Title: Lenard-Balescu thermalization: rigorous derivation from a toy model
Abstract: We study the long-time dynamics of a tagged particle coupled to a background of N other particles, all interacting through long-range forces in the mean-field regime, with the background initially at equilibrium. Starting from a (truncated) toy version of the N-particle BBGKY hierarchy, we show in sufficiently large space dimension that the tagged-particle density converges on the timescale t=O(N) to the solution of a linear Fokker-Planck equation, viewed as a linearization of the Landau equation, thereby proving the slow thermalization predicted by Lenard-Balescu theory.
The approach relies on a rigorous Dyson expansion in terms of Feynman diagrams together with a renormalization scheme that removes leading recollisions. The first main challenge is to control filamentation in the diagrams, for which we appeal to phase mixing and hypoelliptic regularity. Although restricted to a simplified model, our analysis offers some new insight into the problem: the renormalization appears to transform free propagators into hypoelliptic ones, providing a key mechanism to compensate for filamentation. Joint work with Corentin Le Bihan.