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 (Eb>>kBT, Eb is the barrier height, kB 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(lambda(4)), lambda(2)=m/M<<1. The term approximately 1 is equivalent to the Fokker-Planck dissipative operator, and the term approximately 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)Eb/(kBT) 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.