Velocity dependence of friction and Kramers relaxation rates

TL;DR

This paper investigates how velocity-dependent friction influences the escape rates of Brownian particles from deep potential wells, revealing significant effects in underdamped and moderately damped regimes, but minor effects in overdamped conditions.

Contribution

It provides a detailed theoretical analysis of velocity-dependent friction effects on Kramers escape rates using the Rayleigh model, extending understanding beyond the traditional overdamped approximation.

Findings

Velocity dependence affects escape rates significantly in underdamped regimes.

Corrections to Kramers rate are proportional to λ², with larger impact when E_b/(k_B T) is high.

Effects are minor for overdamped particles, more important for under- and moderately damped particles.

Abstract

We study the influence of the velocity dependence of friction on the escape of a Brownian particle from the deep potential well ($E_{b} \gg k_{B}T$, $E_{b}$ is the barrier height, $k_{B}$ is the Boltzmann constant, $T$ is the bath temperature). The bath-induced relaxation is treated within the Rayleigh model (a heavy particle of mass $M$ in the bath of light particles of mass $m\ll M$) up to the terms of the order of $O(\lambda^{4})$, $\lambda^{2}=m/M\ll1$. The term $\sim 1$ is equivalent to the Fokker-Planck dissipative operator, and the term $\sim \lambda^{2}$ is responsible for the velocity dependence of friction. As expected, the correction to the Kramers escape rate in the overdamped limit is proportional to $\lambda^{2}$ and is small. The corresponding correction in the underdamped limit is proportional to $\lambda^{2}E_{b}/(k_{B}T)$ and is not necessarily small. We thus suggest that the effects due to the velocity-dependent friction may be of considerable importance in determining the rate of escape of an under- and moderately damped Brownian particle from a deep potential well, while they are of minor importance for an overdamped particle.