Effective Equations for Large Systems of Interacting Particles or Waves

该项目将专注于理解和预测相互作用的粒子或波的大系统的定性行为。这是一个在数学物理中具有根本重要性的问题，有着广泛的应用：从气体和银河动力学到社会网络，从固体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系统的规模很大，对它们的行为进行确定性描述是不可行的，因此有必要通过一个称为运动论的数学分支，采用平均描述。在这种情况下，研究人员的目标是开发分析和计算工具，通过平均量来描述这种系统的性质，并提供对它们在时间上的演化的统计上的准确预测。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，并在系统大小趋于无穷大时研究它们的渐近行为，目的是找到极限有效的方程，这反过来将揭示在一个大但有限大的系统中观察到的性质。在粒子相互作用的情况下，这是通过波尔兹曼方程实现的，在波浪湍流理论中是通过波动动力学方程来实现的。该项目将扩展到包括：推导和分析玻尔兹曼方程未涵盖的现象的动力学方程，例如非理想气体或混合气体中的多粒子相互作用，非均匀环境中波动动力学方程的推导，以及解决波动湍流理论的局部和全球适定性的问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.