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

数学分析可以帮助理解和推导出从许多粒子的集体行为中产生的有效定律或有效理论。这个项目对这种严格的推导特别感兴趣，在这种情况下，许多粒子与单一力相互作用，比如库仑力，这是自然界的基本电力。了解这些系统的统计行为，以及它们的动力学规律，直接关系到物理学和应用科学的几个领域的基本问题：统计物理学中的库仑气体，天体物理学和等离子体物理学中的等离子体模型，量子力学模型，随机矩阵分析（本身最初是由大原子光谱分析激发的），凝聚态物理学中的相变（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，从分析和概率中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。第一个是得到在任何时间都有效且具有明确错误率的动力学的收敛结果，从而在实践中有用。第二是理解所谓的“双组分等离子体”中著名的Kosterlitz-Thouless相变。这是一种二维气体，由带正电荷和带负电荷的粒子静电相互作用而成。正粒子和负粒子相互吸引，根据温度的不同，它们成对形成塌缩的偶极子（在低温下）或表现为自由电荷（在高温下）。最初令人惊讶的是，根据获得诺贝尔奖的别列津斯基、科斯特利茨和索利斯的预测，存在着第三种物质的中间和新状态，具有非常不寻常的行为，可以用漩涡的形成来解释。关于这一相变还有很多需要严格分析的地方，该项目希望能推进这一理论认识。该项目的广泛影响来自其指导和培训部分、说明性工作、与更广泛受众的交流和外联，以及以各种角色参与社区活动。具有奇异相互作用的系统，特别是库仑系统的有效场或平均场行为，已经在动力学和统计力学的几种情况下得到了理解。在平衡统计力学的情况下，这包括在经典吉布斯测量下检查粒子密度的行为，这是通过大偏差技术和势理论来理解的。在纯排斥性库仑情况下，已经了解了更多，包括平均场极限周围的波动和点的微观行为。该项目进一步扩展了这一理解，证明了在二维库仑情况下与高斯乘法混沌的联系，并通过分析非库仑Riesz排斥相互作用，提出了进一步的挑战。对于由带相反电荷的粒子（然后相互吸引）组成的中性等离子体的情况，人们了解得少得多，这对于二维系统来说是有意义的。特别是，该项目将转向理解这种双组分库仑气体的精细行为，其中一个非常特殊的相变，Berezinski-Kosterlitz-Thouless相变，预计会发生。引入静电和基于大偏差的方法将为解决此类问题提供一种新的方法，替代量子场论的重整化方法，并允许理解临界温度以下和高于临界温度的模型，具有解释相变的偶极子和多极子形成的特征，以及波动的分析。项目的最后一个主要部分转向梯度流动，保守动力学和具有库仑或Riesz排斥或吸引相互作用的系统的牛顿动力学。特别是由于最近的调制能量和调制自由能方法，可以推导出平均场极限，但在此之外的理解要比统计力学设置少得多。该项目将使我们能够了解全局实时收敛是否成立以及何时成立，具有吸引力的情况下的不稳定性问题，以及波动和偏离平均场行为的大偏差，从而提供有关此类动力学的更精确信息。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.