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International Journal of Mo dern Physics B Vol. 21, No. 6 (2007) 955­967 c World Scientific Publishing Company

FOKKER PLANCK EQUATION FOR FRACTIONAL SYSTEMS

VASILY E. TARASOV Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia tarasov@theory.sinp.msu.ru Received 14 March 2006 Revised 6 June 2006 The normalization condition, average values, and reduced distribution functions can b e generalized by fractional integrals. The interpretation of the fractional analog of phase space as a space with noninteger dimension is discussed. A fractional (p ower) system is describ ed by the fractional p owers of co ordinates and momenta. These systems can b e considered as non-Hamiltonian systems in the usual phase space. The generalizations of the Bogoliub ov equations are derived from the Liouville equation for fractional (p ower) systems. Using these equations, the corresp onding Fokker­Planck equation is obtained. Keywords : Fokker­Planck equation; non-Hamiltonian systems; fractal; fractional integral.

1. Intro duction Fractional integrals and derivatives1 have many applications in statistical mechanics and kinetics.2,3 The generalization of the Fokker­Planck equation4 can be used to describe kinetics in the fractal media. It is known that the Fokker-Planck equation can be derived from the Liouville and Bogoliubov equations.5­7 The Liouville equation is obtained from the normalization condition and from the Hamilton equations. The Bogoliubov equations can be derived from the Liouville equation and from the definition of the average value. In this paper, the generalized Fokker­Planck equation is obtained from the Liouville and Bogoliubov equations for fractional (power) systems. For this aim, we use fractional generalizations of the normalization condition and the average values.8­11 In the paper, we suggest the physical interpretation of integrals of noninteger order. The fractional integral is considered as an integral on the fractal or nonintegerdimensional space. This interpretation is connected with the definition of noninteger dimension. We prove that fractional integration can be used to describe processes and systems on fractal. The physical values on fractals can be "averaged," and the distribution of the values on fractal can be replaced by some continuous distribution. To describe the distribution on the set with noninteger dimension, we use the
955


956

V. E. Tarasov

fractional integrals. The order of the integral is equal to the fractal Hausdorff dimension of the set. The consistent approach to describe the distribution on fractal is connected with the mathematical definition of the integrals on fractals.12­15 It was proved12 that integrals on net of fractals can be approximated by fractional integrals. In Ref. 8­11, we proved that fractional integrals can be considered as integrals over the space with noninteger dimension up to a numerical factor. We use the well-known formulas of dimensional regularizations.16 There is an interpretation that follows from the fractional measure of phase space,8­11 which is used in the fractional integrals. The fractional phase space can be considered as a space that is described by the fractional powers of coordinates and momenta. Using this phase space, we can consider some of the non-Hamiltonian systems as generalized Hamiltonian systems.8­11 The fractional systems can be described as exitations of the fractal medium.8­11 In Sec. 2, we consider the Hausdorff measure, the Hausdorff dimension, and the integration on fractals to fix notation and provide a convenient reference. The connections of the integration on fractals and the fractional integrals are discussed. The fractional average values and reduced distribution functions are defined. In Sec. 3, we derive Fokker­Planck equations from the Liouville equation for fractional (power) systems. A short conclusion is given in Sec. 4. 2. Integration on Fractal and Fractional Integration 2.1. Hausdorff measure and Hausdorff dimension Fractals are measurable metric sets with a noninteger Hausdorff dimension. To define the Hausdorff measure and the Hausdorff dimension, we consider a measurable metric set (W, µH ) with W Rn . The elements of W are denoted by x, y , z , . . . , and represented by n-tuples of real numbers x = (x1 , x2 , . . . , xn ) such that W is embedded in Rn . The set W is restricted by the following conditions: (1) W is closed; (2) W is unbounded; (3) W is regular (homogeneous, uniform) with its points randomly distributed. The diameter of a subset E W Rn is diam(E ) = sup{G(x, y ) : x, y E } , where G(x, y ) is a metric function of two points x and y W . Let us consider a set {Ei } of subsets Ei such that diam(Ei ) < i, and W i=1 Ei . Then, we define (Ei , D) = (D)[diam(Ei )]D . (1)

The factor (D) depends on the geometry of Ei . If {Ei } is the set of all (closed or open) balls in W , then (D) = D/2 2-D . (D/2 + 1) (2)


Fokker­Planck Equation for Fractional Systems

957

The Hausdorff dimension D of a subset E W is defined

17

by (3)

D = dimH (E ) = sup{d R : µH (E , d) = } = inf {d R : µH (E , d) = 0} . From definition (3), we obtain (1) µH (E , d) = 0 for d > D = dimH (E ); (2) µH (E , d) = for d < D = dimH (E ). The Hausdorff measure µ
H

of a subset E W is defined


17,18

by (4)

µH (E , D) = (D)

diam(Ei )0 {Ei }

lim

inf

[diam(Ei )]D .
i=1

Note that µH (E , D) = D µH (E , D), where > 0, and E = {x, x E }. 2.2. Function and integrals on fractal Let us consider the functions on W :


f (x) =
i=1

i Ei (x) ,

(5)

where E is the characteristic function of E : E (x) = 1 if x E , and E (x) = 0 if x E. The Lebesgue­Stieltjes integral for (5) is defined by


f dµ =
W i=1

i µH (Ei ) .

(6)

Therefore f (x)dµH (x) =
W diam(Ei )0

lim

f (xi ) (Ei , D)
E
i

= (D)

diam(Ei )0

lim

f (xi )[diam(Ei )]D .
E
i

(7)

It is possible to divide Rn into parallelepipeds E
i1 ···i
n

= {(x1 , . . . , xn ) W : xj = (ij - 1)xj + j , 0 j xj , j = 1, . . . , n} . (8)

Then dµH (x) =
diam(E

lim
i1 ···in

)0

(E
n

i1 ···i

n

, D)
n

=

diam(E

lim
i1 ···in

)0

(xj )
j =1

D /n

=
j =1

d

D /n

xj .

(9)


958

V. E. Tarasov

The range of integration W can be parametrized by polar coordinates with r = G(x, 0) and angle . Then Er, can be thought of as a spherically symmetric covering around a center at the origin. In the limit, function (Er, , D) gives dµH (r, ) =
diam(E

lim
r,

)0

(E

r,

, D) = d

D -1 D -1

r

dr .

(10)

Let us consider f (x) that is symmetric with respect to some point x0 W , i.e., f (x) = const. for all x such that G(x, x0 ) = r for arbitrary values of r. Then the transformation W W : xx =x-x
0

(11)

can be performed to shift the center of symmetry. Since W is not a linear space, (11) need not be a map of W onto itself. Map (11) is measure preserving. Using (10), the integral over a D-dimensional metric space is defined by f dµ
W H

=

2 D/2 (D/2)



f (r)r
0

D -1

dr .

(12)

This integral is known in the theory of the fractional calculus.1 The right Riemann­ Liouville fractional integral is
D I- f (z ) =

1 (D)



(x - z )
z

D -1

f (x)dx .

(13)

Equation (12) is reproduced by f dµ
W H

=

2 D/2 (D) D I f (0) . (D/2) -

(14)

Relation (14) connects the integral on fractal with the integral of fractional order. This result permits to apply different tools of the fractional calculus1 for the fractal medium. As a result, the fractional integral can be considered as an integral on fractal up to the numerical factor (D/2)/[2 D/2 (D)]. Note that the interpretation of fractional integral is connected with the fractional dimension.8­11 This interpretation follows from the well-known formulas for dimensional regularizations.16 The fractional integral can be considered as an integral in the noninteger-dimensional space up to the numerical factor (D/2)/[2 D/2 (D)]. It was proved14 that the fractal space­time approach is technically identical with the dimensional regularization. The integral defined in (7) satisfies the following properties: (1) Linearity: (af1 + bf2 )dµ
X H

=a
X

f1 dµH + b
X

f2 dµH ,

(15)

where f1 and f2 are arbitrary functions; a and b are arbitrary constants. (2) Translational invariance: f (x + x0 )dµH (x) =
X X

f (x)dµH (x)

(16)


Fokker­Planck Equation for Fractional Systems

959

since dµH (x - x0 ) = dµH (x) as a consequence of homogeneity (uniformity). (3) Scaling property: f (ax)dµH (x) = a
X -D X

f (x)dµH (x)

(17)

since dµH (x/a) = a-D dµH (x). It has been shown16 that conditions (15)­(17) define the integral up to normalization.16 2.3. Multi-variable integration on fractal Integral (12) is defined for a single variable, and not multiple variables. It is useful for integrating spherically symmetric functions. This integral can be generalized for the multiple variables by using the product spaces and product measures. Let us consider the measure spaces (Wk , µk , D) with k = 1, 2, 3, and form a Cartesian product of the sets Wk producing the space W = W1 â W2 â W3 . The definition of product measures and the application of the Fubini's theorem provides a measure for W as (µ1 â µ2 â µ3 )(W ) = µ1 (W1 )µ2 (W2 )µ3 (W3 ) . The integration over a function f on the product space is f (r)dµ1 â µ2 â µ3 = f (x1 , x2 , x3 )dµ1 (x1 )dµ2 (x2 )dµ3 (x3 ) . (19) (18)

In this form, the single-variable measure from (12) may be used for each coordinate xk , which has an associated dimension k : dµk (xk ) = 2 k /2 |xk | (k /2)
k -1

dxk ,

k = 1, 2, 3 .

(20)

The total dimension of W = W1 â W2 â W3 is D = 1 + 2 + 3 . Let us reproduce the result (12) from (19). We take a spherically symmetric function f (r) = f (x1 , x2 , x3 ) = f (r), where r 2 = (x1 )2 +(x2 )2 +(x3 )2 . Equation (19) becomes dµ1 (x1 )dµ2 (x2 )dµ3 (x3 )f (x1 , x2 , x3 ) = 2 1 /2 2 2 /2 2 3 /2 (1 /2) (2 /2) (3 /2) â (cos )
1 -1

dr

d
3 -1

d J3 (r, )r f (r) ,

1 +2 +3 -3

(sin )

2 +3 -2

(sin )

(21)

where J3 (r, ) = r2 sin is the Jacobian of the coordinate change. To perform the integration in spherical coordinates (r, , ), we use
/2

sin
0

µ-1

x cos

-1

xdx =

(µ/2)( /2) , 2([µ + ]/2)

(22)


960

V. E. Tarasov

where µ > 0, > 0. Then Eq. (21) becomes dµ1 (x1 )dµ2 (x2 )dµ3 (x3 )f (r) = 2 D/2 (D/2) f (r)r
D -1

dr .

(23)

This equation describes integration over a spherically symmetric function in the D-dimensional space and reproduces result (12). 2.4. Probability distribution on fractal The probability that is distributed in the three-dimensional Euclidean space is defined by P3 (W ) =
W

(r)dV3 ,

(24)

where (r) is the density of probability distribution, and dV3 = dxdy dz for the Cartesian coordinates. If we consider the probability that is distributed on the measurable metric set W with the fractional Hausdorff dimension D, then the probability is defined by the integral PD (W ) =
W

(r)dVD ,

(25)

where D = dimH (W ) = 1 + 2 + 3 , and dV
D

= dµ1 (x1 )dµ2 (x2 )dµ3 (x3 ) = c3 (D, r)dV3 , 8 D/2 |x| (1 )(2 )(3 )
1 -1

(26) . (27)

c3 (D, r) =

|y |

2 -1

|z |

3 -1

There are many different definitions of fractional integrals.1 For the Riemann­ Liouville fractional integral, function c3 (D, r) is c3 (D, r) = |x|1 -1 |y |2 -1 |z |3 -1 , (1 )(2 )(3 ) (28)

where x, y , z are the Cartesian coordinates, and D = 1 + 2 + 3 , 0 < D 3. As the result, we obtain the Riemann­Liouville fractional integral1 in Eq. (25) up to numerical factor 8 D/2 . Therefore, Eq. (25) can be considered as a fractional generalization of Eq. (24). For (r) = (|r|), we can use the Riesz definition of the fractional integrals.1 Then c3 (D, r) = Note that
D 3-

(1/2) |r| 2D 3/2 (D/2)

D -3

.

(29)

lim c3 (D, r) = (4

3/2 -1

)

.

(30)


Fokker­Planck Equation for Fractional Systems

961

Therefore, we suggest to use c3 (D, r) = 2
3-D

(3/2) |r| (D/2)

D -3

.

(31)

Definition (31) allows us to derive the usual integral in the limit D (3 - 0). For D = 2, Eq. (25) gives the fractal probability distribution in the volume. In general, it is not equivalent to the distribution on the two-dimensional surface. Equation (28) is equal (up to numerical factor 8 D/2 ) to the integral on the measurable metric set W with Hausdorff dimension dimH (W ) = D. To have the usual dimensions of the physical values, we can use vector r, and coordinates x, y , z as dimensionless values. 2.5. Fractional average values To derive the fractional analog of the average value, we consider the fractional integral for function f (x). If function f (x) is equal to the distribution function (x), then we can derive the normalization condition. If function f (x) is equal to the multiplication of distribution function (x) and classical observable A(x), then we have the definition of the fractional average value. The fractional generalization of the average value8­11 can be presented by A where
I+ f = y - = (I+ A)(y ) + (I- A)(y ) ,

(32)


1 ()

f (x)dx , (y - x)1-

I- f =

1 ()
8­11

y

f (x)dx . (x - y )1-

(33)

For = 1, Eq. (32) gives the usual average value. The fractional average value (32) can be written A where dµ (x) = Here, we use x = sgn(x)|x| ,


as (34)

=

1 2



[(A)(y - x) + (A)(y + x)]dµ (x) ,
-

|x|-1 dx dx = . () ()

(35)

(36)

where function sgn(x) is equal to +1 for x 0, and -1 for x < 0. Let us introduce notations to consider the fractional average value for phase space. (1) The operator Txk is defined by Txk f (. . . , xk , . . .) = 1 (f (. . . , xk - xk , . . .) + f (. . . , xk + xk , . . .)) . 2 (37)


962

V. E. Tarasov

For k -particle, which is described by coordinates q 1, . . . , m), the operator T [k ] is T [k ] = T
q
k1

ks

and momenta p

ks

(s = (38)

T

p

k1

···T

q

km

T

p

km

.

For the n-particle system, we define the operator T [1, . . . , n] = T [1] · · · T [n]. ^ (2) The operator Ixk is defined by ^ Ixk f (xk ) =
+

f (xk )dµ (xk ) .
-

(39)

Then the fractional integral (34) can be rewritten in the form A


^ = Ix Tx A(x)(x) .
m

^ ^ ^ ^ ^ The integral operator I [k ] = Iqk1 Ipk1 · · · Iqkm Ipk ^ I [k ]f (qk , pk ) = where dµ (qk , pk ) = (())
-2m

is (40)

f (qk , pk )dµ (qk , pk ) ,

dq

k1

dp

k1

· · · dq ^

km

dp

km

. by (41)

For the n-particle system, we use I [1, . . . , n] = I [1] · · · I [n]. The fractional average values for the n-particle system is defined A


^

^

8­11

= I [1, . . . , n]T [1, . . . , n]An .
+

^

In the simple case (n = m = 1), the fractional average value is
+

A



=
- -

dµ (q , p)Tq Tp A(q , p)(q , p) .
8­11

(42)

Note that the fractional normalization condition tion of average values 1 = 1.

is a special case of this defini-

3. Fokker­Planck Equation from Liouville Equation Let us consider a system with n identical particles and the Brownian particle. The distribution function of this system is denoted by n+1 (q, p, Q, P, t), where q = (q1 , . . . , qn ) , p = (p1 , . . . , pn ) , qk = (qk1 , . . . , q
km

), )

pk = (pk1 , . . . , p

km

are coordinates and momenta of the particles; Q = (Qs ) and P = (Ps ) (s = 1, . . . , m) are coordinates and momenta of Brownian particles. The fractional normalization condition8­11 has the form ^ I [1, . . . , n, n + 1]n+1 = 1 , ~ where n+1 = T [1, . . . , n, n + 1] ~
n+1

(43)

(q, p, Q, P, t) .

(44)


Fokker­Planck Equation for Fractional Systems

963

The reduced distribution function for the Brownian particle is ^ B (Q, P, t) = I [1, . . . , n]n+1 (q, p, Q, P, t) . ~ ~ The distribution n+1 satisfies the Liouville equation: ~ n+1 ~ - i(Ln + LB )n+1 = 0 , ~ t where Ln and L
B 8­11

(45)

(46)

are Liouville operators such that
n,m

-iLn =
k,s n,m

k (Gk ) (Fs ) s + qk s pk s

,

(47)

-iLB =
k,s

(gs ) (fs ) + Q Ps s

.

(48)

k The forces Fs and fs , and the velocities Gk and gs are defined by the Hamilton s equations of motion. For the k th particle, dqks = Gk (q, p) , s dt

dps k k = Fs (q, p, Q, P ) , dt
dPs = fs (q, p, Q, P ) . dt

(49)

where k = 1, . . . , n. The Hamilton equations for the Brownian particle are
dQs = gs (Q, P ) , dt

(50)

For simplification, we suppose Gk = ps /m , s k
gs = Ps /M ,

(51)

where M m. Let us use the boundary condition in the form
t-

lim

n+1

(q, p, Q, P ) = n (q, p, Q, T )B (Q, P, t) ,

(52)

where n is a canonical Gibbs distribution function n (q, p, Q, T ) = exp [F - H (q, p, Q)] . Here, H (q, p, Q) is a Hamilton function
n

(53)

H (q, p, Q) = Hn (q, p) +
k=1

UB (qk , Q) ,

(54)

where Hn is a Hamiltonian of the n-particle system, and UB is an energy of interaction between particles and the Brownian particle. If we use Eqs. (49) and (51), then
n,m

Hn (q, p) =
k,s

p2 ks + 2m

U (qk , ql ) .
k
(55)


964

V. E. Tarasov
19

The boundary condition (52) can be realized in the Liouville equation

by the infinitesimal source term

n+1 ~ - i(Ln + LB )n+1 = -(n+1 - n B ) . ~ ~ ~~ t

(56)

^ Integrating (56) by I [1, . . . , n], we obtain the equation for the Brownian particle distribution B ~ + t The formal solution
m s=1

(gs B ) ^ ~ + I [1, . . . , n] Q s

m s=1

(fs n+1 ) = 0. Ps

(57)

19

has the form
0

n+1 (t) = ~
-

d e e

-i (Ln +LB )

B (t + )n , ~ ~

(58)

or
0

n+1 (t) = B (t)n - ~ ~ ~
-

d e e

-i (Ln +LB )

- i(Ln + LB ) B (t + )n . ~ ~ (59)

Substituting (59) into (57), we get B ~ + t
m s=1

(gs B ) ~ + Q s
m

m s=1

B ^ I [1, . . . , n](fs n ) ~ Ps
0

^ - I [1, . . . , n]
s=1

P

s

d e e
-

-i (Ln +LB )

â

- i(Ln + LB ) B (t + )n = 0 . ~ ~

(60)

^ Note that I [1, . . . , n]fs n can be considered as an average value of the force fs . ~ This average value for the canonical Gibbs distribution (53) is equal to zero. The last term can be simplified. Using n 1 = f Q kT s where f
(p) s (p) s n

,

(61)

is a potential force f
(p) s

=-

UB , Q s

(62)

we get -iLB n+1 = ~ Ps f s (gs B ) (fs B ) ~ ~ B + + M kT Q Ps s
(p)t

n .


Fokker­Planck Equation for Fractional Systems

965

It can be proved by interactions that the term (gs B ) ~ B ~ + t Q s in the integral of (60) does not contribute. Then B ~ + t ·
m s=1

(63)

(gs B ) ~ + Q s

m s=1

^ I [1, . . . , n] Ps
-1

0 -

d e fs e

-i (Ln +LB )

n ~

~ (fs B (t + )) + M Ps

fs Ps B (t + ) ~

= 0.

(64)

This equation is a closed integro-differential equation for the reduced distribution function B . Note that force fs can be presented in the form ~ fs = f
(p) s (p) s

+f

(n) s

,

where f is a potential force (62), and f is a non-potential force that acts on the Brownian particle. For the equilibrium approximation P (M k T )1/2 , iLB M -1/2 , iLn m-1/2 , and M m, we can use the perturbation theory. Using the approximation B (t + ) = B (t) for Eq. (64), we obtain the Fokker­ Planck equation for fractional power systems B ~ + t where
1 ss m s=1

(n) s

(gs B ) ~ + Q s

m s=1

P

s

1 M (ss B (t)) ~ + Ps

2 ss

~ Ps B (t)

= 0,

(65)

^ = M I [1, . . . , n]

0 - 0 -

d e fs e

-i L

n

~ fs n ,
(p) ~ s n

(66)


(p)

2 ss

^ = M I [1, . . . , n]

d e fs e

-i L

n

f

.

(67)

1 2 If fs = fs , then ss = ss . Let us consider the one-dimensional stationary Fokker­Planck equation (65) with

(gs B )/ Q = 0 . ~ s Then P


M ( 1 (P )B (t)) ~ + 2 (P )P B (t) ~ P

= 0.

(68)

Obviously, we get the relation ~ M ( 1 (P )B (t)) + 2 (P )P B (t) = const. ~ P Assuming that the constant is equal to zero, we get 2 (P )P [ 1 (P )B (t)] ~ = B (t) , ~ P M (70) (69)


966

V. E. Tarasov

or, in an equivalent form 2 (P )P ln[ 1 (P )B (t)] ~ = . P 1 (P )M The solution is ln[ 1 (P )B (t)] = ~ As the result, we obtain B (t) = ~ N (P )
1

(71)

2 (P )P dP + const. M 1 (P ) 2 (P )P dP , M 1 (P )

(72)

(73)

where the coefficient N is defined by the normalization condition. Equation (73) describes the solution of the stationary Fokker­Planck equation for the fractional (power) system. The special cases of (73) can be derived as done in Ref. 4. 4. Conclusion In this paper, the fractional generalizations of the average value and the reduced distribution functions are used. The generalization of the Liouville and Bogoliubov equations are derived8­11 from the fractional normalization condition. Using these equations, we obtain the Fokker­Planck equation for fractional (power) systems. The Liouville, Bogoliubov, and Vlasov equations for fractional systems8­11 can be considered as equations in the noninteger-dimensional phase space. For example, the systems that live on some fractals can be described by these equations. Note that the fractional systems can be presented as special non-Hamiltonian systems.8­11 Fractional oscillators can be interpreted as elementary excitations of some fractal medium with noninteger mass dimension.8­11 The fractional (power) systems are connected with the non-Gaussian statistics. Classical dissipative and nonHamiltonian systems can have the canonical Gibbs distribution as a solution of the stationary equations.20,21 Using the methods,20,21 it is easy to prove that some fractional dissipative systems can have the fractional analog of the Gibbs distribution (non-Gaussian statistic) as a solution of the stationary equations for fractional systems. References
1. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, New York, 1993). 2. G. M. Zaslavsky, Phys. Rep. 371, 461 (2002). 3. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, Oxford, 2005). 4. V. E. Tarasov, Chaos 15, 023102 (2005). 5. A. Isihara, Statistical Physics (Academic Press, New York, 1971). 6. P. Resibois and M. De Leener, Classical Kinetic Theory of Fluids (John Wiley and Sons, New York, 1977).


Fokker­Planck Equation for Fractional Systems

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