalized coordinates l, and time t. If the conditionsi

~ d"'"

dA k dAm --dqm dqk

(17.3l~

dA m_

+ del - O, 1d -2 dt

lare satisfied, then a generalized potential 0 =

dAk dqm

dAm dqk

----

0 (q, q, t)

exists an~

(17.32

117.3 Fractional temporal derivatives

3851

~f the generalized force A k does not depend on generalized velocities la potential force -aU / aqk ifI

qk, then A k i§

[Equations (17.31) and (17.32) are called the Helmholtz conditions. In these casesJ ~he Hamilton variational principle has the form of the stationary action principlej ~n order to use this principle for a nonholonomic system, we should consider suchl ~rajectones that their mitial conditions satisfy the constraint equation (17.13)J IWe note that nonholonomic constramt (17.13) and non-potential generahze~ ~orce Qk can be compensated such that resultmg generahzed force A k is a general-I lizedpotential force, and system is a Lagrangian system with holonomic constraintsJ

117.3 Fractional temporal derivativesl [he fractIOnal denvative has different defimtIOns (Samko et a1., 1993; Kilbas e~ la1., 2006), and explOltmg any of them depends on the kmd of the problems, imtiall I(boundary) conditions, and the speCifics of the conSidered phYSical processes. Thg ~eft- and right-sided Riemann-Liouville fractional derivatives (Kilbas et al., 2006)1 lare defined b)1 a ()

a'lJI f t

1

d'"

t

= r(m _ a) dtm Ja

f r dr (t _ r)a-m+l

m m-a () all f t ,

= DI

(17.33

Iwhere m - 1 < a < m, and r(z) is the Gamma function. To describe dynamica~ Isystems with long-term memory, we can also use the Caputo fractIOnal denvative~ I(Caputo, 1969, 1967; Caputo and Mainardi, 1971a). Its main advantage is that thel limtial conditIOns take the same form as for mteger-order differential equatIOnsJ [The Riemann-LIOUVille tractIOnal denvative has some notable disadvantages ml lapphcatIOns m mechamcs such as the hyper-smgular improper mtegral, where thel prder of smgulanty is higher than the dimensIOn, and nonzero of the fractIOna~ k1envative of constants, which would entad that diSSipatIOn does not vamsh for ij Isystem m eqmhbnum. The deSire to use the usual mitial value problems for mej ~hamcal systems lead to the use of Caputo tractIOnal denvatives (Kdbas et a1., 2006j fodlubny, 1999) rather than the Riemann-LIOUVille tractIOnal denvativel [The left-sided Caputo tractIOnal denvative (Caputo, 1967; Caputo and Mamardi) ~971b; Kilbas et al., 2006) of order a> 0 is defined byl drD':f(r) = t - r a-m+ 1

r-aDmf(t) a

I

I

,

(17.35

17 Fractional Nonho1onomic Dynamics

1386

Iwhere m - 1 < a < m, and alIa is the left-sided Riemann-Liouville fractional integrall pf order a > 0 that is defined b)j

~ IX

1 II f(t) = qa)

r (tf('r)d'r _

Ja

'r)I-IX'

[The right-sided Caputo fractional derivative (KUbas et al., 2006) of order a > 0 i§ ~efined by

d'rD~f('r)

'r -

t IX-m+ I

=

r-IXDmf(t) I b

I

(17.36

,

Iwhere m - 1 < a < m, and II;: is the right-sided Riemann-Liouville fractional inte1 19ral of order a > 0 that is defined bYI

rI'hese defimtiOns are, of course, more restnctIve than the Riemann-LiOuville fracj ~ional derivatives (Samko et al., 1993; Kilbas et al., 2006) in that require the absoj ~ute mtegrabllIty of the denvatlves of order n. The Caputo fractiOnal denvatIves firs~ ~ompute ordmary denvatIves foIIowed by a fractiOnal mtegral to get the deSire orderl pf fractiOnal denvatIves. The Rlemann-LiOuvlIIe fractIonal denvatIves are compute~ 1m the reverse orderJ ~et f(t) be a function for which the Caputo fractional derivatives (17.35) and! 1(17.36) of order a eXist together with the Riemann-LiOuville fractiOnal denvatIve~ 1(17.33) and (17.34). Then these fractiOnal denvatIves are connected by the relatiOns:1 C DIXf(t) =

a

CDIXf(t) I

a

I

b

=

~IXf(t) _ ~ (t - a)k-IX f(k) (a) I

,

(17.37

(b-t) -IX f(k) (b). i...r(k-a+l)

(17.38

i...

r(k-a+l)

~IXf(t)- m~

t -»

[The Caputo fractiOnal denvatIves (17.35), (17.36) comclde with the Rlemann1 ~iOuvlIIe fractiOnal denvatIves (17.33), (17.34) Ifl

If(a) = (Dif)(a) =

= (D';' If)(a) =

oj

If(b) = (DiJ) (b) =

= (D,;,-lf)(b) =

0.1

the general case, Equations (17.37) and (17.37) mean that D m and alIa, IIr canno~ Ibe considered as commutatIve operatiOnsl [t IS observed that the second terms m Eqs. (17.37) and (17.38) regulanze thel K=aputo fractional derivatives to avoid the potentially divergences from singular inj ~egratiOn at t - o. In additIon, the Caputo fractiOnal dIfferentIatiOn of a constan~ lresults in zerd ~n

117.3 Fractional temporal derivatives

3871

~Dfc=oJ [Note that the Riemann-Liouville fractional derivative of a constant need not be zero] land we have

[f the Caputo tractIOnal denvattve IS used mstead of the RIemann-LIOuvIlle tracj then the mIttal condIttons for tractIOnal dynamIcal systems are thel Isame as those for the usual dynamIcal systems. The Caputo formulatIon of tracj ~ional calculus can be more applicable in mechanics than the Riemann-Liouvill~ ~IOnal denvattve,

Ifommlation

[t is possible to state that the Caputo fractional derivatives allow us to give morel pear mechamcalmterpretatton. At the same ttme we cannot state that the Rlemannj [.:IouvIlle tractIOnal denvattve does not have a phySIcal mterpretatton and that I~ Ishows unphysical behavior. Physical interpretations of the Riemann-Liouville fracj ~ional derivatives are more complicated than Caputo fractional derivatives~ [The RIemann-LIOuvIlle fractIOnal denvatIves can be represented through the LI1 puville fractional derivatives (Samko et aI., 1993; Kilbas et aI., 2006) D% by thel ~quatIOns:

~;;gla f(t) = D~f(t)H(t

- a),1

Iwhere H(t) is the Heaviside function, also called the unit step function, that is a dis~ ~ontmuous functIon whose value IS zero for negatIve argument and one for pOSIttVg largument. The left-SIded LIOUVIlle tractIOnal denvatIve (Samko et aI., 1993; KIlba§ ~t aI., 2006) D~ of order a is defined by the relation:1 (D~f)(t) =

_=;;gla f(t)

F

1

D';'I": af(t)

=

d

m

r(m - a) dt m

/1

-c-co

f( -r)d-r (t - -r)U-m+l '

(l7.39~

land the nght-sIded LIOUVIlle tractIOnal denvatIve (Samko et aI., 1993; KIlbas et aLJ 12006) D a is defined byl I(D~f)(t)

= l;;g::;f(t) = D';'I~ af(t)

t

1

d

m

r(m - a) djI1i

1= 1

f( -r)d-r (t - -r)U-m+l '

(l7.40~

Iwhere m - I < a < m.1 ~f 0 < Re( a) < 1 and f(t) ELI (lR), or 1 ~ P < I/Re( a) and f(t) E Lp(lR), thenl ~he Fourier transforms § of these derivatives (see Theorem 7.1 in (Samko et a1.j ~993) and Theorem 2.15 m (KIlbas et aI., 2006)) are grven byl

Iwhere

17 Fractional Nonholonomic Dynamics

1388

IWe remind of the formula for the Fourier transform of some function /( t) :1

r,%/)(Q))

=

~

k

dx/(t) exp{

-iWf}J

Iwhich is valid for all /(t) E Ll(l~.). If we require /(t) E Lz(l~), then the Parsevall [ormula 11.7/liz = II/liz holds. We note that.7 can be considered as an extensioij pf this Fourier transformation to a unitary isomorphism on Lz(lR)J k\s a result, we obtam that the phySIcal mterpretatIOn of the RIemann-LIOuvIlI~ klerivatives is connected with the frequency dependence of the physical and mechan-I lical values IWe note that the Riemann-Liouville fractional derivatives naturally appear fo~ Ireal phySIcal systems m electrodynamIcs. The dlelectnc susceptIbIlIty of a wldel pass of dlelectnc matenals follows, over extended frequency ranges, a fractIona~ Ipower-Iaw frequency dependence that IS called the unIversal response (JonscherJ ~996, 1999). We prove (Tarasov, 2008b,a) that the electromagnetIc fields m such dI1 ~lectnc medIa are descnbed by dIfferentIal equatIOns WIth RIemann-LIOuvIlle frac1 ~IOnal tIme denvatIves. These fractIOnal equatIons for "unIversal" electromagnetICj Iwaves m dlelectnc medIa are common to a WIde class of matenals, regardless ofj ~he type of phySIcal structure, chemIcal composItIon, or of the nature of the polanz-I Img speCIes.Therefore we cannot state that RIemann-LIOuvIlle fractIOnal denvatIve~ IWIth respect to tIme vanable do not have a phySIcal mterpretatIOn. The phySIcal m1 ~erpretatIOn of these denvatIves m electrodynamIcs IS connected WIth the frequenc)j kIependence of the dlelectnc susceptIbIlIty. As a result, the processes WIth memj pry that are connected WIth dIfferentIal equatIOns WIth RIemann-LIouvIlle fractIona~ kIenvatIves are very Important to phySIcal applIcatIOns, and these denvatIves natu1 Irally appear for real phySIcal systemsl

~ 7.4

Fractional dynamics with nonholonomic constraints

[The baSIC pnnCIple of mechanICS IS the vanatIOnal pnncIple of d'Alembert-Lagrangej

(l7.41~ Iwhere L = L(q,q,t) = T - U is the Lagrange function, T(q,q,t) is kinetic energy,l ~l are virtual displacements. Here we use the summation condition over repeate~ lindexes rrhe fractIOnal generalIzatIOn of vanatIOnal pnncIple of d'Alembert-Lagrangel Ihasthe form

117.4 Fractional dynamics with nonholonomic constraints

38S1

(17.42

IL = L(q, atJ1/Xq, ttJ1f:q) lis the Lagrange function for fractional dynamical system. Note that atJ1/x

t tJ1f: = -DI for a = i

= DI

an~

I

[The Hamilton's principle is obtained by integrating Eg. (17.41) with respect to Iwithin some constant limits to and t] in the formj

~

(17.43 Iwhere we assume that the functions 8l(t) E C2[O,oo] satisfy the conditionsJ

~l(to)=o,

8l(td=O,

k=l, ... ,n.

(17.44~

[TheHaml1ton's prInciple for fractiOnal dynamical systems is obtamed by mtegratmg ~q. (17.42) with respect to t withm some constant hmits to = a and ti = b m thel form:

(17.45 Iwhere conditions (17.44) are satisfied.1 psmg mtegratron by parts, Equation (17.43) givesl

(17.46 ~n Eq. (17.46), we have the time derivatives of coordinate variations d( 8l)/dt. Fo~ OCractiOnal dynamical system, we use the rule of fractiOnal mtegratiOn by parts] [1'0 reahze mathematical transformatiOns m fractiOnal dynamiCs, we should hav~ larule for mtegratiOn by parts for fractiOnal differentiatiOn. In mathematical calculus] ImtegratiOnby parts is a rule that transforms the mtegral of products of functiOns mtol pther mtegrals. For derIvatives of mteger orders, the rule arIses from the productl lrule of dIfferentiatiOn. In fractiOnal calculus, thiS rule connects the left-Sided andl ~he rIght-Sided the Riemann-LiOuvl1le fractiOnal derIvatives of order 0 < a < 1. Le~ Ius give the baSiC result regardmg a fractiOnal generahzatiOn of mtegration by partsJ OCf the functions f(t) and g(t) satisfy the conditionsj

If(t) E I~(Lp(a,b)),

g(t) E I::+(Lq(a,b)),1

Iwhere p? 1, q? 1, andl I

1

1

~+l?-+-, ~

17 Fractional Nonholonomic Dynamics

1390 ~hen

we have the relationl (17.47

[I'he proof of thIS theorem was reahzed m (Samko et aI., 1993) (see Corollary 2 ofj [I'heorem 2.4). EquatIOn (17.47) IS called the tractIonal mtegratIon by parts] IUsmgmtegratIOn by parts, EquatIOn (17.45) gIvesl

(17.48 Iwhere to = a, and tl = b. In Eq. (17.48), we have the time derivatives of coordinat~ Ivariations a[gtUol(t), and t[g!:ol(t)J IWe note that there are only r conditions (17.15), where r < n, to define n varij lations ol. As a result, the Tchetaev's conditions do not uniquely define olJ V? - 1, ... , n. There eXIsts some arbItranness m the determmatIon of denvatIve~ IDjoqk, a[gtUol(t), and t[gf:ol(t). Two equivalent approaches exist on the re~ ~atIOn of these denvatIves WIth vanation of generahzed velOCItIes] ~.

Accordmg to Holder the commutation relatIOn~

IDjol

=

oDil,

k = 1,... .n

(17.49~

lare valid for all generalized coordinates l, i.e., for all k. If constraints (17.13) lare holonomic, i.e., Equations (17.13) are integrable, then the variations;) fs arel lidentically zero. If they are not integrable, then;) is, S = 1, ... , r, are not identicallYI ~ero. We have the relations;) is - 0, S - 1, ... , r, only for special form of variatioili ~efinition (17.49). We note that the variations;) is, may become zero in the casel pf theIr nonhneanty on the strength of the equatIOns of motIon. We note that thel lidentities ;) fs = 0, S = 1, ... , r, and conditions (17.49) are compatible in the casel pf holonomIc systemsl for tractIOnal dynamICS the Holder commutation relatIOns have the formj

(17.50~ (17.51j land Eqs. (17.50) and (17.51) are valid for all generalized coordinates ~, ...,n.

l, k ~

12. Accordmg to Appel and Suslov the commutation relatIOns:1

~ol(t) = 0 :tl(t),

k = 1, ... ,m

(17.52~

39~

117.4 Fractional dynamics with nonholonomic constraints

larevalid only for all independent velocities, i.e., for k = 1, ... , m, where m = n - r j OCn this case, the identitiesi

18is(q,q,t) = 0,

s= 1, H.,rl

lare valtd for the vanatIOn of functtons (17.13). For fractIOnal dynamIcal systems] ~he Appel-Suslov commutation relations have the form (17.50) and (17.51) tha~ lare valid only for independent fractional derivatives a:il? (t) and t:ill: (t)j V<: = 1, ... ,m~

ol

ol

lRemark

!Using the Tchetaev's conditions (17.15) and the definition (17.49) of latton of functIOns (17.13) over vIrtual dIsplacements art:j

IAf dis) 0q, k s=I, ...,r. I's = (dis dqk _!.... dtdqk

Del, the varij

(17.53~

for constraints (17.14) Eqs. (17.53) have the formj (17.54j

(17.55 ~ere

we assume the sum wIth respect to I from 1 to r 1 OC:et Eqs. (17.49) be sattsfied for all generaltzed coordmates. Substttutmg (17.49] linto (17.46), we obtain the HOlder form of Hamilton's principle for nonholonomicj Isystems: 1

to

OL(q,q,t)dt +

1 1

to

Qk(q,q,t) ol dt = 0,

~l(to)=ol(td=O,

k=I,H.,n.

(17.56 (17.57j

[n general, thIS vanatIOnal equatIOn IS nonholonomIc.1f all forces are potentIal, thenl ~q. (17.56) IS a holonomIc equatIOn for nonholonomIc systeml ISubstttutmg (17.50) and (17.51) mto (17.48), we obtam the Holder form o~ ~amIlton's pnnClple for nonholonomIc fractIOnal systems:1 (17.58 IwherecondIttons (17.57) are valtdj [he position of the system on real trajectory l (t) is compared in (17.56) withl ISImultaneous posItton obtamed by movmg from real motIOns posItton by vIrtual1 ~isplacements ol, which define a momentarily configuration. The sequence of disj Iplaced positions l (t) + ol (t) may be considered a roundabout path, which gener~ lallydoes not sattsfy Eqs. (17.13). If the roundabout path sattsfies equations (17.13),1 ~hen the equaltttesj

17 Fractional Nonho1onomic Dynamics

1392

(17.59~

lfs(q+oq,q+oq,t) =0

lare satIsfied. EquatIOn (17.59) glVe§ I

. Is(q,q,t)

aIs oqk + dqkoq aIs ·k + ... =0 j + dqk

[These equalities give 0 fs = 0, that are accurate to smalls of the first order. Notel ~hat these conditions are not satisfied for nonholonomic system. As a result, thel HamIlton's pnnciple (17.56) WIth Qk - 0, k - 1, ... ,n, does not generally representl ~he principle of stationary action potential forces. In the general case, variationa~ ~quation (17.56) cannot be represented byl (17.60 Iwhere L = L(q,q,t), as in the case of holonomic systems. For fractional sysj ~ems, variational equation (17.58) cannot be represented by (17.60) with L ~ IL(q, a§?q, t§!:q)l Rsing (17.56), the equations of motion for nonholonomic system are derived inl ~he form of Lagrange equations with factors Asl

k= l, ...,n.

(17.61~

~quatIOns (17.61) and (17.13) form a closed system of n+r equatIOnsWIth the samel rumber of unknowns. The generalIzed solutIOnof these equations depends on 2n - ~ larbitrary constantsj [The fractional equations for nonholonomic system are derived from (17.58), inl ~he form of fractional Lagrange equations with factors As:1

(17.62 Iwhere k - 1, ... ,nj [.-etus compare the Hamilton's principle (17.56) and (17.58) with the Lagrangcj Iproblem of statIOnary value of the actIOnIntegral (17.60) In the class of curves tha~ Isatisfy Eqs. (17.13). The introduction of indeterminate multiplies /1s(t) reduces tha~ Iprob1em of condItIonal extremum to the Lagrange problem of vanatIOns:1

(17.63~ Iwhere L

= L(q,q,t). The Euler-Lagrange equations for the problem (17.63) arel k= l, ...,n.

(17.64

39~

117.4 Fractional dynamics with nonholonomic constraints

for the fractional case L OChe problem (17.63) arel

= L(q, atJ1taq, ttJ1f:q), the Euler-Lagrange equations fo~

(17.65 Iwhere k - 1, ... ,nl !Obviously that Eqs. (17.61), (17.13) are not equivalent to Eqs. (17.63), (17.13)j ~quatIOns (17.62), (17.13) also are not eqUIvalent to Eqs. (17.63), (17.13). Thel ~lOnequivalence of these two systems of equations does not exclude a possibilitYI Isome of their solutions being the same. Let the general or some particular solutionl ~k(t) of Eqs. (17.61), (17.13) be also a solution of Eqs. (17.64), (17.13) for the samel ItnItIal condtttons. Then the equatIonsj

(17.66~ lare valid. We can multiply Eqs. (17.66) by 8l and summing over all k. Using thel rrchetaev's equation (17.15), we obtam the condttIonj

r s

(17.67~

(!!:..als - al s )8 k =0 dt del del q ,

Iwhich is necessary if two systems have the same solution qk(t). This condition i§ lalso sufficient. For proving this, let us assume that some solution of Eqs. (17.64),1 1(17.13) satisfies (17.67) for any 8l compatible with (17.15). Multiplying EqsJ 1(17.64) by 8l and Eqs. (17.15) by As and summing over all k and s with allowanc~ ~or (17.67) and (17.41), we obtaml

~ diJi1d JL JL als a:t - A Ji1 S

)

k

Qk 8q

I

= 0,

Iwhich shows that the considered solution l(t) also satisfies Eqs. (17.61), an~ 1(17.13). For fractional dynamics, we multiply Eqs. (17.65) by 8l and Eqs. (17.15~ Iby As. Summing over all k and s with allowance for (17.67) and (17.42), we havel

q

[hus condition (17.67) is necessary and sufficient for solution qk(t) of Eqs. (17.6 land(17.13) to be among solutions ofEqs. (17.64) and (17.13). Thus when condition! 1(17.67) is satisfied, the equations of motion (17.61) of nonholonomic system hav~ ~he form of Euler-Lagrange equations (17.64). As a result, the Hamilton's principl~ 1(17.56) for the motion of a nonholonomtc system defined by such solutIOn has thel

1394

17 Fractional Nonho1onomic Dynamics

~haractenstIcs

of the pnncIple of statIOnary actIOn (17.60). For constramts of thel

~orm (17.14) equality (17.67) reduces to conditions J1sA~+s = 0, where k = 1, ... , mj

land we assume the sum with respect to s from 1 to

~ 7.5

r.1

Constraints with fractional derivativesl

k\ phySIcal mterpretatIon of equatIons WIth denvatIves and mtegrals of

nomntege~

prder WIth respect to tune IS connected WIth the memory effects.1 IWe consider the evolution of a dynamical system in which some quantity A (t) i§ Irelated to another quantity B(t) through a memory function M(t) byl

(17.68~ [ThIS operatIOn IS a partIcular case of compOSItIOn products suggested by VItq !Volterra. In mathematics, Equation (17.68) means that the value A(t) is related withl IB(t) by the convolution operationj

IA(t)

=

M(t) *B(t)·1

~quatIOn (17.68) IS a typIcal equatIOn obtamed for the systems coupled to an envI-1 Ironment, WIth envIronmental degrees of freedom bemg averagedl ~et us conSIder the lImItmg cases WIdely used m phYSICS: (1) the absence of thel Imemory; (2) the complete memory; (3) the power-lIke memory. As a result, we havg ~he following special cases ofEq. (17.68)j

[. For a system WIthout memory, the tIme dependence of the memory functIon I§

M(t - r)

= M(t) l5(t-

r),

(17.69~

Iwhere l5(t- -r) is the Dirac delta-function. The absence of the memory meansl ~hat the function A(t) is defined by B(t) at the only instant t. In this case, thel Isystem loses all its states except for one. Using (17.68) and (17.69), we havel

IA(t)

=

l

M(t)8(t - -r)g( -r)d-r = M(t)B(t).

(17.70~

~xpressIOn (17.70) corresponds to the well-known phySIcal process WIth com1 Iplete absence of memory. ThIS process relates all subsequent states to prevIOu~ Istates through the smgle current state at each time t .1 12. If memory effects are mtroduced mto the system, then the delta-functIOn turnsl linto some function with the time interval during which B(t) affects on the func-I ~ion A(t). Let M(t) be the step functionj

IM(t - r)

=

t- 1 [H( r) - Htt - -r)],

(17.71j

3951

117.5 Constraints with fractional derivatives

Iwhere H(t) is the Heaviside function, also called the unit step function. Th~ [Heaviside function H (t) is a discontinuous function whose value is zero for neg-I lative argument and one for positive argument. In Eg. (17.71), the factor t- 1 i§ ~hosen to get normalIzatIon of the memory functIon to umtyj

Il

M(r)dr = 1j

[Then in the evolution process the system passes through all states continuousl)j Iwithout any loss. In this case,1

landthis corresponds to a complete memoryj 13. The power-like memory function:1

IM(t -r)

=

Mo (t - r)£

(17.72j

Iwhere Mo IS a real parameter, IndIcates the presence of the tractIonal denvatIve o~ lintegral. Substitution of (17.72) into (17.68) gives the temporal fractional integrall bf aider eJ (17.73 Iwhere X = r( E )Mo. The parameter X can be regarded as the strength of the per~ ~urbatIOn Induced by the envIronment of the system. The phYSIcal InterpretatIOnl pf the tractIOnal IntegratIOn IS an eXIstence of a memory effect wIth power-lIJ«1 Imemory function. The memory determines an intervallO,tl during which B(r) laffects onA(t)l ~quation

(17.68) is a special case of constraint for A(t) and B(t), where A(t) i~ proportional to M(t) *B(t ). In a more general case, the values A(t) and B(t) be related by the equatIOnj

~irect1y ~an

If(A(t),M(t) *B(t))

=

0,

(17.74~

Iwhere j IS a smooth functIon. For dynamIcal systems relatIon (17.74) defines a con1 Istraint with memory effect. If A(t) is a coordinate q(t) or velocity q(t), and B(t) lis a derivative D;"q(t), then Eq. (17.74) gives the constraint with Caputo fractiona~ ~erivative (KUbas et al., 2006). For a power-like memory function M(t), we repre-I Isent(17.74) as a constraint wIth Caputo tractIOnal denvatIvej

It(q,q, fiDfq)

=

oj

rrhIS relatIonshIp IS a tractIOnal dIfferentIal equatIon. If constraints are realIzed Inl the form:

17 Fractional Nonho1onomic Dynamics

1396

(17.75~

li(A(t),D,!,(M(t) *B(t))) = 0,

Iwhere Dr = d m / dt m and M(t) is a power-like memory function, then we have equaj ~ion with Riemann-Liouville fractional derivative (Kilbas et al., 2006)1

[n the general case, we can have a set of memory functions. For example, m the casel pftwo memory functionsMI (t) and M2(t), the constraint equation can has the form:1

k\s a result, we can use the fractional calculus (Samko et al., 1993; Kilbas et al.l 12006) to describe the motion of systems with the constraints (17.75). We note tha~ ~hese constraints are nonholonomicl

117.6 Equations of motion with fractional nonholonomicl constraints IWe assume that the constramt equations have the formj (17.76j Iwhere the left- and nght-slded Rlemann-LlOuvlIIe tractIOnal denvatives (KI1bas e~ laL, 2006) are used. The constraints WIth Caputo tractIOnal denvatives were consld1 ~red in (Tarasov and Zaslavsky, 2006a)j for non-integer a, the constraints (17.76) are fractional differential equation~ I(Podlubny, 1999). Such constramts are caIIed tractional nonholonomlc constramtsl I(Tarasov and Zaslavsky, 2006a). Smce Eqs. (17.76) have also denvatives of mtegerl prder, we can use the Tchetaev defimtlOn of variation (17.17) and the Lagrang~ ~quatlOns (l7.2I)j IWe assume that the dynamIcal system IS descnbed by the LagrangIan L =i r;t( q, if) - U (q,if), where fractional derivatives are not used. The Lagrange equation~ Iwith the factor As, s = 1, ... , r, have the formj

~ -=;-:st. - -=;st. = Qk + As -=;-:dis , dt

oqk

oqk

oqk

k

=

l, ... ,n,

(17.77~

Iwhere Qk are non-potential forcesl flo slmpbfy the transformatIOns, we conSider r - I, and the Lagrangialli

(17.78~ Iwhere U

= U (q) is a potential energy of the system. Then Eq. (17.77) becomesl

397]

117.6 Equations of motion with fractional nonholonomic constraints

(17.79~

k= l, ... ,n,

IWe assume that the constraint is an integral of motion, i.e., df/dt

= O. Thenl

[Usingthe equalitiesj

IDt1 a::LI t'Ma = D1Dm /m-a = Dm+1 /a t tat tat t landD;

t'Ma+lJ

=a::LIt

= d / dt, Equation (17.80) can be represented asl

ISubstItutiOn of (17.79) into (17.81) gIvesl

from this equation, one can obtain the Lagrange multiplier X j

(17.82~

(17.83~ rrhese equations descnbe a holonomIc system that IS eqUIvalentto the nonholonomKl pne with fractional nonholonomic constraint. For any motion of the system, we hav~ If = o. If the initial values satisfy the constraint conditionl

17 Fractional Nonho1onomic Dynamics

1398

[f(q(O),ti(O),

a~taq(O), t~;:q(O))

= OJ

~hen the solutiOn of Eq. (17.83) descnbes a motion of the system (17.78) wIth fracj tional nonholonomic constraint (17.76)j [f the constramt (17.76) IS lInear WIth respect to the first denvatives iLk. such tha~

(17.84~ ~hen

Rk - ak. and Eqs. (17.83) can be represented asl

(17.85~ Iwherea2 = Lk=l aiai, If the function g is linear with respect to the left-sided Caputol fractiOnal denvative:1

(17.86~ ~n thIScase, the constramt equatiOn IS lInear WIth respect to the mteger denvatIves tiki land the fractional derivatives a~?qk' Substitution of (17.86) into Eq. (17.83) givesl ~he equatiOns of motionj

(17.87~ k\s a result, we obtam the equatiOns of motion wIth RIemann-LiOuvIlle fractiOna~ k1envative of order a + 1. We note that the nonholonomIc systems (17.87) wIth mj ~eger a were considered in Refs. (Tarasov, 2003, 2005b,a) and the systems withl ~ractiOnal a were suggested m (Tarasov and Zaslavsky, 2006a)1

117.7 Example of fractional nonholonomic constraint~ IExample 1.1 ~n

I-dImensiOnal case (n - 1), Equation (17.87) wIth a

> I has the form:1

(17.88~ ~quatiOn

(17.88) can be rewntten m the formj

117.7 Example of fractional nonholonomic constraints

39S1

IDlq(t) + (bI/ado§,aq(t) = Co·1

= Dl o§,a for a> 1, we obtainl

lOsing o§,a

(17.89~

[Equations (17.37) and (17.89) givel

Da- I t

~f

2
()

qt

al () _ Coal

+ bi q t

-

al(D~q)(O) k-a ~ blr(k- a+2)t .

alCI _ mi'l

b, t+ b,

(17.90

< 3, then Eq. (17.90) descnbes the hnear fractIOnal oscl1lator:1 (17.9lj

Iwhere 002 = (aI/bd is dimensionless "frequency", and Q(t) is the forcel

[I'he hnear tractIOnal oscillator IS an object of numerous mvestlgatlons (Gorenflq landMamardl, 1997; Mamardl et aI., 2001; Mamardl and Gorenflo, 2000; MamardlJ ~996; Zaslavsky et aI., 2006; Tarasov and Zaslavsky, 2006b) and (StamslavskyJ 12004, 2005; Achar et aI., 2004, 2002, 2001; Tofighl, 2003; Ryabov and PuzenkoJ 12002; Gafiychuk et aI, 2008) because of dIfferent apphcatIOnsl IRemark.

IWe note that the exact solutIOn (Gorenflo and Mamardl, 1997; Mamardl et aI., 2001 j ~amardl and Gorenflo, 2000) of Eq. (17.91) for 2 < a < 3 I~

(t) = q(O)Ea-l,1 (_oo2 ta- l ) +tq(0)Ea- I,2(_oo2 ta- l )

-1

Q(t - -r)qo( -r)d-r

(17.92) IwhereE a ,{3 (2) is the generalized two-parameter Mittag-Leffler function (DzherbashyanJ ~ 966; Erdelyi et aI., 1981), which is defined byl

land

~oo

17 Fractional Nonho1onomic Dynamics

[he decomposition of (17.92) is (Gorenflo and Mainardi, 1997; Mainardi et a1.j 12001):

q(t) = q(O) [fa,o(t) + ga,O(t)] + tq(O) [fa,l (t) + ga,l(t)]

-l

Q(t - -r)qo(-r)d-r (17.93)

fak(t)

=

,

g a,k (t ) --

(-1)

n

re-rt

io

~icos(n/a) a

2a

r

a r

sin(na)

+ 2ra cos xa + 1

cos tsin('Tr/a) - nk ,. a'

dr, k=O,l

k- 0, . 1

(17.94

for the initial conditions q(O) = 1, and q(O) = 0, Equation (17.93) give§

~(t) =

Ea( _ta)

=

(17.95~

fa,o(t) + ga,o(t) - : Q(t - -r)[ja,o( r) + ga,o( -r)]d-r.

[he first term in (17.95) describes decay in power-law with time while the second! ~erm deSCrIbes decays exponentIally (Gorenflo and MaInardI, 1997; MaInardI et al.J 12001; Mainardi and Gorenflo, 2000; Zaslavsky et al., 2006; Tarasov and Zaslavskyj ~

OO:xample 2J [n the 2-dImenslOnal case (n = 2), EquatIons (17.87) have the formj (17.96

2 D tq2

-

I -F _a_ 2+ 2 2 al a2

_ ~F _ ~ 2+ 2

al

a2

I

r:;;,a+I

2+ 2 a:;Vt

al

a2

qI

_

b2

a2 r:;;,a+l 2+ 2 a:;Vt q2· al a2

[.-etus consider a special case ofEqs. (17.96) and (17.97). If we use

al =

(17.97

0, thenl (17.98

~f al

= 0, and b2 = 0, then Eqs. (17.98) arel

(17.99~ OO:xample 3~ [.-etus consider Eqs. (17.96) and (17.97) with b l = 0 and al =

a2 =

c. Then we hav~

(17.100~

4011

117 8 Fractional conditional extremnm

(17.101~ IOsmg the vanablesj

Iwe can rewrite Egs. (17.100) and (17.101) in the formj

IDZx = -g a~ta+lx+g a~ta+ly,

(17.102~

DZY = F(x,y),

IF(x,y) = FI(ql,q2) - F2(QI,Q2) = FI (x+y,x- y) - F2(X+Y,X- y)j OCf F(x,y) = 0, then Eq. (17.102)

i~

(17.103~

~~a(t _ a)f3 = t

r(f3 + 1) (t _ a)f3- a I r(f3 + 1- a) ,

Iwe obtam

(17.104~

~f

a < 1, then Eq. (17.104) descnbes a system WIth a tractIOnal dampmg termj

117.8 Fractional conditionalextremum [The pnnclple of least actIOn or more accurately pnnclple of statIOnary actIOn I~ la vanatlOnal pnnclple, whIch can be used to obtam the equatIOns of motIon fo~ Iholonomlc and LagrangIan systems. In general, thIS pnnClple cannot be used fofj r.onholonomlc and non-HamIltoman systems. Let us conSIder the statIOnary valu~ pf an actIOnmtegralj

[or the lines that satisfy the constraint equation f(Q,q) multiplier J1 = J1(t), we get the variational equation]

rib

= O.

dt[L(Q,q)+J1(t)f(Q,q)] =0.

Using the

Lagrang~

(17.105~

~02

17 Fractional Nonho1onomic Dynamics

[hen the Euler-Lagrange equations (Rumiantsev, 1978,2000, 1982; Cronstr6m and! IRaIta, 2009) for (17.105) ar~ (17.106 W']ote that these equatIOns consIst of the denvative of Lagrange muItlpher ill lEor the fractional nonholonomic constraintJ (17.107j Iwe can define the Lagrangian a§ (17.108~

IOsmg fractional generahzation of the stationary action pnnciple (Agrawal, 2002J 12006, 2007a,b) that is described by the variational equationj

(17.109~ Iwe obtam the Euler-Lagrange equatIOnsj

ISubstltutIOn of Eq. (17.108) into Eq. (17.110) glve~

=0.

(17.111

[I'hese equatIOns descnbe the fractIOnal condItIOnal extremum] OC:et us conSIder apphcabIhty of statIOnary actIOnpnnciple for systems WIth fracj ~IOnal nonholonomic constramts. The equatIOns of motion are denved from thel WAlembert-Lagrange pnnciple. The fractIOnal condItIOnal extremum can be Ob1 ~amed from the statIOnary actIOnpnnciple. In general, these equatIOns are not eqUlv-1 lalent (Tarasov and Zaslavsky, 2006a)j ~t IS known (Rumiantsev, 1978,2000, 1982; Cronstr6m and RaIta, 2009), that sta1 ~IOnary actIOnpnnciple cannot be denved from the d'Alembert-Lagrange pnncIpl~ ~or WIde class of nonholonomic and non-Hamiltoman systems. ThIS statement wa~ Iproved for the case of nonhnear fractIOnal nonholonomic constraints WIth Caputol klerivatives in (Tarasov and Zaslavsky, 2006a)j [Let us obtain conditions of equivalence of the Hamilton's and stationary actionl Ipnnciples for constraints WIth RIemann-LIOuvIlle fractIOnal denvatives.lf we muI1 ~iply Eqs. (17.77) and (17.111) on the variation 8 qk , then the sum with respect to ~

117.9 Hamilton's approach to fractional nonholonomic constraints

40~

(17.112~

(17.113 [Using the definition of variations (17.17), Equation (17.112) has the forml

(17.114~ ISubstItutiOn of (17.114) mto (17.113) gIves the equatIons:1

Iwhere we use the Tchetaev's condltlonJ

(17.116~ V\s a result, EquatiOns (17.77) and (17.111) for nonholonomic system wIth frac1 nonholonomic constraint (17.107) have the eqUivalent set of solutiOns if thel conditions (17.115) and (17.116) are satisfied] ~iOnal

~7.9

Hamilton's approach to fractional nonholonomid constraints

OC:et us conSIder the Hamtltoman equatiOns for dynamIcal systems WIth fractiOna~ r,onholonomic constramts. Usmg Eq. (17.110), we can define the momenta:1

(17.117~ land the HamlltomanJ

I£(q,p) Iwhere <2 -

L+ J1f. Equations (17.110)

dPk dt

= d£ + dqk

/§!:

= PkQ-5L'j giv~

df J1 d(a§/a qk)

~04

17 Fractional Nonho1onomic Dynamics

[£the system is subjected to the non-potential force Qi, then we have the equationsj

[1'0 simplIfy our calculations, we consider the Lagrangianj

land the fractional nonholonomic constraintl (17.118~

Iwith the left-sided Riemann-Liouville fractional derivative aq}/uq. From Eqs. (17 .117~ land (17.118), we obtaml (17.119~

[I'hen the Hamilton equations arg (17.120~

[To find the Lagrange multiplier J1(t), we multiply Eq. (17.119) on the functions a/J landconSider the sum with respect to kj

(17.122j ijere, we use the constraint (17.118), and the notation A 2 IAk(q, aq}pq). From (17.122), we ge~

= AkAk,

where A k ~

(17.123~ ISubstitutiOn of (17.123) into Eqs. (17.120) and (17.121) give~

~k =

(Okl-

A~~l) PI,

(17.124~ (17.125

117 10 Conclusion

4051

[f Ak = 0, then we have usual equations of motion for Hamiltonian systems. Notel ~hat we derive Hamiltonian equations from Euler-Lagrange equations without usingl ~he Legendre transformation, which is typically usedj

~ 7.~ 0

Conclusionl

fractIOnal dynamIcal systems wIth nonholonomIc constramts and nonholonomHj ~onstramt equatIons wIth denvatIves of non-mteger order are conSIdered. The tracj ~IOnal equatIOns of motIon for nonholonomIc systems we obtam by usmg the vanaj ~ional d' Alembert-Lagrange principle. The fractional nonholonomic constraints arel linterpreted as constraints with power-law long-term memory. fractional nonholo-j InomIc constramts allow us to denve tractIOnal equatIons of motIon by usmg thel OC:agranglans and Hamlltomans that have only mteger-order denvatIvesJ IWe note that nonholonomIc systems have recently been employed to study a wldel Ivanety of problems m the molecular dynamICS (Frenkel and SmIt, 2001). In molec1 lular dynamICS calculatIOns, nonholonomIc systems can be explOIted to generate sta1 bstIcal ensembles as the canomcal, Isothermal-Isobanc and IsokmetIc ensemblesl I(Evans et aI., 1983; HaIle and Gupta, 1983; Evans and MOITlss, 1983; Nose, 1991 j palea and Attard, 2002; Tuckerman et aI., 1999,2001; Mmary, 2003; RamshawJ ~986, 2002) and (Tarasov, 2003, 2005a,b; Tuckerman, 2010). Usmg tractIOnal non-I IholonomIc constramts, It IS pOSSIble to conSIder a tractIOnal extenSIOn of the sta1 ~IstIcal mechamcs of conservatIve Hamlltoman systems to a much broader class o~ Isystems. Let us pomt out some nonholonomIc systems that can be generalIzed byl lusmg the nonholonomIc constraint wIth tractIonal denvatIvesJ [n Refs. (Evans et aI., 1983; HaIle and Gupta, 1983; Evans and MOITlss, 1983j ~ose, 1991), the constant temperature systems wIth mImmal GaUSSIan constram~ Iwere conSIdered. These systems are non-Hamlltoman ones and they are descnbe~ Iby the non-potentIal forces that are proportIOnal to the velOCIty, and the GaussIanl r.onholonomIc constramt. Note that thISconstramt can be represented as an addItIOi1j term to the non-potential force (Tarasov, 2005a)j ~n the papers (Tarasov, 2003, 2005a,b), the canomcal dIstnbutIOn was consIdere~ las a statIOnary solutIOnof the LIOuvIIIe equatIOnfor a WIde class of non-HamIltomanl Isystem. ThIS class IS defined by a very SImple condItIOn: the power of the non-I Ipotentlal forces must be proportIOnal to the velOCIty of the GIbbs phase (elementarYI Iphase volume) change. ThIS condItIOn defines the general constant temperature sysj ~ems Note that the condition is a nonholonomic constraint This constraint leads td ~he canomcal dIstnbutIOn as a statIOnary solutIOnof the LIOuvIIIe equatIOns. For thel IlInear fnctIOn, we denved the constant temperature systems. A general form of thel ~on-potential forces was derived in Ref. (Tarasov, 2005a).1 [n Ref. (Tarasov, 2010) fractIOnal dynamICS of relatIVIstIc partIcle was dIscussedJ perivatives of non integer orders with respect to proper time describe long-terml Imemory effects m relatIVIstIc dynamICS. RelatIVIstIc partIcle subjected to a non-I IpotentIal four-force IS conSIdered as a nonholonomIc system (Tarasov, 2010). Thel

f'l-06

17 Fractional Nonho1onomic Dynamics

ronholonomic constraint in 4-dimensional space-time represents the relativistic inj Ivarianceby the equation for four-velocity uJ1uJ1 + cZ = 0, where c is a speed of ligh~ lin vacuum. In the general case, the fractional dynamics of relativistic particle i§ klescnbed as non-Hamiltoman and diSSipative. Conditions for fractional relativistiCj Iparticle to be a Hamiltoman system were suggested m (Tarasov, 2010).1

lReferencesl Achar, l.W. Hanneken, 1'. Clarke, 2004, Dampmg charactenstics of a fracj ItIOnal oscillator, Physica A, 339, 311-319J !B.N.N. Achar, l.W. Hanneken, 1'. Clarke, 2002, Response characteristics of a fracj Itionaloscillator, Physica A, 309, 275-288j ~.N.N. Achar, l.W. Hanneken, 1'. Enck, 1'. Clarke, 2001, DynamiCS of the fractiona~ oscillator, Physica A, 297, 361-367 ~ p.P. Agrawal, 2002, FormulatIOn of Euler-Lagrange equatIOns for fractIOnal vana1 ItIOnal problems, Journal oj Mathematical Analysis and Applications, 272, 368-1 1379. p.P. Agrawal, 2006, Fractional variational calculus and the transversality condi-I Itions, Journal ofPhysics A, 39, 10375-1O384~ p.P. Agrawal, 2007a, FractIOnal vanatIOnal calculus m terms of Riesz fractIOna~ klenvatives, Journal oj Physics A, 40, 6287-6303.1 p.P. Agrawal, 2007b, Generahzed Euler-Lagrange equatIOns and transversahty con1 klitionsfor FVPs m terms of the Caputo denvative, Journal of VIbratIOn and Conj ~rol, 13, 1217-1237j M. Caputo, 1967, Linear models of dissipation whose Q is almost frequency inj klependent, Part II, Geophysical Journal oj the Royal Astronomical Society, 13J 529-539. M. Caputo, 1969, Elasticita e Dissipazione, Zanichelli, Bologna. In Italian.1 M. Caputo, E Mamardi, 1971a, Lmear models of diSSipatIOn m anelastic sohdsJ lRivista del Nuovo Cimento, SedI, 1, 161-198.1 M. Caputo, E Mamardi, 1971b, A new diSSipatIOn model based on memory mecha1 Imsm, Pure and Applied Geophysics, 91, 134-147J Cronstrom, 1'. Raita, 2009, On nonholonomic systems and vanatIOnal pnnciples) 1J0urnai ofMathematical Physics, 50, 042901; and E-print: arXiv:081O.3611 j IV. V. Dobronravov, 1970, FoundatIOns of Mechamcs of Nonholonomlc SystemsJ IVishayaShkola, Moscow. In Russian.1 IV. V. Dobronravov, 1976, Foundations oj Analytical Mechanics, Vishaya ShkolaJ ~.N.N.

r.

!Moscow In Russian Section I 5J

M.M. Dzherbashyan, 1966, Integral Transform Representations ofFunctions in th~ IComplex Domain, Nauka, Moscowl k\. Erdelyi, W. Magnus, E Oberhettinger, EG. Tricomi, 1981, Higher Transcenden-I ~al Functions, YoU, Kneger, Melbbourne, Flonda, New Yorkl

IR eferences

4071

[D.J. Evans, w.G. Hoover, RH. Failor, B. Moran, A.J.e. Ladd, 1983, Nonequilib-j Irium molecular dynamics via Gauss's principle of least constraint, Physical Re-j Iview A, 28, 1016-102U pJ. Evans, G.P. MOITlss, 1983, The Isothermalhsobanc molecular dynamIcs ensemj Ible, Physics Letters A, 98, 433-436j [D. Frenkel, R Smit, 2001, Understanding Molecular Simulation: From Algorithm~ ro Applications, 2nd ed., Academic Press, New Yorkj ~. Gafiychuk, R Datsko, V. Meleshko, 2008, Analysis of fractional orderl IBonhoeffer-van der Pol oscillator, Physica A, 387, 418-4241 [r.M. Galea, P. Attard, 2002, Constramt method for denvmg noneqmlIbnum molecj lular dynamics equations of motion, Physical Review E, 66, 041207.1 IR. Gorenflo, F. Mainardi, 1997, Fractional calculus: Integral and differential equa-j Itions of fractional order, in Fractals and Fractional Calculus in Continuum Me~ khanics, A. Carpinteri, F. Mainardi (Eds.), Springer, New York, 223-276; and! IE-print arxiv:0805.3823j ~.M. HaIle, S. Gupta, 1983, ExtenSIOns of the molecular dynamIcs sImulatIOnl Imethod. II. Isothermal systems, Journal oj Chemical Physics, 79, 3067-30761 k'\.K. Jonscher, 1996, Universal Relaxation Law, Chelsea Dlelectncs Press, Londonl k'\.K. Jonscher, 1999, Dlelectnc relaxatIOn m solIds, Journal oj Physics D, 32, R571 IR7O: k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj ~ional Dijjerential Equations, ElsevIer, Amsterdam1 f. MamardI, R. Gorenflo, 2000, On MIttag-LeiBer-type functIOns m tractIOnal ev01 IlutIOn processes, Journal oj Computational and Applied Mathematics, 118, 283-1

IZ!N:

f. Mainardi, 1996, Fractional relaxation-oscillation and fractional diffusion-wavd Iphenomena, Chaos, Solitons and Fractals, 7, 1461-1477.1 2001, The fundamental solutIOn of the space1 ItIme tractIonal dIffUSIOn equation, Fractional Calculus and Applied Analysis, 4J 1153-192 f. Mmary, GJ. Martyna, M.E. Tuckerman, 2003, Algonthms and novel applIcatIOn~ Ibased on the IsokmetIc ensemble. I. BIOphySIcal and path mtegral molecular dyj InamIcs, Journal oj Chemical Physics, 118, 2510-25261 IS. Nose, 1991, Constant-temperature molecular dynamICS, Progress oj Theoretica~ IPhysics, Supplement, 103, 1-461 ~.A. Pars, 1964, A Treatise on Analytical Dynamics, Heinemann, London] [. Podlubny, 1999, FractIOnalDlfjerentwl EquatIOns, AcademIC Press, New YorkJ ~.D. Ramshaw, 1986, Remarks on entropy and meversIbIlIty m non-HamIltomanl Isystems, Physics Letters A, 116, 110-114.1 ~.D. Ramshaw, 2002, Remarks on non-Hamlltoman statIstIcal mechamcs, Eur01 !physics Letters, 59, 319-323j IV. V. Rumlantsev, 1978, On HamIlton's pnnClple for nonholonomIc systems, Journall pIApplied Mathematics and Mechanics 42, 407-419; and Hamilton's principlcj Ifor nonholonomIc systems, Prikladnaya Matematika i Mekhanika, 42, 387-3991 lin Russian I

f. MamardI, Yu. Luchko, G. Pagmm,

~08

~.v.

17 Fractional Nonho1onomic Dynamics

Rumiantsev, 1982, On integral principles for nonholonomic systems, Journall 1-8.1 ~.v. Rumyantsev, 2000, Forms of Hamilton's principle for nonholonomic systemsJ IFacta Universitatis. Series Mechanics. Automatic Control and Robotics, 2, 1035-1 11048. http:77facta. jums.m.ac.rs/macar/macar2000/macar2000-02.pdfl lYE. Ryabov, A. Puzenko, 2002, Damped oscillations in view of the fractional os-j ~Illator equatIOn, PhYsical Review B, 66, 184201J IS.G. Samko, A.A. KUbas, 0.1. Marichev, 1993, Integrals and Derivatives of Frac1 klOnal Order and ApplzcatlOns, Nauka i Tehmka, Mmsk, 1987. m Russianj Find FractIOnal Integrals and DerivatIves Theory and ApplzcatlOns, Gordon and] IBreach, New York, 1993J k\.A. Stamslavsky, 2004, FractIOnal oscIllator, Physical Review E, 70, 051103J k\.A. Stamslavsky, 2005, TWistof fractIOnal OSCillatIOns, PhYsica A, 354, 101-11OJ IVB. Tarasov, 2003, ClaSSicalcanomcal distnbution for diSSipativesystems, Modernl IPhysics Letters B, 17, 1219-1226j ~E. Tarasov, 2005a, Stationary solutions of Liouville equations for non-I IHamiItoman systems, Annals oj Physics, 316, 393-4131 IVB. Tarasov, 2005b, ThermodynamiCs of few-particle systems, International Jour1 Inal oj Modern Physics B, 19, 879-8971 IVB. Tarasov, 2008a, Fractional equations of Cune-von Schweidler and Gauss lawsJ IJournalofPhysics A, 20, 145212j IY.E. Tarasov, 2008b, Umversal electromagnetic waves m dielectnc, Journal oj1 IPhysicsA, 20, 175223J IY.E. Tarasov, 2010, FractIOnal dynamiCs of relatiVistic particle, International Jour1 Inal of Theoretical Physics, 49, 293-303.1 ~E. Tarasov, G.M. Zas1avsky, 2006a, Nonho1onomic constraints with fractiona~ derivatives, Journal ofPhysics A, 39, 9797-9815.1 IY.E. Tarasov, G.M. Zaslavsky, 2006b, DynamiCs With low-level fractIOnahty, PhYS1 lica A, 368, 399-4151 W'J.G. Tchetaev, 1932, About Gauss Pnnciple, m Proc. Phys. Math. Soc. of Kazan 1 University, 6 Ser.3. (1932-1933) 68-71 J N.G. Tchetaev, 1962, Stability ofMotion. Works on Analytic Mechanics, Academyl pf SCiences USSR, Moscow, 323-326. In Russianl k\. Tofighi, 2003, The mtnnsic dampmg of the fractIOnal OSCillator, Physica A, 329J 29-34. M.B. Tuckerman, 2010, StatIstIcal MechaniCS and Molecular SImulatIOns, Oxford] IUmversity Press, OxfordJ ~.E. Tuckerman, C.I. Mundy, G.I. Martyna, 1999, On the claSSical statistical me1 ~hamcs of non-HamIltoman systems, Europhysisics Letters, 45, 149-1551 ~.B. Tuckerman, Y Lm, G. Ctccotti, G.I. Martyna, 2001, Non-Hamiltomani Imolecular dynamiCs: Generahzmg Hamiltoman phase space pnnciples to non-I IHami1tonian systems, Journal of Chemical Physics, 115, 1678-1702j p.M. Zaslavsky, A.A. Stanislavsky, M. Edelman, 2006, Chaotic and pseudochaoticj Fittractorsof perturbed fractIOnal oscIllator, Chaos, 16, 0131021

'Pi Applied Mathematics and Mechanics, 46,

~hapter 1~

[Fractional Dynamics an~ ~iscrete Maps with Memory

118.1 Introductionl [The study of nonhnear dynamics m terms of discrete maps is a very important ste~ 1m understandmg the quahtatIve behaviOr of physical systems descrIbed by differ-I ~ntIal equations (Sagdeev et aI., 1988; Zaslavsky, 2005; Chmkov, 1979; Schuste~ land Just, 2005; Collet and Eckman, 1980). Discrete maps lead to a much simplefj ~ormahsm, which is particularly useful m computer simulatiOns. The derIvatives ofj Inon-mteger orders (Samko et aI., 1993; Ml1ler and Ross, 1993; Podlubny, 1999j IKl1bas et aI., 2006) are a natural generahzatiOn of the ordmary differentiatiOn o~ Imteger order. Note that the contmuous hmit of discrete systems with power-Iawl Ilong-range mteractIons gives differential equations with derIvatives of non-mtege~ prders with respect to coordinates (see for example, (Tarasov and Zaslavsky, 2006j [Tarasov, 2006a,b)). Fractional differentiation with respect to time is characterize~ Iby long-term memory effects that correspond to mtrInSiC diSSipative processes ml ~he physical systems. The memory effects to discrete maps mean that the presen~ Istate evolutiOn depends on an past states. The discrete maps with memory werel ~onsidered, for example, m the papers (Fuhnski and Kleczkowskl, 1987; Flck et aI.J ~ 981; Giona, 1991; Hartwich and Fick, 1993; Gallas, 1993; Stanislavsky, 2006) an~ I(Tarasov, 2008d, 2009a,b; Edelman and Tarasov, 2009). The mterestmg questiOn i~ la connection of fractiOnal equatiOns of motion and the discrete maps with memoryj piscrete maps with memory can be derIved (Tarasov, 2008d, 2009a,b) from equa1 ~iOns of motion with fractiOnalderIvatives. In this chapter, we consider discrete mapsl ~hat can be used to study the evolutiOndescrIbed by fractiOnal differential equatiOn~ I(Samko et aI., 1993; Podlubny, 1999; Kilbas et aI., 2006). Note that perturbed by aI IperIodic force, the nonhnear system with fractiOnal derIvative exhibits a new type o~ ~haotIc motion caned the fractiOnal chaotic attractor. The fractiOnal discrete mapsl I(Tarasov, 2008d, 2009a,b; Edelman and Tarasov, 2009) can be used to study a newl ~ype of regular and strange attractors of fractional dynamics of physical systems.1 [n this chapter, fractional equations of motion for kicked systems are consideredj

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

~1O

18 Fractional Dynamics and Discrete Maps with Memory

~orrespondent discrete maps with memory are derived from these equations. [ractional generalizations of the well-known maps are suggestedj

~8.2

Th~

Discrete maps without memory

~n this section, a brief review of discrete maps is considered to fix notations and! IprovIde convement references. For detaIls, see (Sagdeev et aI., 1988; ZaslavskyJ 12005; Chirikov, 1979; Schuster and Just, 2005; Collet and Eckman, 1980)j

IUniversal and standard mapsl OC=et us consIder the equatIon of motIonj (18·1)1 1m whIch perturbatIon IS a perIOdIC sequence of delta-functIOn-type pulses (kIcks] [ollowing with period T = 2n / v, K is an amplitude of the pulses, DZ = d 2 / dt 2 , an~ lG[x] is some real-valued function. This equation can be represented in the Hamilto~ Inian form' co t DJP + KG[x] 0 = O.

L (T - n)

n=O

[t IS well known that dIfferentIal equatIons (18.2) can be represented m the form ofj ~he dIscrete mapj fXn+I -xn = Pn+IT,

Pn+l - Pn = -KTG[xn].

~quatlOns

(18.3)1

(18.3) are called the umversa1 map. For detaIls, see, for example, (ChmkovJ Sagdeev et aI., 1988; Zas1avsky, 2002)1 OC=et us deSCrIbe a standard derIVatIOn of the dIscrete map from equatIOns of moj ~ion (see, for example, Chapter 5 in (Zaslavsky, 2005), or Section 5.1 in (Sagdeev e~ la1., 1988». Between any two kIcks there IS a free motlOnj ~979;

IP - const,

x - pt

+ consq

["he solution of the left side of the n th kickl

fn = p(t

n-

0)

= £---+0+ lim p(nT - £)1

118.2 Discrete maps without memory

Iwhere tn = nT, is connected with the solution on the right side of the kick xUn + 0) j + 0) by Eq. (IS.2), and the condition of continuity xUn + 0) = x(tn - 0). Th~ lintegration of (IS.2) over the interval Un - t,tn + t) give§

IpUn

Ip(tn+O)

=

p(t n -0) -KTG[xn].1

IOsmg notations Xn and Pn, we can derIve the Iteration equationsj

[Xn±I - Xn + Pn±ll',

(18.4)1

IPn±I = Pn -KTG[xn].

(IS.5)1

~quatIOns

(18.4) and (18.5) are called the unIversal mapJ OC=et us consIder another method of derIVatIOn of dIscrete map equations from thel ~Ifferential equatIOns. We can use an eqUIvalence of the dIfferential equation and! ~he Volterra integral equation to obtain the universal mapj

[Theorem 18.1. The Cauchy-type problem for the differential equations.j

ID}x(t) = p(t),

(1s.6)1 (1S.7)1

IWlth the inztzal conditIOns 1

k(O)

= xo, p(O) = PO

(IS.S)1

lis equivalent to the universal map equations in the jorm.j

~n±I =xo+ po(n+ 1)1' -KT2

L G[Xk] (n+ l-k),

(1S.9)1

nl

pn±I =po-KTEG[Xk]'

(1S.10~

1c=-U

IProof. It ISknown that the Cauchy-type problem for the dIfferentIal equatIOnj

~;x(t)

= F[t,x(t)],

0 ~ t ~ tf

(1S.1

q

land the mItial condItIOnsl

k(o)

= Xo,

(D;x)(O) = Po

(1S.12~

lIS eqUIvalent to the Volterra mtegral equation of second kmdj

E(t) =xo+Pot+ ~quation (1S.11)

with the functionj

l

drF[r,x(r)](t-r).

(1S.13~

~12

18 Fractional Dynamics and Discrete Maps with Memory

IF[t,X(t)]

= -KG[x(t)] ~ 8(f -k),

Irepresents Eq. (18.1). Using Eq. (18.13) with function (18.11) for nT < t Iwe obtain

§(t) for the momentum p(t)

=

Xo + Pot - KT

E G[x(kT)](t -

kT).

(18.14~ < (n+ I)T j (18.15~

= D;x(t), Equation (18.15) givesl ll1J

p(t)

=

Po- KT

L G[x(kT)].

(18.16~

[he solutions for the left side of the (n + 1)th kickj lim x(T(n + 1) - e), r'lxn+1 = X(tn+1 - 0) = £--+0+

(18.17~

IPn+1 = p(tn+1 - 0) = lim p(T(n + 1) - e),

(18.18~

£--+0+

Iwhere tn+1 = (n + 1)T, have theform (18.9) and (18.10)1 [I'hIS ends the proof.

q

IRemark. lOSIng Eqs. (18.9) and (18.10), the dIfferences Xn+1 - Xn and Pn+1 - Pn gIve EqsJ 1(18.3) of the unIversal map.1 !Example 1~ -x, then Eqs. (18.3) give the Anosov-type systemj

~f G[x] =

Pn+1 - Pn = K'I'x.;

(18.19~

Pn+1 - Pn - -KT SInXn·

(18.20]

Ixn+1 -Xn = Pn+IT, !Example 2J for Glxl = sinx, Equations (18.3) arel

fXn+1 -Xn - Pn+IT,

[I'hls map IS known as the standard or Chlflkov-Taylor map (Chlflkov, 1979).1

IDissipative standard mapl [Thedissipative standard map (Zaslavsky, 1978; Schmidt and Wang, 1985) is defined!

IQY (18.21]

(18.22]

118.2 Discrete maps without memory

Iwhere we use the

paramete~

Note that a shift Q does not play an important role and it can be put zero (Q = O)j [Thedissipative standard map with Q = 0 can be represented by the equations:1 (18.23~

IPn±1 = -bPn - Z sinXn ·

(18.24~

for b = -1 and Z = K, we get the standard map that is described by Eqs. (18.4) and! 1(18.5) with T = 1. For the parameter~ (18.25~ ~quations (18.23) and (18.24) give Eqs. (18.21) and (18.22) with Q = O. The dissi-j pative standard map is also called Zaslavsky map~ for large q ----+ 00 (for smaIl b ----+ 0), Equations (18.23) and (18.24) wIth Z = -KI Ishnnk to the I -dImensIOnal sme-map:1

IXn±1 =Xn+KsmXn

(l8.26~

Iproposed by Arnold (Arnold, 1965) and studied in many papersj

[Kicked damped rotator ma~ OC:et us conSIder a kIcked damped rotator. The equatIon of motIon for thIs rotator I§

p;x+qDlx = KG [x]

L 8(t-nT).

(18.27~

~

[Theorem 18.2. Equation (18.27) gives the discrete map equations:1

(18.28~ (18.29j IProof. ThIs theorem was proved m SectIOn 2.2 of (Schuster and Just, 2005).

q

[I'hls map IS known as the kIcked damped rotator map. The phase volume shnnksl ~ach time step by a factor exp{ -q}. The map is defined by two important parame-I ~ers,

dISSIpatIon constant q and force amplItude KJ

IRemark lJ (18.28) and (18.29) can be rewntten m the forml

~quatlOns

~14

18 Fractional Dynamics and Discrete Maps with Memory

IXn±1 - Xn +

Pn±Il

g]

[t IS easy to see that these equatIons gIve the Zaslavsk5f map (18.23) and (18.24] IWlth n - 0, I~

IXn =Xn,

e=

Yn = Pn,

K,

T = 1,

Glxl = sinxJ

IRemark 2J ~f we

consider the limit q -+ 00 and K

-+

00

such that q 7K

= 1 and the functionj

P[x] = (r-l)x-rx 2 J ~hen

Eqs. (18.28) and (18.29) gIve the logIstIc mapj

~ote that the evolutIOn of logIstIc maps wIth a long-term memory was lin (Stanislavsky, 2006).1

consldere~

lHenon map, [The Henon map (Henon, 1976; Russel et al., 1980) can be considered as a two-j klimensional generalization of the logistic map (Schuster and Just, 2005). The Henonl Imap IS represented as a map grven by the coupled equatlOns:1

(18.30~ (18.31]

lYn±1 - bx n·

ISubstltutlOnof Eq. (18.31) Into Eq. (18.30) glve§ kn±1

= I - ax~ + bXn-l.

(18.32~

[Ihls equatIOn IS eqUIvalent to the Henon map that IS defined by Eqs. (18.30) an~ 1(18.31). ~ote that the Henon map In the form (18.32) can be denved from the generahze~ k1lsslpatIve mapj (18.33] [Xn±1 - Xn + Pn±ll', (18.34] Iwhere Ibl ::;; 1.1

[Theorem 18.3. Equations (18.33) and (18.34) with K

=

T = 1, and the function.j

118.3 Caputo and Riemann-Liouville fractional derivatives

PJx] = Iglve the Henon map

In

4151

(18.35~

1- (l-b)x-ax 2 ,

the form (18.32)J

IProof. SubstitutIOn of (18.33) m the form Pn+l = Xn+l - Xn mto (18.34) glve~

["hen we havel ftn+l -bXn-l

=

(18.36~

(l-b)x n+G[xn].

q

ISubstItutIOn of (18.35) mto (18.36) gIves the Henon map (18.32).

rrheorem 18.4. Equation oj motion (18.27) withl IG[X] = -1 Iwhere b = - exp{ -q} and T IfquatlOn (18.32)J

=

~b [1 + (1 +b)x+ax2]

,

(18.37~

1, can be represented in the form of the Henon mapl

IProof. Usmg Eq. (18.27) wIth functIOn (18.37), we obtam the map equatIOnsj

1 +b -1- (1 +b)xn-ax2 Xn+l =Xn + -q-Pn n,

I

(18.38~

Iwhere T = 1, and b = - exp{ -q}. As was proved in Section 2.2 of (Schuster and! [ust, 2005), Equations (18.38) and (18.38) are equivalent to the Henon map equation] 1(18.32). q

118.3 Caputo and Riemann-Liouville fractional derivativesl [The left-sIded Caputo fractIOnal denvatIve (Caputo, 1967; Caputo and MamardlJ [971; Gorenflo and Mamardl, 1997; Kdbas et aL, 2006) of order a> 0 IS defined]

IQY c

(X

(

oD t x t

)

n-(X n ( ) Dtx t

= oIt

1

= r( n - a )

it d 'r D~x 'r 0 (t - 'r ) (X-n +1'

(18.40

Iwhere n - 1 < a < n, and oIta is the left-sided Riemann-Liouville fractional integrall pf order a > 0 that IS defined b)1

(18.41~ [he left-sided Riemann-Liouville fractional derivative (Samko et al., 1993; Podj ~ubny, 1999; Kilbas et al., 2006) is defined byl

f'l-16

18 Fractional Dynamics and Discrete Maps with Memory

n

-1 < a:( n.

(18.42) [TheCaputo and Riemann-Liouville fractional derivatives are connected by the equaj tion: (18.43 [TheRiemann-Liouville fractional derivative has some disadvantages in applicaj In fractIonal mechanIcs such as the hyper-SIngular Improper Integral, where thel prder of sIngulanty IS hIgher than the dImenSIOn. Moreover, the RIemann-LIOuvIlI~ fractIOnal denvatIve of a constant need not be zero, and we hav¢1 ~Ions

W'Jonzero denvatIve of constants entaIl that dIsSIpatIon does not vanIsh for a systeml lIn eqUIlIbnum. We note that InItIal value problems for fractIonal equatIons wIthl RIemann-LIOUVIlle denvatIve have unusual form.1 ~t IS easy to see that the second term In Eq. (18.43) regulanzes the Caputo frac1 ~IOnal denvatIve to aVOId the potentIally dIvergence from SIngular IntegratIOn a~ f = O. In addItIOn,the Caputo fractIOnal dIflerentIatIOn of a constant results In zero:1

~Dfc=ol [f the Caputo fractIOnal denvatIve IS used Instead of the RIemann-LIOuvIlle fracj ~IOnal denvatIve, then the InItIal condItIOns for fractIOnal equatIOns are the same a~ ~hose for the usual dIfferentIal equatIOns. The Caputo formulatIOn of fractIOnal caI1 ~ulus can be more applIcable In mechanICS than the RIemann-LIOuvIlle formulatIOnl k\t the same tIme the definItIOn of Caputo fractIOnal denvatIves IS, of course, morg IrestnctIve than the RIemann-LIOuvIlle fractIOnal denvatIve, SInce It reqUIres the abj Isolute IntegrabIlIty of the denvatIve of order n. The Caputo fractIOnal denvatIvg lfirst computes an ordInary denvatIve followed by a fractIOnal Integral to achIeve thel Hesire aider of fractional derivative The Riemann-Liollvj]]e fractional derivative is ~omputed In the reverse orderl [rhe deSIre to use the usual InItIal value problems for mechanIcal systems lead t9 ~he use of Caputo fractIOnal denvatIves rather than the RIemann-LIOuvIlle fractIOna~ ~envatIves. It IS pOSSIble to state that the Caputo fractIOnal denvatIves allow us t9 gIve more clear mechanIcal InterpretatIOn. At the same tIme we cannot state that thel ~Jemann-LIOuvIlle fractIOnal denvatIve does not have a phySIcal InterpretatIOn and! ~hat It shows unphySIcal behaVIOr. PhySIcal InterpretatIOns of the RIemann-LIOuvIll~ ~ractIOnal denvatIves are more complIcated than Caputo fractIOnal denvatIves. Bu~ ~he RIemann-LIOuvIlle fractIOnal denvatIves naturally appear for real phySIcal sysj ~ems In electrodynamICs. We note that the dlelectnc susceptIbIlIty of a WIde clas~ pf dlelectnc matenals follows, over extended frequency ranges, a fractIOnal power1 ~aw frequency dependence that is called the universal response (Jonscher, 1996j

4171

118.3 Caputo and Riemann-Liouville fractional derivatives ~ 999).

As was proved in (Tarasov, 200Sb,a), the electromagnetic fields in such dij media are described by differential equations with Riemann-Liouville fracj ~ional time derivatives. These fractional equations for "universal" electromagneticj Iwaves in dlelectnc medIa are common to a wIde class of matenals, regardless ofj ~he type of physIcal structure, chemIcal composItIon, or of the nature of the poj ~arizing species. Therefore we cannot state that Riemann-Liouville fractional timel klerivativesdo not have a physical interpretation. The physical interpretation of thesel klerivatives in electrodynamics is connected with the frequency dependence of thel ~iIelectnc susceptIbIlIty. As a result, the dIscrete maps WIth memory, whIch are conj InectedWIth dIfferentIal equatIons WIth RIemann-LIOuvIlle tractIonal denvatIves arel Ivery Important to physIcal applIcatIons, and these denvatIves naturally appear fo~ Ireal physical systems. For computer simulation and physical application, it is verYI limportant to take into account the initial conditions for discrete maps with memorYI ~hat are obtained from dIfferentIal equatIons WIth RIemann-LIOuvIlle tractIonal tImel klerivatives IWe note a connectIon between equatIons of motIon WIth the Caputo tractIona~ ~envatIves and equations WIth the RIemann-LIOuvIlle denvatIves.1 ~lectric

[Theorem 18.5. The fractional differential equation:1 rn~D~x = P[t,x,p],

1< a <2

(1S.44~

linvolving the left-sided Caputo fractional derivative ~ Df is equivalent to the equa-I ~

IDZx

= oDZ-ap[t,x,p],

1
< 2,

(1S.45~

Iwith the left-sided Riemann-Liouvillefractional derivative oDZ al Iproof. Using p

= mD}x, Equation (1S.44) can be rewritten in the forml

(1S.46~ I~Df

Ip

= P[t,x,p],

1
< 2.

(1S.47j

fractIOnal integratIOn of (18.47) of order a - I gIvesl

If Ota

I CDa fa 0 t I P=Ot

Ip

[t,x,p. ]

(1S.4sj

lOSing the fundamental theorem of fractIOnal calculus (Tarasov, 2008c)j

Iwe obtain

~(t)

= p(O) + ofta- I P[t,x,p].

(1S.49~

pifferentiation of Eq. (1S.49) give§ I[Jlp=oDz ap[t,x,p],

O<2-a<1,

(1S.50~

~18

18 Fractional Dynamics and Discrete Maps with Memory

Iwhere oD;-a is the left-sided Riemann-Liouville fractional derivative. As a resuld Iwe obtain the fractional equationsj

(18.51~ [DIp ~quations

=

oD;-aF[t,x,p],

I

(18.52~

< a < 2.

q

(18.51) and (18.52) can be rewritten in the form (18.45).

k\s a result, EquatIon (18.44) wIth the Caputo tractIonal derIVatIve IS eqmva-I to Eq. (18.45) with the Riemann-Liouville fractional derivative. Using (18.42),1 ~quation (18.45) has the fonnj ~ent

1< a

~

2. (18.53

[I'he physIcal mterpretatlon of the tractIOnal mtegratIon IS an eXIstence of a memorY] ~ffect wIth power-hke memory functIOn. The memory effect mean that theIr presen~ Istate evolution depends on the force Flr,x(r),p(r)1 at all past phase-space state§ k{escribed by (x( r), p( r)), r E 10,t1J

~8.4

Fractional derivative in the kicked term and discrete map~

[n thISsectIon, we conSIder a tractIOnal generahzatIon of dIfferentIal equatIon (18.1 )1 Iwith Caputo fractional derivative of the order ~ f3 < 1 in the kicked term. W~ pbtam a dIscrete map that corresponds to the suggested tractIOnal equatIon of orderl p ~ {3 < 1. This map can be considered as a generalization of universal mapJ ~et us conSIder a tractIOnal generahzatIOn of (18.1) m the formj

°

0~f3<1,

(18.54

Iwherethe Caputo fractional derivative ~Df i~ 0~f3<1.

for {3

(18.55

°

= fractional equation (18.54) gives Eq. (18.1)J

rrheorem 18.6. The fractional difjerential equation of the kicked system (18.54) isl ~quivalent

to the discrete map.j fxn+I =

Xn

+ Pn+l T ,

(18.56)

118.4 Fractional derivative in the kicked term and discrete maps

4lS1

(18.57 Iwhere the function V2-I3(Z) is defined byl

(18.58~ I

tor Z ?:- 1.

IProof The fractional equation (18.54) can be represented in the Hamiltonian form:1 ~lX=p,

(18.59)

(18.60~ ~etween

any two kicks there IS a free motIon:1 Ip - const,

[Theintegration of (18.60) over (tn

x - pt + const.

+ e, tn+ I - e)

(18.61]

give~

(18.62~

Ip(tn+l - 0)

p(tn + 0).

=

(18.63~

rI'he solutIOn of the left sIde of the n th kickl

IXn = x(t n - 0) = limx(nT - e),

(18.64~

fn

(18.65~

£---+0

=

p(tn - 0) = limp(nT - e), £---+0

Iwhere tn = nT, is connected with the solution on the right side of the kick x(t n + O)j Ip(tn + 0) by Eq. (18.60), and the continuity conditionj (18.66~

rx(tn +0) =x(tn -0). [The integration of (18.60) over the interval (tn - e,tn + e) give~

Ip(tn +0) = p(tn -0) -KTG[~D~x].

(18.67~

IUsmg notations (18.64) and (18.65), and the solutIOn (18.61), we getl

Ip(tn + 0) = p(tn+l - 0)

=

Pn+l1

ISubstItutmg of (18.66) and (18.67) mto (18.62) and (18.63), we denve the IteratIOnl ~quations:

~20

18 Fractional Dynamics and Discrete Maps with Memory

Ixn±1 IPn±1

=

X

n + Pn±ITJ

= Pn -KTG[~Dex].

(18.68~

[fo derive a map, we should express the fractional derivative:1

t

1

n ~ /3 _ 1 -/3 I fDtnx- r(1-f3)}o dr(tn-r) Drx(r j

~hrough the variables (18.64) and (18.65). Using p( r) = Dix( r), this equation canl Ibe represented a§

(18.69 Iwhere tk±1 = tk + T = (k+ I )T, t« = kT, and to = OJ for the interval u; tk±d, Equations (18.61), (18.64) and (18.65) givel

F F

Pk±16((tn -tk)I-/3 - (tn -tk±I)I-/3l Pk±1 S((n -k)I-/3 - (n -k-I)I-/3).

(18.70~

[Using (1- [3)r(I- [3)=r(2- [3) andEq. (18.70), the fractional derivative (18.69~ ~an be represented a~

(18.71

(18.72~ [I'hIS ends the proof.

q

V\s a result, EquatIOns (18.68) take the form of Eqs. (18.56) and (18.57). Thesel define the tractIOnal generalIzatIOn of unIversal map. Let us conSIder thel [ollowing statementj ~quatIOns

[Theorem 18.7. For [3 = 0 and Xo = 0 the fractional universal map, which is define~ ~y Eqs. (18.56) and (18.57), gives the usual universal mapj

118.4 Fractional derivative in the kicked term and discrete maps

42~

IProof. Let us consIder that the fractIOnal unIversal map (18.56) and (18.57) 1/3 = O. Substitution of Eq. (18.56) in the formj

to~

(18.73~ linto iteration equation (18.57)

give~

(18.74 [Then the fractIOnal map (18.56), (18.57) IS defined byl

(18.75 Iwhere 0 ~

f3 < 1. For f3 = 0, we have V2-I3(Z) = 1, an~ u;:]

IL(Xk+I-Xk) =xn-xoj ~

[Then Eq. (18.74) gIvesl

k\s a result, Equations (18.75) with /3 [fhISends of the proof.

= 0 give the usual map for the case Xo = OJ q

~et us consIder some examples of the fractIOnal map that IS defined by Eqsl 1(18.56) and (18.57)~

!Example 1~ OCf Glxl = -x, then Eqs. (18.56) and (18.57) ar~

(18.76 ~quations ~ype

(18.76) can be considered as a fractional generalization of the Anosov-I system.

!Example 2~ sinx, Equations (18.56) and (18.57) giv~

~f G[x] =

~22

18 Fractional Dynamics and Discrete Maps with Memory

(18.77 [This map is a fractional generalization of standard map. The other possible form ofj IEq. (18.77) i§ IXn±1 - X n + Pn±tl (18.78 Iwhere we use Eq. (18.73), T ~ractIOnal standard mapJ

= I, and 0 ~ f3 < 1. These equations can be called thel

OO:xample 3J [The fractional generalization of dissipative standard map (Zaslavsky, 1978; landWang, 1985) can be defined byl

IXn±1 -

X

Schmid~

n + Pn±IJ (18.79

for b = -1, we get the fractIOnal standard map (18.78). ThIS map IS one of possiblel ~ractIOnal generahzatIOns of the dISSIpatIve standard map. In thIS generahzatIOn wei Imtroduce a dIssIpatIon by the change of the vanable Pn ----+ -bpn. EquatIons (18.79] lare not dIrectly connected wIth a tractIonal equatIon of motIon. A generahzatIonl pf the dIssIpatIve standard map that IS denved from the dIfferentIal equatIOn withl ~ractIOnal damped kicks IS suggested m the next sectIOnJ

118.5 Fractional derivative in the kicked term and

dissipativ~

~iscrete map~ ~n thIS sectIOn, a tractIOnal generahzatIOn of dIfferential equatIOn (18.27) for ~ IkIcked damped rotator IS suggested. Let us conSIder the tractIOnal generahzatIOnl pf Eq. (18.27) m the formj

p;x-qDix =

KG[~Dfx] f

Iii"ill

0(1 - nT),

0

~ f3 < 1.

(18.80~

[TheCaputo fractional derivative of the order 0 ~ f3 < I is used in the kicked dampe~ ~erm, l.e., the term of a penodic sequence of delta-functIOn type pulses (kicks). W~ rote that the mmus ISused m the left-hand sIde of Eq. (18.80), where q E lItl [Theorem 18.8. The fractional differential equation (18.80) is equivalent to the disj 'Crete map:

42~

118.5 Fractional derivative in the kicked term and dissipative discrete maps

l-e ql xn+ qT

n+l = e

1

(Pn + KG[r(l _ /3)

Iwhere the junctIOns W2 -

j3 are

(18.8d

Pn+l,

g]

n-l

~ Pk+lW2 - j3 (q,T,n -

k)]),

(18.82

defined b)j

IProof. FractIOnal equation (18.80) can be represented In the Hamiltoman form:1

IDi P - qp Iwhere

=

KG[~Dfx]

f: o(t - nT),

(18.83~

°< f3 <

~etween

0, and q E JRl any two klcksj

(18.84~

ID!p-qp= 0. for t E (tn + O,tn+l - 0), the solution of Eq. (18.84) i~

(18.8Sj ~et

us use the notations t« = nT, andl

IXn = x(t n - 0) = limx(nT - e)J £-----+0

fn = p(tn - 0) = limp(nT - e).

(18.86~

£-----+0

fort E (tn-e,tn+l-e), the general solutionof(18.83) i§

(t)

= Pneq(t-tn) + K

L G[~Dfmx] [ m=O

dre q(t- 7: )o( 'r - mT)

tn-E

(18.87~ lOSIng (18.87), the mtegration of the first equation of (18.83) q1

1- e C j3 Xn+l -X n - --(Pn+KG[oD tn xl). q

I

gIve~

(18.88~

~et us consIder the Caputo fractIOnal derIvative from Eqs. (18.87) and (18.88). I~ lis defined by the equationl

~24

18 Fractional Dynamics and Discrete Maps with Memory

0(13<1 losing p( -r) = Dix( -r), this relation can be rewritten a~

(18.89 Iwhere tk+l = tk + T = (k+ 1 )T, and tk = kT, such that to = OJ for -r E (tk,tk+l), Equations (18.85) and (18.86) giv~

Ip(-r) = p(tk + O)eq('l"-tkJ = P(tk±l - O)e- qTeq('l"-tkJI

be

Ilk+l p(

Pk+leq('l"-tk-T) = Pk±leq('l"-tk+Il j

-r) (tn - -r)-/3 d-r = Pk±llk+l eq('l"-tk+Il (tn - -r)-/3d-rj

IOsmg the vanablej

Iwe obtain

(18.90 Iwhere we use the function:1

IW2- /3 (Q,T ,n ~quatIOns (18.89)

k ) = T 1-/3

10

1

eqT(z-l) (n-k-z)-/3dzj

and (18.90) glvel

(18.91 ISubstltutmg (18.91) into (18.87) and (18.88), we obtaml

(18.92

1-

[1

qT e ( n]) n+l =xn - - - Pn+ KG r(1-f3) LPk+I W2-/3(Q,T,n-k) . q k=O

(18.93

!EquatIOns (18.92) and (18.93) can be represented m the form of Eqs. (18.81) and! 1(18.82). [This ends the proof. q

118.6 Fractional equation with higher order derivatives and discrete map

4251

OO:xample 1J [The iteration equations (18.81) and (18.82) define a fractional generalization ofthel Ikickeddamped rotator map (18.27). We can derive a fractional generalization of thel k!issipative standard map. It we use the condition~

IXn =xn, ~hen

Yn = Pn,

e = K,

T

= 1, G[x] = sinx,

(18.94~

Eqs. (18.81) and (18.82) give the mapl (18.95) 1

T(I- {3)

k n-l

Yk+l W2 - {3 (q , T,n

- k)

(18.96

[These equatlOns can be considered as a fractlOnal generahzatlOn of the Zaslavskyl Imap (18.21) and (18.22) with .Q = 0 and g = g'. This fractional Zaslavsky mapl liS denved from fractional differential equation (18.80) with condition (18.94). Fo~ 1f3 = 0 this map gives the Zaslavsky map (18.21) and (18.22) with g = g' and.Q = OJ

OO:xample 2J IWe can denve a fractional generahzation of the Henon map from equation of motioll] 1(18.80) with IG[x]

= -1 ~b (1 + (1 +b)x+ax2) ,

(18.97~

Iwhere b = - exp{ -q} and T = 1. As a result, this generalization has the form ofj ~qs. (18.81) and (18.82) with the function (18.97). ThiS fractional Henon map is dej Inved from the differential equations with fractional denvative ill the kicked dampe~ ~erm (Tarasov, 2010). For f3 = 0 this map gives the usual Henon mapJ

118.6 Fractional equation with higher order derivatives and! ~iscrete map [.-etus consider a generalization of fractional equation (18.54) in the form:1

D;"x+KG[~Dfx]

L O(f -n) = 0, 0< {3 ~ 1,

(18.98

n=O

Iwhere m is an integer number such that m ? 3. ThiS generahzatlOn of Eq. (18.54) i~ pbtained by D;x ----+ D;"x. Here ~Df is the Caputo fractional derivative of order {3j P< f3 < 1, which is defined by equation (18.55)j

18 Fractional Dynamics and Discrete Maps with Memory

f'l-26

rrheorem 18.9. The fractional difjerential equation of the kicked system (18.98) isl equivaleni to the discrete maps Xn+l

1 m.

m-l

= Xn + ( _l),Pn+l T

m-l

"f.

1 m-s-2 1 s m-l T m - s- l '\' s+1 T1 + T!Pn , Pn+l = (m-s-l)!Pn+l

'='

1 1 1 -I,PnT ,

+ 1=1 z:

(18.99

.

s= 1, ...,m-2, (18.100 (18.lO1~

(18.102

~nd the functions V/;,I (z) are defined b~ (18.103

f3

~nd 1 ~ I ~ m - 1 < m -

~ m, I, m E

Nl

IProo]. Let us define the vanablesj ~'(t)=D:x(t),

s=O, ... ,m-l,

(18.104~

[Then the HamlHoman form of Eq. (18.98) lsi ~lps=ps+l,

Jp m - l

s=0, ... ,m-2,

+KTG[~D;,,-apo]

L

O(f - n) = O.

(18.105~ (18.106

n=O

for t E (In + O,tn+l - 0), Equation (18.98) is D;"x = 0, and the solution can bel Irepresented a~ m;:]]

~(t)=LC,(t-tn)', m?d.

(18.107~

ISubstltutlOn of (18.107) mto (18.104) glve§ IpS(t) = LC1I(l-1) ... (l-s+1)(t-tn)L s ~

(18.108~

118.6 Fractional equation with higher order derivatives and discrete map

for t

4271

= tn, we havel IpS(tn +0) = Csslj

(18.109~ IOsmg Eq. (18.109) and the relatIOnsj

IpS(tn +0) = pS(tn - 0) = ~m-l ( tn

p~,

+ 0) = Pm-l ( tn+1 -

s = 0, ... ,m - 2j

0) = Pn+1 m-Ij

Iwe present Eq. (18.108) in the formj (18.110 Iwhere s - 0, ... ,m - 2. Here, we ust:j

~quatIOn (18.110)

11(l-I) ...(l-s+l)

11

~!

(l-s)!'1

can be rewntten

a~

(18.111 Iwhere s - 0, ... ,m - 2. For t - tn+l, thiS equation glve~ 1 m-s- 1 s pm-IT m-s- 1 + ~ _ps+IT I Pn+1 = (m-s-l)! n+1 i... I! n ,

0

s= ,...,m-

2

k\s a result, the IteratIOn equatIOns art:j s

Pn+1

1

m

I

= (m _ s -1),Pn;1 T . Ip~;l

m s I - =

+

m-s- 1 ~ _ps+ITI i... II. n ,

s=0, ...,m-2

1=0

p~-I-KTG[qnJl

Iwhereqn = ~D~ expo andpO(t) =x(t)J [I'he vanable qn IS defined by the Caputo tractIOnal denvatlvej

~t

can be represented a~ (18.112

~28

18 Fractional Dynamics and Discrete Maps with Memory

Iwhere tk = kT J for t E (tk,tk+d, Equation (18.111) isl 1

m-3

1

)m-2+ ~ _pl+l(t-t )1. P l (t ) = (m-2)! pm-l(t_t n+l k ~ I! n k

(18.113

ISubstituting (18.113) into (18.112), and using!

Iwe obtain

Ta

(18.114~

1

m-l m,m-2( k) n-, (m_2)!1Jn+l Va

F

1= 1, ... ,m-2.

(18.115

[I'hese functIOns can be defined byl

:;,l(z)=jZ (z_y)lya-mdy= -1

r y'(z_y)a-mdy,

io

1=1, ... ,m-2, (18.116

Iwhere 1 :'( [ :'( m -1 < a :'( m, and [,m E N. SubstItutIOn of (18.114) mto (18.112) gives (18.102)1 q [I'his ends the proof. ~ote

that the functIOns that are defined by mtegrals (18.116) can be expressed ml functions. For example, Equation (18.115) with I = 0 give~

~lementary

v:;,O(n - k) = I

1

a-m+l

[(n_k)a-m+l_(n_k_1)a-m+ 1j

l

1

for m - 2 thiS functIOn can be represented a~

°

Va' 2 (n-k)

I

=

--Va(n-k) 1 J a-I

for I = 1, Equation (18.115) givesl

ml (n_k)a-m+ -(n-k-1)a-m+ (n-k+a-m+l) , (n- k) - -'-----'-------'-------'------'-------,---------'a (a-m+l)(a-m+2) .

118.7 Fractional generalization of universal map for 1 < a

~

2

42S1

[The functIOn (18.116) also can be represented through the hypergeometrIc func-I OCIOn (see SectIOn 2.1.3 of (Bateman and ErdelYI, 1953))j

r

r(c)

t' x b- 1(1_xy-b-l

(a,b,c;z) = r(b)r(c-b)Jo

(l-zx)a

I

dx

Iby the relatIonj 1

1+1 zm_a F(m-a,I+1,1+2;z

-I

),

(18.117

Iwhere we use r(l + 1) = I! and r(l +2) = (l + 1)'J [fhIS theorem can be used to obtam a generalIzatIon of the umversal map for thel case a > 2.

118.7 Fractional generalization of universal map for 1 < a :s:; 21 [n thIS sectIOn, a fractIonal generalIzatIon of the dIfferentIal equatIon (18.1) IS sugj gested. This generalization is derived by D;x ----> oDfx, where 1 < a ~ 2. We usel ~he Riemann-Liouville fractional derivative instead of the second derivative D;x.1 [The dIscrete map that corresponds to the fractIOnal equatIOn of order 1 < a ~ 2 I~ ~erIved. ThIS map can be consIdered as a generalIzatIon of the umversal map for thel case 1 < a ~ 2) [.:et us consIder the fractIonal dIfferentIal equatIonj

(18.118~ Iwhere oDf is the Riemann-Liouville fractional derivative, which is defined byl 1(18.42). The dIscrete map that corresponds to thIS fractIOnal dIfferentIal equatIOili pf order 1 < a ~ 2 IS descrIbed by the followmg theoreml

rrheorem 18.10. The fractional dijjerential equation of the kicked system (18.118) lis equivalent to the discrete map.j

(18.119~ IPn+1 = Pn -KTG[xn]'

1 < a ~ 2,

(18.120~

Iwhere nEZ, n? 0, and the/unction Va(z) is defined b~

(18.121j IWlth z ~

1.

~30

18 Fractional Dynamics and Discrete Maps with Memory

Iproof Let us define an auxiliary variable 5'D;-a~=x(t),

S(t) such thatl O~2-a<1,

(1S.122~

Iwhere ~D; a is the Caputo fractional derivativej

r

1r-(r) CD 2-ar-= la-1D1r-= o t ~ 0t t ~ r a-I io dr(t-r)a-2 D~~ . IOsmgLemma 2.22 of (Kdbas et aL, 2006), we ge~ (1S.123~ [J'Iiei:i

IoDfx=D;ol( ax=D;ol( agD; a~ =D;(~(t)-~(O))=D;~. (1S.124j ISubstItutiOn of (18.124) and (18.122) mto Eq. (18.118) gIveij

[I'hIS tractiOnal equatiOn can be represented m the Hamlltoman formj

(1S.125~ Rsing that Eqs. (1S.59) and (1S.59) are equivalent to the discrete map (1S.56),1 I(1S.57), we obtain! (1S.126~

(1S.127 Iwhere 1 < a

~

2, andl (1S.12sj

[I'hese equatiOns define the fractiOnal generalIzatiOn of the umversal map for vanj lables (~n, 1Jn) 1 [The tractiOnal equation (18.118) m the Hamlltoman form can be represented asl

(1S.129~ (1S.130j Iwhere we use oDf

= DI oDf

1J

118.7 Fractional generalization of universal map for 1 < a

~

43~

2

[Equations (18.122) and (18.71) giv~

n = ~D;,,-a~

Ta-I n-I

=

r(a) ~ 1Jk+1 Va(n - k).

(18.131

[Using Eqs. (18.123) and (18.130), we obtainj

Ip = oDf

Ix

= Dl oJfax = Df oJf a gD; a~ = Df(~(t) - ~(O)) = DgJ 1(18.132)

[he defimtlOn of 1] In (18.125) and Eq. (18.132) gIV~

k\s a result, EquatIOns (18.127) and (18.131) glvel

Ta-I n

n+1

=

r(a) ~ Pk+1 Va(n - k+ 1),

1 < a ~ 2,

~n+1 = Pn-KTG[xn]'

Iwhere Va(z) is defined in (18.128)1 [This ends the proof.

(18.133 (18.134~

q

[Equations (18.133) and (18.134) define the fractional umversal map. These equaj are the generalIzatIOn of the map (18.5)1 ~ote that the form of Eq. (18.133) IS defined by both Eq. (18.129) and (18.130)1 [Equation (18.133) cannot be conSidered as an Iteration representation of Eq. (18.130] pnly.lf we use the other form of Eq. (18.129), then Eq. (18.133) IS changedJ ~lOns

[heorem 18.11. The fractional differential equation of the kicked system (18.118) lis equivalent to the discrete map.j

(18.135~ (18.136

IPn+1

= Pn-KTGlxnl,

(18.137]

Iwhere the/unction Sa(z) is defined b}j

ISa(z) Iwith z ~ 1.

= (z-l- l)a

1_ 2za 1+

(z-1)a

(18.138j

~32

18 Fractional Dynamics and Discrete Maps with Memory

IProof. Usmg Egs. (18.133) and (18.134) wIth n = 0, we obtam Eg. (18.135). Egua-I ~ion

(18.133) can be rewritten in the form:1

(18.139~ Iwhere n is an integer number such that n.:? 1. The second form ofEg. (18.133) i§ (18.140 Iwhere we use Va (1 ) = 1. Subtraction of Eqs. (18.140) and (18.139) give§

Iwhere n.:? 1 and thefunction Va(z) is defined in (18.128). Using thefunction Sa(n-I

n we see that Eq. (18.141) gives Eq. (18.136)J

q

[ThIS ends the proof. lRemark that the fractIOnal denvatIve m Eq. (18. I 18) IS not mvanant WIthrespect tq

~ote

~(t) ----+ x(t)

+ const.1

for the RIemann-LIOuvIlle tractIOnal denvatIve, we

hav~

[Therefore the fractional differential equation (18.118) with Glxl Ivanant wIth respect t9 ~(t) ----+ x(t)

+ 2nm,

mE

= sinx,

is not inj

ZJ

k\s a result, the fractIOnal standard map, WhICh IS defined by Egs. (18.119) and! 1(18.120)with Glxl = sinx, is not invariant with respect to the variable replacementsj fxn ----+ X n + 2nm,

m E Zj

~or all

n .:? O. OC:et us conSIder some examples of fractIOnal unIversal map.1

!Example i OC:et us prove that the tractIOnal unIversal map for Imap. Usmg the propertIesj

a-

2 gives the usual

unIversa~

118.7 Fractional generalization of universal map for 1 < a

fS'2(Z) ~quations

=

0,

r(2)

=

~

2

43~

1j

(18.137), (18.136) and (18.135) givel

IPn+! = Pn -KTG[xn]j [This is the universal mapj

[Example 2j OCf Glxl = -x, then this generalization can be represented a§

(18.142~

(18.143~ ~quatIOns ~ype

(18.142) and (18.143) define a fractIOnal generalIzatIOn of the Anosov-I system.

[Example 3j for G[x] = sinx, Equations (18.119) and (18.120) givq

r«:

n+!

n

= r(a) k~tk+!Va(n-k+1),

1 < a ~ 2.

~n+! - Pn - KT SlllXn·

(18.144 (18.145~

[These equatIOns define a fractIOnal standard map, whIch can be caned the fractIOna~ ~hlflkov-Taylor map. We note that thIS map can be defined by the equatIOns:1

(18.146~ (18.147 ~n+!

= Pn -

KT

SlllXn·

(18.148~

IWe note that Eqs. (18.147) and (18.148) are symmetnc wIth respect to X n --+ xn""A 12nm for all n 3 and m E Zj

°

~34

18 Fractional Dynamics and Discrete Maps with Memory

118.8 Fractional universal map for a > 21 [n thiS section, a tractional differential equation (18.118) IS used tor a > 2. Thel kliscrete maps that correspond to the fractional equations are derived. These mapsl ~an be considered as a generalization of the universal map for the case a > 2, i.e.j OChe tractiOnal umversal map (18.126), (18.127) can be generahzed trom 1 < a :'( 21 [0 a > 2. [.:et us consider the tractional equationj

(18.149 Iwhere oDf is the Riemann-Liouville fractional derivative of order a, m- 1 < a :'(mj Iwhlch ISdefined (Samko et al., 1993; Podlubny, 1999; Ktlbas et al., 2006) b5J

a

m m-a

oDt x = D; olt I

~ere we use the notation ~t

1

d

m

r

x( r)dr

x = Ttm _ a) dt" Jo (t _ r)a-I '

m-1 < a:'(

mj

Dr = d m / dt m , and 01{"

a is a fractional integration (Samkol al., 1993; Podlubny, 1999; Ktlbas et al., 2006)1

rrheorem 18.12. The jractional dijjerential equation oj the kicked system (18.149) lis equivalent to the discrete mapj

(18.150~ m-s-

pS+ ~ s Pn+l= n i...

T

T m - s-

I

-II pS+I+ n ( _ -1)1 pm-I n+I' 1=1. m s .

Ip~+l=p~

l-

K T G [xnJ,

S

= 1,..., m - 2 ,

m-1
(18.151

(18.152~

Iwhere the functions V:;,l (z) are defined b~

1=1, ...,m-2, kInd 1 :'( I :'( m - 1 <

(18.153

a :'( m, I, m E Nj

IProof Let us consider Eqs. (18.118) with m -1 ffiat

< a :'(

m. Using ~ = ~(t), suchl

(18.154j Iwe obtain

118.8 Fractional universal map for

~guatIOn

a> 2

4351

(18.118) can be represented a§

D':'~ +KG[~D,:,-a~] ~ 00

oCTt -n) = 0,

m-1 < a < m.

(18.155

II ,et us define W(t)=D:~(t),

s=0, ... ,m-11

[Thenthe Hamiitoman form of Eg. (18.155) I§

°

s= , ... ,m- 21,

ID/1n' /s ="n s+1,

IOsmg that Eqs. (18.105) and (18.106) are eqUIvalent to the dIscrete map (18.99) "1 108.101), we obtaml s 1 m-I m-s-I m-s- 1 s-s-l I n+l = (m_s_1),T/n+l T + l~ li.T/n T,

n::'+/

= T/::' l-KTG[~D~

s=0, ... ,m-2

aT/on

[.-etus use Eg. (18.154) in the form:1 p~

= T/~,

s

= 1, ... ,m-1.1

k'\s a result, we obtaml 1

n = qa-m+1)

n-

m- Ta+l-m+1 I! p~+IV:;,l(n-k

~(~

F

a 1

T -

m-l

m,m-2

(m_2)!Pn+l Va

s pS+ Pn+l= n

m-s~ ~

T

T m - s- 1

pm-l T! pS+I+ n (m-s-1)! n+l'

S

(n-k)),

= 1,...,m - 2 ,

(18.156~ (18.157 (18.158~

[fhIS ends the proof. ~quatIOns

q

(18.156)-(18.158) define a fractIOnal generalIzatIOn of the umversa~ = X n, Equations (18.156)-(18.158) define the fractiona~ ~nosov-type system with a > 2. For G[xnJ = sin r.; we have the fractional standard map for a> 2j rIap for

a> 2. For G[xn]

18 Fractional Dynamics and Discrete Maps with Memory

f'l-36

118.9 Riemann-Liouville derivative and universal map withl lffiemorYi [n the previOUS sectiOns, we consider nonhnear differential equations with Rlemannj [':louvtlle and Caputo tractional denvatives. The problems with lmtial conditions to~ ~he Riemann Liouville fractional derivative are not discussed in these sections Thel luniversal maps with memory can be obtained (Tarasov, 2009a) by using the equiv-I lalence ot the tractional differential equation and the Volterra mtegral equation. I~ lallows us to take mto account the mltial conditions tor tractional differential equaj ~lons. In thiS section, we reduce the Cauchy-type problem tor the differential equaj ~ions with the Riemann-Liouville fractional derivatives to nonlinear Volterra integrall ~quations of second kindj [Let us consider a dynamical system that is described by the fractional differentia~ ~quation

(18.159~

IoDfx(t) = F[t,x(t)]'

°

Iwhere Flt,x(t)1 is a real-valued function, ~ n-1 < a ~ n, andt > 0, and the leftj Isided Riemann-Liouville fractional derivative oDiC is defined for a> by (18.42)J [The function Fit, x( t) I can be interpreted as a force that acting on the system.1 for fractional equation (18.159), we can consider the initial conditionsj (oDf-kx)(O+) =q,

°

(18.160~

k= l, ... ,n.

[he notation (oDf k x) (0+) means that the limit is taken at almost all points of thel Iright-sided neighborhood (0, + e), e > 0, of zero as follows:1

°

roDf-k X ) (0+) = lim oDf-kx(t), ~

k=1, ... ,n-11

1

HO+

[.:etus give the theorem regardmg the equatiOns ot motion mvolvmg the Rlemannj I.:iOuvtlle tractiOnal denvatlve.1 [Theorem 18.13. Let W be an open set in JR and let F[t,xl, where t E (O,t(] andl be a real-valued/unction such that F[t,x] E £(O,t( )for any x E W. Let x(tJ ~e a Lebesgue measurable/unction on (O,t(). The Cauchy-type problem (18.159] I(lnd (18.160) can be reduced to the nonlinear Volterra integraL equation oj secondl

~ E W,

~

xt=

()

Iwhere t

n

E

Ck

k=lT(a-k+l)

t

a-k

1 +-T(a)

1 1

0

F r,x r dt (t-r)l-a'

(18.161

> o.

IProof. ThiS theorem was proved m (Ktlbas et al., 2000a,b) (see also Theorem 3.11 [n Section 3.2.1 ot (Ktlbas et al., 2006)). q

118.9 Riemann-Liouville derivative and universal map with memory

4371

IRemark lJ for a - n - 2, EquatIOn (18.161) gIves (18.13)j IRemark 2J [TheCauchy-type problem (18.159) and (18.160), and the Volterra equation (18.161) lare equivalent in the sense that, if x(t) E L(O,tf) satisfies almost everywhere Eqj 1(18.159) and conditions (18.160), then x(t) satisfies almost everywhere the integrall ~quatIOn (18.161)J [.Jet us consider dynamical system (18.159) in which the force Fit ,x(t) I is a pej IrIodIC sequence of delta-function-type pulses (kIcks) followmg wIth perIod T and! lamplitude K of the pulses. We consider the functionj

[t,x(t)] = -KG[x]

~ O(f -k),

1
~ 2,

(18.162

Iwhere t > 0, and oDf is the Riemann-Liouville fractional derivative defined byl 1(18.42). EquatIOn (18.159) wIth (18.162) can be consIdered as a tractIOnal gener1 lahzatIOn of Eq. (18.1)1

[Theorem 18.14. The Cauchy-type problem for the fractional differential equationl rf the form:

Dfx(t) +KG[x(t)]

~ O(f -k) = 0,

1< a ~ 2,

(18.163

IWlth the in/tzaL condltlOnsJ (18.164j

liS eqUivaLent to the equatlOnj CI

( ) =f(a)t xt

Iwhere nT

a-I

C2

+r(a-l)t

a-2

KT -f(a)

[( )] ( '~=' GxkT t-kT

)a-I

,(18.165

< t < (n+ l)TJ

IProof. Usmg the functIOn (18.162), EquatIOn (18.163) has the form of

(18.159~

IWIth the RIemann-LIOuvIlle tractIOnal derIvative of order a, where 1 < a ~ 2. A~ la result, Equation (18.163) with initial conditions (18.160) of the form (18.164) i§ ~qU1valent to the nonhnear Volterra mtegral equatIOnj

°

Iwhere ~ nT < t gives (18.165)j

108.166]

< (n + 1)T. Then the integration in (18.166) with respect to rl

~38

18 Fractional Dynamics and Discrete Maps with Memory

q

[his ends the proof.

[.Jet us define the momentum p(t) as a Riemann-Liouville fractional derivative ofj prder a-I byl I


Iwhere x( r) is defined for r E (O,t). Using the definition of the [ractional derivative (1S.42) in the form:1

~

2,

(1S.167

Riemann-Liouvill~

(1S.16Sj Iwe have the relationJ

(1S.169~

V\S a result, EquatIOn (18.163) can be represented as tractIOnal equations Ispace of the coordinate x(t) and momentum p(t)j

In

phasel

[Theorem 18.15. The Cauchy-type problem for the fractional differential equationsj (1S.170~

ID: p(t) = -KG[x(t)]

Eo (-f -

(1S.171~

k)

Iwzth the in/tzal condztlOnsJ (1S.172~ lis equivalent to the discrete map equationsj (1S.173

pn+1

= CI

-KT

E G[Xk],

(1S.174~

It=]

IProof. We note that Eq. (IS.165) for x(r) can be used only if r E (nT,t), wherel rT < t < (n+ I)T. To generalize Eq. (1S.165) on the case r E (O,t), we use thel ~eavlSlde step functIOn. Then Eq. (18.165) has the form:1

(r)

CI

ra -

C2 r a -

KT

n

= --r[iXf + r( a _ I) - r(a) ~ G[x(kT)] (r - kT)

a-I

8( r - kT) IOS.175)

118.9 Riemann-Liouville derivative and universal map with memory

43S1

Iwhere -r E (O,t). Using the relations:1 (18.176~ IoD~-ra-1 = ~quations

oD~-ra-2 = 0,

r(a),

1 < a ~ 2j

(18.167) and (18.175) giv~

p(t) =cI-KT

L'" G[x(kT)],

(18.177~

Iwhere nT < t < (n + 1 )T. As a result, the solutions of the left side of the (n + 1)thl lkicks XMI and PMI have the form of Eqs. (18.173) and (18.174). ThIS ends thel Iproof. q IRemark 1J [Thefractional differential equation of kicked system (18.163) in the form of Hamilj ~oman equations (18.170) and (18.171) IS eqUIvalent to the dIscrete map equatIOnsj

(18.178

Ip"+1

=

P" -KTGlx"l,

1
(18.179~

IwherePI = CI, and the function Va(z) is defined byl

IRemark2J IWe note that the map (18.178) and (18.179) WItllj

h

-PI,

C2-Q

Ihasthe form (18.119) and (18.120), where PI - CI. Equations (18.119) and lare obtained by using an auxiliary variable g(t) such tha~

(18.120~

[fhe general mltIal condItIOns (18.172) are not used. In the general case, the fracj ~IOnal dIfferentIal equatIOns (18.170) and (18.171) are eqUIvalent to the dIscrete mapl ~quatIOns (18.178) and (18.179), where the mItIal condItIons (18.172) are taken mtol laccount. The second term of the fIght-hand SIde of Eq. (18.178) IS absent m Eql 1(18.119). Note that usmg -1 < a - 2 < 0, we hav~

[Therefore the case of large values of n is equivalent to C2 = 0.1

~o

18 Fractional Dynamics and Discrete Maps with Memory

IRemark 3J

for a = n = 2, Equations (18.173) and (18.174) give the usual universal map (18.9)1 land (18.10). [n the theorem, the fractional differential equation (18.163) is represented in thel [orm of Hamiltonian equations (18.170) and (18.171), where the momentum is dej lfined by (18.167). In Eq. (18.167), we use the tractIOnal derIvatIve. There IS a second! Ipossibility of defining of momentumj

~(t)

= D}x(t).

(18.180j

[.:et us gIve a dIscrete map that corresponds to thIS defimtIon of the momentumJ

[Theorem 18.16. The Cauchy-type problem for the fractional differential equations.j ~lx(t) = p(t), Dfx(t)

= -KG[x(t)]

L o(~T -k),

(18.181~ 1< a

< 2,

(18.182

k=l

Iwzth the in/tzal condztlOnsJ

(18.183~ lis equivalent to the discrete map equationsj (18.184

(18.185

Iproof. To obtain the momentum (18.180), we use Eq. (18.165). If nT then the differentiation of (18.165) with respect to t give~

nT,

p(t)

=

cIt ar(a-l)

+

c2(a-2)t aKT r(a-l) - r(a-l)

< t < (n-a a-2

n

L

G[x(kT)] (t-kT)

108.186] Iwherewe use the relation r( a) = (a -1)r(a-I), 1 < a :( 2. Using Eqs. (18.165~ land(18.186), we can obtain the solution of the left side of the (n + 1)th kick (18.17~ land (18.18). For 1 < a < 2, we can us(j

(a -2) Iqa-l)

11

qa-2)J

44~

118.10 Caputo fractional derivative and universal map with memory

land Eq. (18.186) gIVes (18.185). As a result, we have Eqs. (18.184) and (18.l85)J [This ends the proof. q ~quatIOns

(18.184) and (18.185) descnbe a generalIzatIOn of Eqs. (18.9) and!

1(18.10). IRemark 1J

[f a = n = 2, and C2 = xo, CI = Po, then Eq. (18.184) gIves (18.9) and (18.10). Tg IprovethIS statement we should use (18.186) mstead of Eq. (18.185)J IRemark 2J

IOsmg the defimtIon of the Rlemann-LIOuvtlle fractional denvatIve (18.42) m thel form (18.168), we havd

~s a result, the Hamiltonian form of the equations of motion with p(t) = D}x(t) Iwill be more complicated than (18.170) and (18.171) with p(t) = oDf-1x(t). Fo~ ~he same reason, the values CI and C2 are not connected with p(O) and x(O). In thel ~ase p(t) = oDf lx(t), we have CI = p(O).1 IRemark 3J

for Glxn I = sinxn , we have the fractional standard map for 1 < a < 2. Comj Iputer SImulatIOns of the fractIOnal standard map were conSIdered m (Edelman and! [I'arasov, 2009). PropertIes of the phase space of the standard map wIth memory arel Imvestigated dependmg on the value of parameter a, whIch IS a fractional order ofj ~he denvative m the equatIOn of motion. The dIscrete systems that are descnbed byl ~he fractIOnal standard map demonstrate (Edelman and Tarasov, 2009) new type o~ lattractors, such that slow convergmg and slow divergmg traJectones, ballIstIc traJec1 ~ones, and fractal-lIke structures. At least one type of fractal-lIke sticky attractors ml ~he chaotic sea can be observed for standard map wIth memoryJ

118.10 Caputo fractional derivative and universal map

wit~

Imemoryl [n some prevIOUS sectIOns, we conSIder nonlmear dIfferentIal equatIOns wIth Caputol ~ractIOnal denvatives. The Cauchy-type problems wIth ImtIal condItIOns for the Caj Iputo fractional denvatIves are not dIscussed m these sections. The dIscrete mapsl IWlth memory can be obtamed (Tarasov, 2009a,b) by usmg the eqmvalence of thel ~ractIOnal dIfferential equatIOn and the Volterra mtegral equatIOn. ThIS eqmvalenc~ lallows us to take into account the initial conditions for fractional differential equaj ~ions. In this section, we reduce the Cauchy-type problem for the differential equaj ~IOns wIth the Caputo fractIOnal denvatIves to nonlInear Volterra mtegral equatIOn~

~2

18 Fractional Dynamics and Discrete Maps with Memory

pf second kind, and then obtain a universal map with memory from these fractionall ~quations.

[Letus consider a dynamical system, which is subjected to the force F[t,x(t)]. W~ lassume that equation of motion of the system IS lInear WIth respect to the Caputol hactIOnal derIvatives of order aJ I~Dax(t) = F[t,x(t)],

(18.187~

°

where ~ m - 1 < a ~ m, and t E [0,t ]. The initial conditions for equation with the aputo fractional derivatives D a are the same as those for the differential equatio bf ofderm' (D1x)(0) = Ck, k = 0, ...,m-1. (18.188~ [Letus give the basic theorem regarding the equation of motion involving the Caputol [ractional derivativeJ rrheorem 18.17. Let F[t,x] for any x EWe ffi. be a function belongs to Cy(O,t!) Iwith ~ r < 1, r < a, where Cy(O,t!) is the weighted space offunctions f[t] givenl ~Jn (O,t!], such thatt Yf[t] E C(O,t!). Then the Cauchy-type problemfor Eq. (18.187~ IWlth conditIOns (18.188) IS equivalent to the nonlinear Volterra Integral equatlOnj

°

m- Ck k

(t)

= k~

k!t

1

r drF[r,x(r)](t-r) a-I ,

+ r(a) Jo

(18.189

Iwhere x(t) E C[O,t(].1 IProof. ThIs theorem was proved In (KI1bas and Marzan, 2004, 2005) (see also (KI1"1 bas et al., 2006), Theorem 3.24). q ~et

us consIder dynamIcal system (18.187) wIth 1 <

a

~

2 In whIch the forcel

IF [t, x(t)] is a periodic sequence of delta- function-type pulses (kicks) following withl IperIod T and amplItude K of the pulses. We consIder the functIOn:1

IF[t,X(t)] = -KG[x(t)]

E8(-f -

k).

(18.190~

ISubstItutIOn of (18.190) Into (18.187) gives the tractIOnal equation of motIOn:1

Dfx(t) +KG[x(t)]

~ 8(-f -k) = 0,

1< a < 2,

(18.191

Iwhere ~Df is the Caputo fractional derivative. The initial conditions for Eq. (18.191~ larethe same as those for the dIfferential equatIOn of second order:1

(18.192~ ~et

us define a momentum by!

44~

118.10 Caputo fractional derivative and universal map with memory

~(t) = D}x(t).1 IOsmgthe defimtIOn of the Caputo derIvative m the formj

Iwe

obtain

rrheorem 18.18. The Cauchy-type problem for the fractional dijjerential equationsj (18.193~

k?!x(t) = p(t), Df-lp(t)

= -KG[x(t)] ~ 8(f -k),

1<

a < 2,

(18.194

Iwith the initial conditionsi

fi(O)

=

Xo,

p(O) =

pol

liS equivalent to the discrete map equatlOnsj

(18.195

(18.196 IProof. We use the eqUIvalence of the Cauchy-type problem (18.187), (18.188) tg ~he nonhnear Volterra mtegral equation (18.189) for function (18.190). Then thel ~auchy-type problem (18.191) and (18.192) IS eqUIvalent to the Yolterra mtegrall ~quatIOn of second kmdj

(18.197 lin the space of continuously differentiable functions x(t) E C 1[O,tf]' If nT I(n+ I)T, then Eq. (18.197) give~ KT (t)=xo+pot- qa) ~(t-kT)a-lG[x(kT)].

< t <j

(18.198

IOsmg (18.193), the differentiatIOn of (18.198) glve§ KT (t)=po-

(

T a-I

n

)L(t-kT)a-2G[x(kT)], =

nT
(18.199

~4

18 Fractional Dynamics and Discrete Maps with Memory

Iwhere we use r( a) = (a -1 )r(a-I). The solution ofthe left side of the (n+ 1)thl IkIck (18.17) and (18.18) can be represented by Eqs. (18.195) and (18.196), wherel Iwe use the condition of continuity x(t n + 0) = x(t n - 0). This ends the proof. q lRemark

[f a = m = 2, then Eqs. (18.195) and (18.196) gIve the unIversal map of the forIllJ 1(18.9) and (18.10) that is equivalent to Eqs. (18.3). The usual universal map is aI Ispeclal case of the suggested UnIversal map wIth memory.1 ~onsider a generalization of fractional equation (18.191) for the case of a > 2j [Then fractional dynamics of the system is described by the equation:1

~Dfx(t) +KG[x(t)] ~ O(f -k) = 0,

m-I < a < m,

(18.200

Iwhere ~Df is the Caputo fractional derivative. The initial conditions for Eq. (I8.200~ larethe same as those for the dIfferentIal equatIon of order m. We define the vanablesl ~(s)(t) = D:x(t), s = 0, 1, ... ,m-I. Then Eq. (18.200) can be rewritten in the forml ~iX(s)(t)=X(s+I)(t),

~Df-m+IX(m-I)(t)= -KG[x(t)]

(18.20I~

s=0,I, ...,m-2,

L O(f -k),

m-I < a < m.

(18.202

k=l

[.:et us gIve the basIc theorem regardIng thIS fractIonal dynamIcal system.1

[Theorem 18.19. The Cauchy type problem for fractional equations (18.201) andl 1(18.202) with the mltlal condltlOns:1

I(D:x)(O)

=

x~),

(18.203~

s = 0, 1, ... ,m-I

lis equivalent to the discrete map equationsj

m-s-l (k+s) x(s)

n+I

=

'"""

J...

~(n+ I/T k k!

KTa-s

n

'""" (n+ I-k)a-I-sG[x]

qa-s)'=

k,

(18.204

Iwhere s = 0, 1, ... ,m - 1.1 IProof. We use the eqUIvalence of the Cauchy-type problem (18.187), (18.188) tq ~he

nonlInear Volterra Integral equatIOn (18.189) for functIOn (18.190). Then thel problem (18.200) and (18.203) IS eqUIvalent to the Volterra Integrall

~auchy-type

~quatIOn: m-l

(t)=

-

~f nT

(k)

K

co

t

L_ ~tk __(_) L r d't'(t-'t')a-IG[x('t')]O(~-k). k! T a _ io T -

< t < (n + 1 )T, then Eq. (18.205)

give~

(18.205

118.11 Fractional kicked damped rotator map

4451

(18.206 [Thes-order denvatIve of (18.206) g1Ve~

(s)(t)=

m-l-s (k+s) ~ ~tk_

'=

k!

KT

qa-s)

n ~(t-kT)a-l-sG[x(kT)],

'=

(18.207

Iwheres = 0, 1, ... ,m-I, nT < t < (n+ I)T, m-I < a < m and we user(z) = (z--j Ur(z- 1). The solution of the left side of the (n + 1) th kick (18.17) and (18.18) canl Ibe represented by Eqs. (18.204), where we use the condition of continuity r(tn-H Pl = rUn - 0), s = 0, 1, ... ,m - 2. This ends the proof. t::j IRemark lJ [f a - m - 2, then Eqs. (18.204) gIve the unIversal map of the form (18.9) and! 1(18.10) that IS eqUIvalent to Eqs. (18.3)J IRemark 2J the case of I < a < 2, m - 2, Equations (18.204) give dIscrete map 108.195) and (18.196) with Pn = x~I) j ~n

~8.11

Fractional kicked damped rotator

equatIOn~

ma~

~n thIS sectIOn, a fractIOnal generalIzatIOn of the dIfferentIal equatIOn for a kIcked! kIamped rotator IS suggested. The dIscrete map that corresponds to the fractIona~ khfferentIal equatIon IS denvedJ [.:et us consIder a kIcked damped rotator. The equatIOn of motIon for thIS rotato~ lis

r;x+qDlx = KG [x]

L o(t-nT).

(18.208~

~

[t is well known that this equation gives (Schuster and Just, 2005) the 2-dimensionail ~

~

n+l = X n +

1- e q

~n+l = e

ql

ql

(Yn + KG [xn])

j

(Yn +KG[xn])J

[.:et us consIder the fractIOnal generalIzatIOn of Eq. (18.208) III the formj

IODfx - qoDfx

= KG[x]

f. o(t - nT),

Iwhere ~ E ~,

1< a

~

2,

{3 = a-I ,I

(18.209~

f:l46

18 Fractional Dynamics and Discrete Maps with Memory

land oDr is the Riemann-Liouville fractional derivative (Samko et al., 1993; Pod~ Ilubny, 1999; Kdbas et al., 2006) defined by (18.42). Note that we use the mmus ml ~he left-hand side ofEg. (lS.209), where q can have positive and negative value.1

[Theorem 18.20. Fractional differential equation (1S.209) is equivalent to the disj rrete map equatlOnsj

1

n+1 =

qa -1)

~ Pk+1 Wa(q,T,n+ 1- k),

(1S.21O (1S.211j

Iwhere the functions Wa are defined b)j

(1S.212

IProof. To prove this theorem, we define an auxiliary variable

g(t) such thatl

I~D;-a~ =x(t),

(1S.213~

Iwhere ~D;-a is the Caputo fractional derivative (1S.55). Using (1S.213) and thel defimtlOn of the RIemann-LIOuvIlle tractIOnal derIvatIve, we obtaml 2 p-a - D 2 p-a c D 2 - a j: I bDax -- D101 xI 101 0 I ,:>, I /1-{3 I /2-a I /2-a cD2- a I bD I{3 X= D101 D101 D101 X= X= 0 I ,:>, j:

rfhe relatIOn:

gIves the eguatlOnsj (1S.214j bDfx = D!(~(t) -

~(O)) = D!~.

(1S.215~

ISubstitution of (1S.214), (1S.215) and (1S.213) into Eg. (1S.209) give§ (1S.216~ ~guation

(1S.216) can be represented in the Hamiltonian form:1

IDll1- ql1 = KG[~D; a~]

L 8(t ~

nT),

1<

a < 2, q E R

(1S.217~

118.12 Fractional dissipative standard map

4471

[Using that fractional equation (18.80) is equivalent to the discrete map 1(18.81) and (18.82), we obtain the discrete map in the formj

~

n+l = ~n +

1 - e ql ~

equation~

I

1Jn+l,

n+l =eqT1Jn+KeqTG[qal_l) nt 1Jk+lWa(Q,T,n-k)] Iwherethe function Wa is defined by (18.212). These equations can be considered a§ la fractional generalization of the kicked damped rotator map for (Sn,1]n). Equatioij 1(18.209) can be represented in the tormj

~D7-1X=p,

DJp-qODfx = KG [x]

[o(t-nT)j ~

for (xn,Pn), Equations (18.213), (18.214) and (18.215) givel

Pn = 1Jn·1 k'\s a result, we havel

1

n+l

=

qa -1) (;/k+l Wa(q,T,n+ 1- k),

(18.218 (18.219j

Iwhere Wa IS defined ~teratIOn

rotator map

~8.12

in

(18.212). ThIs ends the proof.

equations (18.218) and (18.219) define the tractIOnal kicked

q dampe~

(18.208)~

Fractional dissipative standard ma~

k'\ tractIOnal generalIzatIOn ot dIssIpatIve standard map (ZaslavskY, 1978; SchmHfj landWang, 1985) can be defined (Tarasov, 2010) byl 1 < a :'( 2,

(18.220~ (18.22q

Iwhere the parameters are defined by condItIons (18.25). For b = -1 and Z = Kj ~quatIOns (18.220) and (18.221) grve the tractIOnal standard map that IS descflbe~ Iby the equations:1

~8

18 Fractional Dynamics and Discrete Maps with Memory

t»

n

n+l =""f(a) ~Pk+lVa(n-k+ 1), IPn+l

1<

a ~ 2,

= Pn - KT smxnl

Iwith T = 1. Note that this fractional dissipative standard map is not derived from aI [ractional differential equation. This map is derived by Pn ----+ -bpn in the fractiona~ Istandard mapl for small b ----+ 0 Eqs. (18.220) and (18.221) wIth Z = -K shnnk to a 11 ~iImensIOnal mapj

K

n

n+l = r(a) ~ Va(n - k) sinx.,,

1 < a ~ 2.

(1S.222

[Thismap can be considered as a fractional generalization of I-dimensional sine-mapi I(1S.26). The fractional sine-map (1S.222) is characterized by a long-term memorYI Isuch that the present state evolutIOn depends on all past states wIth speCIal forms ofj Iweights function Va (z)·1 fractIOnal dISSIpatIve standard map can be denved from fractIOnal dIfferentIa~ ~quatIOns (Tarasov, 2010). Let us conSIder a fractIOnal dIfferentIal equation for ~ IkIcked damped rotator m the form:1

~Dfx -

qoDfx = K sinx

f. 8(t - nT),

(1S.223~

Iwhere oDf and oDf are derivatives of non-integer order, and q E R This equatio~ Iwj]] he considered withl 11 < a ~ 2, {3 = a - 1J land oDf and oDf are the Riemann-Liouville fractional derivatives (Samko et a1.l ~993; Podlubny, 1999; Kilbas et a1., 2006). The fractional differential equation ofj ~he kicked system (18.223) IS eqUIvalent to the dIscrete mapj

1 n n+l = r(a -1) ~ Pk+l Wa(q,T,n+ 1- k),

~n+l

=

eq1 (Pn + K sinx.},

(1S.224 (1S.225j

Iwhere the functIOns Wa are defined by (18.212). Usmg thIS statement, we can denv~ latractIOnal generalIzatIOn of the dISSIpatIve standard map. If we use the conditIon~

IXn =xn, ~hen

Yn = Pn,

e = K,

T

= 1,

G[x]

= sinx,

(1S.226~

Eqs. (1S.224) and (1S.225) give the map:1

(1S.227~

118.13 Fractional Henon map

44S1 (18.228~

[These equations are a fractional generalization of the Zaslavsky map (18.21) and! 1(18.22) with.Q = O. We note that this fractional dissipative standard map (Tarasovj 12010) is denved trom tractional nonhnear differential equation (18.223) with conj dition (18.226). If we use the variables]

~hen

we have the discrete equationsj

J.e

n+i

n

= qa-I) ~Pk+lWa(q,T,n+I-k), IPn+i - -bPn - Z smXn·

(18.229 (18.230]

[I'hese equations can be considered as a tractional generahzation of map equation~ 1(18.23) and (18.24) with n = O. For a = 2, this tractlOnal diSSipative standard mapl gives the Zaslavsky map (Zaslavsky, 1978) that is descnbed by Eqs. (18.23) and! 108.24). For exp{ q} -+ 0 Eqs. (18.227) and (18.228) shrink to a I-dimensional mapl ~hat can be considered as a tractional generahzation of one-dimensional sme-mapl 1(18.26).

118.13 Fractional Henon map [To denve a tractlOnal generahzatlOn of the Henon map we can use (Tarasov, 2010)1 ~he fractlOnal generahzed diSSipative map and the fact that the Henon map m thel ~orm (18.32) can be denved from the generahzed diSSipative map.1 OC=et US consider a tractional generahzatlOn of map (18.33) and (18.34) m thel form:

t»

n+l

n

= qa) ~Pk+lVa(n-k+I), I < a ~ 2,

(18.231 (18.232]

Iwhere Ihl ~ I. This is the fractional generalized dissipative map. If a = 2, then Eqsl 108.231) and (18.232) give (18.33) and (18.34). If G[x] = sinx, T = I and K = -ZJ ~hen Eqs. (18.231) and (18.232) give fractlOnal diSSipative standard map equatlOn~ 108.220) and (18.221)l flo denve a tractlOnal generahzatlOn of the Henon map, we can use the tractlOna~ ~issipative map (18.231) and (18.232) with K - T - I, and the functlOn (18.35). A~ la result, we obtam the equatlOnsj

(18.233~

18 Fractional Dynamics and Discrete Maps with Memory

f'l-50

~n+l

= -bpn+ 1- (l-b)xn-ax~,

(18.234j

Iwhere Ibl :s;; 1, 1 < a:S;; 2 and Va(z) is defined by (18.128). These equations can bel ~onsIdered as a first fractIonal generalIzatIon of the Henon map (18.30), (18.31)J fractIOnal Henon map (18.233) and (18.234) IS denved from Eq. (18.231) and! 1(18.232), where the dIssIpatIon has been mtroduced mto the IteratIon equatIOn] ~n this case, the fractional equation of motion with dissipation is not used. Equa-I ~ions (18.233) and (18.234) are not derived from fractional differential equation~ pf kIcked damped system. We can suggest the second pOSSIble gettmg of a fracj ~IOnal generalIzatIon of the Henon map. ThIS map can be denved from the fractIona~ ~quatlon of kIcked damped rotator (18.223).1 [Let us consider the fractional differential equation (18.223) that is generalizationl pf Eq. (18.208) by non-integer order derivatives. In orderto derive fractional generj lalization of Henon map, we use Eq. (18.223) with the function (18.37). As a resultj Iwe obtam the equatIonsj 1

n+I = r(a-l) (;/k+IWa(q,T,n+ l-k),

(18.235

(18.236 Iwherethe functions Wa are defined by Eq. (18.212) and q = In( -b). These equation~ klefinea second fractional generalization ofthe Henon map. Equations (18.235) and! 1(18.236) can be called fractIOnal Henon map. ThIS map IS denved from nonlInearl ~hfferentIaI equations WIth denvatIves of non-integer order]

~ 8.~ 4 ~n

Conclusionl

many areas of mechamcs and phYSICS the problems can be reduced to the study o~ maps (see, for example, (Sagdeev et al., 1988; Amschenko, 1990; Nelmarkj land Landa, 1992». A WIde class of these maps can be denved from dIfferentIa~ ~quatIOns of motIon. The specIal case of dIscrete maps has been studIed to descnb~ IpropertIes of regular and strange attractors of these dIfferentIal equatIOns. Under ~ IWIde range of CIrcumstances such maps gIve nse to chaotIC behaVIOr. The suggested! Imaps wIth memory are generalIzatIOn of well-known dIscrete maps. These maps de1 Iscnbe fractIOnal dynamICS of complex phySIcal systems. We note that a WIde clas§ pf the maps can be denved from kIcked fractIOnal dIfferentIal equatIOns (Tarasov J 12008d, 2009a,b). The suggested fractIOnal dIscrete maps can demonstrate a chaotIcl IbehavIOr wIth a new type of attractors (Edelman and Tarasov, 2009). For example) ~he fractIOnal standard map demonstrates (Edelman and Tarasov, 2009) new typel pf attractors, such that slow converging and slow diverging trajectories, ballistiq ~rajectories, and fractal-like structures. The interesting property of these fractiona~ kliscrete maps is a long-term memory. As a result, a present state evolution depend~ ~hscrete

References

4511

pn all past states with the fractional power-law weights functions Va(z), Sa(Z) and! lWa(a,b,c). Note that the fractional maps are equivalent to the correspondent fracj ~ional kicked differential equations. An approximation for fractional derivatives ofj ~hese equations is not used. ThiS fact can be used to study the evolution that is dej Iscribed by fractional differential equations. Computer SimulatiOns of the suggested! kliscrete maps with memory prove that the nonlinear dynamical systems, which arel klescribed by the equations with fractional derivatives, exhibit a new type of chaoticl ~otion. For example, the fractal-like sticky attractors in the chaotic sea can be obj Iservedfor standard map With memory (Edelman and Tarasov, 2009). The fractiona~ ~hscrete maps allow us to study a new type of attractors m fractional dynamiCs ofj Iphysical systemsJ

lReferencesl IV.S. Amschenko, 1990, Complex Oscillations in Simple Systems, Nauka, MoscowJ lIn Russian I IV. Arnold, 1965, Small denommators. I: On the mappings of the Circumference ontol litself, American Mathematical Society Translations, 2, 213-284; and Izvestiyd IAkademii Nauk SSSR, Ser. Matern., 25 (1961) 21-86. In Russian] ~. Bateman, A Erdelyi, 1953, Higher Transcendental Functions, YoU, McGraw1 IHl1l, New Yorkl ~. Caputo, 1967, Lmear models of diSSipatiOn whose Q is almost frequency m1 klependent, Part II, Geophysical Journal of the Royal Astronomical Society, 13j 1529-539 M. Caputo, 1969, Elasticita e Dissipazione, Zanichelli, Bologna. In Italian.1 ~. Caputo, F. Mamardi, 1971, A new diSSipatiOn model based on memory mecha1 Imsm, Pure and Applied Geophysics, 91, 134-147J ~. V. Chmkov, 1979, A umversal mstabihty of many dimenSiOnal OSCillator systemsj IPhysics Reports, 52, 263-379~ f. Collet, J.P. Eckman, 1980, Iterated Maps on the Interval as DynamIcal Systemj IBirkhauser, Baselj ~. Edelman, v.B. Tarasov, 2009, FractiOnal standard map, Physics Letters A, 374J 279-285. ~. Flck, M. Pick, G. Hausmann, 1991, Logistic equatiOn With memory, Physlca~ ~eview A, 44, 2469-2473J V\. Fuhnskl, AS. Kleczkowskl, 1987, Nonhnear maps With memory, Physical IScripta, 35, 119-122J ~.AC. Gallas, 1993, Simulatmg memory effects With discrete dynamical systemsj IPhysica A, 195,417-430; and Erratum, Physica A, 198, 339~ ~. GiOna, 1991, DynamiCs and relaxatiOn properties of complex systems Withmemj pry, Nonlinearity, 4, 991-925j ~. Gorenflo, F. Mamardi, 1997, FractiOnal calculus: Integral and differential equa1 ItiOns of fractiOnal order, m Fractals and FractIOnal Calculus In Continuum Mej

f'l-52

18 Fractional Dynamics and Discrete Maps with Memory

'[hanics, A Carpmten, F. MamardI, (Eds.), Spnnger, New York, 223-276; and! IE-pnnt: afXlv:0805.3823J IK. Hartwich, E. Fick, 1993, Hopf bifurcations in the logistic map with oscillatingl Imemory, Physics Letters A, 177,305-310.1 M. Henon, 1976, A two-dimensional mapping with a strange attractor, Communica-I kion in Mathematical Physics, SO, 69-77J k'\..K. Jonscher, 1996, Universal Relaxation Law, Chelsea Dielectncs Press, Londonl k'\..K. Jonscher, 1999, Dielectnc relaxatIOn m sohds, Journal oj Physics D, 32, R571 IR.'ZII k'\..A. KIlbas, S.A Marzan, 2004, The Cauchy problem for dIfferentIal equatIon~ Iwithfractional Caputo derivative, Doklady Mathematics, 70, 841-845; Translated! IfromDokladjAkademii Nauk, 399, 7-11J k\.A KIlbas, S.A Marzan, 2005, Nonhnear dIfferentIal equatIOns wIth the Caputol IfractIonal denvatIve m the space of contmuously dIfferentIable functIons, Dlfj Iferential Equations, 41,84-89; Translated from Differentsialnye Uravneniya, 41J 182-86. k\.A KIlbas, B. BOnIlla,J.J. TruJIllo, 2000a, Nonhnear dIfferentIal equatIOns of frac1 ItIOnal order IS space of mtegrable functIOns, Doklady Mathematics, 62, 222-226;1 ITranslated from DokladjAkademii Nauk, 374, 445-449. In RussianJ k\.A KIlbas, B. BOnIlla, U. TrujIllo, 2000b, EXIstence and UnIqueness theorems fo~ Inonlinear fractional differential equations, Demonstratio Mathematica, 33, 583-1 iill2: k\.A KIlbas, H.M. Snvastava, J.J. TruJIllo, 2006, Theory and Applications oj Frac1 ~ional Dijjerential Equations, ElsevIer, Amsterdam1 IK.S. MIller, B. Ross, 1993, An IntroductIOn to the FractIOnal Calculus and FracJ, kwnal Differential EquatIOns, WIley, New YorkJ IYu.I. Neimark, P.S. Landa, 1992, StochastIC and ChaotIC OSCIllatIOns, Kluwer AcaJ, ~emIc, Dordrecht and Boston, 500p.; Translated from RUSSian: Nauka, Moscow) 11987. [. Podlubny, 1999, FractIOnal DIfferential EquatIOns, AcademIC Press, New YorkJ p.A Russell, J.D. Hanson, E. Ott, 1980, DImenSIOn of strange attractors, Physlca~ ~eview Letters, 45, 1175-1178j ~.Z. Sagdeev, D.A USIkov, G.M. Zaslavsky, 1988, Nonlinear Physics. From th~ IPendulum to Turbulence and Chaos, Harwood AcademIc, New Yorkl IS.G. Samko, AA KIlbas, 0.1. Manchev, 1993, Integrals and Derivatives oj Frac1 ~wnal Order and ApplIcatIOns, Nauka I TehnIka, Mmsk, 1987, m Russianj ~nd FractIOnal Integrals and Derivatives Theory and Appltcatwns, Gordon and! IBreach, New York, 1993l p. SchmIdt, B.W. Wang, 1985, DISSIpatIve standards map, Physical Review A, 32J 2994-2999.1 RG. Schuster, w.Just, 2005, Deterministic Chaos: An Introduction, 4th ed., Wileyj IVCH, Weinheim, 2005j k\.A. Stanislavsky, 2006, Long-tenn memory contribution as applied to the motionj pf dIscrete dynamIcal system, Chaos, 16, 0431051

References

453

IVB. Tarasov, 2006a, Contmuous lImIt of dIscrete systems WIth long-range mteracj ItIOn, Journal oj Physics A, 39, 14895-14910.1 IVB. Tarasov, 20066, Map of dIscrete system mto contmuous, Journal of Mathemat-I [cal Physics, 47, 092901j IVB. Tarasov, 2008a, FractIOnal equatIOns of Cune-von Schweldler and Gauss laws] Vournal oj Physics: Condensed Matter, 20, 1452121 ~.E. Tarasov, 2008b, Universal electromagnetic waves in dielectric, Journal of! IPhysicsA, 20, 175223J IVE. Tarasov, 2008c, Fractional vector calculus and fractional Maxwell's equatIOnsJ IAnnals ofPhysics, 323, 2756-2778j IVE. Tarasov, G.M. ZaslavskY, 2008d, FractIOnal equations of kicked systems and! Khscrete maps, Journal of Physics A, 41, 435101J ~.E. Tarasov, 2009a, Differential equations with fractional derivative and universa~ Imapwith memory, Journal ofPhysics A, 42, 465102j IVB. Tarasov, 20096, DIscrete map WIth memory from fractional dIfferential equaj Itionof arbitrary positive order, Journal ofMathematical Physics, 50, 122703.1 IV.E. Tarasov, 2010, FractIOnal Zaslavsky and Henon dIscrete maps, Chapter 1, ml ILong-range Interaction, Stochasticity and Fractional Dynamics, A.c.J. Luo, V.I IAfrmmovIch, (Eds.), HEP and Sprmger, 20101 ~E. Tarasov, G.M. Zaslavsky, 2006, Fractional dynamics of coupled oscillators withl Ilong-range interaction, Chaos, 16, 02311 OJ p.M. Zaslavsky, 1978, SImplest case of a strange attractor, Physics Letters A, 69J 1145-147 p.M. Zaslavsky, 2002, Chaos, fractIOnal kmetIcs, and anomalous transport, Physicsl IReports, 371, 461-5801 p.M. Zaslavsky, 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford Unij Iversity Press, Oxford.1

Part VI ~ractional

Quantum Dynamics

~hapter 1~

[Fractional Dynamics o~ IHamiitonian Quantum System~

119.1 Introductionl ~n the quantum mechamcs, the observables are gIven by self-adJomt operators (Mes1 ISlah, 1999; Tarasov, 2005). The dynamIcal deSCrIptIOn of a quantum system IS glVelll Iby a superoperator (Tarasov, 2008b), whIch IS a rule that assIgns to each operator ex1 lactlyone operator. DynamIcs of quantum observable IS descrIbed by the HeIsenberg ~quatIOn. For HamI1toman systems, the mfimtesImal superoperator of the Helsenj Iberg equation is defined by some form of derivation (Tarasov, 2005, 200gb). Th~ linfinitesimal generator (i/h)IH, . I, which is used in the Heisenberg equation, is ij ~erIVatIOn of observables. A derIVatIOn IS a lInear map D, whIch satIsfies the LeIb1 ritz rule D(AB) = (DA)B +A(DB) for all operators A and B. Fractional derivativq ~an be defined as a fractional power of derivative (see Section 5.7 in (Samko et al.j ~993)). We consIder a fractIOnal derIvatIve on a set of observables as a fractIOna~ Ipower of derivative (i/h)IH, . I. It allows us to generalize a notion of quantuml ~amIItoman systems. In thIS case, operator equatIOn for quantum observables I~ la fractIOnal generalIzatIOn of the HeIsenberg equatIOn (Tarasov, 2008a). The sug1 gested fractIOnal HeIsenberg equatIOn IS exactly solved for the HamIItomans of fre~ Iparticle and harmonic oscillator. Fractional power of operators (Balakrishnan, 1960j IKomatsu, 1966; Berens et aL, 1968; YosIda, 1995; Krem, 1971; Martmez and SanzJ 12000) and superoperator (Tarasov, 2008b, 2009) can be used as a possIble approachl ~o descrIbe an mteractIOn between the system and an envIronment. We note tha~ OCractIOnal power of the operator, WhICh IS defined by PIOsson bracket of classIcal1 ~ynamIcs, was consIdered m (Tarasov, 2008c)J [n SectIOn 19.2, the fractIOnal power of derIVatIve and the fractIOnal HeIsenberg ~quatIOn are suggested. In SectIOn 19.3, the propertIes of tIme evolutIOn, whIch I~ ~escrIbed by the fractIOnal equation, are consIdered. In SectIOn 19.4, the fractIOna~ klynamics offree particle is considered. In Section 19.5, we obtain a solution of fracj ~ional Heisenberg equation for linear harmonic oscillator. Finally, a short conclusionj lis given in Section 19.6j

V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

f'l-58

19 Fractional Dynamics of Hamiltonian Quantum Systems

119.2 Fractional power of derivative and Heisenberg eguatioij IQuantum dynamIcs IS descnbed by superoperators (Tarasov, 2008b). A superoper-I lator 2 is a rule that assigns to each operator A exactly one operator 2(A). Fo~ [Hamiltonian H, let L Hbe a superoperator that is defined by the equationj

IQuantum dynamIcs of observables of Hamlltoman system IS descnbed by the operj lator dtfferential equationj (19·1)1 [Equation (19.1) is called the Heisenberg equation for Hamiltonian systems (Mesj ISlah, 1999; Tarasov, 2008b). The time evolutiOn of a Hamlltoman system IS mduce~ Ibythe Hamlltoman HJ [n order to obtam a fractional generahzation of Eq. (19.1), we consIder a concepti pf fractiOnal power for L H . If L H IS a closed hnear superoperator WIthan everywher~ ~ense domain D(LH ) , having a resolvent R(z,LH ) = (zLj - L H ) I on the negativ~ Ihalf-axis, then there exists (Balakrishnan, 1960; Yosida, 1995; Krein, 1971) the su-l Iperoperator:

1

00

I( LH-)a = -smzrz - - dzz a-I R ( -z,LH- )Ly(19.2)1 n 0 [he superoperator (LH ) a is defined on D(LH ) for 0 < a < 1. It is a fractional powe~ pf the LIe left superoperator. Usmg the superoperator (19.2), we can descnbe frac1 dynamIcs of quantum systems by the equationj

~lonal

(19.3)1 Iwhere t and HIli are dimensionless variables. Equation (19.3) is called the fractiona~ [HeIsenberg equatiOnl lRemark

[EquatiOn (19.3) cannot be represented m the formj

IWlth some operator H new . Therefore quantum systems descnbed by (19.3) are no~ [Hamlltoman systems. The systems wIll be called the fractiOnal Hamlltoman SYS1 ~ems. A set of usual Hamlltoman quantum systems IS a specIal case of a set ofj ~ractiOnal systemsJ rrhe Cauchy problem for operator equatiOn (19.1) m whIch the lmtial conditioili lIS given at the time t = 0 by Ao, can be solved. SolutiOn of thIS Cauchy problem canl Irepresented in the form At =
119.2 Fractional power of derivative and Heisenberg equation

45~

[he superoperators t :?: 0 have the semigroup propertyj

Iwhere LI is unit superoperator (LIA = A). The superoperator L"H is a generatinm Isuperoperator, or infinitesimal generator, of the semigroup {
RuA =AVJ

~f we consider the Cauchy problem for fractiOnal Heisenberg equatiOn (19.3) ml Iwhichthe imtlal conditiOn is given by A o, then its solutiOn can be represented m thel

~

IAt(a) =

Iwhere the superoperators

0, have the semigroup property. The semi~ ~roup {

land tractiOnal dynamiCs cannot be considered as HamiHoman.1 a [he superoperators

0, have the propertiesj

k\s a result, tractiOnal dynamiCs that is defined by the equatiOns of form (19.1) i§ ~arkovian. EquatiOn (19. I) descnbes processes without a memory. We can considerl la tractiOnal generahzatiOn in the form:1 (19.4)1

Iwhere ~Df is the Caputo fractional derivatives (Kilbas et al., 2006). In this casej ~ractiOnal dynamiCs is non-Markovian, and Eqs. (19.4) descnbes quantum processesl IWith memoryJ

19 Fractional Dynamics of Hamiltonian Quantum Systems

f'l-60

119.3 Properties of fractional dynamicsl [I'he solutIOn of the Cauchy problem for fractional Heisenberg equation (19.3) i§ ~efined by the fractional dynamical semigroup {
k

(00

= Jo

dsfa(t,s)
t

> 0,

(19.5)1

1

Iwhere the function fa (t, s) is defined byl

a(t,s) lands ~hat

~

0, a, t > 0, and

°

= -I.

2m

l

a ioo

+ dzexp{sz-tz a},

(19.6)

-'00

°< a < I. We note that the branch of

ZU

is taken suchl

Re(zU) > for Re(z) > 0. This branch is a one-valued function in the z-planel

~ut along the negative real aXiS. The convergence of thiS mtegralis ObVIOusly ml Ivirtue of the convergence factor exp{ -tzu}. If the path of integration in (19.6)1 lis represented by the union of two paths rexp{-i8} and rexp{+i8}, wher~ Ir E (0,00), and n12:( 8 :( x, then we getl

ra(t,s)

=

~

1 00

drF(t,s,r),

(19.7)1

f(t,s, r) = exp{srcos 8 - tr U cos( a8)} sin(srsin 8 - tr U sin( a8) + 8)1 ~f we have a solutIOn At of the Heisenberg Eq. (19.1), then Eq. (19.5) gives thel IsolutIOn of fractIOnal equatIOn (19.3) m the forml

(19.8)1 Iwhere t > 0. Equation (19.8) allows us to obtain solutions At(a) of fractiona~ ~eisenberg equation by using solutions As (I) = As of usual Heisenberg equation1 I_ A quantum observable is a self-adjomt operator. It
46~

119.3 Properties of fractional dynamics

IA superoperator as a map from a set of observables into itself should be realj IQuantum dynamics is described by real superoperators. Using the Bochner-I a) IPhillips formula and Eg. (19.7), we can prove that

I_ Using the Bochner-Phillips formula and the property !a(t,s)

~

0, s > 0, it is easyl

~o prove that the superoperators
IA ~ 0, since
I_ Let us consider a Hilbert operator space uft with the scalar productj (AlB) = Tr[A*B]j

k\ superoperator CPt on uft is adjoint to
~f


lis adjoint to
a

I- It is known that CPt is a real superoperator if O} be a semigroup, such that the density matrix operator o, =
[hen the semigroup { CPt( a) ,t > O} describes the evolution of the density operatorj

Iby the fractlOnal eguatlOnj (19.9)1 ~quatlOn

(19.9) IS called thejractlOnal von Neumann equatlOn.1

19 Fractional Dynamics of Hamiltonian Quantum Systems

f'l-62

119.4 Fractional quantum dynamics of free particlel lAs an example of fractIOnal dynamiCs, we consider free I-dimensIOnal particle moj ~ion that is defined by the Hamiltonian H = p 2/2m, where P is dimensionless varH lable, and m- J has the action dimension. Heisenberg equations (19.1) for A = Q an~ IA - P have the formsl

k?!Qt=m-1Pt, [I'he solutions of Eqs. (19.10)

(19.10~

D!Pt=O.

ar~

bt=Qo+m-1tPo

1

(19.11)

Pt=Po·

[The fractional Heisenberg equations for A = Q and A = P

an~

(19.12~ IOsmg the Bochner-Phtlhps formula, we obtam the solutions of fractional 1(19.12) in the formsj

p

t(a)

Iwhere

=

CPt(a) Qo =

loo dsfa(t,S)Qsl 0

equation~

Pt(a) =Poj

o: is given by (19.11). As a result, we getl (19.13j

[I'hese equations descnbe fractional dynamiCs of free particles.1 for a = 1/2, we hav~

land Eqs. (19.13)

giV~

Pt = Po·

(19.14~

OC:et us define the average value and dispersIOn by the equatIOnsj

IUsmg the coordmate representation and the pure statej

(19.15~

46~

119.5 Fractional quantum dynamics of harmonic oscillator

Iwe obtain the average values and dispersions of the operators (19.14) in the formj

< Pt >=

~9.5

pol

Fractional quantum dynamics of harmonic oscillator]

ISecond simple example of tractional dynamics is a hnear harmomc oscillator that iil ~efined by the Hamiitomanj

(19.16~ Iwhere t and P are dimensionless vanables. For A - Q, and A - P, Equation (19.1)1 ~

IDiQt

= ~Pt, DiPt = -mro2QtJ

[The solutions have the form'l

~

= Pocos(rot) - mroQosin(rot).

[I'he tractional dynamiCs of hnear harmomc OSCillator is descnbed by the Wleisenberg equationsj

(19.17~ tractiona~

(19.1Sj Iwhere H is defined by (19.16). Osmg the Bochner-Phtlhps formula, we obtam thel IsolutiOns of Eqs. (19.18) m the form:1

(19.19~ ISubstitutiOn of (19.17) mto (19. 19) givesl

(19.20~ ~

= PoCa(t) - mroQoSa(t),

(19.21]

19 Fractional Dynamics of Hamiltonian Quantum Systems

f'l-64

ICa(t)

=

1'' ' dsfa(t,s) cos(ros),

(19.22~

fa(t)

=

I"J dsfa(t,s)

(19.23~

sin(ros).

~quatlons

(19.20) and (19.21) descrIbe tractIOnal dynamIcs of quantum harmonIcl bsciIIator IWe can consider a = 1/2. In this case, Equations (19.22) and (19.23) givel

- _t_1 F () 1/2

t -

2y

;;;;

1r 0

OO

ds

cos( ros) S

3/2

e

-t 2

/4sj

IWe note that these functIons can be represented through the Macdonald functIoIlj I(see (Prudnikov et aI., 1986), Section 2.5.37.1), which is also called the modified lBessel function of the thifd kind] IUsmg (19.15), we get the average valuesj

r

Qt

>= xoCa(t) + dmpoSa(t)l

f< Pr >= poCa(t) -

mroxoSa(t)J

land the dIspersIOns:1

~t IS easy to see that tractIOnal harmOnIC OSCIllator IS a SImple dISSIpatIve system. Thel IsolutIOns are characterIzed by a dampmg effect for average values of observable~ pf the fractIOnal harmOnIC OSCIllator. The fractIOnal dumpmg IS deSCrIbed by thel Imodified Bessel function of the thifd kind]

[9.6 Conclusioril IWe derIve a tractIOnal generalIzatIOn of the HeIsenberg equatIOn. The derIvatIve 0P1 ~ratIOns of non-mteger order are defined as tractIOnal powers of derIvatIves. Thel Isuggested fractIOnal HeIsenberg equatIOn deSCrIbes a generalIzatIOn of quantuml ~amiltonian system. The solution of the suggested Heisenberg equation with harj monic oscillator Hamiltonian is obtained. Note that solutions of the Cauchy prob-I iem for fractional Heisenberg equation are represented by the superoperators

4651

References

r > 0, which form a semigroup. Therefore the evolution of observables is Marko-I Ivian. This means that the suggested fractional derivatives, which are fractional powj ~rs of derivative, cannot be connected with long-term memory effects. Derivative~ pf non-mteger orders can be used as an approach to deSCrIbe an mteractIOn betweenl ~he quantum system and an enVIronment. ThIS mterpretatIon caused by followmg Ireasons. Using the propertie~

~co fa(t,s) = 1,

fa(t,s) ? 0,

for alI s >

oj

Iwe can assume that fa (t, s) is a density of probability distribution. Then thel !Bochner-PhIllIps formula (19.5) can be conSIdered as a smoothmg of HamIltomailJ ~volution 4>[ with respect to time s > O. This smoothing can be considered as ani Imfluence of the enVIronment on the system. As a result, the parameter alpha can bel lused to deSCrIbe an mteractIOn between the system and the enVIronment. Note tha~ ISchrodmger equatIOn WIth fractIOnal tIme derIvatIves was conSIdered m (NaberJ 12004). The Schrodmger equatIOn WIth fractIOnal power of momentum, whIch canl Ibe conSIdered as a fractIOnal derIVatIve m coordmate representation, was dIscussed! lin (Laskin, 2000, 2002; Guo and Xu, 2006; Wang and Xu, 2007). The fractiona~ guantum dynamICS of Hamlltoman systems m pure states (Laskm, 2000, 2002) canl Ibe conSIdered as a speCIal case of the approach suggested m (Tarasov, 2008b). Wi.j rote that It IS pOSSIble to conSIder quantum dynamICS WIth low-level fractIOnalItyl Iby some generalIzatIOn of method suggested m (Tarasov and Zaslavsky, 2006); seel lalso (Tofighl and Pour, 2007; Tofighl and Golestam, 2008)1

Referencesl ~

Balakrishnan, 1960, Fractional power of closed operator and the semigroup genj them, Pacific Journal ofMathematics, 10, 419-437 j R Berens, P.L. Butzer, U. Westphal, 1968, Representation of fractional powers ofj ImfimtesImal generators of sermgroups, Bulletin oj the American Mathematicall ISociety, 74, 191-196J IS. Bochner, 1949, DIffUSIOn equatIOns and stochastIc processes, Proceedmgs of the, !National Academy ofSciences USA, 35, 369-370j IX.Y. Guo, M.Y. Xu, 2006, Some physical applications of fractional SchrOdingeIj ~quatIOn, Journal oj Mathematical Physics, 47, 0821041 V\.A. KIlbas, H.M. SrIvastava, J.J. TruJIllo, 2006, Theory and Applications oj Frac1 ~ional Dijjerential Equations, ElseVIer, Amsterdam1 R Komatsu, 1966, Fractional powers of operators, Pacific Journal ofMathematicsj 119, 285-346j IS.G. Krem, 1971, Linear Dijjerential Equations in Banach Space, TranslatIOns o~ IMathematIcal Monographs, VoI.29, AmerIcan MathematIcal SOCIety; Translated! Ifrom RUSSIan: Nauka, Moscow, 1967l N. Laskin, 2000, Fractional quantum mechanics, Physical Review E, 62, 3135-3145.1 ~rated by

f'l-66

W'l. LaskIll,

19 Fractional Dynamics of Hamiltonian Quantum Systems

2002, FractIOnal SchrodIllger equatIOn, PhYsical Review E, 66, 05610SJ Martinez, M. Sanz, 2000, The Theory of Fractional Powers of Operators, Else-] lvier, Amsterdam] k\. Messiah, 1999, Quantum Mechanics, Dover, New York, 1152p. Section 8.lOj M. Naber, 2004, 'Erne tractIOnal SchrodIllger equatIon, Journal of Mathematlcall IPhysics, 45, 3339-3352.1 RS. PhIllIps, 1952, On the generatIOn of semIgroups of lInear operators, Pacifid, Vournal oj Mathematics, 2, 343-396.1 k\.P. Prudnikov, Yu.A. Brychkov, 0.1. Marichev, 1986, Integrals and Series, Vol.l.j IElementary Functions, Gordon and Breach, New YorkJ IS.G. Samko, AA KIlbas, 0.1. Manchev, 1993, Integrals and Derivatives of Fracj rional Order and Applications, Nauka i Tehnika, Minsk, 1987. in Russianj !;md Fractional Integrals and Derivatives Theory and Applications, Gordon and! IBreach, New York, 1993J ~E. Tarasov, 2005, Quantum Mechanics: Lectures on Foundations of the TheoryJ gnd ed., Vuzovskaya Kmga, Moscow. In RussIanJ IV.E. Tarasov, 200Sa, FractIOnal HeIsenberg equation, Physics Letters A, 372, 29S4-1 2988. IY.E. Tarasov, 200Sb, Quantum Mechanics oj Non-Hamiltonian and Dissipative SYS1 ~ems, ElseVIer, AmsterdamJ IV.B. Tarasov, 200Sc, FractIOnal powers of denvatIves III claSSIcal mechamcs, Comj Imunications in Applied Analysis, 12, 441-450J IY.E. Tarasov, 2009, FractIOnal generalIzatIOn of the quantum MarkovIan maste~ ~quatIOn, Theoretical and Mathematical Physics, 158, 179-195.1 ~E. Tarasov, G.M. Zaslavsky, 2006, Dynamics with low-level fractionality, Physical lA, 368, 399-415j k\. TofighI, H.N. Pour, 2007, e-expansIOn and the tractIonal OSCIllator, Physlca AJ 1374, 41-45J V\. TofighI, A Golestam, 200S, A perturbatIve study of tractIOnal relaxatIOn phe1 nomena, Physica A, 387, 1807-1817 J IS. W. Wang, M. Y. Xu, 2007, GeneralIzed fractIOnal SchrOdIllger equatIOn wIthl Ispace-time fractional derivatives, Journal ofMathematical Physics, 48, 043502j IK. YosIda, 1995, Functional Analysis, 6th ed., Spnnger, BerlInJ ~.

~hapter2Q

!Fractional Dynamics of Open Quantum Systems

~O.l

Introductionl

IWe can descnbe an open quantum system startmg from a closed Hamlltoman systeml IIf the open system IS a part of the closed system (WeIss, 1993). However sItuatIOnsl ~an anse where It IS dIfficult or ImpossIble to find a Hamlltoman system compns-I Img the gIven quantum system. As a result, the theory of open and non-HamI1tomanl guantum systems can be consIdered as a fundamental generalIzatIon (KossakowskIJ ~972; Davies, 1976; Ingarden and Kossakowski, 1975; Tarasov, 2005, 200gb) ofj ~he quantum Hamlltoman mechamcs. The quantum operatIOns that descnbe dynam-I IICS of open systems can be consIdered as real completely posItIve trace-preservmgj Isuperoperators on the operator space. These superoperators form a completely pOSIj ~Ive semIgroup. The mfimtesImal generator of thIS semIgroup IS completely dIssIpaj ~ive (Kossakowski, 1972; Davies, 1976; Ingarden and Kossakowski, 1975; Tarasovj 12008b). FractIOnal power of operators (Balaknshnan, 1960; Komatsu, 1966; Beren~ ~t aI., 1968; YosIda, 1995; Martmez and Sanz, 2000) and superoperators (TarasovJ 12008b, 2009a) can be used as a possIble approach to descnbe fractIOnal dynamIc~ pf open quantum systems. We consIder superoperators that are fractIOnal powers ofj ~ompletely dISSIpatIve superoperators (Tarasov, 2009a). We prove that the suggested! Isuperoperators are mfimtesImal generators of completely posItIve semIgroups fo~ ~ractIOnal quantum dynamIcs. The quantum MarkovIan equatIOn, whIch mcludesl Ian explIcIt form of completely dISSIpatIve superoperator, IS the most general typel pf MarkoVIan master equatIOn descnbmg non-umtary evolutIOn of the densIty opj ~rator that IS trace-preservmg and completely posItIve for any mItIal condItIOn. AI ~ractIOnal power of mfimtesImal generator can be consIdered as a parameter to dej Iscnbe a measure of screemng of envIronment (Tarasov, 2009a). Usmg the mterac1 ~IOn representatIOn of the quantum MarkovIan equatIOn, we consIder a fractIOna~ Ipower a of non-Hamlltoman part of mfimtesImal generator. In the lImIt a ----+ 0, wei pbtain Heisenberg equation for Hamiltonian systems. In the case a = 1, we have thel lusual quantum Markovian equation. For 0 < a < 1, we have an environmental inj Ifluence on quantum systems. The phySIcal interpretation of the fractIOnal power o~ V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

20 Fractional Dynamics of Open Quantum Systems

f'l-68

linfinitesimal generator can be connected with an existence of a power-like screenin~ bf environmental influenceJ ~n Section 20.2, a brief review of superoperators on an operator space and quan-I ~um operations is defined to fix notations and proVide convement references. Thel ~ractional power of superoperator is considered. In Section 20.3, the tractional genj ~ralization of the quantum Markovian equation for observables is suggested. In Sec-j ~ion 20.4, the properties of fractional dynamical semigroup are described. In Sec-j OClOn 20.5, the tractlOnal Markovian dynamiCs of quantum states is conSidered. Inl ISection 20.6, the fractional non-Markovian quantum dynamics of open quantuml Isystems with long-term memory is discussed. In Section 20.7, the tractional equaj OCion for harmomc OSCillator Withfnction is solved. In Section 20.8, the self-clomngj guantum operations are discussed. Finally, a conclusion is given in Section 20.9j

~O.2

Fractional power of superoperatori

IQuantumtheones conSist of two parts, a kmematics descnbmg the imtial states and! pbservables of the system, and a dynamiCs descnbmg the change of these states and! pbservables With time. In quantum mechamcs, the states and observables are giveIlj Iby operators. The dynamical descnption of the system is given by superoperatorij I(Tarasov, 2008b), which are maps from a set of operators mto itselfl !Let Jft be an operator space, and Jft* be a dual space. Then Jft* is a set o~ lalllinear functionals on .4t. To denote an element of .4t, we use IB) and B. Thel Isymbols (AI and ()) denote the elements of .4t*. We use the symbol (AIB) for ~ Ivalue of the functional (AI on the operator IB) is a graphic junction of the symbolsl I(A I and IB). If .4t is an operator Hilbert space, then (AIB) = Tr[A*B]. We considerl la superoperator as a map 2' from an operator space Jft into itself.1

pefinition 20.1. A superoperator A = 2' on Jft* is adjoint to superoperator 2' onl ~

(A(A)IB) ~or

all BE D(.2) C

= (AI.2(B))

(20.Q

.e and A E D(A) C .4t*1

[The most general state change of a quantum system is a called a quantum oper1 lation (Hellwing and Kraus, 1969, 1970; Kraus, 1971, 1983); see also (Schumacher,1 ~996; Tarasov, 2002b, 2004, 2005, 2008b; Wu et aI., 2007; Oza et aI., 2009). AI guantum operation is a superoperator J;, which maps a density operator P to a denj Isity operator PI = rffr (p ). Any density operator PI = P(t) is self-adjoint (pt = PI )j Ipositive (el > 0) operator with unit trace (Tried = 1). As a result, we have the foIj [owmg reqUirements for a superoperator 6~ to be the quantum operatlOnl [. The superoperator it; is a real superoperator. The superoperator 6~ on an operatorl space .4t is a real superoperator if

t~(A)]*

=

~(A*)I

46~

120.2 Fractional power of superoperator

or all A ED J; c.4, where A * ED J; is adjoint to A. The real superoperato 6'; maps the self-adjoint operator to the self-adjoint operator 6'; = 6'; 12. The superoperator 6'; is a positive superoperator. A non-negative superoperator i§ la map ~ from.4 into.4, such that ~(A2) ;;?: 0 for all A 2 = A*A E D(?r) c .4.1 k\ positive superoperator is a map 6'; from .4 into itself, such that 6'; is non-I regative and J;(A) = 0 if and only if A = O. A density operator p is positive. Ifj IJ; is a positive superoperator, then Pt = J;(p) is a positive operatorL 13. The superoperator 6'; is a trace-preserving map. The superoperator 6'; on an opj ~rator space .4 is trace-preserving if1

prJ;t(I)

=/J

IWe may assume that the superoperator ~ is not only positive but also completelYI Ipositive. ~efinition 20.2. A superoperator ~~ is a completely positive map from an operatorl Ispace At into itself i~ fl

fl

IE LBk~(AkAI)BI ;;?: q V(=lL=ll

~or

all operators A k, Bk E At and all n E

Nl

[To descnbe dynamiCs, we assume that the superoperators l; form a completely! IpOSitive quantum semigroup (Ahckt and Lendi, 1987) such that 11' is an mfimte gen1 ~rator of the semigroup (Lmdblad, 1976a; Ahckt and Lendi, 1987; Tarasov, 2008b)1 IWe also assume that the superoperator 2' adjoint to X is completely dissipativej

~or all Ak,AI E D(£"). The completely dissipative superoperators are infinitesimall generators of completely positive semigroups { eJ>t It> O} that is adjoint of { J;I t ?j Pl. The superoperator £" describes the dynamics of observables of open quantuml Isystems. The evolutiOn of a denSity operator is descnbed by Xl fractional power of operators (Balakrishnan, 1960; Komatsu, 1966; Berens et aLj ~968; Yosida, 1995; Martinez and Sanz, 2000) and superoperators (Tarasov, 2008b,1 12009a) can be used as a possible approach to descnbe fractiOnal dynamiCs of openl ijuantum systems. Let 2' be a closed linear superoperator with an everywhere densel ~omain D(£"). If the resolvent R(-z,£") = (zLj + £") 1, where z > 0, satisfies thel conditiOn: M (20.2)1 IIR(-z,£")II~-, z>O, 1

IZJ

~hen

a fractional power of the superoperator 2' can be defined (Hille and Phillips) Yosida, 1995) byl

~957;

~a = ~ 1'' ' dzza - i R(-z,£")£",

0 < a < 1.

(20.3)1

~70

20 Fractional Dynamics of Open Quantum Systems

[he superoperator !l'u allows a closure. If condition (20.2) holds for a closed suj Iperoperator -2, then -2 a -2{3 = -2a+{3 for a, {3 > 0, and a + {3 < 1.1 [Let -2 be a closed generating superoperator of the semigroup {
INote that the resolvent for the superoperator !l'u can be found by the Kato's forj mula;

Rsing Eg. (20.5), we have that the inegualityj

liS satisfied with the constant M of mequahty (20.2)1 [Let -2 be a closed generating superoperator of the semigroup {
t" dsfa(t,s)
1

t>O

(20.6)1

~orm a semigroup such that -2 a is an infinitesimal generator of

l

a ioo

+ dzexp{sz-tz a},

(20.7)

-'00

Iwhere a, t > 0, s ~ 0, and 0 < a < 1. The branch of ZU is chosen such that Re( ZU) > 01 [or Re(z) > O. This branch is a one-valued function in the z-plane cut along thel r,egative real axis. Using the factor exp{ -tz}U, it can be proved that integral (20.7)1 ~ol1veIges.

[Thefunction fa (t ,s) has the following propertiesl [. For all s > 0, the function fa(t,s) is non-negative: fa(t,s) ~ OJ 12. The function fa (t ,s) satisfies the normalization conditionj

~oo dsfa(t,s) = 1.1

°

13. For t > and x > 0,1 ~.

By denoting the path of integration in (20.7) to the union of two paths r exp{ -i8 }J land r exp{ +i8}, where r E (0,00), and nl2 ~ 8 ~ n, we obtainl

47~

120.3 Fractional equation for quantum observables

(20.8)1

f(t,s, r) = exp{srcos 8 - trU cos( a8)} sin(srsin 8 - trU sin( a8) + 8)J

15. If a = 1/2 and 8 = n, Equation (20.8) givesl

1 n 00

1/2(t,S) = -1

0

dre-Srsin(tvr) =

-----:=-=-= 2

[These properties of fa(t,s) can be used to prove properties of fractional quantuml klynamics.

~O.3

Fractional equation for quantum ohservahlesl

[The dynamICs of open quantum systems can be descnbed m terms of the mfimtes1 Ilmal change of the system. ThIS change IS defined by some form of mfimtesl1 Imal generator. The most general explIcIt form of the mfimteslmal superoperato~ Iwas suggested by Gonm, Kossakowskl, Sudarshan and Lmdblad m (Gonm et aLJ ~976, 1978; Lindblad, 1976a). There exists a one-to-one correspondence betweenl ~he completely posItIve norm contmuous semIgroups and the completely dlSSlpatIvel generatmg superoperators. The followmg structural theorem gives the most generall ~orm of a completely dISSIpatIve superoperator.1

[Theorem 20.1. A generating superoperator 2'v of a completely positive unity1 Ipreserving semigroup {CPt = exp{ -t2'v }I t ~ O} can be represented by the equa-I ItIOn: (20.9)1 Iwhere Lv and Rv are superoperators of left and right multlpllcatwnsj ~vA

- VA,

RvA -AVJ

find the superoperator L H is a left Lie multiplication by H*

=H

such

tha~

(20.10~ IProof This theorem was proved in (Lindblad, 1976a).

q

[Using At = CPt (A), where CPt = exp{ -t2'v }, we obtain the equationj

~At =

-2'vAt.

(20.1l~

~72

~f all

20 Fractional Dynamics of Open Quantum Systems

operators Vk are equal to zero, then <£'0 = L N , and Eq. (20.11) is the Heisenberm for Hamiltonian system. Substitution of (20.9) into (20.11) give§

~quation

(20.12 ~quatlon (20.12)

defines the MarkOVIan dynamIcs of quantum observablesj

IRemark lJ [The form of <£'V is not uniquely fixed by (20.9). The transformationsj

Iwhere ak are arbitrary complex numbers, preserve the form ofEq. (20.9)J IRemark 2J (20.12) gIves an explIcIt form of equatIon for quantum observables, If thel ~ollowmg restrIctIons are satIsfiedj ~quatlon

I- <£'v and A v are bounded superoperators, where A v is adjoint to <£'v 1 I- <£'v and A v are completely dissipative superoperatorsl [n (DaVIes, 1977) thIS result has been extended to a class of quantum IsemIgroup WIth unbounded generatmg superoperatorsj ~n

dynamIca~

order to obtam a generalIzatIOn of quantum MarkOVIan equatIOn (20.12), wei tractIOnal power of quantum MarkOVIan superoperator (20.9) m the form:1

~onsIder a

(

smzrz 2'v ) IX = -

n

1

00

0

dzz IX- I R ( -z,2'v ) 2'v,

O
(20.13

V\s a result, we obtam (Tarasov, 2009a) the tractIOnal quantum MarkOVIan equatIOnj

(20.14~ Iwhere t, Hili and Vk!'vli are dimensionless variables. Equation (20.14) describesl ~he tractIOnal MarkOVIan dynamIcs of quantum observablesJ lRemark3J ~f

Vk = 0, then Eq. (20.14) gives the tractIOnal HeIsenberg equatIOnj

(20.15~ [The superoperator (LH )U is a fractional power of left Lie superoperator (20.10)1

47~

120.4 Fractional dynamical semigroup

IQuantum Markovian equation (20.12) can be represented in the interaction repj Iresentation. Let us define the operatorsj (20.16~

Iwhere U(t)

= exp{ (1/in)H~. Using

(20.16), Equation (20.11) can be written a§

d -Au(t) dt

I

= -2'w - Au (t ),

(20.17~ (20.18~

ISuperoperator (20.18) describes the non-Hamiltonian part of the evolution. Thel [ractional generalization of Eq. (20.17) i§ d

I

dtAu(t)

=

- a Au(t). -(2'w)

(20.19~

~quatIOn (20.19) tS the fractIOnal quantum Markovtan equatIOn m the mteractIOriI IrepresentatIOn. The parameter a can be consIdered as a measure of the mtluence o~ Ian enVIronment. For a-I, we have quantum MarkOVIan equatIOn (20.17). In thel IlImIt a ----7 0, we obtam the HeIsenberg equatIon for observable At of a HamI1tomaIlj Isystem. As a result, we can consIder the phySIcal mterpretatIOn of tractIonal powe~ pf2w as an influence of environment. We have the following cases that can be used! 1m quantum dynamIcsj

I- absence of the envIronmental mtluence (a = 0);1 I- complete envIronmental mtluence ( a - 1)j I- power-lIke envIronmental mtluence (0 < a < 1).1 [The phySIcal mterpretatIOn of Ep. (20.19) can be connected WIth an eXIstencel pf a power-lIke screemng of the envIronmental mtluence on the quantum systeml I(Tarasov, 2009a)1

~O.4

Fractional dynamical semlgroup

[f we consIder the Cauchy problem for Eq. (20.11) m WhICh the mItIal condItIOn I§ gIven at the tIme t - 0 by Ao, then ItS solutIOncan be WrIttenm the form At -
V\s a result, the superoperators <1>t form a semigroup, and the superoperator 2'v is generating superoperator of the semigroup { <1>t It): O} ~

~

~74

20 Fractional Dynamics of Open Quantum Systems

[Let us consider the Cauchy problem for fractional quantum Markovian equationl 1(20.14) with the initial condition given by A o. Then its solution can be represente~ lin the fom,. (a) ( ) = cI>t Ao·I IAt a [he superoperators cI>/ a), t > 0, form a semigroup, which is called the fractiona~ klynamical semigroup. The superoperator (2'v)a is a generating superoperator ofj ~he semigroup {cI>/a) I t ~ O}. The superoperators cI>/a) can be constructed in term~ pf cI>t by the Bochner-Phillips formula (20.6), where fa (t,s) is defined in (20.7). Ifj IA t IS a solutIOn of quantum Markovian equatIOn (20.11), then formula (20.6) givesl ~he solution-

IAt(a) =

I"J dsfa(t,s)A s,

t>

9

pf tractIOnal quantum MarkOVIan equatIOn (20.14).1 Ioefinition 20.3. A linear superoperator cI>t(a) is completely positive if the condi-I bons: IEB;cI>/a) (AiAj)Bj ~ 0 (20.20~ f:.fi

Iholdfor any A;,B; E ::&1

~he following theorem states that the fractional dynamical semigroup { cI>/ a) I t ~

PJ is completely positive (Tarasov, 2009a)1 [Theorem 20.2. If {cI>t It> 0 J is a completely positive semigroup of superoperato~ ~, then the fractional superoperators cI>t(a), which are defined by Eq. (20.6), forml ~ completely positive semigroup {cI>/a) It> oll IProof. Usmg Bochner-PhIllIps formula (20.6), we ge~

~or t

> O. The property fa(t,s)

~

0, s > 0, and the inequalityj

[B;cI>s(AiAj)B j ~

0,1

[;j

gIves (20.20). ThIS ends of the proof.

q

IWe note the followmg corollary] rrheorem 20.3. If 0, is a non-negative one-parameter superoperator, i.e.) ~t(A) ~ 0 for A ~ 0, then the superoperator cI>/a) is also non-negative, i.e.) ~t(a)(A) ~ Ofor A ~ oj

475]

120.5 Fractional equation for quantum states

IProof Bochner-Phillips formula and the property fa(t,s) :?: 0, s> 0, allow us tg Iprove that the superoperator 0 is a non-negativel Isuperoperator. We note that thIS statement can also be proved by usmg BI - I, Al @ lA, and Ai - Bi - 0 (I - 2, ...) m the proof of the preVIOUS theorem. q

pefinition 20.4. Let A * E At' be adjoint to A E At. A real superoperator is a superj pperator
k\ quantum observable is a self-adjoint operator. If
~heorem 20.4. If
~quation

(20.8) means that f!x(t,s) = fa(t,s) is a real-valued function. As a resultJ =
d

~he condition (
~O.5

d

Fractional equation for quantum state~

fractIOnal dynamICS of quantum states can be conSIdered as adJomt to fractIOna~ ~quatIOn for quantum observables. The superoperator CPt that deSCrIbes dynamIc~ pf quantum observables is adjoint to superoperator 0~, which describes evolution o~ guantum states.1

pefinition 20.5. Let
= (AI
adjoin~

(20.21]

[or all BE D(
rrheorem 20.5. If t; is a superoperator adjoint to
~

I

lis adjoint to
=

Jor

dsfa(t,s)

s;A

20 Fractional Dynamics of Open Quantum Systems

f'l-76

IProof Let ~ be an adjoint superoperator to cPt. Then Eq. (20.21) is satisfied. As ~ result, we havd

q

[This ends of the proof.

[t is known that 1&; is a real superoperator if cPt is real. Usmg thiS statementJ ~he Bochner-Phillips formula and Eq. (20.8), it is easy to prove that ~(a) is a reall Isuperoperator, if cPt(a) reall [The evolution of a density operator is described a~

~Po =pd Iwhere { I;; It > O} is a completely positive semigroup. In infinitesimal form, the dy~ r.amics ofthe density operator Pt = J,~Po is described by the equation:1

(20.22~ Iwhere Xv is adjoint to the quantum Markovian superoperator .£'v. The superoperaj ~or Xv can be represented m the formj (20.23 Iwhere Lv and Rv are superoperators of left and nght multiphcationsj ~vA

- VA,

RvA -AVJ

landthe superoperator L H is a left Lie multiphcatiOn (20.10). SubstitutiOn of (20.23] Imto (20.22) give~ (20.24 ~quatiOn

(20.24) descnbes quantum Markovian dynamics of states of open systems.1 ~he semigroup { ~(a) It> O},which is adjoint to fractional dynamical semigrou~ I{ cPt It;;: O},describes the fractional dynamics of the density operatorj

rrhe fractiOnal Markovian equation for the density operator i~

120.6 Fractional non-Markovian quantum dynamics

4771

(20.25~ [This equation defines fractional Markovian dynamics of quantum statesj lRemark ~quatlon

(20.22) wIth Vk - 0 gIves the von Neumann equatIonj

for Vk = 0, Equation (20.25) has the form:1

[This is the fractional von Neumann equationj

~O.6 ~n

Fractional non-Markovian quantum dynamics

general, we can consIder a generalIzatIOn of Eq. (20.12) III the formj

(20.26j Iwhere fiDf is the Caputo fractional derivative (Kilbas et aI., 2006) with respect tg bme, and

OCf a is a non-integer, then Eq. (20.26) defines the non-Markovian fractional dynam-I ~cs of quantum observables. EquatIOn (20.26) descnbes a quantum processes wIthl memory, IWe also can generalIze Eq. (20.24) for the densIty operator such thatl

(20.27 ~quatIOn

(20.27) descnbes non-MarkovIan fractIOnal dynamIcs of quantum states ofj ppen systems. The non-MarkovIan property means that the present state evolutIOlJI ~epends on all past states.1 OCf we consider the Cauchy problem for Eq. (20.26) in which the initial condition] lIS gIven at the tIme t - 0 by Ao, then ItS solutIOncan be represented (Daftardar-Gem ~t aI., 2004) III the formj

~78

20 Fractional Dynamics of Open Quantum Systems

~(a) =

Ear-t a 2'vJJ

[Here E a [2'] is the Mittag-Leffler function (Kilbas et aI., 2006) with the superoper-I laforargumenq

['IJote that (see Lemma 2.23 m (KIlbas et aI., 2006)) the relatIOnl

holds for A E C, a E R and a > OJ [Thesuperoperators CPt ( a), t ~ 0, describe fractional dynamics of open quantuml Isystems. The superoperator 2'v can be considered as an infinitesimal generator ofj ~he fractional dynamical maps CPt ( a) j for a = 1, we havd

ftHl) = EIl-t2'vJ =

exp{ -t2'V}·1

[Thesuperoperators CPt = CPt ( 1) form a semigroup such thatl

[I'hIS property IS reabzed smcg ~xp{ -t2'v} exp{ -s2'v} = exp{ -(t+s)2'v }j ~n

general, we havel

[or a

g N. Therefore the semigroup propertyj

lIS not satIsfied for non-mteger values of a.1 Ks a result, the superoperators CPt (a) with a tf- N cannot form a semigroup. Thi§ Iproperty means that we have a non-MarkovIan evolutIOn of quantum systems. Thel Isuperoperators CPt ( a) describe quantum dynamics of open systems with memory.1

~o. 7

Fractional equations for quantum oscillator with friction!

!Let us consIder an example of fractIOnal dynamIcs of open quantum system. Thg Ibasic assumption is that the general form of a bounded completely dissipative su-j Iperoperator given by the quantum MarkovIan equation holds for an unbounded com1 pletely dissipative superoperator 2'v. We assume that the operators H, and Vk ard

47~

120.7 Fractional equations for quantum oscillator with friction

[unctions of the operators Q and P such that the obtained model is exactly solvabl~ I(Lmdblad, 1976b; Sandulescu and Scutaru, 1987); see also (lsar et al., 1994, 1996)J [herefore we consider Vk = Vk(Q,P) as the first-degree polynomials in Q and P, and! ~he Hamiltonian H = H(Q,P) as the second-degree polynomial in Q and pj

(20.28 Iwhere ak, and bk , k = 1,2, are complex numbers. These assumptIOns mean that thel ~nctIOn force IS proportIOnal to the velOCIty. It IS easy to obtaml

k$?VP = -molQ - j1P - APJ

OC:et us conSIder a matnx representatIOn of the quantum MarkovIan equatIon. Wi.j Hefine the matricesl

(20.29~ [I'hen the quantum MarkovIan equatIon for observables has the formj

(20.30~ Iwhere 2 v A t = MAt. The solution of (20.30) is At =
(20.31~ [I'he matnx M can be represented m the form:1

(20.32j Iwhere F IS a dIagonal matnx. Then we

hav~

(20.33~ r=(-(A+V)

o

0

-(A - v)

~ere we use the complex parameter v, such that v 2

(20.34~

).

= f.12 _

(j)2 ~

IOsmg (20.31), the one-parameter superoperators !Pt can be represented byl

~80

20 Fractional Dynamics of Open Quantum Systems

(20.35 ISubstItutIOn of (20.33) and (20.34) Into (20.35) glVe§

t

-At cosh(vt) + (,u/V) sinh(vt) (l/mv) sinb(vz ) -(moo2 /v) sinh(vt) cosh(vt) - (,u/v) sinh(vt)

=e

Iwheresinh and cosh are hyperbolic sine and cosinej

k\s a result, we obtain At =
Qt = e- At (cosh(vt) +!!.- sinh(vt)) Qo + _e- At sinh(vt)Po,

(20.36

Pt = _~e-At sinh(vt)Qo +e- At (COSh(vt) - ~ sinh(vt) )Po.

(20.37

[I'he tractIOnal quantum Markovian equations for Qt and Pr arg

(20.38~ Iwhere t and Vk/\!Ii are dimensionless variables. The solutions of Eqs. (20.38) arel given by the Bochner-Philhps formulaj

pt(a) =

=


1'' ' dS!a(t,s)Qs, 1"" dS!a(t,s)Ps,

(20.39~ (20.40~

> 0, and the operators Qs and Ps are given In Eqs. (20.36) and (20.37). Thel

~unction!a (t, s) is defined

in (20.7). Substitution of (20.36) and (20.37) into (20.39~

land (20040) givesl

(20.41~ mOO ( Cha(t) - -Sha(t) ,u ) Po, Pr(a) = --Sha(t)Qo+

(20.42

Iwhere we use the notationsl

Fha(t) =

L'" dS!a(t,s)e-AScosh(Vs),

(20.43~

Fha(t) =

1

(20.44~

00

dS!a(t,s)e-ASsinh(vs).

48~

120.7 Fractional equations for quantum oscillator with friction

lAs a result, Equations (20.41) and (20.42) describe a fractional dynamics of quan-I ~lIm harmonic oscillator with friction IWe note that (20.43) and (20.44) with a = 1/2 have the form:1 1

[I'hese functtons can be represented by the Macdonald functton (see (Prudmkov laI., 1986), SectIOn 2.4.17.2) such thatl

e~

Fh l /2(t ) = ~(V+(t) + V_(t))J ISh l / 2 (t) = -~(V+(t) -

V_(t))J

lHere we use the notatIOnJ -

V±(t) I

(t

2±4V)I/4

~

(

K- I / 2 2

V

A (t 2 ± 4V) ) 4

I

1

Iwhere Ku(z) is the Macdonald function (Oldham and Spanier, 1974; Samko et aLJ ~993) and Re(t 2 ) > Re(v), Re(A) > 0.1 ~et ~he

us conSider the tractIOnal quantum non-Markovian equations for Qt and Pt ml form'

5Df Qt =

-2'vQ"

5DfPr = -2'vP"

(20.45~

Iwhere ~Df is the Caputo fractional derivative (Kilbas et aI., 2006) with respect t9 ~ime, and t and Vk / Vh are dimensionless variables. Using matrices (20.29), we canl Irepresent fractional equations (20.45) a§

(20.46~ ~f we conSider the Cauchy problem for Eq. (20.46) m which the Imttal condltton I~ given at the time t - 0 by Ao, then ItS solutIOn can be represented (Daftardar-GeJJ~ ~t aI., 2004) m the formj

VIt

=


[The Mittag-Leffler functIOn (Kdbas et aI., 2006) Withthe matnx argument IS defined! IQy

~82

20 Fractional Dynamics of Open Quantum Systems

for a = 1, we obtain (20.31). Using (20.32), the one-parameter superoperator§ ~ (a) can be represented byl

k'\s a result, we hav~ (20.47~ ISubstitution of (20.33) and (20.34) into (20.47) give§

Ca[A, v,t] + (.u/V)Sa[A, v,t] (1/mv)Sa[A, v,t] -(mw 2 /V)Sa[A, v,t] Ca[A, v,t] - (.u/V)Sa[A, v,t] Iwhere we lise the notationSl

~a[A, v,t] = ~(Ea[( -A + v)t a] parA,v,t] =

Ea[( -A - V)taDj

~(Ea[( -A + v)t a] +Ea[( -A -

V)taDj

Ks a result, we obtain At(a l =
t = -

m: SarA, v,t]Qo+ (Ca[A, v,t] - ~Sa[A, v,t])Po.

(20.49

for a = 1, EquatIOns (20.48) and (20.49) gIVe (20.36) and (20.37). EquatIOn~ 1(20.48) and (20.49) descnbe non-Markovian evolutIOn of quantum coordmate and! fuomentum of open quantum systems. ThIS non-MarkovIan quantum dynamIc~ Iwith a it N cannot be described by a semigroup. Therefore we can consider non-I ~arkovIan evolutIOn(20.48) and (20.49) of quantum OSCIllator WIth fnctIOn as frac1 ~IOnal dynamICS of open systems wIth memory.1

~O.8

Quantum self-reproducing and self-cloning!

[The ImpossIbIhty of Ideally copying (or clomng) of unknown quantum states IS onel pf the basic rules of quantum mechanics (Scarani et aI., 2005). The no-cloning thej prem tells us that quantum states cannot be cloned Ideally for an arbItrary ongma~ 1m-state. The no-clomng theorem of Wootters and Zurek (Wootters and Zurek, 1982)1 ISaId there IS no quantum copy devIce, whIch can copy any unknown quantum purel Istate. The result of the no-clomng theorem was extended to mIXed states in (Bar1 ~um et aI., 1996). They showed that if a particle in an arbitrary mixed state is sen~ linto a device and two particles emerge, it is impossible for the two reduced densitYI

120.8 Quantum self-reproducing and self-cloning

48~

~atrices of the two-particle state to be identical to the input density matrix. Th~ ro-clomng theorem for mIxed states (Barnum et a1., 1996) proves that broadcastmgj pf two noncommuting mixed states is impossible. The no-cloning theorem tells u§ ~hat quantum states cannot be cloned Ideally. There IS a problem how well one canl ~opy quantum states, l.e., how close the copy state can be to the orIgmal state. ThI§ Iproblem was solved m (Buzek and HIllery, 1996), where approxImate clomng sysj OCem was presented. They suggested (Buzek and HIllery, 1996) a umversal quantuml ~Ioning device. This type of device is also called input-state independent. A cloning pf quantum states by probabIhstIc processes IS possIble. The probabIhstIc clomngj ~evice was proposed m (Duan and Guo, 1998). It was shown that a umtary evolutIoIl] OCogether wIth a measurement can exactly clone a fimte set of hnearly mdependentl Istates with certain nonzero probabilitiesj ISelf-reproducing is a process by which a system might make a copy of itself.1 k\ theory of self-reproducmg classIcal automata was suggested by John von Neuj mann (Neumann, 1966). About kinematics of self-reproducing classical system§ Isee (FreItas and Merkle, 2004). The dIscovery of the polymerase actIVIty of thel Iself-sphcmg rIbosomal RNA (RIbo-Nucleic ACId) mtervemng sequence of Tetrahy-I rIena thermophIla tells us that hfe has started from self-rephcatIve RNA sequencesl ~alled rephcases (Cech, 1986; Watson et a1., 1987). Although the rephcase IS a hYP01 OChetIcal RNA molecule, the presence of both "mformatIon" and "functIOn" of selfj IrephcatIon m the same RNA molecule sImphfies the problems of self-reproductIonl pf molecular systems. E.P. Wigner was probably the first, who consIdered (WIgner J ~ 961) the self-reproducmg and self-clomng problem withm the quantum formahsmj ~n some sense, quantum self-clonmg IS a quantum analog of a SImple mathematIca~ Imodel of rephcaseJ [n general, quantum states are descrIbed by denSIty operators. A denSIty operatorl lis a self-adjoint (p* = p), non-negative (p ;;:: 0) operator with unit trace (Tr[p] = 1)1 !pure states can be characterized by the condition e2 = e. The state Ie (t)) of n-qubitl Isystem can be represented byl

Ip(t)) =

L l,u)p,u(t),

(20.50~

lL@:l

Iwhere N

= 4n and pg(t) = (,ulp(t)) are real-valued functions.

~lementA

Here we denote ani of Liouville space (Tarasov, 200gb) by a ket-vector IA). The inner productl

bftwo elements IA) and IB) of £(n) is defined as (AlB) = Tr[A*B]. The basis for

~

liouville space £(n) is defined byl

(20.51~ Iwhere N = 4n, ,ui E {O,1,2,3}, (j,u are Pauli matrices, and (,ul,u') = 8,u,u1. Here ,ul lis in the form,u = ,u14n 1 +...+ ,un-14 + ,un' Using that p(t) is a self-adjoint non-I r.egative operator of unit trace, we hav~

~84

20 Fractional Dynamics of Open Quantum Systems

PO(t) = (Olp(t)) =

I

-12'"

(20.52

land p;;(t) = p,u(t)J [Themost general state change is a quantum operation (see, for example (Tarasov j 1200gb». A quantum operation is a real positive (or completely positive) tracej Ipreserving superoperator g on a Liouville space £(n). A linear quantum operatiod 1£&3 can be represented (Tarasov, 2002b, 200gb) by the equatIOnj ~

W= L L 0",uv l.u)(vl,

(20.53~

in

Iwhere N = 4 n, the matrix 0",uv has the form 0",uv = Tr[CT,uJ( CTv)], and CT,u = CT,u1 ¢j I. . · ® CTlln · quantum self-cloning system (QSCS) is the pair QSCS = {ip; Ip)}. A QSCSI Fan be considered as a quantum operation lp that transforms the input unknownl k{ata Ip') according to some given program Ip):1

K

~Ip) ® Ip')

=

Ip) ® Ip)·

(20.54~

~n general, Ip) can be pure or mixed state. A quantum system is Hamiltonian if thel ijuantum operatIOn £&3 can be represented m the form (f,'iJ p = U*pU where U*U ~ IUU* - I for all states p of the system, otherwise is said to be non-Hamiltonianl [The quantum self-cloning system is defined by a state Ip) of the system and thel guantum operation (20.54), where Ip') is an unknown state, and Ip) is a "proper'l Istate of the system. The proper state is a fixed state of the system. In the laborator)1 Iwe don't know thiS quantum state. Therefore thiS state cannot be dOlling by thel ~xternal copymg machmes. The ideaIIy copymg deVice cannot be constructed for thel Istate that is unknown for us. ThiS is the statement of no-dolling theorem. Note tha~ ~his theorem cannot descnbe a possibihty to self-dOlling of thiS state. The quantuml Iself-dOlling process is a copymg of the state of system by the system of itself msteadl pf a copymg by an external deVice. In order to ideaIIy copymg thiS state by oufj deVice, the state must be known for usJ [The self -reproducmg quantum operatIOn If) is defined byl

(20.55~ I4":==]

IR(n) =

J2I1lp )(01,

j(n) =

L l.u)(.ulj

~

iHere we can assume that a basis to which p belongs is known and p, pi E £(n) J Iwhere dim(£(n)) = 4n . Note that the quantum operation R(n) gives R(n) Ip') = Ip ).1

120.8 Quantum self-reproducing and self-cloning

4851

k\s a result, we have (20.54). ThIs IS the self-clomng quantum system that makes ij ~opy of itself, not copying any changes in timej [Let us consider a quantum self-cloning process such that a system can give itselfj ~opy mcludmg any changes m tIme. ThIs type of quantum self-clomng IS defined byl la state p(t) of the system at t > 0 and the time-dependent quantum operation &(t,t') Isuch that

I&(t ,t')p (r') ® p' (t')

=

P (t' + L1t) ® p(t'),

(20.56~

Iwhere p' is unknown state, and L1t = t - t' is a time of cloning. Here t' is an instan~ pf the begmmng of clomng, t IS an mstant of fimshmg of quantum clomng. Thel pngmal state (source) of the system IS changed dunng of trme of copymgJ [he process (20.56) can be realIzed by the quantum self-clomng system QSCS t =i I{ gp (t,t'), P}, where the self-reproducing quantum operation i~

Here

IR(n)(t,t ')

=

J2i'lp(t'))(OIJ

land s(n)(t,t ') is a quantum operation that describes the change a quantum state ofj ~he quantum mach me such thatl

Is(n)(t,t')lp(t'))

=

Ip(t))1

[The state p (t') in the right-hand side of (20.56) can be considered as a next "young'j generation of the state p (t) 1 ~n Ref. (Tarasov, 20096), we suggested to consIder a quantum nanotechnologyJ IwhIch can allow us to buIld quantum nanomachmes. Quantum nanomachmes canno~ Ibe consIdered only as molecular nanomachmes, lIke as a quantum computer IS no~ pnly molecular computer. Quantum nanomachmes are not only small machmes ofj ranO-SIZe. These machmes should use new (quantum) pnnCIples of work. A quan-I ~um computer IS an example of quantum machme for computatIOn. Quantum selfl pomng machmes can be used for creatIOn of quantum states, and complex struc1 ~ures of quantum states. For example, self-clomng quantum machmes can create SU1 Iperconductmg states of molecular nanOWIres, superflmdmg states of nanomachme§ motIOn, or superradlance states of nanomachmes (or molecular nanoantennas)J [The suggested tractIOnal quantum MarkOVIan and non-MatkovIan dynamIcs aI1 ~ow us to generalIze quantum self-reproducmg and self-clomng. FractIOnalquantuml Inon-MatkovIandynamIcs allows us to take mto account memory effects m quantuml Iself-clomng. A fractIOnal power of the quantum MarkOVIan self-clomng operatIOn§ ~an be consIdered as a parameter to descnbe a measure of screemng of an envI-1 Ironment m self-clonmg processes. FractIOnal quantum self-reproducmg and self=] pomng can be controlled by the parameters of tractIOnal power of superoperatorsj

f'l-86

20 Fractional Dynamics of Open Quantum Systems

120.9 Conclusionl [n quantum dynamIcs of Hamiitoman systems, the InfimtesImal superoperator I§ k1efined by some form of derivation. A derivation is a linear map 2', which satisfie§ ~he Leibnitz rule 2'(AB) = (2'A)B + A (2'B) for all operators A and B. Fractionall klerivative can be defined as a fractional power of derivative. It is known that thel linfinitesimal generator 2' = (17 iii) IH, . I, which is used in Hamiltonian dynamics] lIS a denvatIOn of observables. In general, quantum systems are non-Hamiitomani land 2' is not a derivation. For a wide class of quantum systems, the infinitesimall generator 2' is completely dissipative (Kossakowski, 1972; Davies, 1976; Ingardenl landKossakowskI, 1975; Tarasov, 2008b). We consIder (Tarasov, 2009a) a tractIona~ generahzatIon of the equatIOn of motIon for open quantum systems. A tractIona~ Ipowerof completely dISSIpatIve superoperator can be used In thIS generahzatIOn.1 k\ generahzatIOn of the equatIon for quantum observables IS suggested. In thI§ ~quation, we use superoperators that are fractional powers of completely dissipativel Isuperoperators. We prove that the suggested superoperators are InfimtesImal genera-I ~ors of completely pOSItIve semIgroups. We note that the Bochner-PhIlhops formulal lallows us to obtaIn a tractIOnal dynamIcal descnptIon In terms of solutIOn of non-I ~ractIOnal dynamIcs. PropertIes of thIS semIgroup are consIdered. A tractIonal powe~ pf the quantum MarkovIan superoperator can be consIdered as a parameter to de1 Iscnbe a measure of screemng of an enVIronment(Tarasov, 2009a). Quantum compu-I ~atIOns by quantum operatIOns WIth mIxed states (Tarasov, 2002b) can be controlle~ Iby these parameters. We assume that there eXIst statIOnary states of open quantuml Isystems (Davies, 1970; Lindblad, 1976b; Spohn, 1976, 1977; Anastopoulous and! Halliwell, 1995; Isar et aI., 1996; Tarasov, 2002c,a) that depend on the fractiona~ parameter. IWe prove that solutIOns of the tractIOnal dynamICS of quantum observables an~ Istates are descnbed by the quantum dynamIcal semIgroup. Therefore the evolutIOil] pf observables and states is Markovian (Gardiner, 1985). This means that the sugj gested tractIOnal denvatIves, whIch are tractIOnal powers of operators (Balaknshj ijan, 1960; Komatsu, 1966; Berens et al., 1968; YosIda, 1995; MartInez and SanzJ 12000) and superoperators (Tarasov, 2008b, 2009a), cannot be connected WIth long-I ~erm memory effectsl [I'he suggested fractIOnal quantum MarkovIan equatIOn IS exactly solved for thel Iharmomc OSCIllator WIth hnear fnctIOn. We assume that other solutIOns and prop-I ~rtIes descnbed In (LIndblad, 1976b; Sandulescu and Scutaru, 1987; DaVIes, 1981j IIsar et al., 1994, 1996; Lldar et al., 2001; DIetz, 2002; Tarasov, 2002c,a; Nakazatol ~t al., 2006) can be conSIdered for tractIOnal generahzatIOns of the quantum Marko-I Ivian equation and the Gorini- Kossakowski-Sudarshan equation (Gorini et aI., 1976j

If9T8T. IWe note that It IS Important to conSIder fractIOnal powers of generatIng superj pperators for N-Ievel quantum non-Hamiitoman systems. These systems are de1 Iscnbed by Gonm-KossakowskI-Sudarshan equation (Gonm et al., 1976, 1978) (seel lalso Section 15.11 in (Tarasov, 2008b». Fractional generalizations of the Gorinij IKossakowski-Sudarshan equation to describe the Markovian and non-Markovianl

IR eferences

4871

[ractional dynamics can be studied. The solutions of the Cauchy type problem fo~ ~he fractional equations of finite-dimensional quantum non-Hamiltonian system§ ~an be obtained. For example, interesting results can be obtained for two-level non-I Hamiltonian quantum systems (see Section 15.12 in (Tarasov, 2008b».1

lReferencesl K Ahckl, K. LendI, 1987, Quantum DynamIcal Semlgroups and AppltcatwnsJ ISprInger, Berhn.1 Anastopoulous, U. Halhwell, 1995, Generahzed uncertamty relatIOns and long-I ItIme hmIts for quantum Browman motIon models, Physical Review D, 51, 6870-1

r.

ssss. ~

Balakrishnan, 1960, Fractional power of closed operator and the semigroup genj erated by them, Pacific Journal ofMathematics, 10, 419-437 j IH. Barnum, eM. Caves, eA. Fuchs, R. Jozsa, B. Schumacher, 1996, Noncommut~ ImgmIXed states cannot be broadcast, Physical Review Letters, 76, 2818-28211 IS. Bochner, 1949, DIffuSIOn equations and stochastIc processes, Proceedings oj th~ !NationalAcademy ofSciences USA, 35, 369-370j R Berens, P.L. Butzer, U. Westphal, 1968, Representation of fractional powers ofj ImfimtesImal generators of semigroups, Bulletin oj the American Mathematicall ISociety, 74, 191-196J IV. Buzek, M. HIllery, 1996, Quantum copying: Beyond the no-clomng theorem) IPhysicalReview A, 54, 1844-1852j [T.R. Cech, 1986, A model for the RNA-catalyzed replication of RNA, Proceeding~ oj the National Academy of Sciences USA, 83, 4360-4363.1 IV. Daftardar-GeJJI, A. Babakham, 2004, AnalysIs of a system of tractIOnal dIfferen-1 ItIal equatIOns, Journal oj Mathematical Analysis and Applications, 293, 511-5221 ~.P. DavIes, 1970, Quantum stochastIc processes II, Communlcatwn In Mathemattj cal Physics, 19, 83-105.1 ~.B. Davies, 1976, Quantum Theory of Open Systems, Academic Press, London,1 INew York, San FrancIsco.1 ~.B. DavIes, 1977, Quantum dynamIcal sermgroups and neutron dIffUSIOn equatIOn) IReports in Mathematical Physics, 11, 169-188J ~.B. DavIes, 1981, Symmetry breakmg for molecular open systems, Annales dtj Il'!nstitut Henri Poincare, Section A, 35,149-171.1 IK. DIetz, 2002, AsymptotIc solutIOns of Lmdb1ad equations, Journal oj Physics Aj 135, 10573-105901 ~.M. Duan, G.e Guo, 1998, A probabIhstIc clomng machme for rephcatmg twq Inon-orthogonal states, Physical Letters A, 243, 261-264j KA. FreItas Jr., Re Merkle, 2004, KinematIc Selj-Repltcatlng Machines, Landesl IBioscience, see also http://www.molecularassembler.com/KSRM.htmj Gardmer, 1985, Handbook of Stochastic Methods for Physics, Chemistry andl lNatural SCIences, 2nd ed., SprInger, Berhn1

r. w.

~88

20 Fractional Dynamics of Open Quantum Systems

IV. Gonm, A. KossakowskI, E.CG. Sudarshan, 1976, Completely posItIve dynamIca~ IsemIgroups of N-Ievel systems, Journal of Mathematical PhYsics, 17,821-825.1 ~. Gorini, A. Frigerio, M. Verri, A. Kossakowski, E.C.G. Sudarshan, 1978, Properj ItIes of quantum markovIan master equatIons, Reports In Mathematical PhyslcsJ 113,149-173j IK.E. Hellwmg, K. Kraus, 1969, Pure operatIOns and measurements, Communicationl lin Mathematical Physics, 11, 214-220J IKE. Hellwing, K Kraus, 1970, Operations and measurements II, Communicationl lin Mathematical Physics, 16, 142-147j ~. Hille, R.S. Phillips, 1957, Functional Analysis and Semigroups, American Mathj ~matIcal SOCIety, ProvIdenceJ RS. Ingarden, A. KossakowskI, 1975, On the connectIOn of noneqmhbnum mforj Imation thermodynamics with non-Hamiltonian quantum mechanics of open sysj Items, Annals ofPhysics, 89, 451-485j v\ Isar, A. Sandulescu, H. Scutaru, E. Stefanescu, W. ScheId, 1994, Open quantuml Isystems, International Journal of Modern Physics E, 3, 635-714; and E-printj ijuant-ph70411189J V\. Isar, A. Sandulescu, W. ScheId, 1996, Phase space representatIOn for open quan-I Itum systems WIth the Lmdb1ad theory, International Journal oj Modern Physicsl IB, 10, 2767-2779; and E-print: quant-ph/9605041j k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj ~ional Dijjerential Equations, ElseVIer, AmsterdamJ ~. Komatsu, 1966, FractIOnal powers of operators, Pacific Journal oj MathematicsJ 119, 285-346J v\ KossakowskI, 1972, On quantum statIstIcal mechamcs of non-Hamiitoman sysj Items, Reports in Mathematical Physics, 3, 247-274J IK Kraus, 1971, General state changes in quantum theory, Annals of Physics, 64j 1311-335 IK Kraus, 1983, States, Effects and Operations. Fundamental Notions oj Quantu~ ITheory, Spnnger, BerhnJ IS.G. Krem, 1971, Linear Differential EquatIOns In Banach space, TranslatIOns ofj IMathematIcal Monographs, vo1.29, Amencan MathematIcal SOCIety, Translated! Ifrom RUSSIan: Nauka, Moscow, 19671 Lldar, Z. BIhary, K.B. Whaley, 2001, From completely pOSItIve maps to thel ijuantum MarkOVIan semIgroup master equation, Chemical Physics, 268, 35-53j ~nd E-pnnt: cond-maUOOl1204J p. Lindblad, 1976a, On the generators of quantum dynamical semigroups, Commuj Inication in Mathematical Physics, 48, 119-130.1 p. Lmdblad, 1976b, Browman motIon of a quantum harmomc OSCIllator, Reports i~ Mathematical Physics, 10, 393-406J ~. Martmez, M. Sanz, 2000, The Theory of FractIOnalPowers of operators, Northj Holland MathematIcs StudIes. Vol.187, ElseVIer, AmsterdamJ R Nakazato, Y. Hida, K Yuasa, B. Militello, A. Napoli, A. Messina, 2006, Soluj ItIOn of the Lmdblad equatIOn m the Kraus representation, Physical Review A, 74J 062113; and E-pnnt: quant-ph70606193J

rrx:

References

489

~.

von Neumann, 1966, Theory of Self-Reproducing Automata Source, Umversity ofj lIllinois IK.E. Oldham, J. Spanier, 1974, The Fractional Calculus: Theory and Application~ r.f DijjerentlatlOn and IntegratIOn to Arbitrary Order, AcademIc Press, New YorkJ v\ Oza, A. Pechen, 1. Dommy, V Beltram, K. Moore, H. RabItz, 2009, OptImizatIoIlj Isearch effort over the control landscapes for open quantum systems with Krausj ImapevolutIOn,Journal of Physics A, 42, 205305J RS. PhIllIps, 1952, On the generatIOn of semigroups of lInear operators, Pacifid, IJournalofMathematics, 2, 343-369.1 k\.P. Prudnikov, Yu.A. Brychkov, 0.1. Marichev, 1986, Integrals and Series, Vol.l.j IElementary FunctIOns, Gordon and Breach, New YorkJ IS.G. Samko, A.A. Kilbas, 0.1. Marichev, 1993, Integrals and Derivatives of Fracs rional Order and Applications, Nauka i Tehnika, Minsk, 1987, in Russianj ~nd FractIOnal Integrals and DerivatIves Theory and AppLzcatlOns, Gordon and! IBreach, New York, 1993J v\ Sandulescu, H. Scutaru, 1987, Open quantum systems and the dampmg of colj IlectIve models m deep melastIc collIsIOns, Annals oj Physics, 173, 277-317J IVScaram, S. IblIsdIr, N. GIsm, 2005, Quantum clomng, Review oj Modern Physics) 177, 1225-1256J ~. Schumacher, 1996, Sending entanglement through noisy quantum channelsj IPhysical Review A, 54, 2614-2628J ~. Spohn, 1976, Approach to eqmlIbnum for completely posItIve dynamIcal semI1 groups of N-level systems, Reports in Mathematical Physics, 10, 189-194J ~. Spohn, 1977, An algebraiC condItIOn for the approach to eqmlIbnum of an openl IN-level system, Letters in Mathematical Physics, 2, 33-38j IVB. Tarasov, 2002a, Pure statIonary states of open quantum systems, PhYSIcal Rej Iview E, 66, 056116j IY.E. Tarasov, 2002b, Quantum computer WIth mIxed states and four-valued lOgIC) flournal oj Physics A, 35, 5207-5235J IVB. Tarasov, 2002c, StatIOnary states of dISSIpatIve quantum systems, PhySICS Letj rers A, 299, 173-1781 ~E. Tarasov, 2004, Path integral for quantum operations, Journal ofPhysics A, 37 j 13241-32571 IVB. Tarasov, 2005, Quantum Mechanics: Lectures on Foundations oj the Theory) ~nd ed., Vuzovskaya Kmga, Moscow. In RussianJ ~E. Tarasov, 2008a, Fractional Heisenberg equation, Physics Letters A, 372, 2984-1

I29:BR: IY.E. Tarasov, 2008b, Quantum Mechanics oj Non-Hamiltonian and Dissipative SYS1 ~ems, ElseVIer, AmsterdamJ IY.E. Tarasov, 2009a, FractIOnal generalIzatIOn of the quantum MarkOVIan maste~ ~quation, Theoretical and Mathematical Physics, 158, 179-195.1 IVB. Tarasov, 2009b, Quantum Nanotechnology, InternatIOnal Journal of] Wanoscience, 8, 337-344j

~90

20 Fractional Dynamics of Open Quantum Systems

[.D. Watson, N.H. Hopkins, I.W. Roberts, I.A. Steitz, A.M. Weiner, 1987, Molecj ~lar Biology of the Gene, YoU, 3th ed., Benjamin/Cumming, California, 1103-1 l1l24. 10. WeIss, 1993, Quantum DISSIpatIve SYstems, World SCIentIfic PublIshmg, Smgaj ~ ~.P. WIgner,

1961, The probabIlIty of the eXIstence of the self-reproducmg umt, ml The Logic of Personal Knowledge. Essays presented to Michael Polanyi, Rout-l Iledgeand Paul, London, 231-238j IW.K. Wootters, W.H. Zurek, 1982, A smgle quantum cannot be cloned, Natur~ I(London), 299, 802-803j K Wu, A. Pechen, C. BrIf, H. RabItz, 2007, ControllabIlIty of open quantum sysj Items wIth Kraus-map dynamICS, Journal of Physics A, 40, 5681-5693J IK. YosIda, 1995, Functional Analysis, 6th ed., SprInger, BerlInJ

~hapter2]

IQuantum Analogs of Fractional Derivatives

~1.1

Introductionl

[he fractIOnal denvative has dIfferent defimtIOns (Samko et aL, 1993; KIlbas e~ laL, 2006), and explOItIng any of them depends on the kInd of the problems, ImtIall I(boundary) condItIOns, and the specIfics of the consIdered phySIcal processes. Thel claSSIcal defimtIons are the so-caned RIemann-LIouvIlle and LIOuvIlle denvatIve8 I(Kilbas et al., 2006). These fractional derivatives are defined by the same equation~ pn a finite interval ofJR and of the real axis JR, correspondently. Note that the Caputol land RIesz denvatIves can be represented (KIlbas et aL, 2006; Samko et aL, 1993)1 ~hrough the RIemann-LIOuvIlle and LIOuvIlledenvatIves. Therefore quantIzatIOn o~ RIemann-LIOuvIlle and LIouvIlle fractIonal denvatIves anows us to denve quantuml lanalogs for Caputo and RIesz denvatIves] [t IS wen-known that the denvatIves wIth respect to coordInates qk and momental IPk can be represented as POIsson brackets by the equatIOnsj

ID~kA(q,p) = -{Pk,A(q,p)},1

IDbkA(q,P) = {qk,A(q,p)}1 ~or continuously differentiable functions A(q,p) E C1 (JR 2n ) . Quantum analogs o~ ~hese POIsson brackets are self-adjoInt commutators. The Weyl quantIzatIOn niij I(Tarasov, 2008b, 200Ia,b) gIve§

fW({Pk,A(q,P)})

=

i~ [nW(Pk), nW(A)]~

fW({ qk,A(q,p)})

=

~ [nW(qk), nW(A)]~

Iwhere IA,EI = AE - EA. As a result, we have tha~

(21.1)1 V. E. Tarasov, Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

~92

21 Quantum Analogs of Fractional Derivatives

Iwhere Qk = nW(qk) and Pk = nW(Pk), can be considered as quantum analogs ofj ~erivatives and Then quantum analogs of derivatives of integer order ~ pan be defined by the products of L~ and L Qk. For example, a quantum analog o~

D1k

D1k'

ID1pbl has the form -LpkLQ1 = (l/n?[Pk, [Q/, . ]]. For the derivative D~k' we havel 1(-It(LpJn. ~he quantum analogs of the derivatives D~k and Dbk are commutators (21.1) an~

1(21.2). It IS Important to find analogs of RIemann-LIOuvIlle and LIouvIlle tractIona~ klerivatives for quantum theory. To obtain these analogs of Riemann-Liouville fracj OCIOnal denvatIves, whIch are defined on a fimte mterval of JR, we can use a represen-I ~ation of these derivatives for analytic functions. In this representation the Riemannj OC:IouvIlle denvatIve IS a senes of denvatIves of mteger orders. It allows us to use thel ~orrespondence between the mteger denvatIves and the self-adjomt commutator~ I(Tarasov, 2008a). To define a quantum analog of the Liouville fractional derivative,1 IwhIch ISdefined on the real aXIS JR, we can use the representatIOn of Weyl quantIza-1 ~IOn by the Founer transformatIOn (Tarasov, 2008b). Quantum analogs of fractIOna~ ~envatIves gIve us a notIon that allows one to consIder quantum processes that arel ~escnbed by tractIonal dIfferentIal equatIons at classIcal level. Usmg the Weyl quan-I ~Izatlon methods, we can reabze quantIzatIon of nonddferentIable functIons that arel ~efined on a phase space. Note that an mterestmg approach to quantum fractal con1 IstructIOn was proposed by WOJcIk, BIalymckI-BIrula, and ZyczkowskI m (WoJcIk e~ lal., 2000). [n SectIon 21.2, Weyl quantIzatIon and ItS propertIes are descnbed. In SectIOnl 121.3, we consIder a Weyl quantIzatIon of RIemann-LIouVIlle tractIonal denvatIveJ ~n SectIOn 21.4, the quantIzatIOn of LIOuVIlle fractIOnal denvatIve IS reabzed byl lusmg Founer transformatIOn. In SectIOn 21.5, we define a Weyl quantIzatIOn o~ IWeIerstrass nonddferentIable functIOns. Fmally, a short conclusIOn IS given m Sec1 tion 21.6.

~1.2

Weyl quantization of differential operatorsl

OC:et us consIder the dIfferentIal operatofj

w=

2'[q,p,D~,D~]

(21.3)1

pn a phase space ]R2n, which is a function of the phase space coordinates qk, PkJ ~ = 1, ... ,n, and the derivatives D1k and Dbk' k = 1, ... ,n. Quantization is usuallYI lunderstood as a procedure, where any real-valued function A (q, p) is associated withl la relevant quantum observable, i.e., a self-adjoint operator A(Q, P). Correspondenc~ Ibetween operators A = A(Q,P) and symbols A(q,p) is completely determined byl ~he formulas that express the symbols of operators QA, AQ, PA, AP in terms o~

49~

121.2 Weyl quantization of differential operators

~he symbols of the operator A. It is well-known (Tarasov, 2008b) that the Weyll guantization trw is defined b)j

!Letus define the left operators LA and Ifommlas'

'?A B(q,p) =

Lf acting on phase space functions by thel

{A(q,p),B(q,p)}J

ILfB(q,p) =A(q,p)B(q,p).

(21.8)1

from these defimtiOns, we obtaml (21.9)1

~qkA(q,p) = D1k A(q,P),

LpkA(q,p) = -D~kA(q,p).

(21.10~

~hen the operator 2[q,p,D~,Db] will be represented b~

psing the superoperators L~ and L~, which are defined byl

(21.11~ (21.12~ Iwe rewnte Eqs. (21.4)-(21.7) m the tormj

fw(LtkA) = Lj;/i,

trW(L;k A) = LJ-i,Aj

ISince these relations are valid for any A = trw(A), we can define (Tarasov, 2001b,1 12008b); see also (Tarasov, 2001 a,c) the quantization of operators L~ and Lik1 ~efinition

21.1. The Weyl quantization trw of qk and Pk gives the operatorsi

~94

21 Quantum Analogs of Fractional Derivatives

~he Wey I quantization of the operators L ~ and L ~k is defined by the equation:1 frw(L;t) = L~k'

Jrw(L~)

(21.13~

= L Qk,

prw(LtJ = L~,

Jrw(C;J = LI7c . ~quatlons (21.13) and (21.14) define the Weyl quantIzatIon of the pperator 2[q,p,D1,D1JJ

(21.14~ ddferentIa~

[Theorem 21.1. Weyl quantization Jrw associates the differential operator 2[q,p,1 ID~,D1] on the function space and the superoperator 2[L~,Lt, -Lp ,LQ ], actingl km the operator spacej

IProof

q

This theorem was proved in (Tarasov, 2008b).

IRemark lJ [I'he Weyl quantIzatIon of polynomIal operator (21.3) IS defined by the formula:1

IRemark2J IWe note that the commutation relations for the operators L~, L~k and L~k' L~ coin1 Fide. Then the ordering of L~ and L~k in the superoperator 2[L~,Lt, -Lp ,LQ ] i~ luniquely determined by ordering in 2[Lt ,Lt, -L/i ,L;]j

~1.3

Quantization of Riemann-Liouville fractional

derivative~

[n order to realIze Weyl quantIzatIOn of RIemann-LIOuvIlle tractIOnal derIvatIve, wei luse a fractional analog of the Taylor series. The fractional derivative oD~ on 10,bl 1m the RIemann-LIOuvIlle form IS defined by the equatIOn:1

DaAx x

Iwhere m - 1 < a ~

()-

I

-d

m

r(m-a)dxm

lX (x_y)a-m+l' A Y dy 0

(21.15

m.1

[Theorem 21.2. If A(x) is an analytic function for x E (O,b), then the fractionall I(lerivative (21.15) can be represented in the jormj

(21.16~

a n a =

I

(,)

r(a+l) I r(n+l)r(a-n+l)r(n-a+l)'

121.3 Quantization of Riemann-Liouville fractional derivatives

495]

IProof. ThIs theorem was proved III (Samko et aL, 1993) (see Lemma 15.3 III (Samkol ~t al., 1993)). q ~n order to define Weyl quantization of Riemann-Liouville fractional derivative,1 Iwe consider representation (21.16) in the phase space. If A(q,p) is an analytic func-I ~ion on the phase space ]R2n, then we can use (21.16) for the Riemann-Liouvill~ ~ractIOnal denvatIves wIth respect to qk and PkJ

rrheorem 21.3. The Weyl quantization of the Riemann-Liouville fractional deriva1 Itlves (21.21) and (21.22) gIves the superoperatorsj

(21.17~

(21.18~ If A(q,p) is an analytic function on the phase space ]R2n, then the Riemann~ pouville fractional derivatives with respect to qk and Pk can be represented (21.16) lin the toon'

IProof

(21.19~ (21.20~

Iwhere k - 1, ... ,n. The Weyl quantIzatIOn IS defined by relatIOns (21.13) and (21.14)J [Theretore tractIOnal denvatIves (21.19) and (21.20) must be represented through thel pperators L~ and L~k' which are defined by (21.9) and (21.10). Using the operators IL~ and Equations (21.19) and (21.20) are rewritten in the form:1

L;k'

pD~kA(q,p)

=

L a(n, a)(L~)n a (-Lpk)nA(q,p)l ~

rthese equations hold for all analytic functions A(q,p) on the phase space ]R2n. As ~ Iresult, the tractIOnal denvatIves are defined bYI (21.2q

pD~k =

L a(n, a)(Lty-a (L;Y·

(21.22~

~

~he WeyI quantization of the operators L ~ and L ~k is defined by Eqs. (21.13) an1 1(21.14). As a result, we obtain relations (21.17) and (21.18). q

21 Quantum Analogs of Fractional Derivatives

f'l-96

lRemark

[Equations (21.17) and (21.18) can be considered as definitions of the klerivation superoperators on an operator spacej

fractiona~

[Example. ~t

is not hard to prove tha~

Iwhere n ?: 1, and a ?: O~

~1.4

Quantization of Liouville fractional

derivativ~

[To define a quantization of LIOuvIlle fractIOnal denvatIve, we use the Founer trans1 ~onn operator!Y. The Fourier transformA(a) of some function A(x) ELI (R) i~

A(a) =!Y {A (x)}

=

~/2 2n

r dxA(x) exp{ -iax}

JJR

~f A(x) ELI (lR), then the Parseval formula IIAI12 = IIAI12 holds. Let § be an exten-I Ision of this Fourier transformation to a unitary isomorphism on L2(lR). We definel ~he operatorsj

Iwhere the functions Lk(a), kEN, are measurable. These operators fonn a commu-I ~ative algebra. If .2'12 is an operator associated with L12(a) = LI (a)L2(a), then wei IJ.lli.Ye [.Jet us define a fractional derivative as an operator D~ such tha~ (21.23j

for A(x) E L2(lR), Equation (21.23) gives the integral representationj

~A(x) = [t

IS

2n

l2

dadx' (ia)U A(x') exp{ia(x-x')}.

easy to prove that Eq. (21.24) representatIOnj

~IOuvllle mtegral

IS

(21.24

eqUIvalent to the well-known Rlemannj

497]

121.5 Quantization of nondifferentiable functions

dm

=-:---,- - -

dx'"

jX dx' -;---~:-;-­ -00

[I'hIS representatIOn cannot be used to quantIzatIon] ~n order to define Weyl quantization of Liouville fractional derivative, we conj Isider the representation (21.24) in the phase space. Let A (q,p) be a function ofj IL2(]RZ) on the phase space ]Rz. Then Eq. (21.24) is represented in the formJ D aDf3A(q p)= { dadbdqdp (ia)a(ib)f3A(q' p')ek(a(q-q')+b(p-p')). (21.25 JJF,4 2nli 2 ' q p ,

[TheWeyl quantization of A(q,p) is defined b~

IUsmg thIS equation, we define the Weyl quannzation of (21.25) m the form:1

i "h(a(Q-qI) +b(P- pI))

[his equation can be considered as a definition of and D~ on a set of quantuml bbservables k\ generalization ofWeyl quantization of A(q,p) can be defined byl

DO

( dadbdqdp AF(Q,P) = JJE.4 2nli 2 F(a,b)A(q,p) exp

i

"h(a(Q-qI) +b(P- pI))

(21.27) OCf F(a,b) = I, then (21.27) defines the Weyl quantization. For F(a,b) = cos(ab/2li)J Iwe have the Rivier quantization. Therefore Eq. (21.26) can be considered as a genj ~ral quantization of A(q,p) with the functionj

Note that thIS functIOn has zeros on the real a, b aXIs.1

~1.5 ~n

Quantization of nondifferentiable function~

1872, Karl Weierstrass gave an example of a functIOn wIth the non-mtuitrve prop-I of being everywhere continuous but nowhere differentiable (Weierstrass, 1895j Hardy, 1916). The graphs of the Weierstrass functions can be considered as fracj OCals (Mandelbrot, 1983; Frame et aL, 2010; Falconer, 1990, 1985). Fractals have ij IWIde applIcatIOn in phYSICS (Mandelbrot, 1983; Feder, 1988; Kroger, 2000; Potapov J ~rty

~98

21 Quantum Analogs of Fractional Derivatives

12005; Berry, 1996). We note a general approach to quantum fractal constructIOilJ ~hat was proposed by Wojcik, Bialynicki-Birula, and Zyczkowski in (Wojcik et al.J ~

[.JetW(x) be a function on R Under certain circumstances the graph { (x, W (x)) j

k E lR~ regarded as a subset of the (x,y)-coordinate plane may be a fractal. If W(x]

Ihas a continuous derivative, then it is not difficult to prove that the graph has dimen-I Ision 1. However it is possible for a continuous function to be sufficiently irregularl ~o have a graph of dimension strictly greater than 1. The well-known example i~

IW(x) =

E a(s-2)k sin(akx)J lk==O

Iwhere 1 < s < 2, and a > 1. ThiS function has the box-countmg dimenSion D - sJ The well-known Weierstrass function j

IW(x) = Eansin(bnx)j ~

Iwhere 0

< a < 1 < b, ab > 1, is an example of a contmuous, nowhere dtfferentiabli.j

~unction.

The box-countmg dimenSIOn of itS graph is the non-mteger number:1

§=2-1~~ IWe can conSider the wave functionJ

Iwhere a = 2,3, .., and s E (0,2). In the physically interesting case of any finite MJ ~he wave function PM(t,X) is a solution of the Schrodinger equation. The limit casej

IWith the normahzatIOn constantl

liS contmuous but nowhere differentiable. In analogy With the theory of distnbutIOnJ litcan be conSidered as a solutIOn of the Schrodmger equatIOn m the weak sense. I~ Iwas shown that the probability density P(t,x) = IP(t,xW has a fractal nature an~ ~he surface P(t,x) has the box-counting dimension D = 2 + 0.5s1 [Thecomplex Weierstrass function has the form:1

49«;J

121.5 Quantization of nondifferentiable functions

Wo(x)

=

(1 - a2)- 1/2 L akexp{2nibkx}J ~

Iwhere b > 1 is a real number, and a = b D 2, 1 < D < 2. It can be proved that thi§ [unction is continuous, but it is not differentiable. Note that Wo(x) is continuous andJ k!ifferentlable if D < 1J [.Jet us consider this function on the phase space lR.2, and x = q or x = p. Using ~he operatorsj ~A(q,p) = ~he

qA(q,p),

LtA(q,p)

=

pA(q,p)J

complex WeIerstrass functIon can be representek! m the operator formj

IWo(Lt)

=

(1 - a2)- 1/2

txn

L ak exp{2nibkLt}.

(21.28~

[I'he baSIC assumptIon IS that the general propertIes of the Weyl quantIzatIon (Tarasov J 12008b, 20mb) given by the equatIOnsj

prw(Lt) = L6,

nw(Lt)

=

L6J

fW(~Ak(q,P)) = ~nw(Ak(q,p)l lare also vahk!for mfimte sums. Then the Weyl quantIzatIon of (21.28) gIve~

~O(L~) = nw(Wo(Lt))

=

(l-a 2 )

1/2

L akexp{2nibkL~},

(21.29~

~

IWO(Lt)

=

nw(Wo(Lt))

=

(l_a 2)- 1/2 L akexp{2nibkLt},

(21.30~

~

Iwhere Q = nw(q) and P = nw(p), and the superoperators L6, Lt are defined byl 1(21.11).As a result, Equations (21.29) ank! (21.30) k!efine the Weierstrass superoper-I latorfunctions Wo(L~) and Wo(Lt) on an operator algebra. In the Wignerrepresenta-I ~IOn of quantum mechamcs these superoperators are representek! by the Welerstras§ ~unctions Wo(q) and Wo(p) on the phase space lR.21 IWe can formally conSIder the operatorsj

(21.3q ""

IWo(L;) = (l_a 2)- 1/2 Lakexp{2nibkL;}, ~

Iwhere

(21.32~

fSOO

21 Quantum Analogs of Fractional Derivatives

ILqA(q,p)

=

{q,A(q,p)}

=

D1A(q,p),

LpA(q,p)

=

{p,A(q,p)}

=

-D~A(q,p)J

IOsmg the functlon§

W(q) = exp{ Aq}, Iwe

lJI(p)

=

exp{ Ap},

q,p

E

lRj

obtain

[Then lJI(x) are eigenfunctions of the operators Lx with the eigenvalues ±X. Th~ Ibox-counting dimension of its spectrum graphs (X, ±X) is I. The operators (21.31] land(21.32) gIV~

lWo(Lq )1JI(q)

=

Wo(X)IJI(q),

Wo(L p )1JI(p)

=

Wo( -X)IJI(p )J

k\s a result, the Weierstrass functions WO(±A) are eigenvalues of the operator~ 1(21.31) and (21.32) with the eigenfunction lJI(x). Then the spectrum graphsl I(X, Wo(±X)) of these operators are fractal sets. The box-counting dimensions o~ ~hese graphs are non-mteger numbersJ psmg the formulas:1

Iwe can realIze the Weyl quantlzatIOn of the operators (21.31) and (21.32). As Iresult, we obtam the superoperators of the formj

~o(LQ) = Jrw(Wo(L;;))

=

(1- a 2)- I/2

~

txn

L akexp{2JribkLQ},

(21.33~

~

[VO(L p) = Jrw(Wo(L;))

=

(1- a2 ) -

1/ 2

L akexp{2JribkL p},

(21.34j

~

Iwhere Q = Jrw(q) and P = Jrw(p). The superoperators LQ and Lp are defined byl 1(21.12). EquatIOns (21.33) and (21.33) define the WeIerstrass superoperator func-I ~ions Wo(L~) and Wo(Lt) on a set of quantum observables. In the Wigner represen-I ~atIOn of quantum mechamcs these superoperators are represented by the operator~ lWo(Lq) and Wo(L p) with the fractal spectrum graphs (X,Wo(±X)).1

121.6 Conclusionl IOsmg the Weyl quantIzatIOn (Tarasov, 200la,b,c, 2008b), and the representatIOn ofj ~ractIOnal denvatlve for analytICfunctIOns (Samko et al., 1993), we obtam (Tarasov J 12008a) quantum analogs of the Riemann-Liouville and Liouville derivatives. Th~ raputo and RIesz denvatlves can be represented (KIlbas et al., 2006; Samko et aLJ ~993) through the RIemann-LIOuVIlle and LIOUVIlle denvatlves. Therefore quantuml

References

5011

lanalogs of Riemann-Liouville fractional and Liouville derivatives allow us to deriv~ ~orrespondent analogs for Caputo and Riesz derivatives. Quantization of fractiona~ klerivatives gives us a notion that allows one to consider quantum processes that arel klescnbed by fractIonal dIfferentIal equatIons at claSSIcal levelJ IQuantum analogs of fractIonal denvatIves (Tarasov, 2008a) allow us to consIderl la generalIzatIon of the notIon of fractIOnal Hamlltoman system (Tarasov, 2005). Inl ~his case, a wide class of quantum non-Hamiltonian systems (Tarasov, 2008b) canl Ibe considered as fractional Hamiltonian systems. Using this approach, we can studYI la WIde class of quantum analogs of determImstIc dynamIcal systems WIth regularl land strange attractors (Amschenko, 1990; Nelmark and Landa, 1992). Note tha~ guantum analog of the Lorenz system (Lorenz, 1963; Sparrow, 1982) was suggested! lin (Tarasov, 2001a, 2008b)J

Referencesl IV.S. Amschenko, 1990, Complex Oscillations in Simple Systems, Nauka, Moscow] lIn Russian I M.V. Berry, 1996, Quantum fractals in boxes, Journal ofPhysics A, 29, 6617-6629.1 IK.E Falconer, 1990, Fractal Geometry. MathematIcal FoundatIOns and Appltca-I rions, WIley, ChIchester, New YorkJ IK.E Falconer, 1985, The Geometry oj Fractal Sets, Cambndge OmversItYI IPress,Cambndgej ~. Feder, 1988, Fractals, Plenum Press, New York, London) M. Frame, B. Mandelbrot, N. Neger, 2010, Fractal GeometryJ Ihttp://classes.yale.edu7fractal~ p.H. Hardy, 1916, WeIerstrass's non-dIfferentIable functIOn, Transactions oj th~ IAmerican Mathematical Society, 17, 301-325 J k\.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fracj kwnal Dijjerentwl EquatIOns, ElseVIer,Amsterdam] WI. Kroger, 2000, Fractal geometry In quantum mechamcs, field theory and SpInI Isystems, Physics Reports, 323, 81-181j ~.N. Lorenz, 1963, DetermInIstIc nonpenodIc flow, Journal oj the Atmospheric Sci1 ~nces, 20, l30-14U lB. Mandelbrot, 1983, The Fractal Geometry of Nature, Freeman, New York] IYu.I. Nelmark, P.S. Landa, 1992, StochastICand ChaotICOscIllatIOns, Kluwer Acaj klemIc, Dordrecht and Boston; Translated from RUSSIan: Nauka, Moscow, 1987) V\.A. Potapov, 2005, Fractals in Radiophysics and Radiolocation, 2nd ed., Omver-I ISItetskaya Kmga, Moscow. In RussIanJ IS.G. Samko, A.A. KIlbas, 0.1. Marlchev, 1993, Integrals and DerivatIves of Fracj rwnal Order and Appltcatwns Nauka I Tehmka, MInsk, 1987. In RUSSIan; and! IFractional Integrals and Derivatives Theory and Applications, Gordon and! IBreach, New York, 1993J

fS02 ~.

21 Quantum Analogs of Fractional Derivatives

Sparrow, 1982, The Lorenz Equations: Bifurcation, Chaos, and Strange Attracj Springer, New York.1 ~.E. Tarasov, 2001a, Quantization of non-Hamiltonian and dissipative systemsJ IPhysics Letters A, 288, 173-182j IV.B. Tarasov, 2001 b, Weyl quantization of dynamtcal systems wtth flat phase spaceJ Moscow University Physics Bulletin, 56, 5-lOJ ~.E. Tarasov, 2001c, Quantization of non-Hamiltonian systems, Theoretica~ IPhysics, 2, 150-160J ~E. Tarasov, 2005, Fractional generalization of gradient and Hamiltonian systemsJ IJournalofPhysics A, 38, 5929-5943j IV.E. Tarasov, 200Sa, Weyl quantization of fractional denvatives, Journal of Mathej Imatical Physics, 49, 102112J ~.E. Tarasov, 2008b, Quantum Mechanics ofNon-Hamiltonian and Dissipative SYS1 ~ems, Elsevter, Amsterdaml f. Weierstrass, 1895, Uber kontinuierliche funktionen eines reellen arguments, diel Ifur kelllen wert des letzteren elllen bestimmten dtfferential quotienten besttzenJ lIn Mathematische Werke II, Mayer-Muller, Berhn, 71-741 p. WOJctk, I. Bta1ymckt-Btru1a, K. Zyczkowskt, 2000, Ttme evolutiOn of quantuml Ifractals, Physical Review Letters, 85, 5022-5026; and E-pnnt: quant-ph100050601 ~ors,

IIndex

Euler equations, 61, 821 plpha-mteraclOn, 16G

~alaknshnan formula, 470 Ibalance of energy, 58, 60 Ibalance of mass, 51, 54 Ibalance of momentum, 56, 57 ~emoulh mtegral, 65 Ibl-Llpschltz conditIOn, 15 ~Iot-Savart law, 96 Bochner-Phillips formula, 470 ~ogolyubov hierarchy equations, 341 1B0rei sets, 12

~antor

dust, 15 fractIOnal denvatIve, 246, 248 ~ole-Cole exponent, 359 Eomplete memory, 395 Fompletely positrve map, 469 Fompletely positive superoperator, 474 ~oulomb's law, 96 ~une-von Schweldler law, 364 ~aputo

f:!enslty of states, 22, 31, 42, 50, 90 ~imensional regularization, 27 dipole moment, 104 f:!lsslpatIve standard map, 412

electric susceptibility, 361

1504

Indexl Mitag-Leffler function, 2351 Mittag-Leffler functIOn, 257, 366, 368, 392 moment of inertia, 751

actional quantum Markovian equation, 472 ractional reaction-diffusion equation, 187, 189 ractional semigroup, 459 ractional stability, 328 ractional Stokes' formula 258 ractional variation 317 321 326 328 ractional Vlasov e uation, 344 ifractional volume integral, 254 [ractional von Neumann equation, 477 ifractional wave equation, 287, 370, 372 [undamental theorem of fractional calculus, 1247

pmzburg-Landau equation, 115, 117, 215 priinwald-Letnikov fractional derivative, 201 riinwa - etm ov- iesz mteraction, 208 radient system, 294 reen's formula, 242, 254

Navier-Stokes equations, 63 nearest-neighbor interaction, 18 no-cloning theorem, 482 nonholonomic constraint, 383

Poincare theorem, 26§ Pomcare-Cartan I-form, 3161 power-like memory, 3951 PSI-senes, 23q

~adrupole

moment, 1051 quantum self -reproducmg, 4821

R

IHOIder conditIOn, 13, 15 Hamiitoman system, 3051 ausdorff dimension, 14 ausdorff measure, II eaviside function, 364, 395 elmholtz conditions, 308 odge star operator, 280

Ie, 317, 321, 32

nteractIon representatIon, isometry, 13

T Tchetaev condition 38 transform operatIon, 16~ onent,359

liouville equation, 147 Liouville fractional integral, 6, 362 [.:ipschitz condition, 13 IIong-range interaction, 181, 184, 187, 190 ~orenz equations, 303

WI

Weierstrass function, 202, 498, 49~ Welerstrass-Mandelbrot functIon, 204 WeyI quantIzatIOn, 493, 49~ Wnght functIOn, 37~

M

ZI

fuagnetohydrodynamics equations, 110 mass dimenSIOn, l':~ ~engersponge, IG

ZasIavsky map, 4131

INonlinear Physical Science I(Series Editors: Albert C.l. Luo, Nail H. Ibragimov)

~aiI. H. Ibragimovl Vladimir. F. Kovalev: Approximate and Renor1 mgroup Symmetries

k\bdul-Majid Wazwaz: Partial Differential Equations and Solitaryl IWaves Theoryl

cr Luo: Discontinuous DynamIcal Systems on Time-varymg

[Albert lDomams

[Anjan Biswasl Daniela Milovicl Matthew Edwards: MathematIcall [Iheory of Dispersion-Managed Optical Solitons Meike Wiedemann! Florian P.M. Kobn /Harald Rosner ! Wolfd gang R.L. Hanke: Self-organization and Pattern-formation in Neu-l lronal Systems under Conditions of Variable Gravityl [Vasily E. Tarasov: Fractional Dynamic§ IVladimir V. Uchaikin: FractIOnal Derivatives m PhYSIC~ k\lbert C.l. Luo: Nonlinear Deformable-body Dynamic§ ~vo

Petras: FractIOnal Order Nonlmear System§

[Albert C.l. Luo I Valentin Afraimovich (Editors): HamIltomalli Chaos Beyond the KAM Theory! k\lbert C.l. Luo I Valentin Afraimovich (Editors): Long-range Inj ~eraction, Stochasticity and Fractional Dynamics k\lbert C. l. Luo I lian-Qiao Sun(Editors): Complex Systems withj fractIOnahty, TIme-delay and SynchromzatIOnl Feckan Michal: BIfurcatIOn and Chaos m Discontinuous and luous Systemsl

Contm~