Population Fluctuations at Equilibrium and Kramers' Rate of Diffusive Barrier Crossing

For diffusive dynamics, Kramers' expressions for the transition rates between two basins separated by a high barrier has been rederived in many ways. Here, we will do it using the Bennett-Chandler method which focuses on the time derivative of the occupation number correlation function that describes fluctuations of the basin populations at equilibrium. This derivative is infinite at t = 0 for diffusive dynamics. We show that on a slightly longer time scale, comparable to the time it takes the system to fall off the barrier, this time derivative is proportional to the spatial derivative of the committor evaluated at the barrier top. The committor or splitting probability is the probability that the system starting on the barrier ends up in one basin before the other. This probability can be found analytically. By asymptotic evaluation of the relevant integrals, we recover Kramers' result without having had to rely on his formidable physical intuition.