Finite volume corrections and decay of correlations in the canonical ensemble

TL;DR

This paper analyzes finite volume effects and decay of correlations in a classical particle system with finite-range interactions, providing bounds on free energy errors and correlation decay rates in the canonical ensemble.

Contribution

It offers explicit estimates for finite volume corrections and decay of correlations, extending the understanding of thermodynamic limits in classical particle systems.

Findings

Finite volume free energy error bounded by surface-to-volume ratio.

Correlation contributions from ideal gas are of order 1/N.

Interaction contributions decay exponentially with distance.

Abstract

We consider a classical system of $N$ particles confined in a box $\Lambda\subset\mathbb{R}^d$ interacting via a finite range pair potential. Given the validity of the cluster expansion in the canonical ensemble we compute the error between the finite and the infinite volume free energy and estimate it to be bounded by the area of the surface of the box's boundary over its volume. We also compute the truncated two-point correlation function and find that the contribution from the ideal gas case is of the order $1/N$ while the contribution of the interactions is of the order $1/|\Lambda|$ plus an exponentially small error with the distance.