Escape by diffusion from a double-well potential across a barrier

Escape by diffusion from a double-well potential across a barrier is studied on the basis of the Smoluchowski equation in one dimension. By comparison with exact results for a piecewise parabolic potential a reduced description is constructed in terms of a set of rate equations for the occupation probabilities of the two wells. The rate equations contain memory terms and a source term for the rate of return from the outer space. The reduced description yields a quite accurate approximation for the two example potentials studied.