Thermally activated crossing of a sharp potential barrier

Abstract

The methods of nonequilibrium statistical mechanics are used to determine the rate at which particles in contact with a heat bath are able to escape from a potential well over a sharp barrier. Specifically, the rate constant for the process is found by computing the smallest nonzero eigenvalue of the governing Fokker–Planck equation. Singular perturbation techniques are used to obtain approximate expressions for the appropriate eigenfunction, and substitution into a variational formula then provides the desired eigenvalue. Explicit results are presented for cases of moderate and high friction, and the latter, in particular, is in sharp disagreement with the findings of earlier investigators: it indicates that the approximation afforded by the simpler Smoluchowski equation is not valid, even if the friction constant is arbitrarily large. Semianalytical calculations of the rate constant provide support for the new theory.