Hilbert's Sixth Problem: From Particles to Waves

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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.