Many-particle Systems with Singular Interactions: Statistical Mechanics and Mean-field Dynamics

数学分析可以帮助理解并从许多粒子的集体行为中得出有效的定律或有效的理论。这个项目特别感兴趣的是，在许多粒子与奇异力相互作用的情况下，这种严格的推导，例如库仑力，这是自然的基本电动力。了解这类系统的统计行为及其动力学规律，直接关系到物理学和应用科学中的几个基本问题：统计物理中的库仑气体，天体物理和等离子体物理中的等离子体模型，量子力学模型，随机矩阵的分析(其最初的动机是分析大原子的光谱)，凝聚态物理中的相变(超导体和超流体)，以及生物学、社会科学和神经网络中的集体行为。最近已经取得了进展，从分析和概率角度提出了新的工具来分析这些问题，无论是否具有随机性，但仍有许多工作要做。该项目特别侧重于两个方向。第一种方法是获得对所有时间都有效并且具有显式错误率的动力学收敛结果，因此在实践中是有用的。二是对所谓“双组分等离子体”中著名的Kosterlitz-Thouless相变的理解。这是一种由带正电荷和带负电荷的粒子通过静电相互作用形成的二维气体。正粒子和负粒子相互吸引，根据温度的不同，它们成对形成坍塌的偶极子(在低温下)或表现为自由电荷(在高温下)。最初令人惊讶的是，根据贝尔津斯基、科斯特利茨和索利斯的诺贝尔奖获得者预测，物质存在第三种、中间和新的状态，其相当不寻常的行为可以用漩涡的形成来解释。关于这一阶段转变还有许多需要严格分析的地方，该项目希望推动这一理论理解。该项目的更广泛影响来自其指导和培训部分、说明性工作、与更广泛受众的沟通和外联，以及参与社区的各种角色。具有奇异相互作用的系统的有效或平均场行为，特别是库仑相互作用，已经在动力学和统计力学的几种情况下被理解。在平衡统计力学的情况下，这包括检查粒子密度在正则吉布斯测量下的行为，这已经通过大偏差技术和势能理论得到了理解。在纯粹排斥库仑的情况下，人们已经了解了更多，包括平均场极限附近的波动和点的微观行为。该项目进一步扩展了这一理解，证明了与二维库仑情况下的高斯乘法混沌的联系，并通过分析非库仑Riesz排斥相互作用，这提出了进一步的挑战。对于中性等离子体中的相反电荷粒子(然后相互吸引)的情况，人们了解的要少得多，这作为一个二维系统是有意义的。特别是，该项目将转向了解这种双组分库仑气体的精细行为，其中预测会发生一种非常特殊的相变，即别津斯基-科斯特利茨-索利斯相变。将基于静电和大偏差的方法引入到这一主题中，将提供一种新的方法来解决这类问题，替代量子场论的重整化方法，并允许理解临界温度以下和以上的模型，用偶极和多极的形成来解释相变的特征，并分析涨落。该项目的最后一个主要部分转向具有库仑或Riesz排斥或吸引相互作用的系统的梯度流、守恒动力学和牛顿动力学。特别是由于最近的调制能量和调制自由能方法，平均场极限可以得到，但除此之外，人们对它的理解比在统计力学环境中要少得多。该项目将使我们了解全球时间收敛是否以及何时成立，吸引力情况下的不稳定问题，以及与平均场行为的波动和大偏差，从而提供关于此类动态的更准确的信息。该奖项反映了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。

Mathematical analysis can help understand and derive effective laws or effective theories emerging from the collective behavior of many particles. This project is particularly interested in such rigorous derivations in the case where the many particles are interacting with singular forces, such as the Coulomb force, which is the fundamental electric force of nature. Understanding the statistical behavior of such systems, as well as their dynamical laws, is directly related to fundamental questions in several areas of physics and applied science: the Coulomb gas in statistical physics, models of plasmas in astrophysics and plasma physics, quantum mechanics models, analysis of random matrices (itself initially motivated by the analysis of the spectrum of large atoms), phase transitions in condensed matter physics (superconductors and superfluids), but also collective behavior in biology, social sciences, and neural networks. Recent progress has been made bringing forward new tools from analysis and probability to analyze such questions, with or without randomness, but much remains to be done. The project focuses in particular on two directions. The first is obtaining convergence results for dynamics that are valid for all time and with an explicit error rate, thus useful in practice. The second is in understanding the famous Kosterlitz-Thouless phase transition in the so-called "two component plasma". This is a two-dimensional gas made of positively and negatively charged particles with electrostatic interaction. Positive particles and negative particles attract and, depending on the temperature, they pair into collapsed dipoles (at low temperature) or behave as free charges (at high temperature). What was an initial surprise is that, according to the Nobel-prize winning prediction of Berezinsky, Kosterlitz, and Thouless, a third, intermediate and new state of matter exists, with quite unusual behavior that is explained by the formation of vortices. Much remains to be rigorously analyzed about this phase transition, and the project hopes to advance this theoretical understanding. The broader impacts of the project stem from its mentoring and training component, expository work, communication and outreach to broader audiences, as well as involvement with the community in various roles. The effective or mean-field behavior of systems with singular interactions, in particular Coulombic ones, has been understood for several situations of dynamics and statistical mechanics. In the case of equilibrium statistical mechanics, this consists in examining the behavior of the particle density under the canonical Gibbs measure, and this has been understood via large deviations techniques and potential theory. In the purely repulsive Coulomb case, much more has been understood, including the fluctuations around the mean-field limit and the microscopic behavior of the points. The project further extends this understanding by proving the connection to the Gaussian Multiplicative Chaos in the 2D Coulomb case, and by analyzing non-Coulomb Riesz repulsive interactions, which present further challenges. Much less has been understood about the case of a neutral plasma of oppositely charged particles (which then attract), which makes sense as a two-dimensional system. In particular, the project will turn to understanding the fine behavior of such a two-component Coulomb gas, in which a very particular phase transition, the Berezinski-Kosterlitz-Thouless phase transition, is predicted to happen. Bringing in an electrostatic and large deviations-based approach to this topic will provide a new approach to such problems, alternate to the renormalization methods of quantum field theory, and allow to understand the model below and above the critical temperature, with characterizations of the formation of dipoles and multipoles which explain the phase transition, and analysis of the fluctuations. The last main part of the project turns to gradient flow, conservative dynamics and Newtonian dynamics of systems with Coulomb or Riesz repulsive or attractive interactions. Thanks in particular to the recent modulated energy and modulated free energy methods, the mean-field limit can be derived, but much less has been understood beyond this than in the statistical mechanics setting. The project will allow us to understand whether and when global-in-time convergence holds, questions of instability in the case with attraction, and fluctuations and large deviations away from the mean-field behavior, thus providing much more precise information on such dynamics.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.