Modern view of activated rate processes: unidirectional fluxes at equilibrium, correlation functions, and splitting probabilities
More than 80 years ago Kramers published a paper calculating how fast a Brownian particle escapes from a potential well over an activation barrier. Since then Kramers' model has been widely adopted by nuclear physics, biophysics and chemical physics communities as a description of activated barrier crossing. From a chemical kinetics perspective, Kramers' theory provides a mapping from continuous dynamics to discrete-state chemical kinetics. Motivated by recent developments, this Perspective provides a rigorous way of performing such a mapping, explaining why and how Kramers' theory works from several points of view. Specifically, we consider transitions of a Brownian particle between two potential wells corresponding to the ``reactant'' and the ``product'' of a chemical reaction. A central unifying idea is to divide the equilibrium ensemble of possible states of the system into two sub-ensembles corresponding to the reactant and product states and then to consider fluxes between these sub-ensembles. Importantly, naive separation based on the location measured relative to the barrier top does not result in a mapping that is physically tenable, and instead the past of the trajectory should be considered. Thus constructed reactant and product ensembles provide an internally consistent description of the problem when also viewed from two different perspectives: one based on the definition of the rate as a conditional transition probability per unit time and the other based on the relaxation modes of the time-evolution operator governing the dynamics.