Speaker: Zaher Hani, University of Michigan
Mathematical wave kinetic theory
Wave kinetic theory, also known as wave turbulence theory, is the wave counterpart of Boltzmann’s classical kinetic theory, where colliding particles are replaced by interacting waves. The first incarnations of this theory first appeared around 1930 in the context of crystal vibrations (by Peierls) and quantum statistical physics (by Nordheim and Uhling-Uhlenbeck). Starting from the 1960’s the theory metamorphosed into a general paradigm for understanding the nonequilibrium statistical physics of wave systems, from oceanography to plasma physics.
The mathematical study of wave kinetic theory is a relatively young subject that featured an explosion of activity in the past decade. It culminated in a series of joint works with Yu Deng, in which we gave the rigorous foundations of wave kinetic theory: deriving the wave kinetic equation and proving propagation of chaos for short times. This was the counterpart of Lanford’s theorem in the classical particle theory. In 2023, we extended the derivation to long times, for as long as the solution of the kinetic equation exists. This was the first result of its kind, be it for particle or wave systems, which naturally led to our subsequent work deriving the classical Boltzmann equation on arbitrarily long times (with Xiao Ma). In this talk, we shall give an introduction to wave kinetic theory, review some of this progress, and discuss some future directions.
Where: Sapienza Università di Roma, Dipartimento di Matematica, Aula L
