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所控制的，比如静电和引力。这些力有些是吸引的，有些是排斥的。例如，晶体的形成，原子的周期性排列，可以非常粗略地解释为排斥力与结合力的结合。这个项目采用数学和一般的观点来看待一类这样的现象：给定一个由N个点（或粒子）组成的系统，具有特定的排斥力（通常是静电学中遇到的库仑排斥力，或其他与两点之间的距离成反比的相互作用），加上一个约束力，人们想要描述当点的数量N变得非常大时，系统的典型宏观和微观行为，还包括可能的热效应（预计温度会增加系统的无序性）。PI的研究与重要的物理模型具体相关：超导体中漩涡的排列，大原子能级的研究（大随机矩阵的谱），与磁性相关的理论物理模型，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，原子团簇或流体力学等问题有更松散的联系。项目的第一个主题是库仑气体在一个外部势和相关模型的统计力学。这是由随机矩阵，分数量子霍尔效应，甚至近似理论驱动的。人们感兴趣的是描述许多粒子的宏观（平均场）和微观排列，当它们的数量N趋于无穷大时，以及它们如何依赖于温度和势，特别是某些特征是否普遍（即独立于势），以及随着温度的变化是否存在相变。PI及其合作者最近的工作通过证明二维库仑气体中粒子分布的波动收敛于高斯自由场，以及将微观尺度上的极限点过程表征为最小化某个速率函数的大偏差原理结果，对这些问题进行了深入的研究。根据这些结果，人们预计当温度趋于0时，该系统将“结晶”成三角形晶格。先前开发的方法为处理几个重要的相关问题开辟了道路：高维库仑气体的情况，更一般的相互作用的情况，局部统计的普遍性，极限点过程的存在性，以及对其远程相关性的描述。第二个主题是金兹堡-朗道超导模型中的涡旋，其中包含引入无序的固定术语，最后一个主题是推进通过排斥性奇异相互作用的许多粒子的最简单设置的平均场动力学分析，这是一个众所周知的难题。

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.