Abstract
Activated rate processes of a reactive coordinate in a symmetric double well coupled to a harmonic mode are studied in the limit of large damping. The transition rate is given by the least nonvanishing eigenvalue of the corresponding two-dimensional Smoluchowski equation. This eigenvalue is numerically determined for four different temperatures and various different coupling and anisotropy parameters and compared with the value of the Rayleigh quotient for the trial function following from the Kramers-Langer theory. Deviations between the numerically exact rates and Kramers-Langer theory are due to finite-barrier heights and may become very large in the case of a slow harmonic mode, i.e., large anisotropy. As long as these deviations are not too large, the rate expression obtained from the Rayleigh quotient is in excellent agreement with the numerically exact results. The stochastic separatrix is numerically determined as the node of the eigenfunction corresponding to the least nonvanishing eigenvalue and compared to results from a perturbation theory.
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