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

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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

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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 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 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)


18

Iwhere a

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

] 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


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 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:


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
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


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


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, 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.


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


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)


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


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~


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_ 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


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


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


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


= 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


~.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


(2.26)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


[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


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


~.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,


[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


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


[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


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


[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


~(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


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


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
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


(5.23)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


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


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


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


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.


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


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


!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


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


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


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)

VL\ {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

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

0 are defined (KUbas et al., 2006) byl

~erivatives aDr;

x>a,1 x -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


(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


[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 0, 0 < IAal < 00, and!

1178


[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 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)),

02,

(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


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 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


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 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


[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


~.

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


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


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


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


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 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


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
. 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

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


~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

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


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


(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

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


~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


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 ~.


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 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


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~


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; = 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

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


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
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


(15.14 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


[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


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


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


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

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 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 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 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

= 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 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


[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 {

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 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


~(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 = 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


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


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


[.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


~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


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


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


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 ~.


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

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