APDE seminar@BCAM: On the rigorous derivation of the Wave Kinetic Equation from the periodic cubic NLS

In a system formed of many objects (particles, molecules, electrons, ...), instead of computing the evolution of each individual object described by elementary laws of physics, a kinetic equation predicts the distribution of such objects. The most famous example is likely Boltzmann’s equation, which describes the evolution of the density of gas molecules, each of which moves according to Newton’s laws. An analogue theory exists for a wave system described by a dispersive PDE, whose “average” behavior is described by a wave kinetic equation (WKE). The classic derivation of these kinetic equations is a heuristic limiting procedure, but it is a major pending task is to make it rigorous.

This talk will be a very informal introduction to the derivation of the Wave Kinetic Equation. I will discuss its heuristic derivation from the periodic cubic NLS and the main challenges to make it rigorous, give the most recent results, and show how the problem has very relevant probabilistic, combinatorial and number theoretic components.