Vol. 2, No. 1, 2021

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Lenard–Balescu correction to mean-field theory

Mitia Duerinckx and Laure Saint-Raymond

Vol. 2 (2021), No. 1, 27–69

In the mean-field regime, the evolution of a gas of N interacting particles is governed in first approximation by a Vlasov type equation with a self-induced force field. This equation is conservative and describes return to equilibrium only in the very weak sense of Landau damping. However, the first correction to this approximation is given by the Lenard–Balescu operator, which dissipates entropy on the very long timescale O(N). We show how one can derive rigorously this correction on intermediate timescales (of order O(Nr) for r < 1) close to equilibrium.

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interacting particle system, mean-field limit, fluctuations around thermal equilibrium, tagged particle, thermalization, Lenard–Balescu kinetic equation, BBGKY hierarchy, cumulant expansion, linearized Vlasov operator, Markovian limit
Mathematical Subject Classification 2010
Primary: 35Q70, 35Q82, 82D10
Received: 22 November 2019
Revised: 17 September 2020
Accepted: 19 October 2020
Published: 16 March 2021
Mitia Duerinckx
Université Paris-Saclay, CNRS
Laboratoire de Mathématiques d’Orsay
Université Libre de Bruxelles
Département de Mathématique
Laure Saint-Raymond
Unité de mathématiques pures et appliquées
École Normale Supérieure de Lyon