Cluster Expansion and Autocorrelation Functions: Self-Diffusion Coefficient

Abstract

A density series of the autocorrelation function for the self‐diffusion coefficient of an isotropic fluid is obtained by means of a cluster expansion (different from that used by Ernst, Haines, and Dorfman) of the resolvent operator corresponding to the Liouville operator for the system. Nonsingular potentials are assumed for the interaction potential between pairs of the particles. It is shown that the self‐diffusion coefficient, and thus transport coefficients in general, can be put into a form similar to that of equilibrium statistical mechanics, when the resolvent operator is expanded in terms of clusters just as the configuration integral is expanded in terms of the cluster integrals in equilibrium statistical mechanics. The cluster functions may also be expanded into irreducible cluster functions. This decomposition of the clusters into the irreducible clusters is derived from the integral equations for the collision operators corresponding to the clusters, and makes it possible to establish formally, to an arbitrary order of density, Zwanzig's procedure of inverting the resolvent operator, G_N (ε). The density coefficients of the inverted operator, B_m+1 (ε), are given in terms of the irreducible cluster functions and would exist as $\text{ε→ 0}$ , provided that the collision operators fulfill certain analytic properties. On the assumption of an exponential potential for a two‐dimensional gas, B₃ (ε) is estimated to be finite in the very weak interaction limit as $\text{ε→ 0}$ .