Evaluating the partition function for systems with long range interactions

TL;DR

This paper presents a method to evaluate the partition function of systems with long-range interactions by expanding around the mean-field solution and including Gaussian corrections, providing a systematic approach to account for fluctuations.

Contribution

It introduces a functional integral approach and a saddle point expansion to accurately compute the partition function beyond mean-field theory for long-range interacting systems.

Findings

Partition function expressed as a functional integral over density fields

Saddle point corresponds to mean-field solution

Gaussian corrections improve the approximation accuracy

Abstract

We express the partition function for an equilibrium system of interacting particles in the canonical ensemble as a functional integration over the particles' density field. We outline a method to evaluate the partition function by expanding around a saddle point. The saddle point is shown to be the solution of the equivalent mean-field theory. Leading corrections to the mean-field theory takes the form of a Gaussian integral.