A toy model for the Lenard-Balescu problem

We are interested in a system of N particles in the mean-field regime, that is, with weak (1/N) but long-range (first-order) interactions. Over short timescales (first order), this system is described by the Vlasov equation, which is conservative. Over much longer timescales (N order), however, the Lenard-Balescu theory predicts the relaxation of the system towards equilibrium (in the strong sense, with entropy dissipation). Formally, this slow relaxation is explained by the correlations between the particles. However, no rigorous justification has been obtained to date: the only results are consistency proofs (derivation at time 0).

In this talk, we revisit the problem starting from a simplified model, inspired by the linear version of the problem (due to Duerinckx and Saint-Raymond), in which the BBGKY hierarchy is truncated to an arbitrary order. Starting from a perturbative approach in Feynman diagrams, using ideas of renormalization, and hypoelliptic estimates for renormalized propagators, we manage to reach the kinetic time (of order N) for this model.