Thermal approximation of the equilibrium measure and obstacle problem

Abstract

We consider the probability 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 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 to the equilibrium 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 $\beta^{-1/2}$.