Thermal approximation of the equilibrium measure and obstacle problem

TL;DR

This paper studies how a thermal equilibrium measure, derived from a free energy functional involving Coulomb interactions, converges to the classical equilibrium measure as temperature decreases, providing explicit correction estimates.

Contribution

It offers quantitative convergence estimates and correction terms for the thermal equilibrium measure approaching the classical equilibrium measure in Coulomb gas models.

Findings

Explicit convergence rates in strong norms

Correction terms in powers of 1/β

Analysis of boundary layer tail behavior

Abstract

We consider the probability measure minimizing a free energy functional equal to the sum of a Coulomb interaction, a confinement potential and an entropy term, which arises in the statistical mechanics of Coulomb gases. In the limit where the inverse temperature $\beta$ tends to $\infty$ the entropy term disappears and the measure, which we call the "thermal equilibrium measure" tends to the well-known equilibrium measure, which can also be interpreted as a solution to the classical obstacle problem. We provide quantitative estimates on the convergence of the thermal equilibrium measure to the equilibrium measure in strong norms in the bulk of the latter, with a sequence of explicit correction terms in powers of $1/\beta$, as well as an analysis of the tail after the boundary layer of size $\beta^{-1/2}$.