Introduction to wave turbulence theory

par Enguerrand Brun

lundi 15 déc. 2025, 16:30 → 17:30 Europe/Paris

S435 (UMPA)

The sixth Hilbert problem is about treating "in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.". In particular, the question of how to go from microscopic laws to mesoscopic laws in nature is fundamental, and physicist have for long derived mesoscopic laws for many body systems. The point of Hilbert's sixth problem is to justify these derivations mathematically rigorously. One example would be a system of N particles of small radius r which obey the fundamental laws of Newton. We take the number of particles to infinity and their radius to 0 to derive Boltzmann equations. Of course, N and r have to dependent one on the other for the derivation to work. Wave turbulence is about making similar derivations but replacing the many body systems by a "many wave" system. The microscopic law will be a Schrödinger equation and our mesoscopic one a kinetic equation. We will only give the heuristic of the proof and not go into the details of the analysis.