a fractional diffusion equation to describe l&y flights

7/25/2019 A Fractional Diffusion Equation to Describe L&y Flights

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23 February 1998

PHYSICS LETTERS A

ELSl3VIER

Physics Letters A 239 (1998) 13-16

A fractional diffusion equation to describe L&y flights

A S Chaves a,b,

D epart amento de Fisi ca. Insti tut e de Cihci as Exatas. Uni versidade Federal de M inas Gerai s.

CJ? 702 CEP 30161-970 Belo Hor izont e MG. Brazi l

h Depart amento de Fisica Uni versidade de Brasnia. C.P. 04455. CEP 70910-900 Brasil ia DE Brazi l

Received 3

March 1997;

revised manuscript received 15 October 1997; accepted for publication 21 November 1997

Communicated by A.R.

Bishop

Abstract

A fractional-derivatives diffusion equation is proposed that generates the Levy statistics. The fractional derivatives are

defined by the eigenvector equation a: eax =

an eoX

and for one dimension the diffusion equation in an isotropic medium

reads &n = (D/2) (a, + a,) n + u n, 1 < ru Q 2. The equation is based on a proposed generalization of Ficks law which

reads j = -(D/2)

(Oy-* -

'7 ;'

n + vn.

The diffusion equation is also written for an anisotropic medium, and in this

case it generates an asymmetric Levy statistics. @ 1998 Published by Elsevier Science B.V.

PACS 02.50.-r; 02.50.Ey; 02.60.Nm; 05.4O.+j

Non-Gaussian probability distributions of stochas-

tic variables have been a subject of interest for a long

time

[

l-31. More recently, these distributions have

been applied to the description of many physical pro-

cesses, including turbulent flow

[

41, diffusion in com-

plex systems [ 51, chaotic dynamics of classical con-

servative systems

[

61, and others

[

71.

In

this Letter I propose a generalization of Ficks

law in which the particle current is proportional to

the fractional derivatives of the particle density. With

the generalized Ficks law I obtain a fractional deriva-

tives equation which generates the Levy distribution.

The proposed diffusion equation might be useful for

the investigation of the mechanism of superdiffusion.

With the present approach I also show that superdif-

fusion in a medium which is not symmetric with re-

spect to space inversion is also asymmetric. This is

very distinct from the behavior of normal diffusion,

which always presents space inversion symmetry. An

important consequence of the asymmetry in superdif-

fusion is that linear transport in such conditions is not

symmetric with respect to the reversion of the bias.

Besides the scientific interest, this could also result in

novel devices.

Ordinary diffusion is an important process de-

scribed by a Gaussian distribution. In one dimension,

the probability density P(x, t) of a particle, ini-

tially (t = 0) at x = 0, being at x the instant t is

P(x, t) = (4TDt)-1/2exp( -x2/4Dt), where

D

is

the Einstein diffusivity constant. A main feature of

the process is the linear relation between the mean

square displacement and time, namely (x2) = 2Dt. In

anomalous diffusion one might find (x2) cx tY, y

1,

or else (x) might be a divergent integral for

t f 0

My interest here is in this latter process, so-called

Levy flights, and the associated distribution, which

is the Fourier transform of p(k, t) = exp( -atlkj),

with (Y < 2. Ordinary diffusion is generated as the

0375-9601/98/$19.00 @ 1998 Published by Elsevier Science B.V. All rights reserved

PII SO375-9601(97)00947-X


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solution of the Fokker-Planck equation

dfP(X, t> =

-v P(n, t> + Da j P n , ) ,

(1)

for the initial condition P ( X, 0) = S(n) , where a non-

zero mean flow speed u was allowed for the particles,

I propose a generalization of Eq. ( 1) using fractional

derivatives ~?,a,1 < a 6 2, in substitution for $, as a

generator of the Levy flights.

In contrast to the Levy flights, the ordinary diffu-

sion equation is well understood and can be deduced

from the BoItzmann-Gibbs statistics. Another way of

looking at the problem is to examine the physical basis

of Eq. ( 1)

.

If one combines the continuity equation

V.j+$=O

with Ficks empirical law

(2)

J

= -DVn+vn,

(31

one obtains Eq. ( 1) for dimension d = 1. Hence,

Ficks iaw can be deduced for a system near equi-

librium which is described by the Boltzmann-Gibbs

statistics. Furthe~ore, as Eq. (2) is a fundamental

law for conservative particles, it is inevitable to mod-

ify Ficks Iaw in order to obtain anomalous diffusion.

I take this approach in this Letter. There is a very

distinct way of looking at the problem. For a distri-

bution is which, for small time t, the first and sec-

ond moments are proportional to t and all the other

moments decrease faster than t, it is possible to de-

duce the Fokker-Pianck equation in terms of those

first moments [2]. That is a matter of ma~ematical

self-consistency in which the mechanism is not rele-

vant. As the second moment of a Levy distribution is

infinite, the method fails in this case. Zaslavsky [6]

extended this traditional approach by using fractional

derivatives and obtained a fractionaf Fokker-Planck

equation which generates the Levy distribution. He

used the Riemann-Liouville fractional derivatives as

defined in Ref. [ 81,

where (D-, = xTa-i

,fr( -cY), r is the gamma func-

tion and x+ is a generalized function defined on the

positive semi-axis x >, 0.

In this Letter I use the fractional derivative linear

operator defined by the eigenvector equation.

(5)

where LY, and a are complex numbers This form of

derivation can be applied to any function which can

be expanded as a sum of exponentials and therefore

to any distribution. The multiplication

rul

(6)

follows immediately from the definition. It is oppor-

tune here to notice that by using the definition of frac-

tional derivative given by Eq. (4) one obtains

U-1 e-

1 -

>

-(--a)

x>o

and this is different from an n for fractional CY. hus,

we are referring to two distinct de~nitions of fractional

derivatives.

Next, I propose to generalize Ficks law to the form

j= -~(V~-l-V_;i)~+mr,

1


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A.S.

Chaves/Physics Letters A 239 1998) 13-16

where the kernel P(x, t) is

J

k

P(x, ?) = ; cos k( x - or)

x exp [cos (a;) Dfk]

.

(13)

This is a L&y distribution. The condition (Y > 1 in

Eq. (8) guarantees the convergence of the integral in

Eq. ( 13). The condition (Y < 2 is necessary to guar-

antee that P( X, t) 3 0, as required by a probability

distribution

[

1,2]

.

In the reciprocal space defined for k > 0, Eq. (10)

becomes

a@ D

at=2 -[(ik)+(-ik)]@-iku@. ( 14)

Seshadri and West

[

91 first studied the L&y distribu-

tion as a solution to a similar equation for the charac-

teristic function. They investigated the distribution of

particle speeds p( u, t) and defined the Fourier trans-

form

* dk

P(L), f) =

J

&( k, t) e?.

(15)

The differential equation for 4( k, t) is

(16)

where A is an attenuation coefficient for the particle

speeds. Integrating Eq. ( 16), they obtained the char-

acteristic function of the LCvy statistics. The formula-

tion of the diffusion equation in the form of Eq. (10)

is essential for the generalization to anisotropic media,

which is our next step and a main focus of this Letter.

In Eq. (9) we considered that

D

is an isotropic

generalized diffusivity. In the most general case of

a medium that will be an unjustified assumption

and Eq. ( 10) must be modified to become the skew

Fokker-Planck equation

(17)

where the left (Dr) and right (0,) diffusivities are

distinct. The associated kernel is

OOdk

P(X.f) =

J

p

(k, t) ek(x-) ,

--ca

(18)

Fig. I. Plot of

P .r, r)=

iT(dk/rr) expl k cos(av/2)

I

cos[kx + ksin(mr/2)]

for a =

1.25, 1.5, 1.75 and 1.99. This

corresponds to P( x, t) defined in Eqs.

(

18)

and ( 19)

for L= 0.

D,=Oand t=l.

where the characteristic function is

(

19)

This kernel has obviously three properties necessary

to be accepted as a distribution:

Its characteristic function has the properties

(a) P(-k,t) =P*(k,r),

(b) ~(0, 1) = 1,

(c) p(k,r+ t) =~(k,t)p(k,t)

7

which means that

(a) P(x,t) E Iw,

(b) J:aP(X>t)dx= 1,

(c) P(x, t) = J-, P(n - .Y, )P(y, t - r) d.v,

i.e. P(x, t) is a real normalized function which

obeys the

Bachelier-Smoluchowsky-Chapman-

Kolmogorov chain equation (see for instance

Ref.

[

21) . It remains to prove that P (x, t) is posi-

tive. Some numerical inspection convinced me that

this is true, although a mathematical proof has not

been done. Fig.

I

shows the behavior of P(x, t) for

Dft = 2, D, = 0, u = 0, and four values of LY.

A very important property of

P(x, t)

given by

Eqs.

(

18) and (19) is that, except for cy = 2,

P(x - ut, t) + P( -x + vt, t). This asymmetry

in the diffusion is exclusively due to the fractional

derivatives. Ficks law for an anisotropic medium is


4/4

16

AS. Chmes/Physics Letters A 239 1998) 13-16

j=-D.Vn+un,

20)

where D is a second rank tensor and consequently

is invariant under space inversion. In the generalized

Ficks law,

V,

and

D

are

substituted by entities

which do not have a well-defined tensorial character.

Fig. 1 illustrates the asymmetry of the anomalous dif-

fusion in an asymmetric medium. Notice that when cy

increases to values near 2, the distribution becomes

more and more symmetric, even though the medium

is highly asymmetric.

It should be noticed that the integer derivative of a

function f(x) is a local property in the sense that it

can be obtained if one knows the function in an in-

finitesimal range around X. That is not true for the

fractional derivative, because f(x) must be known in

a finite range, so that one can make the analytical con-

tinuation and then to take the derivative. This suggests

that this non-locality in the fractional derivative diffu-

sion equation is what causes the anomaly in the diffu-

sion. Thus, superdiffusion takes place if the medium

cannot be characterized by local properties. This is ex-

actly what happens with a fractal. The connection be-

tween non-locality of the fractional derivative and the

Levy flights has already been discussed by Shlesinger

et al. [6] and others [ lo].

Recent works [ 111 have established a link between

the Levy flights and a generalization of statistical me-

chanics in which the entropy is given by [ 121 S, =

ka( 1 - C;py)/(q - l), q E R. Hence, it should be

possible to justify Eq. (8) with basis on that entropic

form. For q ----f 1, Si =

-kB xi pi

In pi, and in this

limit the established Ficks law should be recovered.

The results obtained by Tsallis et al. [ 1 ] in fact sug-

gest that the normal Ficks law should be recovered

for q < 513.

I am grateful to J.F. Sampaio, M.L. OCarroll and

J.M. Figueiredo for several discussions, to J.F. Sam-

paio for the production of the plot in Fig. 1, and to

C. Tsallis for his interest in this work and calling my

attention to the work by Zaslavsky in Physica D. This

work was supported by the Conselho National de De-

senvolvimento Cientifico e Tecnologico.

References

III

121

L31

141

151

161

[71

[81

r91

[lOI

L111

1121

P. Levy,

Theeone

de IAddition des Variables Alietoires

(Guthier-Villars, Paris, 1937).

E.W. Montroll, B.J. West, in: Fluctuation Phenomena, eds.

E.W. Montroll, J.L. Lebowitz (North-Holland, Amsterdam,

1979).

B.B. Mandelbrot, J.W. van Ness, SIAM Rev. 10 ( 1968) 423;

B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman,

New York, 1983).

M.F. Shlesinger, J. Klafter, B.J. West, Physica A 140 ( 1986)

212;

M.F. Shlesinger, B.J. West, J. Klafter, Phys. Rev. Lett. 58

(1987) 1100.

A. Ott, J.P. Bouchaud, D. Langevin, W. Urbach, Phys. Rev.

Lett. 65 (1990) 2261;

J.P Bouchaud, A. Georges, Phys. Rep. 195 ( 1991) 127.

M.F. Shlesinger, G.M. Zaslavsky, J. Klafter, Nature 363

(1993) 31;

J. Klafter, G. Zumofen, A. Blumen, Chem. Phys. 177 ( 1993)

821;

G.M. Zaslavsky. D. Stevens, H. Weitzner. Phys. Rev. E 48

(1993) 1683;

G.M. Zaslavsky, Physica D 76 ( 1994) 110;

J. Klafter, G. Zumofen, Phys. Rev. E 49 (1994) 4873.

H. Sphon, J. Phys. I (Paris) 3 (1993) 64;

T.H. Solomon, E.R. Weeks, H.L. Swinney, Phys. Rev. Lett.

71 (1993) 3975;

F. Bardou, J.P. Bouchaud, 0. Emile, A. Aspect, C. Cohen-

Tannoudji, Phys. Rev. Lett. 72 (1994) 203.

I.M. Gelfand, G.E. Shilov, Generalized Functions, Vol. 1

(Academic Press, New York, 1994).

V. Seshadri, B.J. West, Proc. Nat. Sci. 79 (1982) 4501;

B.J. West, V. Seshadri, B.J. West, Physica 113 A (1982)

203.

A. Compte, Phys. Rev. E 53 (1996) 4191;

P. Allegrini, P. Grigolini, B.J. West. Phys. Rev. E 54 (1996)

4760.

D.H. Zanette, PA. Alemany, Phys. Rev. Lett. 75

(

1995) 366;

C. Tsallis, S.V.F. Levy, A.M.C. Souza, R. Maynard, Phys.

Rev. Lett. 75 (1995) 3589.

C. Tsallis, J. Stat. Phys. 52 (1988) 479.

a fractional diffusion equation to describe l&y flights

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