Diffusive barrier crossing rates from variationally determined eigenvalues

Kramers' procedure for calculating the rate of activated processes involves partitioning space into reactant, barrier, and product regions by introducing two dividing surfaces. Then, a nonequilibrium steady state is established by injecting particles on one surface and removing them when they reach the other. The rate is obtained as the ratio of the steady-state flux between the surfaces and the population of the initial well. An alternative procedure that seems less artificial is to estimate the first non-zero eigenvalue of the operator that describes the dynamics and then equate its magnitude to the sum of the forward and backward rate constants. Here, we establish the relationship between these approaches for diffusive dynamics, starting with the variational principle for the eigenvalue of interest and then using a trial function involving two adjustable surfaces. We show how Kramers' flux-over-population expression for the rate constant can be obtained from our variationally determined eigenvalue in the special case where the reactant and product regions are separated by a high barrier. This work exploits the modern theory of activated rate processes where the committor (the probability of reaching one dividing surface before the other) plays a central role. Surprisingly, our upper bound for the eigenvalue can be expressed solely in terms of mean first-passage times and the mean transition-path time between the two dividing surfaces.