Analysis of Droplets in Lattice Systems with Long-range Kac Potentials

TL;DR

This paper studies the formation and properties of droplets in a lattice gas with long-range interactions, linking equilibrium configurations to large deviation principles and identifying conditions for droplet formation.

Contribution

It introduces a detailed analysis of droplet formation in lattice systems with Kac potentials, connecting large deviations to mesoscopic free energy minimizers.

Findings

Determined the critical density for droplet formation.

Characterized the nature of droplets as a function of temperature and system size.

Linked equilibrium configurations to large deviation principles.

Abstract

We investigate the geometry of typical equilibrium configurations for a lattice gas in a finite macroscopic domain with attractive, long range Kac potentials. We focus on the case when the system is below the critical temperature and has a fixed number of occupied sites.We connect the properties of typical configurations to the analysis of the constrained minimizers of a mesoscopic non-local free energy functional, which we prove to be the large deviation functional for a density profile in the canonical Gibbs measure with prescribed global density. In the case in which the global density of occupied sites lies between the two equilibrium densities that one would have without a constraint on the particle number, a "droplet" of the high (low) density phase may or may not form in a background of the low (high) density phase. We determine the critical density for droplet formation, and the nature of the droplet, as a function of the temperature and the size of the system, by combining the present large deviation principle with the analysis of the mesoscopic functional given in CCELM