Gamma Calculus Beyond Villani and Explicit Convergence Estimates for Langevin Dynamics with Singular Potentials

Abstract

We apply Gamma calculus to the hypoelliptic and non-symmetric setting of Langevin dynamics under general conditions on the potential. This extension allows us to provide explicit estimates on the convergence rate (which is exponential) to equilibrium for the dynamics in a weighted $$H^1(\mu )$$ sense, $$\mu $$ denoting the unique invariant probability measure of the system. The general result holds for singular potentials, such as the well-known Lennard–Jones interaction and confining well, and it is applied in such a case to estimate the rate of convergence when the number of particles N in the system is large.