Hilbert's Sixth Problem: From Particles to Waves

希尔伯特第六个问题：从粒子到波

• 批准号: 2350242
• 负责人: Zaher Hani
• 金额: $ 39.83万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350242&HistoricalAwards=false
• 关键词: Hilbert; Sixth; Problem; Particles; Waves

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.

希尔伯特（Hilbert）于1900年提出的第六个问题，要求对统计物理学的宏观定律进行严格的数学推导，该定律是由麦克斯韦（Maxwell）和鲍尔茨曼（Boltzmann）于19世纪提出的，从动力学的微观定律（又名第一原理）开始。该问题的经典设置与粒子系统有关，这些系统根据经典力学定律碰撞。在更现代的统计物理学理论中也出现了同样的问题，在统计物理学的理论中，颗粒被根据一些汉密尔顿波型部分微分方程相互作用的波代替。波浪统计物理学的这种理论通常以“波湍流理论”的名义出现，因为它们在理解波浪系统中的湍流行为方面起着核心作用。这在许多科学领域（例如量子力学，海洋学，气候科学等）都有应用。从广义上讲，该项目的目的是推进数学，因此对科学的理解，对这种湍流理论的理解，并在数学物理学中解决了统计力学基础上的数学物理学的一些长期概念。该项目为研究生提供了研究培训机会。即使在经典环境中，希尔伯特的第六个问题仍然是一项艰巨的任务，仅在短时间内解决了。该项目旨在提供长期解决方案，从而为这个长期开放的问题提供了最终答案。从牛顿的定律开始，其金额为Boltzmann的动力学方程式提供了严格的推导，然后是宏观流体模型（Euler's and Navier-Stokes方程）的推导。同时，项目提案在波湍流理论的设置中类似的理由。在那里，首席研究员（PI）试图为各种科学意义的波浪系统提供相应的“波动力学方程”的长期推导。从非线性Schrödinger方程作为非线性波系统的主要模型开始，随后将对其他波浪系统进行类似的投资，例如许多粒子量子系统以及来自海洋和气候科学的一些模型。最后，该项目将研究与波湍流理论的湍流方面有关的数学问题。 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 precious of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.