Coulomb Gases and Vortex Systems: Two-Dimensional Physics and Beyond

库仑静电相互作用是自然界中的基本作用力之一，支配着基本带电粒子的相互作用。例如，恒星本质上是等离子体--即根据库仑力，带正电荷的离子和带负电荷的电子相互吸引和排斥的气体。晶体的形成是原子的周期性排列，可以通过排斥静电(库仑)力与结合力的耦合来大致解释。在物理学中，“双组分等离子体”是一大群带正电荷的粒子和同样多的带负电荷的粒子；“一组分等离子体”是一大群带正电荷的粒子，负粒子被均匀的中和背景所取代。这样的系统也被称为“库仑气体”，它们在理论物理中非常重要。物理学家预测，在二维库仑气体中，一旦达到一定的临界温度，物质的新中间状态--由相反带电粒子的偶极形成--应该存在于完全有序(固体)和无序(液体)状态之间。这是统计物理学中的一种新型相变。它是2维特有的，也是二维物理的一个重要例子。本项目特别关注在统计力学(即温度)的框架下，利用数学分析和概率论的技术，分析2维及更高维的大库仑气体，包括单组分和双组分。引发这些问题的还有对大原子能级(大型随机矩阵谱)、超导体和玻色-爱因斯坦凝聚体中的涡旋、分数量子霍尔效应的研究，以及生物学或流体动力学中更宽松的问题。该项目还包括在著名的Ginzburg-Landau模型的背景下对超导体中的涡旋进行数学分析的组成部分，这是二维物理的另一个相关实例，并在3维空间对这一现象进行了调查。该项目为研究生提供了研究培训机会。这个项目是关于具有库仑相互作用的大型点或线系统的数学分析。这样的系统在物理模型中普遍存在，从量子力学、统计力学到等离子体物理和凝聚态物理，但也存在于随机矩阵理论和近似理论中。该提案的主要部分涉及经典库仑气体的统计力学。在第一部分中，它涉及带正电的点数。主要问题是在较宽的温度范围内，研究温度对点组态的局部结构的影响，以及了解它们可能的普遍特征。虽然在维度2中已经了解了很多，但要了解哪些性质在更高维度上持续存在，特别是理解电荷涨落的阶数及其可能的高斯性质，以及系统和自由能展开的长期关联，仍有许多工作要做。该提议还旨在通过研究Riesz气体的相同问题来了解有多少特征是库仑相互作用特有的。第二部分涉及具有相反电荷的二维库仑气体(或双组分等离子体)。这是一个非常重要的模型，与发生类似相变的XY模型有关，该模型本身是凝聚态物理中二维系统的近似，包括约瑟夫森结阵列和薄的无序超导颗粒膜。这种BKT转变的特征是偶极子的出现和从指数衰减到代数衰减的关联的转变。许多这种行为正在等待进一步的数学证明，这就是该项目将试图在双组分库仑气体的背景下解决的问题。最后，提案的最后一部分涉及具有库仑相互作用的线，这种相互作用在超导和超流的三维模型中以涡旋的形式出现。线的几何形状使分析比对点的分析精细得多。虽然许多起源于几何测量理论的数学工具已经被用来处理这一方面，但线的库仑相互作用，外加磁场下涡线起始的表征，以及它们的集体行为仍然需要进一步的研究。该项目的目标是用严格的数学证明和工具来研究所有这些问题，这些工具将分析、偏微分方程、变分和概率相结合。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The Coulomb electrostatic interaction is one of the fundamental forces in nature, governing how elementary charged particles interact. For example, stars are essentially plasmas - that is gases of positively charged ions and negatively charged electrons attracting and repelling one another according to the Coulomb force. The formation of crystals, which are periodic arrangements of atoms, can be roughly explained via repulsive electrostatic (Coulomb) forces coupled with a binding force. In Physics, a "two-component plasma" is a large collection of positively charged particles and as many negatively charged particles; a "one-component plasma" is a large collection of positively charged particles with the negative particles replaced by a uniform neutralizing background. Such systems are also called "Coulomb gases," and they are quite important in theoretical physics. Physicists predicted that in two-dimensional Coulomb gases, a new intermediate state of matter - formed of dipoles of oppositely charged particles - should exist between the totally ordered (solid) and disordered (liquid) state, once a certain critical temperature is reached. This was a new type of phase transition in statistical physics. It is specific to dimension 2 and an important example of two-dimensional physics. This project is particularly concerned with the analysis of large Coulomb gases in dimension 2 and higher, both one-component and two-component, in the framework of statistical mechanics (that is, with temperature), and using the techniques of mathematical analysis and probability theory. Also motivating such questions are the study of energy levels of large atoms (spectrum of large random matrices), vortices in superconductors and Bose-Einstein condensates, the fractional quantum Hall effect, and more loosely questions in biology or hydrodynamics. The project also contains a component on the mathematical analysis of vortices in superconductors in the context of the famous Ginzburg-Landau model, another and related instance of two-dimensional physics, with the investigation of this phenomenon in dimension 3. This project provides research training opportunities for graduate students. This project is about the mathematical analysis of large systems of points, or lines, with Coulomb interactions. Such systems are ubiquitous in physics models, ranging from quantum mechanics, statistical mechanics to plasma physics and condensed matter physics, but also in random matrix theory and approximation theory. The main part of the proposal concerns the statistical mechanics of classical Coulomb gases. In a first part it deals with positively charged points. The main questions are to investigate the effect of temperature on the local structure of the point configurations, in broad temperature regimes, as well as to understand their possibly universal features. While a lot has been understood in dimension 2, much remains to be done to understand which properties persist in higher dimension, in particular to understand the order of the charge fluctuations and their possibly Gaussian nature, as well as the long-range correlations of the system and free energy expansion. The proposal also aims at understanding how much of the features are specific to the Coulomb interaction by investigating the same questions for Riesz gases. A second part concerns the two-dimensional Coulomb gas with opposite charges (or two-component plasma). This is a very important model, related to the XY model in which a similar transition happens, itself an approximation for two-dimensional systems in condensed matter physics including Josephson junction arrays and thin disordered superconducting granular films. This BKT transition is charcterized by the emergence of dipoles and the transition from exponentially decaying to algebraically decaying correlations. A lot of this behavior is awaiting further mathematical proofs and this is what the project will try to address in the context of the two-component Coulomb gas. Finally the last part of the proposal concerns lines with Coulomb interactions, which arise as vortices in three-dimensional models from superconductivity and superfluidity. The geometry of the lines makes the analysis much more delicate than for points. While many mathematical tools originating in geometric measure theory have been developed for dealing with that aspect, the Coulomb interaction of the lines, the characterization of the onset of vortex lines under an applied magnetic field, and their collective behavior still need to be further investigated. The project's goal is to study all these questions with rigorous mathematical proofs and tools that will mix analysis, PDE, calculus of variations, and probability.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.