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

## TL;DR

This paper establishes explicit exponential convergence rates to equilibrium for Langevin dynamics with singular potentials, providing practical estimates based on growth conditions and local Poincaré constants, applicable to multi-particle systems.

## Contribution

It extends convergence analysis to singular potentials under general growth conditions, offering explicit, measurable convergence rates with estimates depending on system parameters.

## Key findings

- Exponential convergence to equilibrium is proven under broad growth conditions.
- Explicit convergence rate estimates are derived in terms of system parameters.
- Results apply to systems with singular interactions and polynomial confining 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 $N$.