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ARTICLE IN PRESS

Physica A 383 (2007) 291 - 308 www.elsevier.com/locate/physa

Fractional dynamics of systems with long-range space interaction and temporal memory
Vasily E. Tarasov
a

a,b,

Ó, George M. Zaslavsky

a,c

Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA b Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia c Department of Physics, New York University, 2-4 Washington Place, New York, NY 10003, USA Received 8 January 2007 Available online 27 April 2007

Abstract Field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau and nonlinear Schrodinger equations. As another example, dynamical ? equations for particle chains with power-law interaction and memory are considered in the continuous limit. The obtained fractional equations can be applied to complex media with/without random parameters or processes. r 2007 Elsevier B.V. All rights reserved.
Keywords: Fractional derivatives; Fractional equations; Long-range interaction; Power-law memory

1. Introduction Complex media, with its important applications and underlying microscopic processes, is far from simplicities uniform gases, liquids, or solids. The most typical features of the new physical objects and/or processes are fractality of their structure and intrinsic dynamics or kinetics. Observation of fractality of the basic processes began fairly long ago (see, for review, Refs. [1,2]). Typically the complexity of systems is linked to long-term memory, long-range interactions, non-Markovian kinetics, and particularly Levy-type processes (Levy flights) [3]. The literature on this subject is vast. Let us mention some of the most related references, where the indication of the complexity can lead, in one or another way, to the fractional description of the dynamic and/or kinetic processes with fractional time [4-6]; systems of many coupled elements [7,8]; colloidal aggregates and chemical reaction medium [9]; wave processes [10-12]; porous media [13]; quantum mechanics and quantum field theory [14-16]; plasma physics [17-19]; magnetosphere [20]; random processes and random walks [21-25]; fractional diffusion and Brownian motion [6,26,27]; weak and strong turbulence [11,28,29]; fractional kinetics and chaos theory [30] (see, for review, Refs. [31,32]).
ÓCorresponding author. Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia. Tel.: +7 95 9395989; fax: +7 95 9390397. E-mail address: tarasov@theory.sinp.msu.ru (V.E. Tarasov).

0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.04.050


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It seems that the basic formal tool to be applied is the description of the processes by fractional equations, i.e., by the ones that contain fractional derivatives or integrals [33-36]. The theory of derivatives of noninteger order goes back to Leibnitz, Liouville, Riemann, Grunwald, and Letnikov [33,34]. Derivatives and integrals of fractional order have found many applications in recent studies in physics because of their continually growing numerous applications. Usually, onset of fractional derivatives (integral) is linked to different power-type asymptotic interactions or temporal memories. Depending on what kind of specific features characterize the physical object, the fractional derivative (integrals) can be with respect to time or space coordinate. In the description of particles transport, when the dynamics is chaotic, the fractional derivatives emerge in space and time simultaneously as a natural reflection of scaling properties of the phase space dynamics [30,31]. The diffusion described by the fractional equations is called anomalous. The occurrence of such derivatives could also be related to the space-time decay [37,38], i.e., to pure dynamical processes without kinetics or diffusion. Particularly it was shown in Refs. [39-43] how the long-range interaction between different oscillators can be described by fractional differential equations in the continuous medium limit. Another way to connect the fractional equations with specific dispersion laws of the media was considered in Refs. [10,11,44]. The goal of this paper is to provide a systematic approach to the onset of fractional equations as a result of existence of long-range interaction in a corresponding space and long-range time memory in the system of fields or particles depending on what kind of physical objects are considered. The notions of long-terms memory or interaction can be exactly specified by power laws in time for a memory function and power-law interaction between different elements of the medium. It is of importance to understand the conditions when the fractional derivatives (integrals) occur since it allows us to involve powerful tools of fractional calculus. In Section 2, we consider the variation of the action functional that describes a field with memory and longrange interaction. The long-time memory and long-range interaction can be introduced through power-like kernels of the action functional. The corresponding powers are defined by the exponent a (for space) and b (for time), which in general can be fractional. The Euler-Lagrange equations lead to the equation with fractional ?a; bî-derivatives. In Section 3, the obtained results are used for derivation of ?a; bî-generalization of the Ginzburg-Landau and nonlinear Schrodinger equations (NSES). In Section 4, we consider chains of particles ? with long-range interaction and memory function. Applying the results of Section 2, we derive the continuous limit of the particle dynamics equations. In two Appendices, we provide some brief information on the Riemann-Liouville, Caputo and Riesz fractional derivatives used in paper, and n-dimensional generalization of the final fractional equations. 2. Action functional and its variation 2.1. Action functional Let us define the action functional as Z Z 1 1 2 2 S Í u Ì dx d y qt u?xîg0 ?x; yîqt0 u?yî? qr u?xîg1 ?x; yîqr0 u?yîÐ V ?u?xî; u?yîî . 2 2 R R

(1)

Here x Ì ?t; rî, t is time, r is coordinate, and y Ì ?t0 ; r0 î. The integration is carried out over a region R of the two-dimensional space R2 to which x belong. The field u?xî is defined in a two-dimensional region R of R2 . We assume that u?xî has partial derivatives qu? t ; r î qu?t; rî ; q r u ? xî Ì , qt qr which are smooth functions with respect to time and coordinate. Here are three examples of this action. (a) If qt u?xî Ì g0 ?x; yî Ì Ðg1 ?x; yî Ì d?x Ð yî, V ?u?xî; u?yîî Ì V ?u?xîîd?x Ð yî, (2)


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then we get the usual action Z 1 1 d2 x Íqt u?xî2 Ð Íqr u?xî2 Ð V ?u?xîî . S Í u Ì 2 2 R (b) If g0 ?x; yî Ì Ðg1 ?x; yî Ì d?x Ð yîC 1 ?D; rî, V ?u?xî; u?yîî Ì V ?u?xîîd?x Ð yîC 1 ?D; rî, where C 1 ?D; rî Ì jrjDÐ1 G?Dî ?0oDo1î,

then we obtain Z Z 1 1 S Í u Ì dt dl D Íqt u?xî2 Ð Íqr u?xî2 Ð V ?u?xîî , 2 2 where dl D Ì C 1 ?D; rî dr. This action defines the field u?xî in a medium with the fractional Hausdorff dimension D [45]. (c) If V ?u?xî; u?yîî Ì V ?u?xîîd?x Ð yî, g0 ?x; yî Ì g0 d?r Ð r0 îK0 ?t; t0 î, g1 ?x; yî Ì g1 d?t Ð t0 îK1 ?r; r0 î, then it follows from Eqs. (1) and (3), Z Z Z 1 S Í u Ì g0 dr dt dt0 qt u?t; rîK0 ?t; t0 îqt0 u?t0 ; rî 2 R R R Z Z Z Z Z 1 0 0 0 dt dr dr qr u?t; rîK1 ?r; r îqr0 u?t; r îÐ dt drV ?u?t; rîî, ? g1 2 R R R R R and the time and space dependent kernels are separated in the terms with derivatives. We will be interested in a homogeneous case K1 ?r; r0 î Ì K1 ?r Ð r0 î, and an algebraically decaying kernel K1 with a power tail, i.e., K1 ?lrî Ì ?lî
1Ða

(3)

?4î

K1 ?rî

?1oao2î.

(5)

Similarly, we can consider K0 ?t; t0 î Ì K0 ?t Ð t0 î for 0ot0 ot as a homogeneous function of order 1 Ð b: K0 ?lt0 î Ì l
1Ðb

K0 ?t0 î

?0obo2; 0ot0 otî.

(6)

Relation (5) means that we have power-law long-range interaction in the system. Eq. (6) indicates the memory effects with power-law memory function, which can be regarded as the influence of the environment. Just this case of the power-law dependences of K0 ?tî and K1 ?rî, Eqs. (5) and (6), will be considered to derive the field equations with fractional derivatives.


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2.2. Gateaux differential and variation of action The field equations will be derived by using the Gateaux differential [46-48] of S Íu at the point u?xî, which is defined as the limit d SÍu ? ehÐ SÍu S Íu ? eh , (7) dS Í u; h Ì Ì lim e !0 de e eÌ0 and which exists for fairly smooth integrable functions h?xî Ì du?xî. The Gateaux derivative is slightly different from the Frechet derivative dF SÍu; h, where
khk!0

lim

k S Íu ? hÐ SÍuÐ dF S Íu; hk Ì 0. khk

(8)

The Gateaux derivative is more general concept than Frechet derivative. If a function is Frechet differentiable, it is also Gateaux differentiable, and dS Íu; h is a linear operator. However, not every Gateaux differentiable function is Frechet differentiable. In general, unlike other forms of derivatives, the Gateaux derivative is not linear with respect to h?xî. Action (1) for u ? eh is Z Z 1 S Íu ? eh Ì d2 x d2 y qt ?u?xî? eh?xîîg0 ?x; yîqt0 ?u?yî? eh?yîî 2 R R 1 ? qr ?u?xî? eh?xîîg1 ?x; yîqr0 ?u?yî? eh?yîî Ð V ?u?xî? eh?xî; u?yî? eh?yîî . 2 This expression up to the order e has the form Z Z 1 1 d2 x d2 y qt h?xîg0 ?x; yîqt0 u?yî? qt u?xîg0 ?x; yîqt0 h?yî S Íu ? eh Ì SÍu? e 2 2 R R 1 1 ? qr h?xîg1 ?x; yîqr0 u?yî? qr u?xîg1 ?x; yîqr0 h?yî 2 2 qV ?u?xî; u?yîî qV ?u?xî; u?yîî h?xîÐ h?yî ?Ñ Ñ Ñ . Ð qu?xî qu? yî In the second, fourth and sixth terms of the right-hand side, we change the variables x 2 y. Then Z Z 1 2 2 S Íu ? eh Ì SÍu? e dx d y qt h?xîÍg0 ?x; yî? g0 ?y; xîqt0 u?yî 2 R R 1 qÍV ?u?xî; u?yîî ? V ?u?yî; u?xîî ? qr h?xîÍg1 ?x; yî? g1 ?y; xîqr0 u?yîÐ h?xî ? ÑÑÑ . 2 qu?xî It is convenient to introduce the functions K 0 ?x; yî Ì 1Íg0 ?x; yî? g0 ?y; xî, 2 K 1 ?x; yî Ì 1Íg1 ?x; yî? g1 ?y; xî, 2 U ?u?xî; u?yîî Ì V ?u?xî; u?yîî ? V ?u?yî; u?xîî. Then the variation of action is S Íu ? ehÐ S Íu dS Íu; h Ì lim e!0 e Z Z qU ?u?xî; u?yîî 2 2 Ì h?xî . dx d y qt h?xîK 0 ?x; yîqt0 u?yî? qr h?xîK 1 ?x; yîqr0 u?yîÐ qu?xî R R Using the relations qt h?xîK 0 ?x; yîqt0 u?yî Ì qt Íh?xîK 0 ?x; yîqt0 u?yî Ð qt ÍK 0 ?x; yîqt0 u?yîh?xî, (9)


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qr h?xîK 1 ?x; yîqr0 u?yî Ì qr Íh?xîK 1 ?x; yîqr0 u?yî Ð qr ÍK 1 ?x; yîqr0 u?yîh?xî, qt qt0 u?yî Ì qr qr0 u?yî Ì 0, and the boundary condition Íh?yî we get Z dS Í u; h Ì
R qR

Ì 0, Z
R

d xh?xî

2

qU ?u?xî; u?yîî d y Ðqt ÍK 0 ?x; yîqt0 u?yîÐ qr ÍK 1 ?x; yîqr0 u?yîÐ . qu?xî
2

(10)

For the symmetric potential U ?u?xî; u?yîî Ì U ?u?xîîd?x Ð yî, Eq. (10) transforms into Z Z Z qU ?u?xîî 2 2 2 dS Í u; h Ì Ð d xh?xî d y qt K 0 ?x; yî qt0 u?yî? d y qr K 1 ?x; yîqr0 u?yî? . qu?xî R R R The dynamical equation follows from the stationary action principle dS Í u; h Ì 0 for any h. The field u Ì u?xî, which leads to a minimum or saddle values of S Íu, describes the space-time evolution. For action (1), the stationary principle gives Z Z qU ?u?xîî Ì 0. (12) d2 y qt K 0 ?x; yî qt0 u?yî? d2 y qr K 1 ?x; yî qr0 u?yî? qu?xî R R It is an integro-differential equation, which allows us to derive field equations for different cases of the kernels K 0 ?x; yî and K 1 ?x; yî. 2.3. Special cases Let us consider here two special cases: (a) system without memory and with local interaction in space, (b) field with power-law memory and long-range interaction. (a) In absence of memory and for local interaction kernels (9) are defined at the only instant t and point r, i.e., K 0 ?x; yî Ì g0 d?x Ð yî; K 1 ?x; yî Ì g1 d?x Ð yî qU ?u?t; rîî Ì 0. qu?t; rî

(11)

with some constants g0 and g1 . Then Eq. (12) gives g0 q2 u?t; rî? g1 q2 u?t; rî? t r For g0 Ì 1, g1 Ì Ð1, and U ?u?t; rîî Ì Ð cos u?t; rî, we get the sine-Gordon equation q2 u?t; rîÐ q2 u?t; rî? sin u?t; rî Ì 0. t r (13)

(b) In this example, we show how time and space variables can be separated leaving a possibility to consider the system with power-law memory and long-range interaction. Let K 0 ?x; yî and K 1 ?x; yî have the form K 0 ?x; yî Ì d?r Ð r0 îK0 ?t; t0 î, K 1 ?x; yî Ì d?t Ð t0 îK1 ?r; r0 î, (14) (15)


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where x Ì ?t; rî, and y Ì ?t0 ; r0 î. Then Eq. (12) can be presented as Zt ?t; rî? Z r ?t; rî? where Z Z t ? t ; rî Ì
Ð1 ?1

qU ?u?t; rîî Ì 0, qu?t; rî

(16)

dt0 qt K0 ?t; t0 î qt0 u?t0 ; rî,
?1

(17)

Z Zr ?t; rî Ì

dr0 qr K1 ?r; r0 î

Ð1

qu? t ; r 0 î qr 0

(18)

with separated spatial and temporal kernels. Till now, the kernels K0 ?t; t0 î and K1 ?r; r0 î were not defined. Their specific choice to present a long-term memory and long-range interaction will be in the next two subsections. 2.4. Power-law memory Consider the kernel qt K0 ?t; t0 î of integral (17) in the interval t0 2 ?0; tî such that ( M?t Ð t0 î; 0ot0 ot; 0 qt K0 ?t; t î Ì 0; t 0 4 t ; t 0 o 0: Then Z Z t ? t ; rî Ì
0 t

(19)

dt0 M?t Ð t0 î qt0 u?t0 ; rî Ì M?tîÓ qt u?t; rî.

(20)

As the result, we have the evolution field equation in which the quantity Z t ?t; rî is related to another quantity qt0 u?t0 ; rî through a memory function M?tî. Eq. (20) is a typical non-Markovian equation obtained in studying of systems coupled to an environment, where environmental degrees of freedom being averaged. For a system without memory, we have M?t Ð t0 î Ì d?t Ð t0 î, and Z Z t ? t ; rî Ì
0 t

(21)

d?t Ð t0 îqt0 u?t0 ; rî dt0 Ì qt u?t; rî,

(22)

i.e., the function Z t ?t; rî is defined by qt u?t; rî at the only current instant t. Consider now the power-like memory function M?t Ð t0 î Ì g0 1 G?1 Ð bî ?t Ð t0 î
b

?0obo1î,

(23)

where g0 is a constant that can be presented as a strength of perturbation induced by the environment, and G?1 Ð bî is the Gamma function. Substitution of Eq. (23) into Eq. (20) gives Zt g0 Z t ? t ; rî Ì ?t Ð t0 îÐb qt0 u?t0 ; rî dt0 Ì g0 C Db u?t; rî ?0obo1î, (24) 0t G?1 Ð bî 0 where C Db is the left fractional Caputo derivative [35,36]. 0t For the kernel qt K0 ?t; t0 î in integral (17) such that (0 M ?t0 Ð tî; tot0 o0; 0 qt K0 ?t; t î Ì 0; t0 40; t0 ot;

(25)


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where M0 ?t0 Ð tî Ì we get Z t ? t ; rî Ì ?Ð1îg00 G?1 Ð bî Z
t 0

?Ð1îg00 1 0 Ð tî G?1 Ð bî ?t

b

?0obo1î,

(26)

q t0 u ? t 0 ; r 0 î ?t0 Ð tî
b

dt0 Ì g

0C 0t

Db u 0

?0obo1î,

(27)

which is the right fractional Caputo derivative [35,36]. In general, the kernel K0 ?t; t0 î can include positive and negative intervals of time. Then 8 0 0 > M?t Ð t î; 0ot ot; < 0 qt K0 ?t; t0 î Ì M ?t0 Ð tî; tot0 o0; > : 0; 0otot0 ; t0 oto0;

(28)

where M?t Ð t0 î and M0 ?t0 Ð tî are defined by Eqs. (23) and (26). Then, we get a linear combination of left and right Caputo derivatives Z t ? t ; rî Ì g
C 00

Db u?t; rî? g t

0C 0t

Db u?t; rî 0

?0obo1î.

(29)

As a result, Eq. (12) consists of fractional time derivatives, and it will be written in Section 2.6. We also will be interested in the case when ( M?t Ð t0 î; 0ot0 ot; 0 K0 ?t; t î Ì 0; t0 4t; t0 o0 or 8 0 > M?t Ð t î; < 00 K0 ?t; t0 î Ì M ?t Ð tî; > : 0; 0ot0 ot; tot0 o0; 0otot0 ; t0 oto0

(30)

(31)

(compare to Eqs. (19) and (25)), with the functions M, M0 as in Eqs. (23) and (26). Substitution of Eqs. (19) and (25) into Eq. (17), and integration by parts gives, similarly to Eqs. (24) and (29), Z t ? t ; rî Ì g or Z t ? t ; rî Ì g
C 00 C 00

D

b?1 t

u?t; rî ?0obo1î
0C 0t b?1 0

(32) ?0obo1î

D

b?1 t

u?t; rî? g

D

u?t; rî

(33)

with the same Eq. (16). Depending on different kernels (19), (25) and (30), (31), we obtain field equations with different order of time derivatives (see in Section 2.6). The Caputo fractional derivatives can be linked to fractional powers of variable s for the corresponding Laplace-transformed equation. It is known [35,49,50] that the Laplace transform of the Caputo fractional derivative is Z1 mÐ 1 X eÐst ÍC Db u?t; rî dt Ì sb v?s; rîÐ sbÐqÐ1 u?qî ?0; rî, (34) 0t
0 sÌ0

where m Ð 1obpm, qq u? t ; r î , qt q and v?s; rî is the Laplace transform of u?t; rî: Z1 eÐst u?t; rî dt. v?s; rî Ì u?qî ?t; rî Ì
0

(35)


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Note that formula (34) involves the initial conditions u?qî ?0; rî as integer derivatives u?qî ?t; rî with respect to time. Therefore we can put the initial conditions in a usual way. The functions u?t; rî satisfy the condition Z1 eÐst ju?t; rîjo1. (36)
0

For 0obp1, Eq. (34) has the form Z1 eÐst ÍC Db u?t; rî dt Ì sb v?s; rîÐ s 0t
0

bÐ1

u?0; rî.

(37)

Inversion of Eq. (37) gives Z 1 Cb Dt u?t; rî Ì est Ísb v?s; rîÐ s 0 2pi Br

bÐ1

u?0; rî ds,

(38)

where Br denotes the Bromwich contour. The final equation of u?t; rî will be written in Section 2.6. 2.5. Nonlocal interaction Consider the kernel K1 ?r; r0 î of integral (18) as K1 ?r; r0 î Ì C?jr Ð r0 jî Ì Ð g1 1 cos?pa=2îG?2 Ð aî jr Ð r0 j
aÐ1

?1oao2î

(39)

that describes the power-law interaction. Then, we obtain Z ?1 q2 u? t ; r 0 î qa dr0 C?jr Ð r0 jî Ì g1 u? t ; r î , Zr ?t; rî Ì qr 0 2 qjrja Ð1

(40)

where the fractional Riesz derivative with respect to coordinates is introduced [33,36] (see also Appendix A). The connection between the Riesz fractional derivative and its Fourier transform is known [33] F: qa Ð! Ð jkja , qjrja Z
?1

(41)

where F is defined by f~?kî Ì ?F f î?kî Ì and F
Ð1

f ? rî e
Ð1

Ðikr

dr,

(42)

is an inverse Fourier transform Z 1 ?1 ~ f ?kî eikr dk. f ?rî Ì ?FÐ1 f~î?rî Ì 2p Ð1

(43)

The fractional Riesz derivatives describe properties of fractal media or complex media with fractional dispersion law (see for example in Ref. [44]). 2.6. Field equations with fractional derivatives Substitution of Eqs. (29) and (39) into Eq. (16) gives the fractional field equation g
C 00

Db u?t; rî? g t

0C 0t

Db u?t; rî? g 0

1

qa qU ?u?t; rîî Ì 0 ?1oao2; 0obo1î. u?t; rî? qu?t; rî qjrja

(44)

This equation describes the field of the system with power-law memory and long-range interaction. Depending on the situation, g0 or g00 could be zero or not. For example, the potential a b U ?u?t; rîî Ì u2 ?t; rî? u4 ?t; rî 2 4


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in Eq. (44) gives the fractional time-dependent generalization of the Ginzburg-Landau equation (GLE) g
C 00

Db u?t; rî? g t

0C 0t

Db u?t; rî? g 0

1

qa u?t; rî? au?t; rî? bu3 ?t; rî Ì 0 qjrja

?1oao2; 0obo1î.

(45)

In the case of the time kernel (31), Eq. (44) is replaced by g
C 00 b Dt ?1 u?t; rî? g 0C 0t

D

b?1 0

u?t; rî? g

1

qa qU ?u?t; rîî Ì 0 ?1oao2; 0obo1î. u?t; rî? qu?t; rî qjrja

(46)

This equation has increased by one the order of time derivative and can be applied to the wave propagation in media with fractional dispersion law. Particularly in the case when only the right time derivative should be used (one-directional wave propagation) and the potential is U ?u?t; rîî Ì Ð cos u?t; rî, Eq. (46) gives the fractional sine-Gordon equation
C 0

D

b?1 t

u?t; rîÐ

qa u?t; rî? sin u?t; rî Ì 0 ?1oao2; 0obo1î, qjrja

(47)

where we put g0 Ì 1, g00 Ì 0 and g1 Ì Ð1, and which is a generalization of Eq. (13) for noninteger derivatives with respect to time and coordinate. Finally, let us simplify the notation and write down Eqs. (44) or (46) as g qb qa u?t; rî? g1 u?t; rî? U 0 ?u?t; rîî Ì 0, qt b qjrja (48)

where qb =qtb stays for left, right, or both Caputo derivatives (in the latter case, the constant g can be different for different derivatives), and 0obo2, 0oao2, and U 0 ?uî Ì qU =qu. Let us comment that the choice of the derivative qb =qtb depends on the type of initial conditions and the processes, and other than Caputo derivative can appear. In Appendix B, we present a generalization of Eqs. (43) and (46) for the n-dimensional coordinate case. 3. Fractional GLE Since the variable x in Eq. (1) is not specified, one can apply a similar technique to other problems, defined by the extremum of a functional with long-range interaction. As an example, consider a free energy functional for a model of Ginzburg-Landau equation (GLE) that consists of long-range interaction. The fractional generalization of the Ginzburg-Landau equation (FGLE) was suggested in Ref. [44]. This equation can be used to describe the processes in complex media [51,52]. Some properties of FGLE are discussed in Refs. [40,53,54]. It is known [55] that the stationary GLE gDu Ð au Ð bu3 Ì 0 can be derived as the variational Euler-Lagrange equation dF Í u Ì0 du? r î for the free energy functional ! Z 1 b4 2 2 F Í u Ì F 0 ? g?quî ? au ? u dr, 2R 2 (49)

(50)

where qu Ì qu?rî=qr, and the integration is over a region R. Here F 0 is a free energy of the normal state, i.e., F Íu for u Ì 0.


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Consider the thermodynamic potential (free energy functional) F Íu for the nonequilibrium state of a medium with power-law nonlocal interaction. The generalized free energy functional has the form Z Z F Í u Ì F 0 ? dr dr0 F?u?rî; u?r0 î; qu?rî; qu?r0 îî, (51)
R R

where the generalized density of free energy 1 a2 b F?u?rî; u?r0 î; qu?rî; qu?r0 îî Ì gK1 ?r; r0 îqk u?rîqk0 u?r0 î? u ?rî? u4 ?rî d?r Ð r0 î r r 2 2 4 has the kernel K1 ?r; r0 î defined as in Eq. (39). The variational equation (49) gives g qa u?rî? au?rî? bu3 ?rî Ì 0 qjrja ?1oao2î, (53) (52)

which can be called the a-FGLE. This equation can be easily generalized for the three-dimensional variable r (see Appendix B). In the nonstationary case, Eq. (49) should be replaced by qu?t; rî dF Í u Ì qt du? t ; r î (54)

(see Ref. [56]), where there is no explicit time memory effects. To put such memory into Eq. (54), we can write Zt qu?t0 ; rî dF Íu?t; rî , (55) dt0 M?t Ð t0 î Ì qt 0 du?t; rî 0 and to assume for M?t Ð t0 î the power-law form of Eq. (23). Then we arrive to a nonstationary generalization of ?a; bî-FGLE qb u? t ; r î qa Ìg u?t; rî? au?t; rî? bu3 ?t; rî ?0obo1; 1oao2î, qt b qjrja (56)

where qb =qtb is used for Caputo derivative while any other fractional derivative can be applied by modifying the memory kernel M?tî, and initial conditions. It is worthwhile to compare Eq. (56) to its counterpart NSE i qu Ì gDu ? au ? bjuj2 u qt (57)

with u Ì u?t; rî, and complex a, b. In the case of D? instead of D, Eq. (57) also known as parabolic equation for wave propagation. Generalization of Eq. (57) for the case of fractional space derivative and nonlocal interaction (a-NLS) was considered in Refs. [44,54]: i qu Ì Ðg?ÐDî qt
a=2 a=2

u ? au ? bjuj2 u

?1oao2î,

(58)

where the fractional Laplacian is defined through the Fourier transform and Riesz derivatives [33]: F: ?ÐDî Ð k2 îa=2 . !? (59)

Similar to Eq. (56) generalized ?a; bî-NLS equation has the form qb u Ì Ðg?ÐDî qt b
a=2

u ? au ? bjuj2 u,

(60)

where qb =qtb is now Riemann-Liouville derivative with the Fourier transform F: qb Ð ioîb , !? qtb (61)

and the memory function is working through the parameter b.


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It is convenient also to interpret Eq. (60) through the nonlinear dispersion law by applying to Eq. (60) Fourier transform in both time and space. Then it gives with the help of Eqs. (59) and (61) ?ioîb Ì Ðg?k2 î
a=2

? a ? bjuj,

(62)

which was derived for b Ì 1in Refs. [44,54]. Onset of fractional time derivative in Eq. (60) can stop self-focusing of waves, steepening of the solution, developing of a singularity. These phenomena need a special analysis. Eq. (60) can be easily generalized for the anisotropic case qb u Ì Ðg? ?ÐD? îa? =2 u Ð gk ?ÐDk îak =2 u ? au ? bjuj2 u (63) qt b with a corresponding anisotropic dispersion equation instead of Eq. (62) (see also Refs. [44,54] for b Ì 1). 4. Discrete system with memory and long-range interaction 4.1. Equation for discrete chains In this section, we show how the obtained results of Section 2 can be applied to discrete systems, for example chains of interacting particles. Long-range interaction has been a subject of a great interest for a long time. Thermodynamics of a model of classical spins with long-range interactions has been considered in Refs. [57-61]. The long-range interactions have been widely studied in discrete systems of lattices as well as in their continuous analogues: solitons in onedimensional lattice with the Lennard-Jones-type interaction [62]; kinks in the Frenkel-Kontorova model [63]; time periodic spatially localized solutions (breathers) [64,65]; energy and decay properties of discrete breathers in the framework of the Klein-Gordon equation [66], and discrete NSEs [18]. A remarkable property of the dynamics described by the equation with fractional space derivatives is that the solutions have power-like tails. Similar features were observed in the lattice models with power-like long-range interactions [41,64,65,67-69]. Long-range interaction can be relevant to the systems such as neuron populations [70] and Josephson junctions [71]. The synchronization of chaotic systems with power-law long-range interactions were considered in Refs. [40,71,72]. A model of coupled map lattices with coupling that decays in a power law was considered in Refs. [72-75].Notea power-law decay of structure factor for geometry of colloids aggregates [9]. It will be shown how long-range coupling of particles and memory function with power tails can reveal a new type of particle equations with fractional derivatives and the connection of these equations to their continuous media counterpart. Consider an one-dimensional chain of interacting oscillators that can be described by the action Z ?1 Z ?1 ?1 X _ _ S Í u Ì dt dt0 L?un ?tî; un ?t0 î; un ?tî; un ?t0 îî, (64)
Ð1 Ð1 nÌÐ1

where un are displacements of the oscillators from the equilibrium and L is a Lagrangian. If _ _ _ L?un ?tî; un ?t0 î; un ?tî; un ?t0 îî Ì L?un ?tî; un ?tîî d?t Ð t0 î, then we have the chain without memory. Let us introduce a generalization of Eq. (64) with the action 0 1 ! Z ?1 Z ?1 ?1 ?1 X1 X _ _ K0 ?t; t0 îun ?tîun ?t0 îÐ V ?un ?tî; un ?t0 îî Ð S Í un Ì dt dt0 @ U ?un ?tî; um ?t0 îîA. 2 n;mÌÐ1 Ð1 Ð1 nÌÐ1
ma n

(65) In the same way as in Section 2, let us separate the kinetic energy from the long-range interaction and potential parts: U ?un ?tî; um ?t0 îî Ì 1g0 J a ?jn Ð mjî ?un ?tîÐ um ?tîî2 d?t Ð t0 î, 4 V ?un ?tî; un ?t0 îî Ì V ?un ?tîî d?t Ð t0 î. (66) (67)


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Note that Eqs. (67) and (66) in action (64) are equivalent to U ?un ?tî; um ?t0 îî Ì Ð1 g0 J a ?jn Ð mjî un ?tîum ?tî d?t Ð t0 î, 2 ~ V ?un ?tî; un ?t0 îî Ì ?V ?un ?tîî ? 1gu2 ?tîî d?t Ð t0 î, 2n where ~ gÌg
0 ma0

(68) (69)

X

J a ?jmjî.

The second term in the right-hand side of Eq. (69) removes the infinity of the interaction Eq. (68) in the continuous medium limit. The interparticle interaction J a ?jn Ð mjî in Eq. (66) is defined by J a ?jn Ð mjî Ì 1 jn Ð mja
?1

? a4 0î .

(70)

Some other examples of functions J a ?nî can be found in Ref. [43]. Using Eqs. (19) and (23), for the kernel K0 ?t; t0 î in Eq. (65), i.e., 8 g0 < ?t Ð t0 îÐb ; 0ot0 ot ?0obo1î; 0 qt K0 ?t; t î Ì G?1 Ð bî : 0; t0 4t; t0 o0; we obtain the corresponding Euler-Lagrange equations
C 0 ?1 X 0
mÌÐ1 man

(71)

Db un ?tî? g t

J a ?jn Ð mjî Íum ?tîÐ un ?tî ? F ?un ?tîî Ì 0,

(72)

where F ?uî Ì qV ?uî=qu. A continuous limit of Eq. (72) can be defined by a transform operation from un ?tî to u?x; tî [39-43]. First, ^ define un ?tî as Fourier coefficients of some function u?k; tî, k 2 ÍÐK =2; K =2, i.e., ^ u?t; kî Ì
?1 X nÌÐ1

un ?tî e

Ðikxn

Ì FD fun ?tîg,

(73)

where xn Ì nDx, and Dx Ì 2p=K is a distance between nearest particles in the chain, and Z 1 ?K =2 ^ un ?tî Ì dk u?t; kî eikxn Ì FÐ1 fu?t; kîg. D^ K ÐK =2 Secondly, in the limit Dx ! 0 (K ! 1) replace un ?tî Ì ?2p=K îu?xn ; tî ! u?x; tî dx, xn Ì nDx Ì 2pn=K ! x. In this limit, Eqs. (73) and (74) are transformed into the integrals Z ?1 ~ dx eÐikx u?t; xî Ì Ffu?t; xîg Ì lim FD fun ?tîg, u?t; kî Ì
Ð1 Dx ! 0

(74) and

(75)

u?t; xî Ì

1 2p

Z

?1

~ ~ dk eikx u?t; kî Ì FÐ1 fu?t; kîg Ì lim FÐ1 fu?t; kîg. D^
Dx!0

(76)

Ð1

Applying Eq. (73) to Eq. (72) and performing limit (75), we obtain q b u ? t ; xî qa u?t; xî ? ga ? F ?u?t; xîî Ì 0 qtb qjxja where ?0obo2; 1oao2î, (77)

pa (78) 2 is the renormalized constant. The Caputo time derivative is written in a simplified form qb =qtb , and the value of b depends on the choice of memory function. Eq. (77) can be generalized to a nonlinear long-range ga Ì 2g0 ?Dxîa G?Ðaî cos


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interaction. Consider, instead of Eq. (72),
C 0

D b un ? g t

?1 X 0
mÌÐ1 man

J a ?jn Ð mjî Íf ?um îÐ f ?un î ? F ?un î Ì 0,

(79)

where f ?uî is a function of u. For example, f ?uî Ì u2 or f ?uî Ì u Ð gu2 . Then the corresponding continuous limit for the same J a ?jn Ð mjî as Eq. (70) leads to the time-space fractional equation qb u?t; xî qa ? ga f ?u?t; xîî ? F ?u?t; xîî Ì 0 ?0obo2; 1oao2î. qtb qjxja (80)

Eqs. (77) and (80) for b Ì 1 were considered in Refs. [39-43]. Generalization to 0obo2 significantly extends the area of their applications. A physical motivation is that a dynamical process typically reveals fractional features simultaneously in space and time. Such situation just was considered in chaotic dynamics [30,31]. Now we have such a possibility far beyond the fractional kinetics. An evident generalization of Eq. (80) is for the interparticle interactions with two or more different kernels. For example one can consider regular terms without long memory together with a term with long memory: qb u?t; xî qs u?t; xî qa u?t; xî ? gs ? ga ? F ?u?t; xîî Ì 0 qtb qjxjs qjxja with some gs and ga and integer s. 5. Conclusion Starting from a variation of the action functional, we consider different type of kernels that define the character of particle interaction and the influence of an environment on the memory function. The main stress is on the long-range interaction and memory that occur in complex media. For the case when the interaction or memory function have power-law structure the system can be described by the equation of motion with fractional derivatives qb =qtb and qa =qjxja depending on the power of interaction and memory function. We have discussed how different types of the derivatives and possible values ?a; bî may occur with respect to the type of memory and interaction. The final equations of motions can be considered as a new kind of tool to study dynamics with space-time distributed interactions. A number of examples of such kind of systems can be found in the reviews [1,31] related to random or chaotic processes. The study of this paper shows that the list of possible applications of fractional equations can be naturally expanded to include nonchaotic and nonrandom dynamics as well. Acknowledgements Authors thank M.F. Shlesinger for valuable remarks. This work was supported by the Office of Naval Research, Grant No. N00014-02-1-0056, and the NSF Grant No. DMS-0417800. Appendix A. Fractional derivatives The fractional derivative has different definitions [33,34], and exploiting any of them depends on the kind of the problems, initial (boundary) conditions, and the specifics of the considered physical processes. The classical definition is the so-called Riemann-Liouville derivative [33,34]. The left and right Riemann-Liouville derivatives for an interval Ía; b are defined by Zx 1 qn u?zî dz a , a Dt u?xî Ì n G?n Ð aî qx a ?x Ð zîaÐn?1 ?0obo2; 1oao2î (81)


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t

Da u?xî Ì b

?Ð1în qn G?n Ð aî qxn

Z
x

b

u?zî dz ?z Ð xîaÐn

?1

,

(82)

where n Ð 1oaon. End-points a, b can be extended to Ð1, 1 if the integral exists. Due to reasons, concerning the initial conditions, it is more convenient to use the Caputo fractional derivatives [35]. Its main advantage is that the initial conditions take the same form as for integer-order differential equations. The Caputo fractional derivatives are Z x ?nî 1 u ?zî dz Ca , a Dx u?xî Ì G?n Ð aî a ?x Ð zîaÐn?1
C x

?Ð1în D u?xî Ì G?n Ð aî
a b

Z
x

b

u?nî ?zî dz , ?z Ð xîaÐn?1

(83)

where u?nî ?zî Ì dn u?zî=dzn , and n Ð 1oaon. The Caputo fractional derivatives can be defined through the Riemann-Liouville derivatives [36] by ! nÐ1 X ? x Ð aî k Ca a u ?k î ? a î , a Dx u?xî Ì a Dx u?xîÐ k! kÌ0
C x

D u?xî Ì

a b

x

D

a b

! nÐ1 X ?b Ð xîk ?k î u ? bî , u?xîÐ k! k Ì0

(84)

where n Ð 1oaon. These equations give
a

Da u?xî Ì x

C a

Da u?xî? x

nÐ1 X ?x Ð aîkÐa u ?k î ? a î , G?k Ð a ? 1î kÌ0 nÐ1 X ?b Ð xîkÐa u?kî ?bî. G?k Ð a ? 1î kÌ0

(85)

x

Da u?xî Ì b

C x

Da u?xî? b

(86)

The Riesz fractional derivative of order a are Ða Ñ qa 1 D u?xî? Da u?xî , u?xî Ì Ð Ð 2 cos?pa=2î ? qjxja where aa1; 3; 5 .. . ; and Da are Riemann-Liouville fractional derivatives with infinite limits: Ö Zx 1 qn u?zî dz Da u?xî Ì , ? n G?n Ð aî qx Ð1 ?x Ð zîaÐn?1 Da u?xî Ì Ð ?Ð1în qn G?n Ð aî qxn Z
x 1

(87)

u?zî dz ?z Ð xîaÐn

?1

.

(88)

Substitution of Eqs. (88) into Eq. (87) gives Z x qa Ð1 qn u?zî dz a u?xî Ì aÐn n 2 cos?pa=2îG?n Ð aî qx qjxj Ð1 ?x Ð zî

Z
?1

?1 x

?

?Ð1în u?zî dz . ?z Ð xîaÐn?1

(89)

The Fourier transform of the fractional derivatives [33,36] is a q ~ u?xî ?kî Ì Ðjkja u?kî, F qjxja ~ F?Da u?xîî?kî Ì ?Öikîa u?kî, Ö

(90) (91)


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where F is defined by ~ u?kî Ì ?Fuî?kî Ì

Z

?1

u?xî e
Ð1

Ðikx

dx.

(92)

The inverse Fourier transform is Z 1 ?1 Ð1 ~ ~ u?xî Ì ?F uî?xî Ì u? k î e 2p Ð1 Appendix B. n-Dimensional case

ikx

dk.

(93)

The generalization of action (1) for the case r 2 Rn ,where x Ì ?t; rî,and r Ì ?x1 ; .. . ; xn î,gives thefieldequation g
C 00

Db u?t; rî? g t

0C 0t

Db u?t; rî? 0

n X k Ì1

g

k

qa qU ?u?t; rîî Ì 0. u?t; rî? qu? t ; r î qjxk jak

(94)

For the case r 2 Rn , there exists the other possibility to define the kernels M t ?x; yî and M r ?x; yîÌ fM k ?x; yî; k Ì 1; .. . ; ng. We can consider Z n X qu?t; r0 î Z r ?t; rî Ì dn r0 M k ? r; r0 î , (95) qx0k Rn kÌ1 where M k ?r; r0 î is Riesz kernel [33]: 1 M k ?r; r0 îÌ K ak ?r Ð r0 î Ì g n ? ak î Here ak 40 (ak an; n ? 2; n ? 4; .. . n 8 > 2a pn=2 G?a=2î=G > > < gn ?aîÌ 1; > > > ?Ð1î?nÐaî=2 2aÐ1 pn=2 : ( j r Ð r0 j
ak Ðn

; ln jr Ð r j;
0

ak Ð na0; 2; 4; .. . ; ak Ð n Ì 0; 2; 4; ... :

Ðjr Ð r j

0 ak Ð n

), and Ð a ; 2 h a Ð ni G?a=2î !; 2

aan ? 2k; na Ð 2k; n ÌÐ2k; aan ? 2k: (96)

Note that the multivariable Riesz integral Z 1 u?t; r0 î dr0 ?I a uî?t; rî Ì , gn ?aî Rn jr Ð r0 jnÐa where a40, can be presented as convolution: Z a K a ?r Ð r0 î u?t; r0 î dn r0 , ?I uî?t; rî Ì
R
n

(97)

(98)

with the Riesz kernel K a ?rî. It allows us to write Eq. (95) as Z r ?t; rî Ì
n X kÌ1

I

a

k

qu? t ; r î . qxk

(99)

The fractional Riesz integrals of orders ak (k Ì 1; .. . ; n) in the field equations describe the fractal media. Then the field equation is g
C 00

Db u?t; rî? g t

0C 0t

Db u?t; rî? 0

n X k Ì1

I

a

k

qu?t; rî qU ?u?t; rîî Ì 0. ? qxk qu?t; rî

(100)

If M k ?r; r0 î in Eq. (95) is an operator such that M k ?r; r0 îu?t; r0 î Ì g
k

1 ?Dlr0 uî?t; rî , d n;l ?ak î jr0 jn?ak

(101)


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where d n;l ?aîÌ
l X ?Ð1îkÐ1 l ! 2Ða p1?n=2 k G?1 ? a=2îG??n ? aî=2î sin?ap=2î kÌ0 ?l Ð kî!k! a

is normalized multiplier [36], and ?Dlr0 uî?t; rîÌ
l X kÌ0

?Ð1î

kÐ1

l! u?t; r Ð kr0 î ?l Ð kî!k!

is symmetrized difference [36], then Zr ?t; rîÌ
n X kÌ1

g

k

qak qu?t; rî . qjrjak qxk

As a result, we have g
C 00

Db u?t; rî? g t

0C 0t

Db u?t; rî? g 0

k

n X qak qu?t; rî qU ?u?t; rîî Ì 0, ? qu?t; rî qjrjak qxk k Ì1

(102)

which is the field equations with n fractional Riesz multivariable derivatives. References
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