On quasi-continuous approximation in classical statistical mechanics

TL;DR

This paper introduces a quasi-continuous approximation method for classical statistical mechanics systems, showing that the approximate correlation functions converge to the true functions as the partition size shrinks to zero, applicable to both two-body and many-body interactions.

Contribution

The paper establishes the convergence of a new approximation scheme for correlation functions in infinite particle systems with superstable interactions, extending to many-body potentials.

Findings

Approximate correlation functions converge to true functions as partition size approaches zero.

Convergence holds for both two-body and many-body interaction potentials.

The method applies for any positive inverse temperature and fugacity.

Abstract

A continuous infinite system of point particles with strong superstable interaction is considered in the framework of classical statistical mechanics. The family of approximated correlation functions is determined in such a way, that they take into account only such configurations of particles in $\mathbb{R}^d$ which for a given partition of the configuration space $\mathbb{R}^d$ into nonintersecting hyper cubes with a volume $a^d$ contain no more than one particle in every cube. We prove that these functions converge to the proper correlation functions of the initial system if the parameter of approximation $a\rightarrow 0$ for any positive values of an inverse temperature $\beta$ and a fugacity $z$. This result is proven both for two-body interaction potentials and for many-body case.