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

密闭系统中的粒子，如容器中气体的原子或分子，通过排斥或吸引相互作用，随着粒子靠近，相互作用的强度增加。原则上，经典物理学和量子物理学的方程允许在任意时间段内完全确定系统中每个粒子的行为。在实践中，粒子的数量以及系统的复杂性超出了最好的计算资源的能力。本项目旨在通过统计学的角度大幅降低计算复杂度，重点关注在给定时间找到系统中某一粒子在空间的某一位置并以某一速度运动的概率。这些结果有望直接应用于物质状态的建模，如玻色-爱因斯坦凝聚体或等离子体，以及具有粒子行为的系统，如流体或超导体中的涡流。该项目将为数学和物理交叉领域的新一代研究人员提供指导和培训机会。该项目的第一部分涉及具有逆功率势的粒子系统的平均场极限，例如库仑或Riesz型。研究者的目的是确定定量收敛所需的极限方程的最小规则性假设，收敛在更现实的动态噪声设置中是否有效，平均场近似保持的最佳时间尺度，以及收敛的快速速度。第二部分涉及超临界平均场标度制度，牛顿第二定律的奇异极限或半经典Schrödinger方程，导致理想流体欧拉方程的动力学推广。目标是通过分析和数值方法确定该极限有效性的最佳范围，建立在单运动情况下，极限方程简化为不可压缩欧拉方程，并利用与等离子体物理学中的准中性极限的联系。测量收敛性的一个重要量是调制能量熵或自由能，它与库仑和Riesz气体统计力学中出现的重正化能量有关。研究这些量及其沿输运场的变化，得到了换向子型泛函不等式，为独立兴趣的谐波分析建立了新的联系。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。

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.