Probability distribution escape time calculations

The conditions are such that the particle is originally in a potential hole, but it may escape in the course of time by passing over a potential barrier. The analytical problem is to calculate the escape probability as a function of the temperature and of the viscosity of the medium, and then to compare the values so found with the ones of the activated state method. For sake of simplicity, Kramers studied only the one-dimensional model, and the calculation rests on the equation of diffusion obeyed by a density distribution of particles in the. phase space. Definite results can be obtained in the limiting cases of small and large viscosity, and in both cases there is a close analogy with the Cristiansen treatment of chemical reactions as a diffusion problem. When the potential barrier corresponds to a rather smooth maximum, a reliable solution is obtained for any value of the viscosity, and, within a large range of values of the viscosity, the escape probability happens to be practically equal to that computed by the activated state method. 

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