Effective Equations for Large Systems of Interacting Particles or Waves

该项目将侧重于理解和预测相互作用的粒子或波的大系统的定性行为。这是数学物理中一个非常重要的问题，有着广泛的应用：从气体和星系动力学到社会网络，从固态物理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规模很大，对它们的行为的确定性描述是不可行的，因此有必要通过称为动力学理论的数学分支求助于平均描述。在这种情况下，研究者的目标是开发分析和计算工具，通过平均数量来描述这些系统的特性，并提供统计上准确的预测它们的时间演变。该项目将为本科生和研究生提供培训和指导机会。动力学理论背后的思想是确定一个大系统的宏观性质，并研究它们的渐近行为，随着系统的大小趋于无穷，目的是找到限制有效方程，这反过来将揭示在一个大而有限的系统中观察到的性质。在相互作用粒子的情况下，这是由玻尔兹曼方程实现的，在波动湍流理论中是由波动动力学方程实现的。本项目将扩展该项目，包括：玻尔兹曼方程未涵盖的现象的动力学方程的推导和分析，例如非理想气体或气体混合物中的多粒子相互作用，非齐次环境下波动动力学方程的推导，以及解决波动湍流理论中的局部和全局适定性问题。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。

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.