Volume 65A, number 1 PHYSICS LETTERS 6 February 1978

EFFECT OF INITIAL PAIR CORRELATIONS ON THE KINETIC EQUATION

DESCRIBING A TWO DIMENSIONAL NON-RELATIVISTIC CLASSICAL GAS

Mahendra PRASAD Clarendon Laboratory, Oxford OX1 3PU, UK and Department of Theoretical Physics, Oxford OX1 3NP, UK

Received 8 December 1977

The effect of initial pair correlations on the kinetic equation describing a two-dimensional classical non-relativistic gas is investigated. It has a maximum at time t = 0, vanishes in the large time limit as t -2 , and ceases to influence the dynamics after the collision duration.

The time evolution of a system of particles is governed by a chain of coupled equations, known as the B o g o l i u b o v - B o r n - G r e e n - K i r k w o o d - Y v o n (BBGKY) hierarchy [1]. Exact solutions of these equations are often not possible and approximate decoupling me- thods have been developed by various authors [1 ] for the resolution of the hierarchy. These methods differ in physical ideas introduced in the diagram analysis for treating the collision and correlation between any two particles of the system.

The main effect o f a two-body potential on the evolution of a one-body density matrix at any time t is to introduce correlation between the particles after the initial setting of the system at t = 0 and to describe a collision dynamics at a later time r (0 < r < t). In fact this is the philosophy of the connected diagram method for obtaining a closed equation for the evolu- tion of the one-body density matrix as described by Fujita [2]. This equation, although closed, does not offer analytic solution. The collision term is non-linear and the correlation term remembers the state of the system at t = 0 and is therefore non-Markovian. These terms are most suitably described in terms of diagrams, in particular in the asymptotic time limit the collision term reduces to Boltzmann's collision term. The effect of the initial correlation term on the kinetic equation for a three-dimensional classical gas has been studied by Lee et al. [3]. They found that the correlation term ceases to influence the dynamics after a time of the

same order as the duration of the collision. Also the decay of the correlation was found to be independent o f the shape of the short-range two-body central po- tential.

The purpose of the present paper is to examine these questions for a two-dimensional classical, nbi~-ieiati- - vistic gas interacting via a two-body short-range central potential.

Let us consider a system of N non-relativistic distin- guishable particles moving in a plane and interacting with each other via a two-body central potential. The hamiltonian of the system is given by

N

n = ~ + x ~ V(Ir i - rjl). (1) i=1 ~ i</"

r and p are two-dimensional vectors, m is the mass of a particle (extension to two different masses is imme- diate) and X is a coupling parameter. The evolution of the one-body density matrix n(t) depends on the two- body density matrix n(212)(t) through the following hierarchy equation:

( a /a t + i/~o) n(t) = - i k t r (2) [~r (12)n~12)(t)], (2)

where/~0 and 1?(12) are the Liouville operators corre- sponding to h 0 and V (12) respectively, tr (2) means the single particle trace with respect to the second particle. The interaction term on the fight hand side of the above equation has been analysed by Fujita [4] through proper

Volume 65A, number 1 PHYSICS LETTERS 6 February 1978

[- "~

I i I I & ]'

Fig. 1. Lowest-order correlation diagram, whose expectation value in the momentum representation is given by eq. (5).

connected diagrams. This analysis yields the following result:

~-~ i/~ 0 n ( t ) = ~ g I ln+ ~ d [In ~ lx . (3)

The two terms on the right hand side of the above equa- tion are called the "collision" and "initial pair correla- t ion" terms and are defined through the proper con- nected diagrams. The ×'s in the above equation are given as initial condition and are defined through the following set of equations:

n(1) = X(1), n (12) = n(1)n(2) + X~212) ,

n~ 123) = n(1)n(2)n(3) + n(1)X(223)

+ n(2)X~ 13) + n (3)',,(12) + v(123) (4) "~2 "~3 "

The collision term in eq. (3) has been shown to re- duce to the well-known Boltzmann collision term.

The lowest order correlation diagram is defined through fig. 1, and its contribution is given by the operator

--i X tr(2) [ IT"(12)exp {--i t(/~ (1) +/~(2))} X(12)] .

In the momentum representation its expectation value is given by (5)

R (p, t) = <p[ tr(2)[ V (I 2)exp { - i t(/~ (01)+/~(02))} X (1 2) 1 [p).

In order to calculate the above matrix element one uses the p-k representation introduced by Prigogine and coworkers [5], and we obtain the following result:

R(p, t) = - i X fd2p ' fd2q V(q)

X exp [ - i t { ( p +q)2 _p2+(p,_q)2_p'2}/2m ]

X (p + q, p ' - q [ X (1 2)[p, p, > + c.c . , (6)

V(q) being the Fourier transform of the interaction potential.

Let us assume X2 to be an equilibrium correlation operator given by

X (12)= - X f d ~ 1 exp [-/31 {h(1) + h ( 2 ) u u ) + 2a.] V (12) . (7) 0

Inserting eq. (7) into eq. (6) one gets

R(p, t) - i X 2 f d2p' f d2q = fd~llV(q)12 o

X exp [ ( - i t -[Jl){(p+q)2-p 2 + ( p ' _ q)2 _p'2}/2m ]

X fo(P)Io(P') + c.c. , (8)

where fo(P) = e'~-~P 2/2rn is the Maxwell-Boltzmann distribution function. In the limit of large time, the most important contribution comes from the region where (p + q)2 _ p2 + ( p ' _ q)2 _ p'2 = 0; then the/31 integration can be replaced by/3. This will be a good approximation when the temperature is large. Then we have

R(p, t) = -iX2[3fo(P) f d2p ' f d2q[V(q)[2fo(p') (9)

X exp [ - i t {(p + q)2 _ p2 + (p ' _ q)2 _ p'2}/2 m ] + c.c.

The integration over p' can be easily effected and the result is given by

R(p, t) = - iX2fo (p ) 27rme ~ fd2qlV(q)l 2

× [- l,,q=+"°.° m m - - + c . c . .

Let us assume that the interaction between the two particles be described by a gaussian force

Vo V(r) = exp (-r2/d2), (11)

zrd 2

V 0 and d being the strength and the range of the poten- tial respectively (V 0 > 0 for repulsive potentials, V 0 < 0 for attractive potentials). Then the Fourier transform of this potential is given by

V(q) = [Vo/(2zr)2] exp [-q2d2/4]. (12)

Substituting this expression in eq. (I0), the q-inte- gration can be easily carried out and the result is given by

Volume 65A, number 1

v°2 I, + R(p,t)=ik227rmea(-~n)4 o ( P ) ( ~ it+m ~ - ~ , t2 ] - 1

p2t2 t 2 -1

Thus we see that the initial correlation term has a maximum at t = 0, the initial time point and vanishes. in the large time limit as t -2 . This decay is slower than that for a three-dimensional system, investigated by Lee et al. [3]. Also when the time is given by t d = d/u~, mu~ --- k~T, the effect o f the correlation term ceases to

PHYSICS LETTERS 6 February 1978

influence the dynamics. This turns out to be a general feature of eq. (3).

References

[ 1 ] N. Bogoliubov, Problems of a dynamical theory in statistical physics (OCIS, Moscow, 1946).

[2] S. Fujita, J. Phys. Soe. Japan 26 (1969) 505. [3] D. Lee, S. Fujita and F. Wu, Phys. Rev. A2 (1970) 854. [4] S. Fujita, Physica 64 (1973) 1, 23; 69 (1973) 355; 76 (1974)

373,395. [5] P. R6sibois and I. Prigogine, Physica 28 (1962) 1;

P. R6sibois, Physica 27 (1961) 541.

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