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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
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     title = {Thermal approximation of the equilibrium measure and obstacle problem},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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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/

[1] Romain Allez; Jean-Philippe Bouchaud; Alice Guionnet Invariant beta-ensembles and the Gauss-Wigner crossover, Phys. Rev. Lett., Volume 109 (2012), 094102, 5 pages | DOI

[2] Scott Armstrong; Sylvia Serfaty Local Laws and Rigidity for Coulomb Gases at any Temperature, Ann. Probab., Volume 49 (2021) no. 1, pp. 46-121 | MR | Zbl

[3] Scott Armstrong; Sylvia Serfaty; Ofer Zeitouni Remarks on a constrained optimization problem for the Ginibre ensemble, Potential Anal., Volume 41 (2014) no. 3, pp. 945-958 | DOI | MR | Zbl

[4] Robert J. Berman From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit, Math. Z., Volume 291 (2019) no. 1-2, pp. 365-394 | DOI | MR | Zbl

[5] Fabrice Bethuel; Haïm Brézis; Frédéric Hélein Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differ. Equ., Volume 1 (1993) no. 2, pp. 123-148 | DOI | MR | Zbl

[6] Thierry Bodineau; Alice Guionnet About the stationary states of vortex systems, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 35 (1999) no. 2, pp. 205-237 | DOI | Numdam | MR | Zbl

[7] Luis A. Caffarelli The obstacle problem revisited, J. Fourier Anal. Appl., Volume 4 (1998) no. 4-5, pp. 383-402 | DOI | MR

[8] Emanuele. Caglioti; Pierre-Louis. Lions; Carlo Marchioro; Mario Pulvirenti A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Commun. Math. Phys., Volume 143 (1992) no. 3, pp. 501-525 | DOI | MR | Zbl

[9] Gustave Choquet Diamètre transfini et comparaison de diverses capacités, 1958 (Technical report, Faculté des Sciences de Paris) | Numdam

[10] Alessio Figalli; Xavier Ros-Oton; Joaquim Serra Generic regularity of free boundaries for the obstacle problem (2019) (https://arxiv.org/abs/1912.00714)

[11] David Gilbarg; Neil S. Trudinger Elliptic partial differential equations of second order. Reprint of the 1998 edition, Classics in Mathematics, Springer, 2001 | DOI

[12] Adrien Hardy; Gaultier Lambert CLT for Circular beta-Ensembles at High Temperature (2019) (https://arxiv.org/abs/1909.01142)

[13] Maxime Hauray; Stéphane Mischler On Kac’s chaos and related problems, J. Funct. Anal., Volume 266 (2014) no. 10, pp. 6055-6157 | DOI | MR | Zbl

[14] Michael K.-H. Kiessling Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math., Volume 46 (1993) no. 1, pp. 27-56 | DOI | MR | Zbl

[15] Thomas Leblé; Sylvia Serfaty Large deviation principle for empirical fields of log and Riesz gases, Invent. Math., Volume 210 (2017) no. 3, pp. 645-757 | DOI | MR | Zbl

[16] Joachim Messer; Herbert Spohn Statistical mechanics of the isothermal Lane-Emden equation, J. Stat. Phys., Volume 29 (1982) no. 3, pp. 561-578 | DOI | MR

[17] Cassio Neri Statistical mechanics of the N-point vortex system with random intensities on a bounded domain, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 21 (2004) no. 3, pp. 381-399 | DOI | Numdam | MR | Zbl

[18] Nicolas Rougerie; Sylvia Serfaty; Jakob Yngvason Quantum Hall phases and plasma analogy in rotating trapped Bose gases, J. Stat. Phys., Volume 154 (2014) no. 1-2, pp. 2-50 | DOI | MR | Zbl

[19] Edward B. Saff; Vilmos Totik Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften, 316, Springer, 1997 | DOI

[20] Sylvia Serfaty Coulomb gases and Ginzburg-Landau vortices, Zürich Lectures in Advanced Mathematics, 70, European Mathematical Society, 2015 | DOI | Numdam

[21] Sylvia Serfaty Gaussian Fluctuations and Free Energy Expansion for 2D and 3D Coulomb Gases at Any Temperature (2020) (https://arxiv.org/abs/2003.11704)

[22] Sylvia Serfaty; Joaquim Serra Analysis and PDE, Anal. PDE, Volume 11 (2018) no. 7, pp. 1803-1839

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