Nonequilibrium statistical operator and generalized kinetic equations

Volume 49A, number 3

PHYSICS LETTERS

9 September 1974

NONEQUILIBRIUM STATISTICAL OPERATOR AND GENERALIZED KINETIC EQUATIONS K. BIALAS-BORGIEL, A. PAWLIKOWSKI and E. ZIPPER

Instytut FizykL UniwersytetSlaski, Katowice, Poland Received 30 July 1974 A new method of constructing the nonequilibrium statistical operator (NSO) leading to the kinetic equations is suggested. The connections with other methods are discussed. Using the orthogonal operator expansion method we derive the form of the NSO directly from the Liouville equation:

d I n p(t)

-i

~(t) In p(t)

(1)

where

We introduce the projection operator P by the relation: 171

P = ~ Or(Or, )

(6)

I=1

and write

~ ( t ) = [~(t), ],

~f(t) = ~fo + 7(0V

(2)

7(t)V is a small perturbation, 7(0 is a c-number coefficient. Let us define the scalar product in the linear space F o f the observables:

In p(t) = P In p(t) + ( I - P ) In p(t).

(7)

From (1) and (7) as in [4] we have: In p(t) = -(C(t) + B ( t ) )

(8)

where m

(A,B) = Tr {p'A+B}

(3)

C(t) = k~=l Xk(t)O k - ( l - P ) In Po'

(9)

and introduce the orthonormal set of the operators

01 ...Ore:

t

(Oi, O/) = 6i/

(4)

for which we assume [3] :

s(t) = -i

(5)

k= 1 tkOk,

f dt ' S(t,t)(1-P) '

( t ') O k h k (t')

to

+ (S(t, to)- 1 ) ( l - e ) In Po,

(10)

(Ok, In p(t)),

(11)

m

oo t

If

hk (t) = -

t

S(t, t') = Texp { - i f dr m does need to be equal to the dimension of the space F.

(12)

to Po is the initial statistical operator.

To obtain the kinetic equations for (Ol)t = Tr {OlP(t)} we expand p(t) up to the terms linear in 7(t)V: e_(t ) t p(t) ~, ~ 1 - i j{dt' e x p [ - i ~ f o ( t - t ' ) ] l?(t')C(t')}: Tr e -(t) to For such V that PVP = O, these equations have the form: m

t

d (Ol) t = - i ~ ap.(Oz.) t - iT(t)([O l, Vi )c + k=l

(13)

~

,.

fdt' ~l(t)~/(t')([V(-t+t'), [0t, ¢1 ]>c',

(14)

to 225

Volume 49A, number 3

PHYSICS LETTERS

9 September 1974

where

V(-t+t')=exp[-i~fo(t-t')]V , (...)c=(Tre-C)-lTr{e-C...). Eq. (14) will be the kinetic equation for

(Ol)t,

(15)

if

(01)t = ( 0 l ) c.

(16)

It can easily be shown, that the relation (16) is fulfilled for the operator (13), the same with which the kinetic equation (14) was derived. After multiplying (7) by Oko' and taking the trace we obtain linear, exact equations for ),k(t): m

~Xk(t ) =--i /-1 ~(Ok,~(t)O])),l.(t)-i(Ok,~(t)S(t, to)(1--P)lnOo ) rn

t

-/~=1-t! dt ' (Ok' ~f(t)S(t,t')(1-P)~f(t)O/)~/(t').

(17)

We can solve eqs. (17), and then, because of (16), compute m

(Ol)t = Tr {0 l e x p [ - ~ ~,k(t)Ok +(1--/') In Po] ) { Tr e-C) -1 k=l

(18)

The kinetic equations obtained by this method have the same form as those obtained by Zubarev [1]. One of the differences with the Zubarev method is the equations for parameters ~k(t). The present method seems to be more general than the Shimizu [2] one, because we use it to consider equally well phenomena very far from equilibrium and those which occur in systems being in thermal contact with a reservoir. It seems, that the Shimizu method is valid in the high temperature approximation only. One of the authors (A.P.) wishes to thank Professor D.N. Zubarev for a discussion.

References [1 ] [2] [3] [4]

226

D.N. Zubarev, Nonequilibriumstatistical thermodynamics (Nauka, Moscow, 1971) (in Russian). T. Shimizu, J. Phys. Soc. Japan 28 (1970) 811. S.V. Peletminskiand A.A. Yatsenko, JETP 53 (1967) 1327. R. Zwanzig, J. Chem. Phys. 33 (1960) 1338.