Large systems with repulsive interactions in statistical mechanics, condensed matter physics and PDE

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Nature is governed by interaction forces between particles, such as the electrostatic and gravitational forces. Some of these forces are attractive, some are repulsive. For instance, the formation of crystals, which are periodic arrangements of atoms, can be very roughly explained via repulsive forces coupled with a binding force. This project takes a mathematical and general view on a class of such phenomena: given a system of N points (or particles) with a specific repulsive interaction (typically the Coulomb repulsive force encountered in electrostatics, or other interactions which are in inverse power of the distance between two points), together with a confining force, one would like to describe the typical macroscopic and microscopic behavior of the system as the number of points N gets very large, and possible thermal effects are included (temperature being expected to add disorder to the system). The research of the PI is concretely related to important physics models: the arrangements of vortices in superconductors, the study of energy-levels of large atoms (spectrum of large random matrices), theoretical physics models related to magnetism, but also more loosely connected to questions in biology, astrophysics, plasma physics, Bose-Einstein condensates, atomic clusters or hydrodynamics.The first topic of the project is the statistical mechanics of Coulomb gases in an external potential and related models. This is motivated by random matrices, the fractional quantum Hall effect, and even approximation theory. One is interested in describing the macroscopic (mean-field) and microscopic arrangements of the many particles as their number N tends to infinity, and how they depend on temperature and the potential, and in particular whether some features are universal (i.e. independent of the potential) and whether there are phase transitions as the temperature varies. Recent works of the PI and collaborators have given insight into these questions with a proof that the fluctuations of the distribution of particles in a two-dimensional Coulomb gas converge to a Gaussian Free Field, and a Large Deviation Principle result which characterizes the limiting point processes at the microscopic scale as minimizing a certain rate function. With these results, one expects that the system should "crystallize" into a triangular lattice as the temperature tends to 0.The methods previously developed open the way to treating several important related questions: the case of higher-dimensional Coulomb gases, the case of more general interactions, the universality of the local statistics, the existence of a limiting point process, and the description of its long-range correlations. The second topic is that of vortices in the Ginzburg-Landau model of superconductivity, with pinning terms that introduce disorder and the final topic is to advance the analysis of mean-field dynamics for the simplest setting of many particles interacting via a repulsive singular interaction, a notoriously difficult question.