Volume 69A, number 6 PHYSICS LETFERS 22 January 1979 GENERALIZED KINETIC EQUATIONS WITH MEMORY J. LUCZKA Institute of Physics, Silesian University, Katowice, Poland Received 20 November 1978 Applying Zubarev’s nonequilibrium statistical operator the integro-differential kinetic equations are derived. The con- nections with other equations are discussed. We shall consider a system in a nonequilibrium state, which can be described by the expectation values <P~ = Tr [p (t)Pk] of a limited set of operators ~k• The choice of the operators ~k is definite from the form of the hamiltonian H of the system and its symmetry [1]. The nonequilibrium statistical operator (NSO) p(t) can be constructed in such a way that it depends on the operators ~k and functions (Pk)t only. In order to fmd the func- tions (Pk>t one must solve a set of integro-differential equations for (Pk>t, which can be obtained by averaging with p (t) the equations of motion for the operators ~k• In this paper we utilize Zubarev’s NSO, which satisfies the Liouville equation with an infinitesimal source [2]: dp(t)/dt + iLp(t) = —e [p(t) — p 2(t)] (1) and is subject to the initial condition p(—°°) = pg(—°°). The limit ~Omust be taken after thermodynamic limit. Here L = [H, ...] /h, pQ(t) = Q— 1(t) exp [—~kFk (t)Pk] is the local equilibrium statistical operator, Q(t) = Tr {exp [~ F~ (t)Pk ] } takes care of the normalization of PQ (t), and Fk(t) are the thermodynamic variables conjugate to (Pk>t in the nonequilibrium thermodynamics sense. They are determined so that: (Pk)t=(Pk)~. (2) (~••)Z’ denotes the average over the NSO, (...)~denotes the local average. The formal solution of eq. (1) has the form: p(t) = exp [(t 0 — t)(e + iL)] p(t0) + ef dt~ exp [(t1 — t)(e + iL)] PQ (t1), (3) to where we choose t0 = ~oo• The expectation value of an arbitrary operator A calculated with NSO (3) is given by: i dF (t ) a(A(t — t ))t1 (A)t = (A>Q~— f dt1e~t1_O ~[HA(t t1)J>~1 + 1 8F~(t1) ) (4) To obtain (4) we have integrated by parts the second term of the right-hand side of eq. (3). Let us consider a quantum-mechanical system with the hamiltonian [1]: HH0+H1, [HO,Pk]—EakflPfl, (5) 393