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Large-Scale Behavior of Interacting Particle Systems

相互作用粒子系统的大规模行为

基本信息

批准号：
1811723
负责人：
Mykhaylo Shkolnikov
金额：
$ 21万
依托单位：
Princeton University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811723&HistoricalAwards=false
关键词：
Large Scale Behavior Interacting Particle

项目摘要

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.

相互作用粒子系统形成了一个数学框架，用于在科学和工程的各种领域中出现的具有相互作用组件的系统。例如，金融机构通过借贷相互作用，大脑中的神经元通过电信号同步，物理粒子通过它们自身产生的（例如电磁）电势相互作用。在许多情况下，所涉及的系统是大的，例如，在人脑的给定部分中的神经元的数量在一百万的数量级上。该项目的目标是推进这种大型相互作用粒子系统的定量理解，依赖于随机和确定性偏微分方程。该项目将解决描述相互作用粒子系统的大尺度行为的方程以及由此产生的爆破。更具体地说，爆破Stefan问题的抛物型偏微分方程的限制粒子密度的粒子系统相互作用，通过命中时间将调查的概率和分析技术相结合。特别是，爆破的确切时间将被确定，并建立一个合适的解决方案的存在性和唯一性超过第一次爆破的时间。此外，高斯近似的碰撞时间在相互作用的粒子系统将使用限制随机偏微分方程。最后，解决方案，后者所产生的相互作用的粒子系统将被证明是一致的熵解决方案自然产生的偏微分方程的分析。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。