Документ взят из кэша поисковой машины. Адрес оригинального документа : http://theory.sinp.msu.ru/~tarasov/PDF/Chaos2004.pdf
Дата изменения: Wed Dec 17 19:03:27 2003
Дата индексирования: Mon Oct 1 21:31:11 2012
Кодировка:

CHAOS

VOLUME 14, NUMBER 1

MARCH 2004

Fractional generalization of Liouville equations
Vasily E. Tarasova)
Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia

Received 15 August 2003; accepted 23 October 2003; published online 16 December 2003 In this paper fractional generalization of Liouville equation is considered. We derive fractional analog of normalization condition for distribution function. Fractional generalization of the Liouville equation for dissipative and Hamiltonian systems was derived from the fractional normalization condition. This condition is considered as a normalization condition for systems in fractional phase space. The interpretation of the fractional space is discussed. © 2004 American Institute of Physics. DOI: 10.1063/1.1633491

We call a fractional equation a differential equation that uses fractional derivatives or integrals. Fractional derivatives and integrals have found many applications in recent studies of scaling phenomena. We formulate fractional analog of main integro-differential equation to describe some scaling process--Liouville equation. Usually used for scaling phenomena Fokker Planck equation can be derived from Liouville equation. Therefore it is interesting to consider a fractional generalization of the Liouville equation. To derive fractional equation for a distribution function we must consider a fractional analog of the normalization condition for distribution function. Most of the fractional equations for distribution function does not use correspondent normalization condition. Therefore these equations ,,with fractional coordinate derivatives... can be incorrect equations. In this paper fractional Liouville equation for dissipative systems is derived from the normalization condition. The coordinate fractional integration for this normalization condition is used.
I. INTRODUCTION

tion a differential equation that uses fractional derivatives or integrals. To derive fractional equations for a distribution function we must consider a fractional analog of the normalization condition for distribution function. Fractional Liouville equation for dissipative systems is derived from the normalization condition. In this paper, the coordinate fractional integration for normalization condition is used. This condition is considered as a normalization condition for systems in fractional phase space. If any fractional equation for distribution function does not use correspondent normalization condition, then this equation with fractional coordinate derivatives can be incorrect. In Sec. II the normalization condition for distribution function and notations are considered. In Sec. III we derive the Liouville equation from the normalization condition. In Sec. IV, the physical interpretation of fractional normalization condition is considered. Finally, a short conclusion is given in Sec. V.

II. NORMALIZATION CONDITION

Fractional derivatives and integrals have found many applications in recent studies of scaling phenomena.15 The main aim of most of these papers is to formulate fractional integro-differential equations to describe some scaling process. Modifications of equations governing physical processes such as the Langevin equation,6 diffusion equations, and Fokker Planck equation have been suggested713 which incorporate fractional derivatives with respect to time. It was shown in Ref. 14 that the chaotic Hamiltonian dynamics of particles can be described by using fractional generalization of the Fokker Planck Kolmogorov equation. In Ref. 14, coordinate fractional derivatives in the Fokker Planck equation were used. It is known that Fokker Planck equation can be derived from Liouville equation.15,16 Therefore it is interesting to consider a fractional generalization of Liouville equation and Bogoliubov hierarchy equations. We call a fractional equaa

Let us consider a distribution function ( x , t ) for x R 1 . Let ( x , t ) L 1 ( R 1 ) , where t is a parameter. Normalization condition has the form x , t dx 1. This condition can be rewritten in the form
y

x , t dx
y

x , t dx 1,

1

where y ( , ). Let ( x , t ) L p ( R 1 ) , where 1 p 1/ . Fractional integrations on ( , y ) and ( y , ) are defined17 by I y,t 1
y

x , t dx , y x1 x , t dx . y1

2

I

y,t

1
y

x

3

Electronic mail: tarasov@theory.sinp.msu.ru 123

Using these notations, Eq. 1 has the form
© 2004 American Institute of Physics

1054-1500/2004/14(1)/123/5/$22.00

Downloaded 17 Dec 2003 to 199.98.104.161. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp

124

Chaos, Vol. 14, No. 1, 2004

Vasily E. Tarasov

I

1

y,t

I

1

y,t

1.

where d / dt is a total time derivative d dt t F
t

Using definitions 2 and 3 we can get the fractional analog of normalization condition 1 , I y,t I y,t 1.

x

.
t

The function xt ,t 4 d ln x dt
1 t

Equations 2 and 3 can be rewritten in the form I y,t 1
0

x

1

xt x0

y

x , t dx .

This leads to the normalization condition 1
0

describes the velocity of phase space volume change. Equation 9 is a fractional Liouville equation in the Hamilton picture. If the equation of motion has the form dx dt
t

x

1

y

x,t

y

x , t dx 1.

5

Ft x , is defined by
1 t

If we denote ~ x,t and d x x
1

then the function y x,t y x,t 6 xt ,t dx , 7

d ln x dt 1

ln
t

x x

t 0

1 dx x t dt

dx t . x t dt

then condition 1 has the form ~ x,t d
0

As the result we have 8 xt ,t 1F xt
t

x

1. ct in 6 , we get the sum ct x t , t .

Ft . xt

Note that substituting y ~ xt ,t ct x t , t

This sum can be considered as a sum of right and back waves of the distribution functions.

The normalization in the phase space is derived by analogy with a normalization in the configuration space. The fractional normalization condition in the phase space ~ q,p,t d
0 0

q,p

1,

10

III. LIOUVILLE EQUATION

where d d

( q , p ) has the form q,p d qp
2

Let us consider a domain B 0 for the time t 0. In the Hamilton picture we have ~ xt ,t d x
t

q d
1

q

dq dp
2

Bt

B0

~ x 0 ,0 d
t

x0 . x t ( x 0 ) , where x 0 is a

dq dp .

11

Using the replacement of variables x Lagrangian variable, we get ~ xt ,t x
1 t

The distribution function ~ ( q , p , t ) in the phase space is defined by ~ q,p,t q q q,p q,p p,t p,t q q q,p q,p p,t p,t .

B0

xt dx x0

0

B0

~ x 0 ,0 x

1 0

dx 0 .

Since B 0 is an arbitrary domain we have ~ xt ,t d or ~ xt ,t x
1 t

Let us use the well-known transformation dq t dp
t t0

x

t

~ x 0 ,0 d

x0 ,

qt ,p

t0

dq 0 dp 0 ,

12

xt x0 xt x0

~ x 0 ,0 x

1 0

where q t , p nant . qt ,p
t0

is Jacobian which is defined by the determiqt ,p q0 ,p q kt / q p kt / q q kt / p p kt / p

Differentiating this equation in time t , we obtain d~ x t , t x dt or d~ x t , t dt xt ,t ~ xt ,t 0, 9
1 t

det

t 0

det

l0 l0

l0 l0

.

~ xt ,t

d x dt

1 t

xt x0

0,

Using ~ t d tion ~ qtp
t 2 t 1

( q t , p t) ~ 0 d dq t dq

( q 0 , p 0 ) , we get the relaq0p
0 1

t

~

0

2

dq 0 dq 0 .

13

Using 12 , we have condition 13 in the form

Downloaded 17 Dec 2003 to 199.98.104.161. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp

Chaos, Vol. 14, No. 1, 2004

Fractional generalization

125

~ t qtp

t

1

qt ,p

t0

q0p

0

1

~0.

14

q,p

1q q,F 1 .

1

G q,p

p

1

F q,p

G,p

1

Let us write condition 14 in more simple form ~ t qt ,p
t 0 2

23

q0p

0

1

~0.

15

The time derivatives of this equation lead to the fractional Liouville equation d~ q t , p t , t dt where the function qt ,pt ,t qt ,pt ,t ~ qt ,pt ,t is defined by qt ,p
1 t 0

This relation allows to derive for all dynamical systems 22 . It is easy to see that the usual nondissipative system dq dt
t

pt , m

dp dt

t

f qt ,

24

0,

16

has the -omega function q,p 1 mq p
1

p

2

mq f q

d q ,p dt t

t

0

d ln q t , p dt

t

0

.

and can be called a fractional dissipative system. For example, the linear harmonic oscillator ( f ( q ) m 2q ) q,p 1 mq p
1

17 In the usual notations we have qt ,pt ,t d ln det dt qt ,p q0 ,p
t 0

p

2

m

2

22

q,

is a fractional dissipative system. . 18
IV. INTERPRETATION

Using well-known relation ln det A Sp ln A, we can write the -omega function in the form dq
t

dt where A,B

,p

t

qt ,

dp

t

dt

,

A q

B p

A p

B . q

The fractional normalization condition can be considered as a normalization condition for the distribution function in a fractional phase space. In order to use this interpretation we must define a fractional phase space. The first interpretation of the fractional phase space is connected with fractional dimension. This interpretation follows from the well-known formulas for dimensional regularizations:18 x d nx 2
n /2

In the general case ( 1 ) the function is not equal to 0 ) for Hamiltonian systems. If 1, we have zero ( 0 only for non-Hamiltonian systems. It is easy to see that any system which is defined by the equations dq
t

n /2

xx
0

n1

dx .

25

Using Eq. 25 , we get that the fractional normalization condition 8 can be considered as a normalization condition in the fractional dimension space /2 2
/2

dt

pt , m

dp

t

dt

f qt ,

19

~ x,t d x 1

26

0. This system has the -omega function equal to zero can be called a fractional nondissipative system. For example, a fractional oscillator is defined by the equation dq
t

p

t

dt

m

,

dp

up to the numerical factor ( /2)/(2 /2 ( )). The second interpretation is connected with the fractional measure of phase volume. The parameter defines the space with the fractional phase volume V
B

t

dt

m

2

qt .

20 d q,p .

The -omega function can be rewritten in the form qt ,pt ,t 1 q
1 t

dq dt ,

t

p

1 t

dp dt

t

dq t ,p dt

t 1

It is easy to prove that the velocity of the fractional phase volume change is defined by dV dt q,p,t d
B

dp qt , dt

t 1

q,p .

21

where .,. 1 is usual Poisson bracket. If the Hamilton equations have the form dq dt
t

Note that the volume element of fractional phase space can be realized by fractional exterior derivatives19
n n

G qt ,pt ,

dp dt

t

d F qt ,pt , 22
k1

dq

k

q

k

a

dp
k k1

k

p

k

b

,
k

27 in the form

then the -omega function is defined by

Downloaded 17 Dec 2003 to 199.98.104.161. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp

126

Chaos, Vol. 14, No. 1, 2004

Vasily E. Tarasov
n

dq dp

2

4 2
2

H q ,p
k1 1

1 g 2

kl

q, p pkp

l

Uq.

34

1 1

qp

1

d qd p.

28

Note that the Hamiltonian 34 defines a nonlinear onedimensional sigma-model20,21 with metric g
kl

The system can be called a fractional dissipative system if a fractional phase volume changes, i.e., 0. The system which is a nondissipative system in the usual phase space, can be a dissipative system in the fractional phase space. The fractional analog of the usual conservative Hamiltonian nondissipative system is defined by the equations dq
k

q,p

m

1 2( pk

1) kl

.

dt

gk p , m

dp

k

dt

fk q .

29

The usual nondissipative systems 24 are dissipative in the fractional phase space. In the general case, the fractional system is a system in the fractional phase space. We shall say that a system is called a fractional system if this system can be described by the fractional powers of coordinates and momenta, q
k

It is easy to see that fractional systems 33 can lead to the non-Gaussian statistics. The interest in and relevance of fractional kinetic equations is a natural consequence of the realization of the importance of non-Gaussian statistics of many dynamical systems. There is already a substantial literature studying such equations in one or more space dimensions. Note that the classical nonlinear dissipative systems can have canonical Gibbs distribution as a solution of stationary Liouville equations for this dissipative system.22 Using the methods,22 it is easy to prove that some of fractional dissipative systems can have fractional canonical Gibbs distribution q,p Z T exp H q ,p kT ,

q

k

,

p

k

p

k

,

where k 1, ..., n . The fractional systems allow to consider the interpretation of the fractional normalization condition which is used to derive the fractional Liouville equation. The fractional normalization condition for the distribution function can be considered as a normalization condition for the systems in the fractional phase space. The Hamilton equations for the fractional system have the form dq
k

as a solution of the fractional Liouville equations p t
k

m

q

k

p

Fk q ,p
k

0.

35

Here the function H ( q , p ) is defined by 33 .

V. CONCLUSION

p

k

dt

m

,

dp

k

dt

Fk q ,p .

30

Obviously, that the equation dp k / dt F k can be rewritten in the fractional form. Multiplying both sides of this equation p 1 F k . However the by p 1 , we obtain dp k / dt equation dq k / dt p k / m cannot be rewritten in the fractional form dq k / dt p k / m . The fractional conservative Hamiltonian system is described by the equation dq
k

H q ,p p
k

dt or dq
k

,

dp

k

H q ,p q
k

dt

,

31

dt

qk ,H

,

dp

k

dt

pk ,H

.

32

Here H H ( q , p ) is a fractional analog of the Hamiltonian function
n

H q ,p
k1

p

2 k

2m

Uq .

33

The fractional system can be considered as a nonlinear system with

Derivatives and integrals of fractional order have found many applications in studies of scaling phenomena.15 In this paper we formulate fractional analog of main integrodifferential equation to describe some scaling process-- Liouville equation. We consider the fractional analog of the normalization condition for the distribution function. Fractional Liouvile equation for dissipative systems is derived from the normalization condition. In this paper, the coordinate fractional integration for the normalization condition is used. This fractional normalization condition can be considered as a simulating unconventional environment for systems with fractional dimensional phase space or phase space with fractional powers of coordinates and momentums. Note that the adoption of fractal formalism yields properties that the ordinary formalism would produce only in the case where the system is made non-Hamiltonian by the presence of an environment, whose influence can be mimicked by means of friction, for instance. Suggested fractional Liouville equation allows to formulate the fractional equation for quantum dissipative systems23 by methods suggested in Refs. 24 and 25. In general, we can consider this dissipative quantum systems as quantum computer with mixed states.26 These dissipative quantum systems can have stationary states.27 Stationary states of dissipative quantum systems can coincide with stationary states of Hamiltonian systems.28

Downloaded 17 Dec 2003 to 199.98.104.161. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp

Chaos, Vol. 14, No. 1, 2004

Fractional generalization
14 15

127

ACKNOWLEDGMENT

This work was partially supported by the RFBR Grant No. 02-02-16444.
1 2 3

16

17

4 5 6 7 8 9 10 11

12 13

T.F. Nonnenmacher, J. Phys. A 23, L697 1990 . W.G. Glockle and T.F. Nonnenmacher, J. Stat. Phys. 71, 741 1993 . R. Metzler, W.G. Glockle, and T.F. Nonnenmacher, Physica A 211, 13 1994 . M.F. Shlesinger, J. Stat. Phys. 36, 639 1984 . G.M. Zaslavsky, Phys. Rep. 371, 461 2002 . E. Lutz, Phys. Rev. E 64, 051106 2001 . M. Giona and H.E. Roman, J. Phys. A 25, 2093 1992 . H.E. Roman and M. Giona, J. Phys. A 25, 2107 1992 . W. Wyss, J. Math. Phys. 27, 2782 1986 . W.R. Schneider and W. Wyss, J. Math. Phys. 30, 134 1989 . W.R. Schneider, Dynamics and Stochastic Processes. Theory and Application, Lecture Notes in Physics. Vol. 355 Springer-Verlag, Berlin, 1990 , pp. 276 286. G. Jumarie, J. Math. Phys. 33, 3536 1992 . H.C. Fogedby, Phys. Rev. Lett. 73, 2517 1994 .

18

19 20 21 22 23

24 25 26 27 28

G.M. Zaslavsky, Physica D 76, 110 1994 . A. Isihara, Statistical Physics Academic, New York, 1971 , Appendix IV. P. Resibois and M. De Leener, Classical Kinetic Theory of Fluids Wiley, New York, 1977 , Sec. IX.4. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Integrals and Derivatives of Fractional Order and Some of its Applications Nauka i Tehnika, Minsk, 1987 ; Fractional Integrals and Derivatives Theory and Applications Gordon and Breach, New York, 1993 . J.C. Collins, Renormalization Cambridge University Press, Cambridge, 1984 , Sec. 4.1. K. Cottrill-Shepherd and M. Naber, J. Math. Phys. 42, 2203 2001 . V.E. Tarasov, Mod. Phys. Lett. A 26, 2411 1994 . V.E. Tarasov, Phys. Lett. B 323, 296 1994 . V.E. Tarasov, Mod. Phys. Lett. B 17, 1219 2002 . V.E. Tarasov, Mathematical Introduction to Quantum Mechanics MAI, Moscow, 2000 . V.E. Tarasov, Phys. Lett. A 288, 173 2001 . V.E. Tarasov, Moscow Univ. Phys. Bull. 56, 5 2001 . V.E. Tarasov, J. Phys. A 35, 5207 2002 . V.E. Tarasov, Phys. Lett. A 299, 173 2002 . V.E. Tarasov, Phys. Rev. E 66, 056116 2002 .

Downloaded 17 Dec 2003 to 199.98.104.161. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp