Gamma calculus beyond Villani and explicit convergence estimates for Langevin dynamics with singular potentials

Abstract

This paper studies convergence to equilibrium for second-order Langevin dynamics under general growth conditions on the potential. Although we are principally motivated by the case when the potential is singular, e.g. when the dynamics has repulsive forces and/or interactions, the results presented in this paper hold more generally. In particular, our main result is that, given (very) basic structural and growth conditions on the potential, the dynamics relaxes to equilibrium exponentially fast in an explicitly measurable way. The ``explicitness" of this result comes directly from the constants appearing in the growth conditions, which can all be readily estimated, and a local Poincar\'{e} constant for the invariant measure $\mu$. This result is applied to the specific situation of a singular interaction and polynomial confining well to provide explicit estimates on the exponential convergence rate $e^{-\sigma}$ in terms of the number $N \geqslant 1$ of particles in the system. We will see that $\sigma \geqslant c/(\rho \vee N^p)$, where $\rho>0$ is the local Poincar\'{e} constant for $\mu$ and $c>0, p\geqslant 1$ are constants that are independent of $N$.