Escape rate for nonequilibrium processes dominated by strong non-detailed balance force

Quantifying the escape rate from a meta-stable state is essential to understand a wide range of dynamical processes. Kramers' classical rate formula is the product of an exponential function of the potential barrier height and a pre-factor related to the friction coefficient. Although many applications of the rate formula focused on the exponential term, the prefactor can have a significant effect on the escape rate in certain parameter regions, such as the overdamped limit and the underdamped limit. There have been continuous interests to understand the effect of non-detailed balance on the escape rate; however, how the prefactor behaves under strong non-detailed balance force remains elusive. In this work, we find that the escape rate formula has a vanishing prefactor with decreasing friction strength under the strong non-detailed balance limit. We both obtain analytical solutions in specific examples and provide a derivation for more general cases. We further verify the result by simulations and propose a testable experimental system of a charged Brownian particle in electromagnetic field. Our study demonstrates that a special care is required to estimate the effect of prefactor on the escape rate when non-detailed balance force dominates.