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Thermal approximation of the equilibrium measure and obstacle problem
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 4, pp. 1085-1110.

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 β tends to 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 β -1 , as well as an analysis of the tail after the boundary layer of size β -1/2 .

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 β 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 β, de même qu’une analyse des queues apparaissant après une couche limite de taille β -1/2 .

Published online:
DOI: 10.5802/afst.1714
Keywords: Equilibrium measure, obstacle problem, Coulomb gases, potential theory
Scott Armstrong 1; Sylvia Serfaty 1

1 Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Scott Armstrong and Sylvia Serfaty},
     title = {Thermal approximation of the equilibrium measure and obstacle problem},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1085--1110},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Scott Armstrong; Sylvia Serfaty. Thermal approximation of the equilibrium measure and obstacle problem. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 4, pp. 1085-1110. doi : 10.5802/afst.1714. https://afst.centre-mersenne.org/articles/10.5802/afst.1714/

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