FIRST-ORDER GLOBAL ASYMPTOTICS FOR CONFINED PARTICLES WITH SINGULAR PAIR REPULSION

• 成果类型: Article
• 署名作者: Chafai, Djalil; Gozlan, Nathael; Zitt, Pierre-Andre
• 署名单位: Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Gustave-Eiffel
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP980
• 发表日期: 2014
• 页码: 2371-2413
• 关键词: exact statistical-mechanics; 2-dimensional euler equations; large deviations; diffusing particles; dimensional system; stationary flows; coulomb forces; energy points; potentials; STATES

摘要

We study a physical system of N interacting particles in R-d, d >= 1, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as N tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension d > 2, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as N tends to infinity. In the more specific case of Coulomb interaction in dimension d > 2, and when the external field is a convex or increasing function of the radius, then the equilibrium measure is supported in a ring. With a quadratic external field, the equilibrium measure is uniform on a ball.