Mean-Field and Singular Limits of Deterministic and Stochastic Interacting Particle Systems

确定性和随机相互作用粒子系统的平均场和奇异极限

• 批准号: 2206085
• 负责人: Matthew Rosenzweig
• 金额: $ 19.5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2023-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206085&HistoricalAwards=false
• 关键词: Mean; Field; Singular; Limits; Deterministic

Particles in confined systems such as the atoms or molecules of a gas in a container interact either through repulsion or attraction, with interactions increasing in strength as particles become close together. In principle, the equations of classical and quantum physics allow complete determination of the behavior of each particle in the system for arbitrary periods of time. In practice, the number of particles, and therefore the complexity of the system, ranges beyond the capabilities of the best computing resources. This project aims to achieve a substantial reduction in computational complexity through a statistical point of view, focused on the probability of finding at a given time a particle in the system at a certain position in space and moving with a certain velocity. The results are expected to be directly applicable to the modeling of states of matter such as Bose-Einstein condensates or plasmas, and of systems with particle-like behavior, as vortices in fluids or superconductors. The project will provide mentoring and training opportunities for a new generation of researchers at the intersection of mathematics and physics. The first part of the project concerns the mean-field limit of systems of particles with inverse power potentials, for instance of Coulomb or Riesz type. The investigator aims to determine the minimal regularity assumptions on the limiting equation needed for quantitative convergence, whether convergence is valid in the more realistic setting of noise in the dynamics, the optimal time scales for the mean-field approximation to hold, and the sharp rate of convergence. The second part deals with the supercritical mean-field scaling regime, a singular limit of Newton’s second law or the semiclassical Schrödinger equation leading to a kinetic generalization of Euler’s equation for an ideal fluid. The goal is to identify the optimal range for the validity of this limit through analytical and numerical means by building on progress for the monokinetic case where the limiting equation reduces to the incompressible Euler equation and drawing on a connection to the quasineutral limit in plasma physics. An important quantity for measuring convergence is a modulated energy-entropy or free energy, which is related to renormalized energies appearing in the statistical mechanics of Coulomb and Riesz gasses. Studying these quantities and their variations along transport fields leads to functional inequalities of commutator type, establishing new connections to harmonic analysis of independent interest.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.

在容器中，气体的原子或气体的分子等密封系统中的颗粒通过排斥或吸引力相互作用，并且随着颗粒的近距离相互作用，强度的相互作用会增加。原则上，经典物理和量子物理的方程式可以完全确定系统中每个粒子的行为任意时间。实际上，粒子的数量以及系统的复杂性，范围超出了最佳计算资源的功能。该项目旨在通过统计的角度实现计算复杂性的大幅度降低，重点是在给定时间在空间中某个位置处找到粒子并以一定速度移动的可能性。预计该结果将直接适用于物质状态（例如Bose-Einstein凝结物或等离子体）以及具有颗粒样行为的系统，作为流体或超导体中的涡流。项目将为新一代研究人员提供数学与物理学交集的新一代研究人员的指导和培训机会。该项目的第一部分涉及具有反功率电势的粒子系统的平均场限制，例如库仑或里斯类型。研究者旨在确定定量收敛所需的极限方程的最低规律性假设，是否在动力学中噪声的更现实设置中有效，最佳的时间尺度，含义场近似的最佳时间尺度以及较高的收敛速率。第二部分介绍了超临界平均尺度缩放制度，牛顿第二定律的奇异极限或半经典的Schrödinger方程，从而导致Euler对等效性对于理想流体的等效性。目的是通过基于单基因剂的进度来确定该限制通过分析和数值手段的最佳范围，在这种情况下，限制方程将减少到不可压缩的Euler方程并绘制与等离子物理学中的准中性极限的连接。测量收敛性的重要数量是调制能量渗透或自由能，该能量与库仑和瑞斯气体的统计机理中出现的重新归一化能量有关。研究这些数量及其沿运输领域的变化导致换向器类型的功能不平等，建立了与独立兴趣的谐波分析的新联系。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子和更广泛的影响标准来评估通过评估来诚实地对支持进行评估。