Overdamped limits for Langevin dynamics with position-dependent coefficients via $L^2$-hypocoercivity

TL;DR

This paper presents a straightforward derivation of the overdamped limit for Langevin dynamics with position-dependent coefficients using hypocoercivity estimates, clarifying the noise-induced drift and extending to related models.

Contribution

The authors introduce a new hypocoercivity-based approach for deriving overdamped limits in Langevin dynamics with position-dependent coefficients, providing clearer insights and broader applicability.

Findings

Derived the overdamped approximation using hypocoercivity estimates.

Clarified the origin of the noise-induced drift term.

Extended the approach to kinetic models with position-dependent mass matrices.

Abstract

This note provides a simple derivation of the overdamped approximation for kinetic (or underdamped) equilibrium Langevin dynamics, in cases where certain coefficients depend on the position variable. The equivalent small-mass limit of these dynamics, known as the Kramers--Smoluchowski approximation, in the case of a state-dependent friction coefficient, has been previously studied by a variety of approaches. Our new approach uses hypocoercivity estimates, which may be of interest in their own right, and lead to a very direct derivation, providing in particular a clear explanation of the ``noise-induced drift'' term in the overdamped equation in the case of a state-dependent friction term. Using the same approach, we also treat effective kinetic dynamical models derived from a coarse-graining approximation of a high-dimensional system, as well as a class of kinetic dynamics with position-dependent mass matrices. All of these models are relevant to applications in computational chemistry. We finally identify a mistake in a related work, and suggest a solution.