Escape by diffusion from a parabolic well across a parabolic barrier

The problem of escape of a particle by diffusion from a parabolic potential well across a parabolic barrier adjacent to free space is studied on the basis of the one-dimensional Smoluchowski equation for the space- and time-dependent probability distribution. For the model potential the Smoluchowski equation is solved exactly by a Laplace transform with respect to time for the initial condition that at time zero the probability distribution is given by a thermal equilibrium distribution in the well. In the limit of a high barrier the rate of escape is given by an asymptotic result due to Kramers. For a potential barrier of moderate height there are significant corrections to the asymptotic result. The probability that the particle resides in the well–barrier region at time tt decays with a 1/t long-time tail.