Construction, ergodicity and rate of convergence of N-particle Langevin dynamics with singular potentials

Abstract

We construct N-particle Langevin dynamics in \({\mathbb{R}^d}\) or in a cuboid region with periodic boundary for a wide class of N-particle potentials Φ and initial distributions which are absolutely continuous w.r.t. Lebesgue measure. The potentials are in particular allowed to have singularities and discontinuous gradients (forces). An important point is to prove an L p -uniqueness of the associated non-symmetric, non-sectorial degenerate elliptic generator. Analyzing the associated functional analytic objects, we also give results on the long-time behaviour of the dynamics, when the invariant measure is finite: Firstly, we prove the weak mixing property whenever it makes sense (i.e. whenever {Φ < ∞} is connected). Secondly, for a still quite large class of potentials we also give a rate of convergence of time averages to equilibrium when starting in the equilibrium distribution. In particular, all results apply to N-particle systems with pair interactions of Lennard–Jones type.