Convergence of density expansions of correlation functions and the Ornstein-Zernike equation

TL;DR

This paper proves the convergence of multi-body correlation functions and the direct correlation function in a density expansion, establishing their relation via the Ornstein-Zernike equation in the thermodynamic limit.

Contribution

It introduces a convergent power series for the direct correlation function in the canonical ensemble and connects it to the Ornstein-Zernike equation.

Findings

Convergence of the correlation function as a power series in density.

Characterization of coefficients via sums over two-connected graphs.

Validation of the Ornstein-Zernike equation in the thermodynamic limit.

Abstract

We prove convergence of the multi-body correlation function as a power series in the density. We work in the context of the cluster expansion in the canonical ensemble and we obtain bounds uniform in the volume and the number of particles. In the thermodynamic limit, the coefficients are characterized by sums over some class of two-connected graphs. We introduce the "direct correlation function" in the canonical ensemble and we prove that in the thermodynamic limit it is given by a convergent power series in the density with coefficients given by sums over some other class of two-connected graphs. Furthermore, it satisfies the Ornstein-Zernike equation from which quantified approximations can be derived.