Hilbert's Sixth Problem: From Particles to Waves

希尔伯特的第六个问题于1900年提出，要求从微观动力学定律（又名第一原理）出发，对麦克斯韦和玻尔兹曼在19世纪提出的统计物理学宏观定律进行严格的数学推导。这个问题的经典设定是关于粒子系统根据经典力学定律发生碰撞。同样的问题也出现在更现代的统计物理理论中，根据哈密顿波型偏微分方程，粒子被相互作用的波所取代。这种关于波动的统计物理理论通常被称为“波动湍流理论”，因为它们在理解波动系统中的湍流行为方面起着核心作用。这在许多科学领域都有应用，如量子力学、海洋学、气候科学等。从广义上讲，这个项目的目标是推进对这种湍流理论的数学和科学理解，并在统计力学的基础上解决数学物理中一些长期存在的猜想。本项目为研究生提供研究训练机会。即使在经典的背景下，希尔伯特的第六个问题仍然是一个艰巨的任务，它只在很短的时间内得到了解决。该项目寻求提供其长期解决方案，从而为这个长期存在的开放性问题提供最终答案。这相当于从牛顿定律出发，给出玻尔兹曼动力学方程的严格推导，然后推导宏观流体模型（欧拉方程和纳维-斯托克斯方程）。同时，该项目在波浪湍流理论的背景下提出了类似的理由。在那里，首席研究员（PI）也试图为各种科学兴趣的波浪系统提供相应的“波浪动力学方程”的长期推导。从非线性Schrödinger方程作为非线性波系统的基本模型开始，接下来将对其他波系统进行类似的研究，如多粒子量子系统和一些来自海洋和气候科学的模型。最后，本计画将探讨波浪湍流理论中有关湍流方面的数学问题。在那里，PI打算使用上述波浪动力学方程的严格推导，结合对这些方程的解的分析，来理解波浪系统的湍流现象，如能量级联和Sobolev范数的增长。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。

Hilbert’s sixth problem, posed in 1900, asks for a rigorous mathematical derivation of the macroscopic laws of statistical physics, formulated by Maxwell and Boltzmann in the nineteenth century, starting from the microscopic laws of dynamics (aka first principles). The classical setting of this problem pertains to particle systems which collide according to the laws of classical mechanics. The same problem emerges in more modern theories of statistical physics, where particles are replaced by waves that interact according to some Hamiltonian wave-type partial differential equation. Such theories of statistical physics for waves often go by the name of “wave turbulence theory”, because they play a central role in understanding turbulent behaviors in wave systems. This has applications in many areas of science such as quantum mechanics, oceanography, climate science, etc. Broadly speaking, the goal of this project is to advance the mathematical, and hence scientific, understanding of such turbulence theories, and settle some longstanding conjectures in mathematical physics on the foundations of statistical mechanics. The project provides research training opportunities for graduate students.Even in its classical setting, Hilbert’s sixth problem remains a formidable task, that has only been resolved for short times. The project seeks to provide its long-time resolution, thus giving a final answer to this longstanding open problem. This amounts to giving the rigorous derivation of Boltzmann’s kinetic equation starting from Newton’s laws, followed by the derivation of the macroscopic fluid models (Euler’s and Navier-Stokes equations). In parallel, the project proposes similar justifications in the setting of wave turbulence theory. There too, the Principal Investigator (PI) seeks to provide the long-time derivation of the corresponding “wave kinetic equations” for various wave systems of scientific interest. Starting with the nonlinear Schrödinger equation as a prime model for nonlinear wave systems, this will be followed by similar investigations for other wave systems, like many-particle quantum systems and some models coming from ocean and climate science. Finally, the project will investigate mathematical problems related to the turbulence aspects of wave turbulence theory. There, the PI intends to use the above rigorous derivation of the wave kinetic equations, combined with an analysis of solutions to those equations, to understand turbulence phenomena for wave systems, such as energy cascades and growth of Sobolev norms.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.