Droplet minimizers for the Gates-Lebowitz-Penrose free energy functional

TL;DR

This paper analyzes the structure of constrained minimizers of the non-local Gates-Lebowitz-Penrose free-energy functional at low temperatures, identifying conditions for droplet formation and relating macroscopic minimizers to microscopic configurations.

Contribution

It determines the critical density for droplet formation and characterizes the droplet's nature as a function of density and system size.

Findings

Identifies the critical density for droplet formation.

Describes the nature of droplets depending on parameters.

Connects macroscopic minimizers to microscopic configurations.

Abstract

We study the structure of the constrained minimizers of the Gates-Lebowitz-Penrose free-energy functional ${\mathcal F}_{\rm GLP}(m)$, non-local functional of a density field $m(x)$, $x\in {\mathcal T}_L$, a $d$-dimensional torus of side length $L$. At low temperatures, ${\mathcal F}_{\rm GLP}$ is not convex, and has two distinct global minimizers, corresponding to two equilibrium states. Here we constrain the average density $L^{-d}\int_{{\cal T}_L}m(x)\dd x$ to be a fixed value $n$ between the densities in the two equilibrium states, but close to the low density equilibrium value. In this case, a "droplet" of the high density phase may or may not form in a background of the low density phase, depending on the values $n$ and $L$. We determine the critical density for droplet formation, and the nature of the droplet, as a function of $n$ and $L$. The relation between the free energy and the large deviations functional for a particle model with long-range Kac potentials, proven in some cases, and expected to be true in general, then provides information on the structure of typical microscopic configurations of the Gibbs measure when the range of the Kac potential is large enough.