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: 2345533
• 负责人: Matthew Rosenzweig
• 金额: $ 19.5万
• 依托单位: Carnegie-Mellon 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: 2023-07-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2345533&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.

受限系统中的粒子（例如容器中气体的原子或分子）通过排斥或吸引力相互作用，随着粒子变得靠近，相互作用的强度增加。原则上，经典和量子物理方程可以完全确定行为。在实践中，粒子的数量以及系统的复杂性超出了最佳计算资源的能力，该项目旨在通过以下方式实现计算复杂性的大幅降​​低。从统计学的角度来看，关注的是概率在给定时间发现系统中空间特定位置并以特定速度移动的粒子，其结果预计可直接应用于物质状态（例如玻色-爱因斯坦凝聚态或等离子体）以及系统的建模。该项目将为新一代研究人员提供数学和物理交叉领域的指导和培训机会。颗粒与逆幂势，例如库仑或里斯类型研究者的目的是确定定量收敛所需的极限方程的最小正则性假设，收敛在更现实的动力学噪声设置中是否有效，最佳时间尺度。第二部分涉及超临界平均场标度制度，这是牛顿第二定律或半经典的奇异极限。薛定谔方程导致了理想流体的欧拉方程的动力学推广，其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，以单动力学情况为基础，其中极限方程为不可压缩欧拉。方程并借鉴等离子体物理学中准中性极限的联系，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会导致换向器类型的功能不等式，从而与独立利益的谐波分析建立新的联系。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。