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

自然界是由粒子之间的相互作用力所支配的，例如静电力和引力。有些力是吸引力，有些是排斥力。例如，晶体是原子的周期性排列，其形成可以通过排斥力与结合力的耦合来非常粗略地解释。这个项目需要对一类这样的现象的数学和一般的看法：给定一个N点系统具有特定排斥相互作用的（或粒子）（通常是静电学中遇到的库仑排斥力，或与两点之间的距离成反比的其他相互作用），连同约束力，当点数N变得非常大时，人们希望描述系统的典型宏观和微观行为，并且包括可能的热效应（预期温度会增加系统的无序）。PI的研究具体涉及重要的物理模型：超导体中涡旋的排列，大原子能级的研究（大随机矩阵的谱），与磁性有关的理论物理模型，但也与生物学，天体物理学，等离子体物理学，玻色-爱因斯坦凝聚，该项目的第一个主题是库仑气体在外部势和相关模型中的统计力学。这是由随机矩阵，分数量子霍尔效应，甚至近似理论的动机。人们感兴趣的是描述许多粒子的宏观（平均场）和微观排列，因为它们的数量N趋于无穷大，以及它们如何依赖于温度和势，特别是某些特征是否是普适的（即独立于势），以及随着温度的变化是否存在相变。PI和合作者最近的工作已经深入了解了这些问题，证明了二维库仑气体中粒子分布的波动收敛于高斯自由场，以及大偏差原理的结果，该结果将微观尺度上的极限点过程描述为最小化某个速率函数。这些结果表明，当温度趋于0时，系统将“结晶”成三角形晶格。这些方法为处理高维库仑气体、更一般相互作用、局域统计的普遍性、极限点过程的存在以及长程关联的描述等重要问题开辟了道路。第二个主题是超导的金兹伯格-朗道模型中的涡旋，引入无序的钉扎项，最后一个主题是推进对最简单的多粒子通过排斥奇异相互作用相互作用的平均场动力学的分析，这是一个非常困难的问题。

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.