Large-Scale Behavior of Interacting Particle Systems

相互作用的粒子系统为在科学和工程的各个领域中出现的具有相互作用成分的系统形成了一个数学框架。例如，金融机构通过借贷相互作用，大脑神经元通过电信号同步，物理粒子通过自身产生的（例如电磁）电位相互作用。在许多情况下，所涉及的系统是很大的，例如，在人类大脑的给定部分中，神经元的数量大约是一百万个。该项目的目标是依靠随机和确定性偏微分方程，推进对这种大型相互作用粒子系统的定量理解。该项目将处理描述相互作用粒子系统的大规模行为和由此产生的爆炸的方程。更具体地说，抛物型偏微分方程的Stefan问题的爆破将通过概率和分析技术的结合来研究，这些问题控制着粒子系统中通过撞击时间相互作用的极限粒子密度。特别是，将确定爆炸的确切时间，并在第一次爆炸之后确定一个合适的解决方案的存在性和唯一性。此外，将利用极限随机偏微分方程推导出相互作用粒子系统中碰撞时间的高斯近似。最后，由相互作用的粒子系统产生的后者的解将被证明与在偏微分方程分析中自然产生的熵解一致。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。

Interacting particle systems form a mathematical framework for systems with interacting components that arise in a variety of fields in science and engineering. Examples include, among many others, financial institutions interacting with each other via borrowing and lending, synchronization of neurons in the brain via electrical signals, and physical particles interacting through the (e.g. electromagnetic) potential they themselves generate. In many situations, the systems involved are large, e.g. the number of neurons in a given part of the human brain is on the order of a million. The goal of the project is to advance the quantitative understanding of such large interacting particle systems, relying on stochastic and deterministic partial differential equations. The project will tackle equations describing the large scale behavior of interacting particle systems and the blowups arising thereby. More specifically, blowups in Stefan problems for parabolic partial differential equations governing the limiting particle density in particle systems interacting through the hitting times will be investigated by a combination of probabilistic and analytic techniques. In particular, the exact timing of blowups will be determined, and the existence and uniqueness of a suitable solution will be established beyond the time of the first blowup. In addition, Gaussian approximations for the hitting times in interacting particle systems will be derived using the limiting stochastic partial differential equations. Finally, solutions to the latter arising from interacting particle systems will be shown to coincide with the entropy solutions arising naturally in the analysis of partial differential equations.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.