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

统计力学、凝聚态物理和偏微分方程中具有排斥相互作用的大型系统

基本信息

批准号：
1700278
负责人：
Sylvia Serfaty
金额：
$ 19.5万
依托单位：
New York University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-06-01 至 2020-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700278&HistoricalAwards=false
关键词：
Large systems repulsive interactions statistical

项目摘要

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.

自然界受到粒子之间相互作用力的控制，例如静电力和重力。这些力量有些是有吸引力的，有些是令人排斥的。例如，晶体的形成是原子的周期性排列，可以通过排斥力与结合力相结合来非常粗略地解释。该项目对一类此类现象采用数学和一般观点：给定一个由 N 个点（或粒子）组成的系统，该系统具有特定的排斥相互作用（通常是静电学中遇到的库仑排斥力，或与两点之间的距离），加上限制力，我们想描述当点的数量 N 变得非常大时系统的典型宏观和微观行为，并且包括可能的热效应（预计温度会增加无序性）这系统）。 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）、与磁相关的理论物理模型，但也与磁学有更松散的联系。生物学、天体物理学、等离子体物理学、玻色-爱因斯坦凝聚体、原子团簇或流体动力学问题。该项目的第一个主题是外部势能下库仑气体的统计力学和相关模型。这是由随机矩阵、分数量子霍尔效应、甚至近似理论推动的。人们感兴趣的是描述许多粒子的宏观（平均场）和微观排列，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电势，特别是某些特征是否具有普遍性（即独立于势）以及随着温度变化是否存在相变。 PI和合作者最近的工作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，以及表征极限点过程的大偏差原理结果在微观尺度上最小化某个速率函数。有了这些结果，人们预计当温度趋于 0 时，系统应该“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、局部统计的普遍性、极限点过程的存在及其远程相关性的描述。第二个主题是超导性金兹堡-朗道模型中的涡旋，其中包含引入无序的固定项，最后一个主题是推进平均场动力学分析，以实现许多粒子通过排斥奇异相互作用相互作用的最简单设置，这是一个众所周知的难题。