Velocity dependence of friction and Kramers relaxation rates

We study the influence of the velocity dependence of friction on the escape rate of a Brownian particle from the deep potential well (E(b)≫k(B)T, E(b) is the barrier height, k(B) is the Boltzmann constant, and 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≪M) up to the terms of the order of O(λ⁴), λ² = m/M≪1. The term ∼1 is equivalent to the Fokker-Planck dissipative operator, and the term ∼λ² is responsible for the velocity dependence of friction. As expected, the correction to the Kramers escape rate in the overdamped limit is proportional to λ²and is small. The corresponding correction in the underdamped limit is proportional to λ²E(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