Effective Equations for Large Systems of Interacting Particles or Waves

相互作用的粒子或波的大型系统的有效方程

基本信息

批准号：
2206618
负责人：
Ioakeim Ampatzoglou
金额：
$ 15.36万
依托单位：
New York University
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：
2022-05-01 至 2024-03-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206618&HistoricalAwards=false
关键词：
Effective Equations Large Systems Interacting

项目摘要

The project will focus on understanding and predicting the qualitative behavior of large systems of interacting particles or waves. This is a problem of fundamental importance in mathematical physics, with a wide range of applications: from gas and galactic dynamics to social networks, and from solid state physics and plasma theory to water waves and oceanography. With the size of the systems of interacting particles or waves being large, a deterministic description of their behavior is not feasible, and it is necessary to resort, via a branch of mathematics called kinetic theory, to an averaging description. In this context, the investigator aims to develop analytical and computational tools to describe the properties of such systems by means of averaging quantities and provide a statistically accurate prediction of their evolution in time. The project will offer training and mentoring opportunities for undergraduate and graduate students. The idea behind kinetic theory is to identify macroscopic properties of a large system and to study their asymptotic behavior as the size of the system tends to infinity, with the intent of finding limiting effective equations, which in turn will reveal properties observed in a system of large but finite size. In the case of interacting particles, this is achieved by the Boltzmann equation, in wave turbulence theory by the wave kinetic equation. This project will expand this program to include: the derivation and analysis of kinetic equations for phenomena not covered by Boltzmann equation, for instance multiple particle interactions in non-ideal gases or mixture of gases, the derivation of wave kinetic equations in the inhomogeneous setting, as well addressing the question of local and global well-posedness in wave turbulence theory.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.

该项目将着重于理解和预测大型相互作用粒子或波浪的定性行为。这是数学物理学中基本重要性的问题，具有广泛的应用：从气体和银河动力学到社交网络，从固态物理学和等离子体理论到水波和海洋学。由于相互作用的粒子或波的系统大小很大，因此对其行为的确定描述是不可行的，并且有必要通过称为动力学理论的数学分支来求助于平均描述。在这种情况下，研究人员旨在开发分析和计算工具，以平均数量来描述此类系统的性质，并在统计上准确地预测其时间的演变。该项目将为本科生和研究生提供培训和指导机会。动力学理论背后的想法是识别大系统的宏观特性，并研究其渐近行为，因为系统的大小倾向于无穷大，目的是找到限制有效方程，这又会揭示在大型但有限大小的系统中观察到的属性。在相互作用的粒子的情况下，这是通过波动力学方程在波湍流理论中通过Boltzmann方程实现的。该项目将扩展该程序，包括：玻尔兹曼方程未涵盖的现象的动力和分析，例如在非理想气体或气体的混合物中的多个粒子相互作用，在非综合性中的波动力学方程的衍生物的衍生物，以及统计范围内的统计范围，以及统计范围的问题。值得通过基金会的智力优点和更广泛的影响审查标准来通过评估来支持。