The Kirkwood closure point process: A solution of the Kirkwood-Salsburg equations for negative activities

TL;DR

This paper proves the existence of the Kirkwood closure process for stable and regular pair potentials, extending previous results, and shows it is a Gibbs process satisfying a Kirkwood-Salsburg type equation.

Contribution

It demonstrates that stability and regularity of the pair potential are sufficient for the existence of the Kirkwood closure process, generalizing prior conditions.

Findings

Existence of Kirkwood closure process under stability and regularity.

The process is Gibbs for locally stable potentials.

Kernel satisfies a Kirkwood-Salsburg type equation.

Abstract

The Kirkwood superposition is a well-known tool in statistical physics to approximate the $n$-point correlation functions for $n\geq 3$ in terms of the density $\rho $ and the radial distribution function $g$ of the underlying system. However, it is unclear whether these approximations are themselves the correlation functions of some point process. If they are, this process is called the Kirkwood closure process. For the case that $g$ is the negative exponential of some nonnegative and regular pair potential $u$ existence of the the Kirkwood closure process was proved by Ambartzumian and Sukiasian. This result was generalized to the case that $u$ is a locally stable and regular pair potential by Kuna, Lebowitz and Speer, provided that $\rho$ is sufficiently small. In this work, it is shown that it suffices for $u$ to be stable and regular to ensure the existence of the Kirkwood closure process. Furthermore, for locally stable $u$ it is proved that the Kirkwood closure process is Gibbs and that the kernel of the GNZ-equation satisfies a Kirkwood-Salsburg type equation.