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

具有奇异相互作用的多粒子系统：统计力学和平均场动力学

• 批准号: 2247846
• 负责人: Sylvia Serfaty
• 金额: $ 70.48万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247846&HistoricalAwards=false
• 关键词: Many; particle; Systems; Singular; Interactions

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.

数学分析可以帮助理解和得出从许多粒子的集体行为中产生的有效定律或有效的理论。在许多粒子与奇异力相互作用的情况下，例如库仑力，这是自然的基本电力。了解此类系统的统计行为及其动力学定律直接与物理和应用科学领域的基本问题直接相关：统计物理学中的库仑气体，天体物理学中的等离子体模型和等离子体物理学，量子力学模型，量子力学模型，分析随机矩阵（最初是由大原子光谱分析的动机），冷凝物理物理学（超导体和超流体）的相变，以及生物学，社会科学和神经网络中的集体行为。最近的进展已取得了新的工具，从分析和可能性分析此类问题的可能性，有或没有随机性，但仍有很多工作要做。该项目特别关注两个方向。首先是获得有效的动力学的收敛结果，并具有明确的错误率，因此在实践中很有用。第二个是理解所谓的“两个组件等离子体”中著名的kosterlitz-无尽的相变。这是由带静电相互作用的积极和负电荷颗粒制成的二维气体。正颗粒和负颗粒会吸引，根据温度，它们将其成叠成倒置的偶极子（在低温下）或作为自由电荷（在高温下）的表现。最初的惊喜是，根据诺贝尔·普雷尔（Nobel-Prible）的胜利预测，贝雷赞斯基（Berezinsky），科斯特利兹（Kosterlitz）和thouless（第三个，中间和新的物质状态）的预测存在，其行为非常不寻常，这是通过涡流的形成来解释的。关于这一阶段过渡还有很多严格的分析，该项目希望进步这一理论理解。该项目的更广泛影响源于其指导和培训组成部分，说明性工作，沟通和向更广泛的受众群体以及各种角色与社区的参与。 在几种动态和统计力学的情况下，已经理解了具有奇异相互作用的系统的有效或平均场行为，尤其是库仑相互作用。在平衡统计力学的情况下，这在于检查规范吉布斯度量下粒子密度的行为，这已经通过大偏差技术和潜在理论来理解。在纯粹的排斥库仑病例中，已经了解了更多的理解，包括围绕平均场极限的波动和点的显微镜行为。该项目通过证明在2D库仑案中与高斯乘法混乱的联系进一步扩展了这种理解，并通过分析非库仑riesz拒绝相互作用，这提出了进一步的挑战。关于相反电荷颗粒的中性血浆（然后吸引）的情况，人们对此的理解少得多，这是一个二维系统。特别是，该项目将转向理解这种两分量库仑气体的良好行为，在这种情况下，预计会发生非常特殊的相变，即berezinski-kosterlitz-无尽的相变。引入基于静电和较大的基于较大的偏差方法将为此类问题提供新的方法，替代量子场理论的重新归一化方法，并允许理解下方和高于临界温度的模型，并具有地层的特征。解释相变和波动分析的偶极子和多极。该项目的最后一部分转向具有库仑或Riesz排斥或有吸引力的相互作用的系统的梯度流，保守的动力学和牛顿动力学。特别要感谢最近的调制能量和调制的自由能法，可以得出平均场限制，但比在统计力学设置中所理解的要少得多。该项目将使我们能够理解是否以及何时何时存在及时的融合，具有吸引力的情况下的不稳定问题，并且波动和与均值场行为的巨大偏差，从而为这种动态提供了更加精确的信息。该奖项反映了NSF的法定使命，并通过使用基金会的知识分子和更广泛的影响审查标准进行评估而被认为值得支持。