Equations for space-time and time correlation functions and proof of the equivalence of results of the Chapman–Enskog and time correlation methods

A study is made of the problem of determining the space-time or time correlation functions in a many-body classical system. The general correlation function of dynamical variables of a binary type (it is precisely these functions that are encountered in applications) is expressed in terms of the first two terms of a sequence of functions that depend on an increasing number of arguments and satisfy Bogolyubov's chain of equations with known initial data. In the lowest order in the density the correlation function can be expressed solely in terms of the first function of the sequence. This function is the solution of the initial-value problem for the linearized Boltzmann equation. An investigation is made of the initial-value problems for the correlation functions that determine the transport coefficients of simple and multicomponent gases. This investigation renders it possible to give a simple, complete, and rigorous proof of the results of the Chapman–Enskog and correlation function methods. The proof is based on the well-known properties of the linearized collision operator and it is possible to avoid the divergences encountered in other investigations.