TL;DR
This paper reviews Coulomb point systems across physics and mathematics, focusing on mean-field limits and microscopic analysis to understand macroscopic behavior and crystallization phenomena.
Contribution
It introduces methods to analyze systems beyond the mean-field approximation, linking microscopic details with macroscopic models.
Findings
Derivation of effective models from Coulomb interactions
Analysis of fluctuations beyond mean-field limit
Insights into temperature effects and crystallization
Abstract
Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and give rise to a variety of questions pertaining to calculus of variations, Partial Differential Equations and probability. We will review these as well as "the mean-field limit" results that allow to derive effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order beyond the mean-field limit, giving information on the system at the microscopic level. In the setting of statistical mechanics, this allows for instance to observe the effect of the temperature and to connect with crystallization questions.
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