Exact solution of the diffusion in a bistable piecewise linear potential

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Abstract
An exact solution is obtained for the diffusion in the symmetric bistable W-shaped potential, by using the Laplace transform method. The results thus obtained are used to elucidate the status of the Kramers theory, usually formulated in the limit of large barrier height, as well as to find corrections to the Kramers approximation, and their relation to the pattern of singularities in the complex plane of the Laplace-transformed time variable. The probability mass within a given attraction basin is studied in detail. It is found to satisfy a non-Markovian dynamics in the general case, which reduces to a simple rate equation in the Kramers regime.

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