Thermal approximation of the equilibrium measure and obstacle problem

Armstrong, Scott; Serfaty, Sylvia

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 4, pp. 1085-1110.

On considère la mesure de probabilité qui minimise une énergie libre égale à la somme d’une interaction coulombienne, d’un potentiel de confinement et d’un terme d’entropie, et qui apparaît en mécanique statistique des gaz de Coulomb. Dans la limite où la température inverse $\beta$ tend vers l’infini, le terme d’entropie disparaît et la mesure, que l’on appelle “mesure d’équilibre thermique”, tend vers la mesure d’équilibre habituelle qui peut également être interprétée comme solution du problème de l’obstacle classique. On obtient des estimées quantitatives de convergence de la mesure d’équilibre thermique vers la mesure d’équilibre dans des normes fortes à l’intérieur du support de cette dernière, avec une série de termes correctifs explicites en puissances inverses de $\beta$, de même qu’une analyse des queues apparaissant après une couche limite de taille ${\beta }^{-1/2}$.

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 ${\beta }^{-1}$, as well as an analysis of the tail after the boundary layer of size ${\beta }^{-1/2}$.

Mots-clés : Equilibrium measure, obstacle problem, Coulomb gases, potential theory

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