Phase transitions and coarse-graining for a system of particles in the continuum

TL;DR

This paper extends the proof of liquid-vapor phase transition in particle systems to include hard-core interactions, using coarse-graining and cluster expansion techniques to analyze phase behavior in mean-field and finite-range interactions.

Contribution

It introduces a method to incorporate hard-core interactions into phase transition proofs using coarse-graining and cluster expansion, extending previous results.

Findings

Established phase transition for mean field limit

Proved phase transition with long but finite interaction range

Developed a density model via cluster expansion

Abstract

We revisit the proof of the liquid-vapor phase transition for systems with finite-range interaction by Lebowitz, Mazel and Presutti and extend it to the case where we additionally include a hard-core interaction to the Hamiltonian. We establish the phase transition for the mean field limit and then we also prove it when the interaction range is long but finite, by perturbing around the mean-field theory. A key step in this procedure is the construction of a density (coarse-grained) model via cluster expansion. In this note we present the overall result but we mainly focus on this last issue.