Hilbert’s sixth problem asks for a mathematically rigorous justification of the macroscopic laws of statistical physics from the microscopic laws of dynamics. The classical setting of this problem asks for the justification of Boltzmann’s kinetic equation from Newtonian particle dynamics. This justification has been proven for short times, starting with the work of Lanford in 1975, but its long time justification remains one of the biggest open problems in kinetic theory.

If classical colliding particles are replaced with interacting waves, one formally obtains what is known as "wave kinetic theory”, which is sometimes also called "wave turbulence theory". This theory of statistical physics for waves has been developed, starting in the late 1920s, for wave systems that arise in various scientific disciplines like many-particle quantum physics, oceanography, climate science, etc. The central mathematical problem there is also the justification of a kinetic equation, known as the wave kinetic equation, starting from the Hamiltonian PDE that governs the corresponding microscopic system. In this talk, we shall describe the state of the art of this problem, leading to a most recent joint work with Yu Deng (USC), in which we give the first instance of a long time justification of a nonlinear (particle or wave) collisional kinetic limit.