Fractional Fokker–Planck equation

Chaos, Solitons and Fractals 11 (2000) 791±798

www.elsevier.nl/locate/chaos

Fractional Fokker±Planck equation S.A. El-Wakil *, M.A. Zahran Department of Physics, Faculty of Science, University of Mansoura, Mansoura, Egypt Accepted 11 August 1998

Abstract By using the de®nition of the characteristic function and Kramers±Moyal Forward expansion, one can obtain the Fractional Fokker±Planck Equation (FFPE) in the domain of fractal time evolution with a critical exponent a (0 < a 6 1). Two di€erent classes of fractional di€erential operators, Liouville±Riemann (L±R) and Nishimoto (N) are used to represent the fractal di€erential operators in time. By applying the technique of eigenfunction expansion to get the solution of FFPE, one ®nds that the time part of eigenfunction expansion in terms of L±R represents the waiting time density W…t†, which gives the relation between fractal time evolution and the theory of continuous time random walk (CTRW). From the principle of maximum entropy, the structure of the distribution function can be known. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction Di€usion in disordered systems such as random fractal structure does not follow the classical laws which describe transport in ordered crystalline medium and this leads to many anomalous physical properties [1]. Some typical examples of transport properties in fractured [2], porous rocks [3,4] and the anomalous density of states in randomly diluted magnetic systems [5]. The problem of di€usion in disordered media is a part of the general problem of the transport in disordered media. When dealing with the di€usion or random walks, probability densities are of great importance. Consider for example the probability density q…r; t† of ®nding a random walker at r and time t, having started at the origin at t ˆ 0. This enables one to calculate the average moments, Z …1:1† hrk …t†i ˆ rk q…r; t† dk r; where k is the number of spatial dimensions. In uniform Euclidean systems the mean-square displacement R ˆ hr2 …t†i is proportional to time t, hr2 …t†i  t, for any number of spatial dimensions. However in disordered systems, this law is not valid in general. Rather the di€usion law becomes anomalous [6], hr2 …t†i  t2=dw

…1:2†

with dw P 2. Also the probability density q…r; t† has a non-Gaussian shape [1], u

q…r; t†  tÿdf =dw e‰const:…r=R† Š ;

…1:3†

where r=R  1, t ! 1, u ˆ dw =…dw ÿ 1†, and df is the fractal dimension. In view of the possible general validity of Eq. (1.3), one is tempted to search for a generic approach describing di€usion on disordered

*

Corresponding author.

0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 8 ) 0 0 2 0 5 - 7

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S.A. El-Wakil, M.A. Zahran / Chaos, Solitons and Fractals 11 (2000) 791±798

systems, e.g. fractals. The main purpose of the present paper is to introduce the fractional Fokker± Planck equation (FFPE) in which the time derivative is replaced with a derivative of fractional order …a ˆ 2=dw †. The de®nitions of the characteristic function (CF) and Kramers±Moyal forward expansion are used to derive FFPE [7]. Two classes of fractional integral operators are used to represent the fractional di€erential operator in time i.e. L±R and N-de®nitions. An exact analytical solution for the fractional equation is obtained by employing the technique of eigenfunction expansion. Only with the L±R fractional operator, we ®nd that there exists a precise and rigorous relation between the FFPE and the continuous time random walk CTRW. So the corresponding waiting time density (WTD) W(t) is obtained in closed form in terms of generalized Mitteg±Le€er function. Using the maximum entropy principle, the distribution function shows the same slowing down behavior of Eq. (1.3). Our paper is organized as follows. Section 2 contains the essential of fractional di€erential operators. Next, Section 3 is devoted to the derivation of the FFPE. In Section 4, we apply the eigenfunction technique for FFPE for two di€erent classes of fractional di€erential operators. 2. Fractional de®nitions There are many de®nitions for the generalized di€erintegration operators, which can mainly be divided into two main classes. The ®rst one depends on the so-called Riemann's classic operator de®nition for iterated integral which is the so-called Liouville±Riemann fractional de®nition. The other class depends on the Cauchy theorem of complex integration. We will give the de®nitions of two main classes for fractional integration. 2.1. Liouville±Riemann (L±R) fractional integral operator [8] This de®nition is coming from the generalization of Riemann's classic (integer number) integral transform operator, Z t 1 ÿn D f …t† ˆ ds…t ÿ s†nÿ1 f …s†; …2:1† 0 t C…n† 0 where C…q† is Gamma function and f …t† the continuous over ‰0; 1Š. If n is not an integer, but any number may be still be meaningful and is de®ned for q > 0 as the Liouville±Riemann q 6ˆ 1; 2; 3 . . . ; then 0 Dÿq t fractional operator, Z t 1 ÿq ds…t ÿ s†qÿ1 f …s† …2:2† 0 Dt f …t† ˆ C…q† 0 and L±R de®nition for fractional di€erentiation as 0 Dmt …m > 0† and is given by, m 0 Dt f …t†

ˆ

dn ÿn‡m f …t†; 0 Dt dtn

ÿn ‡ m < 0

…2:3†

i.e. Within the context of the L±R fractional calculus, the operation `fractional di€erentiation' can be decomposed into a `fractional integration' 0 Dtÿn‡m followed by an ordinary di€erentiation, where n is the least positive integer than dn =dtn : 2.2. The class of de®nition depends on Cauchy's theorem [9] Caschy's theorem for the function of single variable is, Z n! f …n† f n …z† ˆ dn n 2 N [ f0g; z 2 D; 2pi c …n ÿ z†n‡1

…2:4†

S.A. El-Wakil, M.A. Zahran / Chaos, Solitons and Fractals 11 (2000) 791±798

793

Fig. 1.

where f …z† is the analytic function in domain D, which is surrounded with a piecewise smooth closed Jordan curve c, in n plan. Nishimoto [9] extends the above formula to de®ne the fractional derivative. If f …z† is an analytic function (regular) function and it has no branch point inside C fC‡ ; Cÿ g and on C, and Z Z ‡ C…m ‡ 1† f …n† C…m ‡ 1† 0 ÿ…m‡1† Cÿ fm …z† ˆ dn ˆ g f …z ‡ g† dg m‡1 2pi 2pi Cÿ …n ÿ z† ÿ1 …n 6ˆ z; 0 6 arg…n ÿ z† 6 p; m 62 zÿ1 †; Z Z ‡ C…m ‡ 1† f …n† C…m ‡ 1† 0 ÿ…m‡1† C‡ fm …z† ˆ dn ˆ g f …z ‡ g† dg m‡1 2pi 2pi C‡ …n ÿ z† ÿ1

…2:5†

…n 6ˆ z; 0 6 arg…n ÿ z† 6 2p; m 62 zÿ1 † fÿn ˆ Cfÿn ˆ LimCfn …n 2 z‡ ; C ˆ fCÿ ; C‡ g†; m!ÿn

where Cÿ and C‡ are integral curves as shown in Fig. 1, (i.e., Cÿ is a curve along the cut joining two points z and ÿ1 ‡ iIm…z†, and C‡ is a curve along the cut joining two points z and 1 ‡ iIm…z†, then fm ˆ fC‡ fm ; C‡ fm g is the fractional derivative of order m of the function f …z†, m 2 R and z 2 C. 3. The derivation of fractional Fokker±Planck equation (FFPE) From the de®nition of the one-dimensional characteristic function (CF) as the Fourier transform of the transition probability density (TPD) [7], Z 1 0 dx eiK…xÿx † P …x; t ‡ s j x0 ; t†; …3:1† P …K; x0 ; t; s† ˆ ÿ1

where K P 0. P …K; t† is known as the intermediate scattering function in the theory of liquids [10], and represented the correlation of density±density ¯uctuation of wave vector in the ¯uid. The above formula can be put in the form of, Z 1 X 1 …iK†n n 0 …x ÿ x0 † P …x; t ‡ sjx0 ; t† dx P …K; x ; t; s† ˆ n! ÿ1 nˆ0 or, P …K; x0 ; t; s† ˆ 1 ‡

n 1 X …iK† Mn …x0 ; t; s†; n! n‡1

where Mn -nth is the moments of the particles which is de®ned by, Z n 0 Mn …x ; t; s† ˆ …x ÿ x0 † P …x; t ‡ sjx0 ; t† dx: Since the CF is the Fourier transform of the TPD and vice versa, the TPD is given by, Z 1 1 0 0 P …x; t ‡ sjx ; t† ˆ eÿiK…xÿx † P …K; x0 ; t; s† dK; 2p ÿ1

…3:2†

…3:3†

…3:4†

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S.A. El-Wakil, M.A. Zahran / Chaos, Solitons and Fractals 11 (2000) 791±798

1 P …x; t ‡ sjx ; t† ˆ 2p 0

Z

1

ÿ1

( e

ÿiK…xÿx0 †

) n 1 X …iK† 0 Mn …x ; t; s† dK: 1‡ n! nˆ1

Use the identities, n  Z 1 1 o n iK…xÿx0 † …iK† e dK ˆ ÿ d…x ÿ x0 † 2p ÿ1 ox

…3:5†

…3:6†

and, d…x ÿ x0 †f …x0 † ˆ d…x ÿ x0 †f …x†: Therefore, Eq. (3.5) yields, " 0

P …x; t ‡ sjx ; t† ˆ 1 ‡

1  X nˆ1

o ÿ ox

…3:7† n

# Mn …x0 ; t; s† 0 d…x ÿ x † : n!

…3:8†

From the de®nition of Kramers±Moyal forward expansion that the probability density q…x; t ‡ s† at time t ‡ s and the probability density at time t are connected by …s P 0†, Z …3:9† q…x; t ‡ s† ˆ P …x; t ‡ sjx0 ; t†q…x; t† dx0 therefore Eq. (3.8) after using the above de®nition (3.9) becomes, # n Z " 1  X o Mn …x; t; s† ÿ d…x ÿ x0 †q…x; t† dx0 q…x; t ‡ s† ˆ 1‡ ox n! n‡1 and Eq. (3.10) upon integration, n 1  X o Mn …x; t; s† ÿ q…x; t ‡ s† ÿ q…x; t† ˆ q…x; t†: ox n! nˆ1

…3:10†

…3:11†

For anomalous stochastic Brownian motion, the increment x…t ‡ s† ÿ x…t† is given by [11] M1 ˆ hx…t ‡ s† ÿ x…t†i ˆ f …x; t†sa

0 < a61

and the mean-square displacement, M2 ˆ h…x…t ‡ s† ÿ x…t††2 i ˆ g…x; t†sa substituting for M1 and M2 into Eq. (3.11), one gets, q…x; t ‡ s† ÿ q…x; t† ˆ ÿ

o 1 o2 f …x; t†sa q…x; t† ‡ g…x; t†sa q…x; t†: ox 2 ox2

…3:12†

Divide Eq. (3.12) by sa and de®ne fq…x; t ‡ s† ÿ q…x; t†g=sa ˆ oat as the fractional di€erential operator with respect to time t, so Eq. (3.12) can represent the fractional time evolution of the FFPE, oat q…x; t† ˆ ÿ

o 1 o2 f …x; t†q…x; t† ‡ g…x; t†q…x; t†: ox 2 ox2

…3:13†

The structure of the probability distribution function by applying the maximum entropy principle takes the form, ! a 2 …x ÿ f s † ÿ1=2 exp ÿ ; …3:14† q…x; t† ˆ …2pgsa † 2gsa

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which may give the same slowing down behavior of Eq. (1.3). This result depends on the explicit form of g…x; t† and f …x; t†. It is well known that fractional Brownian [12], has hx…t ‡ s ÿ x…t†i ˆ 0; h‰x…t ‡ s† ÿ x…t†Š2 i  s2H , where 0 < H < 1=2 is called fractional Brownian ``Antipersistence'' [13]. It can model the ``subdi€usion'' process [14], which is slower di€usion than the normal di€usion, for 1=2 < H < 1 is called fractional Brownian ``Persistence'' [13] which can model the superdi€usion process [14], which is faster than the normal di€usion. Therefore the FFPE reduces to, o2H t q…x; t† ˆ

1 o2 g…x; t†q…x; t†; 2 ox2

…3:15†

where, a ˆ 2H . Taking g…x; t† ˆ 4HD0 , Eq. (3.15) is the same e€ective Fokker±Plank equation (EFPE) obtained by [15], which has the form, oq2H …x; t† o2 q2H …x; t† ˆ 2HD0 t2H ÿ1 ot ox2 and its solution with initial condition q…x; 0† ˆ d…x† is,  p 2 2H q…x; t† ˆ 1= 4pD0 t2H e…ÿx =4D0 t † :

…3:16†

…3:17†

The above distribution function is the exact structure used to describe the fractional Brownian motion.

4. Solution of FFPE using eigenfunction expansion technique For the FFPE (3.13), assuming that the two given functions f and g depend on the position x only, i.e. f ˆ f …x† and g ˆ g…x†. The FFPE becomes, oat q…x; t† ˆ ÿ

o 1 o2 f …x†q…x; t† ‡ g…x†q…x; t†: ox 2 ox2

By using the change of variable as, Z x s 2D1 yˆ dv; g…v† x0

…4:1†

…4:2†

where D0 is a positive constant, the FFPE (4.1) takes the form; oat p…y; t† ˆ ÿ

o o2 …q…y†q…y; t†† ‡ D0 2 q…y; t†; oy oy

…4:3†

where,

s  2D0 1 og…x† q…y† ˆ f …x† ÿ g…x† 4 oy

and the transformed FFPE reads,    o oF …x† o2 a q…x; t† ‡ D0 2 q…x; t†; ot p…x; t† ˆ ÿ ox ox ox where F …x† is the potential, Z x F …x† ˆ ÿ q…v† dv:

…4:4†

…4:5†

…4:6†

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S.A. El-Wakil, M.A. Zahran / Chaos, Solitons and Fractals 11 (2000) 791±798

Now, we look for the solution in the form q…x; t† ˆ u…x†W…t†. Therefore, Eq. (4.5) becomes,    o oF …x† o2 a W…t†/…x† ‡ D0 2 W…t†/…x†: ot W…t†/…x† ˆ ÿ ox ox ox

…4:7†

The time part of the above equation will give the fractal time evaluation of fractional Brownian motion, oat W…t† ˆ ÿkW…t†

…4:8†

k is the constant of separation. The fractional operator oat can be represented in two di€erent classes of fractional di€erential operators. 4.1. The time part of FFPE in terms of L±R de®nition In the L±R fractional de®nition, Eq. (4.8) is written as, a 0 Dt W…t†

ˆ ÿkW…t†:

…4:9†

Using the Laplace±Millen technique [16] to solve (4.9) as follows, Z 1 eÿut W…t† dt W…u† ˆ L…W…t†; u† ˆ

…4:10†

as the Laplace transform, where the Mellin transform is given by Z 1 W…s† ˆ M…W…t†; s† ˆ tsÿ1 W…t† dt:

…4:11†

0

0

The connection between the Laplace and Millen is, M…W…t†; s† ˆ

1 M…L…W…t†; u†; 1 ÿ s†; C…1 ÿ s†

…4:12†

where C…s† is the gamma function. Therefore Eq. (4.9) in Laplace domain yields W…u† ˆ

W0 ; … k ‡ ua †

where W0 is a constant. By using Eq. (4.12) for Eq. (4.13) gives, 8 ÿ1 s
ÿ
9 < 1 s = C ÿ ‡ C 1 ÿ 1 ÿs=a a a a a …k† : W…s† ˆ …W0 =k† : a…kÿ1=a † ; C…1 ÿ s† Inverting Eq. (4.14) into a time domain leads to the solution [17],   W0 aÿ1 1=at …0; 1=a† 12 t H11 k W…t† ˆ ; …0; 1=a†…1 ÿ a; 1† a 11 is de®ned as Fox function [18], where H12   …a1 ; a1 † . . . …aN ; aN †; …aN ‡1 ; aN ‡1 † . . . …aP ; aP † MN MN HPQ …z† ˆ HPQ z …b1 ; b1 † . . . …bM ; bM †; …bM‡1 ; bM‡1 † . . . …bQ ; bQ †

and de®ned by the contour integration, Z 1 A…m†B…m† MN dm; …z† ˆ HPQ 2pi L C…m†D…m†

…4:13†

…4:14†

…4:15†

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with, A…m† ˆ

M Y kˆ1

B…m† ˆ

N Y

C…bk ÿ bk m†;

C…1 ÿ ak ‡ ak m†;

kˆ1

C…m† ˆ

Q Y kˆ1

D…m† ˆ

P Y

C…1 ÿ bk ‡ bk m†;

C…ak ÿ ak m†:

kˆ1

Using the series expansion for the Fox function [18], one can obtain, 1 X 1 …ÿkta †n : W…t† ˆ ktaÿ1 C…na ‡ a† nˆ0

…4:16†

Showing that W…t† behaves as, W…t†  tÿ1‡a

…4:17†

for small t ! 0, while for t ! 1 it will give the asymptotic expansion of W…t† as, W…t†  tÿ1ÿa :

…4:18†

This result shows that the waiting time distribution has an algebraic of the kind usually considered in the theory of random walk [19]. The space part of fractional Fokker±Planck Eq. (4.7) is    o oF …x† o2 …4:19† /…x† ‡ D0 2 /…x† ˆ ÿk/…x†: ÿ ox ox ox For a special initial condition q…x; t† ˆ d…x ÿ x0 †, the general expansion of transition density obtained from the solution of FFPE using the completeness condition [7], n X …ÿksa † : …4:20† /n …x†/n …x0 †W0 ktaÿ1 P …x; t ‡ sjx0 ; t† ˆ exp……F …x0 †=2† ÿ …F …x†=2†† C…na ‡ a† n 4.2. The time part of the FFPE in terms of the Nishimoto (N) de®nition Knowing that, Nishimoto (N) fractional de®nition depends on the theory of complex integration. The time part in fractional form, Na W…t† ˆ ÿkW…t†

…4:21†

the general solution of Eq. (4.21), see Ref. [9], W…t† ˆ

m0 X

 1=a  Bn exp t keip…1‡2n† ;

…4:22†

nˆ0

where m0 is ®nite for a ˆ rational number and in®nite for a ˆ irrational number. According to the above result, some special solutions of Eq. (4.22) are founded in Table 1. The general transition density function for the FFPE in N-de®nition becomes,   X 1=a  /n …x†/n …x0 †Bn exp s keip…1‡2n† : P …x; t ‡ sjx0 ; t† ˆ eF …x† n

…4:23†

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S.A. El-Wakil, M.A. Zahran / Chaos, Solitons and Fractals 11 (2000) 791±798

Table 1 a Orders

Fractional equation

Solution

1/2 1/3 1 2/3

N1=2 W…t† ˆ ÿkW…t† ! N1=3 W…t† ˆ ÿkW…t† ! N1 W…t† ˆ ÿkW…t† ! N2=3 W…t† ˆ ÿkW…t† !

W…t† ˆ B0 et…ke † ip 3 W…t† ˆ B0 et…ke † t…keip † W…t† ˆ B0 e ip 3=2 3ip 3=2 W…t† ˆ B0 et…ke † ‡ B1 et…ke †

ip 2

4.3. Discussion The FFPE with the fractal time evolution can be used to describe the anomalous behavior of the transport process through the fractal medium with the mean-square displacement h… x…t ‡ s† ÿ x…t††2 i  g…x; t†sa . All orders L±R de®nition give a supper slow (decaying) behavior for W…t†. The obtained solution or the waiting time distribution function coincidence with the form of the kind usually considered in the theory of random walk. On the other hand, some order of N de®nition such as a ˆ 1=2; 1=4; 1=6 . . . cannot give the same decaying behavior for W…t†. The de®nitions L±R and N coincide with the standard solution when a ˆ 1. Therefore, when we apply the fractional calculus to any physical transport phenomena, we must pay much attention to this fact, whatever N de®nition as a mathematical tools being more elastic and easier than L±R de®nition, but gives incorrect behavior for the relaxation phenomena that occur in the fractal medium. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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