Analysis of Classical Statistical Mechanics by Means of Collective Coordinates

TLDR

Collective coordinates are employed to approximate a three‑dimensional classical many‑body system, assuming known two‑body correlation functions. A self‑consistent integral equation derived from this approximate statistical state is solved in a virial expansion to determine the two‑body correlation function and the system’s thermodynamic properties. The approach reproduces the first three virial coefficients exactly and the fourth almost exactly for hard‑sphere systems.

Abstract

The three-dimensional classical many-body system is approximated by the use of collective coordinates, through the assumed knowledge of two-body correlation functions. The resulting approximate statistical state is used to obtain the two-body correlation function. Thus, a self-consistent formulation is available for determining the correlation function. Then, the self-consistent integral equation is solved in virial expansion, and the thermodynamic quantities of the system thereby ascertained. The first three virial coefficients are exactly reproduced, while the fourth is nearly correct, as evidenced by numerical results for the case of hard spheres.