Overdamped limits for Langevin dynamics with position-dependent coefficients via L2-hypocoercivity

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 [1] and suggest a solution.