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Journal of Mathematical Physics

Green's Functions and Double-Time Distribution Functions in Classical Statistical Mechanics

John C. Herzel

Published Aug 1, 1967• 6 citations
Publisher

Abstract

This paper rederives the Bogoliubov/Sadovnikov classical-equilibrium-correlation Green's function hierarchy by using the double-time theory of Rostoker. Thus the traditional variational technique is avoided.
DOI
10.1063/1.1705404

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The present paper is concerned with the retarded many-time thermodynamic Green's function method in classical statistical mechanics. In the formulation of the nonlinear response of a dynamical variable, unaveraged Green's functions are specified and used to discuss the physical meaning of many-time Poisson-bracket nests. Equilibrium ensemble averages of these Green's functions are relevant to the classical Kubo response theory, the latter of which is interpreted from the viewpoints of both the classical ``Heisenberg'' and ``Schrödinger'' pictures. A method for evaluating the many-time thermodynamic Green's functions based on their definition in terms of the resolvent operator is suggested and discussed in relation to the linear dielectric function and the anharmonic oscillator. The many-time Green's function equations of motion are shown to be equivalent to identities based on the resolvent operator. It is shown how microscopic equations of motion can be used to derive the Bogoliubov-Sadovnikov (B-S) hierarchy from the lowest-order many-time Green's function equation of motion.
The basic mathematical properties of Green's functions used in statistical mechanics as well as the equations defining these functions and the techniques of solving these equations are reviewed. An approach is presented called the Correlated Iteration Theory, which has been developed by the author. This approach differs from all other known variants of perturbation theory for Green's functions by the combination of two factors: the systematic formulation of an algorithm for obtaining subsequent approximations and the consistent consideration of interparticle correlations at each step of the procedure.
It is shown that the correct lowest-order two-particle correlation function solution to the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy is given via a Green's function whose Fourier transform satisfies the analyticity requirements for causality. The Bogoliubov adiabatic approximation is shown to imply a solution which is acausal. Extension to the sss-particle correlation function is discussed.
Thermodynamic perturbation theory is described in terms of temperature-dependent Green functions. The relationship of these functions with time-temperature-dependent Green functions and with correlation functions is discussed and some of their analytic properties are derived. The chain of coupled equations forming the equations of motion of the Green functions is derived and is written in terms of functional derivatives. Two examples of the use of Green functions, discussions of the sharpness of the Fermi surface and of neutron scattering from liquid helium, are given. The use of Green functions in the theory of relaxation phenomena is described.