Phase space distribution function approach to the Kramers problem. III. Anharmonic potentials Original Research Article

In a recent series of papers we showed how a phase space distribution function approach could be combined with the method of reactive flux to obtain the rate of barrier crossing as a function of solvent friction in the intermediate to high friction limit. Those studies dealt with both the Markovian and non-Markovian cases, but were restricted to analytic results for parabolic barriers. Here we extend the approach to anharmonic barriers. The guiding approximation is to assume that the phase space distribution for each initial velocity, starting at the barrier top, remains Gaussian for all time, with Gaussian parameters given by time-dependent mean field equations. We expect this approximation to be accurate for short times, up to the “Ehrenfest” time; if this time exceeds the “plateau” time — the time for the distribution to reach its asymptotic partitioning — the quality of the results should be high. There are no adjustable parameters, although some reasonable criterion is needed for ending the integration of the mean field equations to prevent divergence. Numerical results for the linear cusp and the quartic potential show that the method is quite accurate for dimensionless frictions ≤1, although the accuracy degrades for higher frictions.