Method of two-time Green's functions in molecular hydrodynamics

Abstract

occur in the equations for the relaxation functions as initial data, are the susceptibilities associated with the derivatives of the thermodynamic quantities. Thus, a theory based on equations of hydrodynamic type encompasses a fairly full set of properties of the investigated physical system and for this reason has been widely used [5-8]. However, such an approach has one shortcoming which means that it is not consistently microscopic, i.e., based solely on equations of motion. The point is that the static susceptibilities used as initial data are taken from outside (from experiment or from independent calculations) and cannot be obtained from the hierarchy of equations for the relaxation functions. The situation is different, for example, in the case of commutator Green's functions. Because of the lowering of the order of the correlation functions in the equal-time commutators, which in this case are initial data, a self-consistent calculation of all the quantities occurring in the equations is possible. The aim of the present paper is to construct a scheme of calculations making it possible in the case of the relaxation functions too to calculate the correlation functions occurring in the equations without going outside the framework of the employed hierarchy. This is achieved by means of equations that relate the moments of the mass operators occurring in the equations of hydrodynamic type for the Green's functions to the static susceptibilities. The scheme developed in the present paper, which we describe as consistently microscopic, combines the advantages of the hydrodynamic approach and the microscopic theory both on equations of motion of a system of interacting particles. First, the equations obtained still have a form close to that of hydrodynamic equations and reduce readily to them in the limit of low frequencies and long waves. Second, we have the possibility of calculating in a self-consistent manner and in a unified approximation not only equilibrium characteristics of the system (susceptibilities) but also transport coefficients on the basis of "first principles" without going outside the framework of the employed system of equations. In the first section, we give the basic equations for the Green's functions and relaxation functions employed in the paper. In the second section, using the usual hydrodynamic basis, we obtain generalized hydrodynamic equations and expressions for the Green's functions. We establish the connection between the transport coefficients and the correlation functions of the flux densities. We obtain the connection between the static susceptibilities and the moments of the flux correlation functions. In the third section, the mass operators are expressed in terms of higher correlation functions, which make it possible to separate more readily the free motion from the interaction. The obtained equations of molecular hydrodynamics permit