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力之一，控制着基本带电粒子如何相互作用。例如，恒星本质上是等离子体--即带正电的离子和带负电的电子根据库仑力相互吸引和排斥。晶体的形成是原子的周期性排列，可以通过与结合力耦合的排斥静电（库仑）力来粗略地解释。在物理学中，“双组分等离子体”是大量带正电的粒子和同样多的带负电的粒子的集合;“单组分等离子体”是大量带正电的粒子的集合，其中带负电的粒子被均匀的中和背景所取代。这种系统也被称为“库仑气体”，它们在理论物理学中非常重要。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，一种新的物质中间态--由带相反电荷的粒子的偶极子形成--应该存在于完全有序的（固体）和无序的（液体）状态之间。这是统计物理学中的一种新型相变。它是二维物理学的一个重要例子。该项目特别关注在统计力学（即温度）的框架内，使用数学分析和概率论技术，分析2维及更高维的单组分和双组分大库仑气体。激发这些问题的还有对大原子能级的研究（大随机矩阵的谱），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，以及生物学或流体力学中更松散的问题。该项目还包括在著名的金兹伯格-朗道模型的背景下对超导体中的涡旋进行数学分析的部分，金兹伯格-朗道模型是二维物理学的另一个相关实例，并在三维中调查这一现象。该项目为研究生提供了研究培训机会。这个项目是关于大型系统的点，或线，与库仑相互作用的数学分析。这种系统在物理模型中无处不在，从量子力学，统计力学到等离子体物理和凝聚态物理，以及随机矩阵理论和近似理论。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，它涉及带正电荷的点。主要的问题是调查温度的影响上的局部结构的点配置，在广泛的温度制度，以及了解他们可能的普遍功能。虽然在2维中已经了解了很多，但仍有许多工作要做，以了解哪些性质在更高的维度中持续存在，特别是了解电荷波动的顺序及其可能的高斯性质，以及系统和自由能膨胀的长程相关性。该提案还旨在通过研究Riesz气体的相同问题来了解库仑相互作用的特征。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XY模型相关，XY模型中发生了类似的转变，XY模型本身是凝聚态物理学中二维系统的近似，包括约瑟夫森结阵列和薄的无序超导颗粒膜。这种BKT转变的特点是偶极的出现和从指数衰减到代数衰减的相关性的过渡。许多这种行为正在等待进一步的数学证明，这就是该项目将在双组分库仑气体的背景下尝试解决的问题。最后，该建议的最后一部分涉及库仑相互作用的线，这些线在超导性和超流性的三维模型中作为漩涡出现。线的几何形状使分析比点的分析更加精细。虽然许多数学工具起源于几何测量理论已经开发了处理这方面，库仑相互作用的线，在施加磁场下的涡旋线的发病特征，以及他们的集体行为仍然需要进一步研究。该项目的目标是通过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、变分法和概率结合在一起。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值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.