Asymptotic behavior of Green's functions and correlation functions in the statistical theory of nonequilibrium processes

Abstract

Expressions are obtained for the asymptotic Green's and correlation functions of arbitrary operators which are not integrals of the motion. The derivation of the expressions presupposes only the existence of a set of operators (coarse-grain variables) such that the variation in time of their mean values determines the slow evolution of the system. It is shown that the correlation functions of the coarse-grain variables, the fluxes, and their "random" components can be expressed in terms of the same transport coefficients. The nonequilibrium statistical operator method is used to derive dynamical equations with sources for the asymptotic Green's functions of the coarse-grain variables. In the statistical theory of nonequilibrium processes [1, 2], one can, using asymptotic solutions of the Liouville equation, derive macroscopic equations describing the slow variation in time of the coarse-grain variables which specify the nonequilibrium state of the system. The transport coefficients in these equations can be expressed in terms of the correlation functions of the random fluxes. It is also of interest to consider the "inverse" problem of obtaining the low-frequencylas ymptotic behavior of the Green's functions and the correlation functions of various dynamical variables using the equations for the coarse-grain variables [3-5]. On the one hand, this asymptotic behavior is universal, since at large times the evolution of the system is basically determined by the variation of the coarse-grain variables. On the other hand, the asymptotic Green's functions relate the low-frequency component of the fluctuation spectrum of the dynamical variables, which is measured, for example, in inelastic neutrorl scattering experiments, to the macroscopic transport coefficients. There exist two main approaches to the calculation of the low-frequency asymptotic behavior of Green's functions. The method based on introducing into the Hamiltonian additional terms describing the interaction of the system with external fields was proposed by Bogolyubov [3] and developed later in various papers [6-10]. Sources appear in the hydrodynamic equations, and the variations with respect to them of the hydrodynamic quantities are related to the asymptotic Green's functions. The indirect linear response method of Kadanoff-Martin [4, 11] consists of comparing the expressions for the response of the system to a small perturbation with the relaxation of the system to thermal equilibrium after an external field has been switched off. It is assumed that the relaxation of the system can be described by the equations of phenomenological hydrodynamics. This method was applied, for example, to a Bose superfluid in [12] and to magnetic systems in [13]. Both approaches make it possible to obtain expressions for the asymptotic Green's functions of the coarse-grain variables themselves if the macroscopic equations are known. However, it is obvious that once the slowly varying coarse-grain variables have been chosen (the set of them is determined by the physics of the considered problem) the asymptotic behavior of the Green's functions of all operators must follow from the microscopic theory without recourse to the phenomenologieat equations. In the present paper, using the Liouville equation, we obtain general expressions for the asymptotic behavior of the Green's functions and the time correlation functions of arbitrary operators that are not integrals of the motion. In the special case when the coarse-grain variables are these operators the asymptotic behavior of the Green's functions corresponds to the results of the Kadanoff-Martin theory. At the end of the paper, Zubarev's nonequilibrium statistical operator method [1, 14] is used in a different approach to the calculation of the asymptotic Green's functions based on the introduction of sources into the dynamical equations for the coarse-grain variables. It is shown in particular that sources can be introduced in the standard manner for an arbitrary set of coarseInstitute of Civil Aviation Engineers, Moscow. Translated from Teoreticheskaya i Matematicheskaya