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First-order global asymptotics for confined particles with singular pair repulsion. (English) Zbl 1304.82050

Summary: We study a physical system of \(N\) interacting particles in \(\mathbb{R}^{d}\), \(d\geq 1\), 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.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
31B99 Higher-dimensional potential theory
60F10 Large deviations
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