The Existence of Pair Potential Corresponding to Specified Density and Pair Correlation

TL;DR

This paper proves that for sufficiently small density and pair correlation functions, there exists a pair potential and activity level that produce these correlations in a Gibbs distribution, addressing an inverse problem in statistical mechanics.

Contribution

It establishes the existence of a pair potential and activity corresponding to specified small density and pair correlation functions, solving an inverse problem.

Findings

Existence of pair potential for given small correlations

Construction of Gibbs distribution with prescribed correlations

Extension of known results to inverse correlation problem

Abstract

Given a potential of pair interaction and a value of activity, one can consider the Gibbs distribution in a finite domain $\Lambda \subset \mathbb{Z}^d$. It is well known that for small values of activity there exist the infinite volume ($\Lambda \to \mathbb{Z}^d$) limiting Gibbs distribution and the infinite volume correlation functions. In this paper we consider the converse problem - we show that given $\rho_1$ and $\rho_2(x)$, where $\rho_1$ is a constant and $\rho_2(x)$ is a function on $\mathbb{Z}^d$, which are sufficiently small, there exist a pair potential and a value of activity, for which $\rho_1$ is the density and $\rho_2(x)$ is the pair correlation function.