2025/11/13 by Mitia Duerinckx, Duerinckx, Mitia, Corentin Le Bihan +1 · 1 citation
Materials Science · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Material Dynamics and Properties #Mathematical Physics (math-ph) #Micro and Nano Robotics #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.2511.10778
openalex publication_date 2025/11/13 · openalex created_date 2025/11/18 · openalex updated_date 2026/07/28
We study the long-time dynamics of a tagged particle coupled to a background of N other particles, all interacting through long-range pairwise forces in the mean-field scaling, with the background initially at thermal equilibrium. Starting from the N-particle BBGKY hierarchy, we introduce a simplified (truncated) hierarchical model and show, in sufficiently large spatial dimension, that the tagged-particle density converges, on timescales t∼ N, to the solution of a linear Fokker-Planck equation, viewed as the linearization of the Landau equation. This provides, in a simplified setting, a rigorous derivation of the slow thermalization predicted by Lenard-Balescu theory. Our approach relies on a rigorous Dyson expansion in terms of Feynman diagrams and on a novel renormalization scheme that removes leading recollisions. The main technical challenge is to control the effect of phase-space filamentation within the diagrams, which we achieve by combining phase mixing and hypoelliptic regularity. Although restricted to a simplified model, our analysis offers new insight into Lenard-Balescu thermalization: notably, the renormalization appears to transform free propagators into hypoelliptic ones, providing a key mechanism that compensates for filamentation.