Breakdown of the Overdamped Approximation in Fluctuating Environments

Abstract

The overdamped approximation is widely used to describe Brownian motion in complex environments, but its validity becomes nontrivial when the friction coefficient itself fluctuates in time. We investigate this problem by comparing underdamped and overdamped Langevin dynamics subject to the same fluctuating friction and satisfying the fluctuation--dissipation relation at a common temperature. We show that environmental averaging and inertial elimination generally lead to different long-time transport when environmental fluctuations are fast compared with velocity relaxation. For rapidly fluctuating friction, the underdamped dynamics is governed by the arithmetic mean friction and yields $D_{\rm eff}^{\rm under}=k_{\rm B}T/\langle\gamma\rangle$, whereas the overdamped dynamics gives $D_{\rm eff}^{\rm over}=k_{\rm B}T\langle\gamma^{-1}\rangle$. For a two-state Markov friction, we derive the finite-time effective diffusion coefficient exactly and identify the full crossover between these two regimes, controlled by the competition between velocity relaxation and environmental switching. We further establish the fast-fluctuation result for general stationary friction processes and verify it for a continuous log-Ornstein--Uhlenbeck environment, for which the characteristic crossover time can also be determined independently. Our results reveal a noncommutativity between environmental averaging and inertial elimination, establish a timescale-dependent criterion for the validity of overdamped dynamics in temporally heterogeneous environments, and show that rapid environmental fluctuations can enhance, rather than suppress, the consequences of inertia.