Master equation representation of Fokker-Planck operators in the energy diffusion regime: strong collision versus random walk processes

The localized function formalism, introduced to transform diffusion equations with multistable potentials into discrete master equations, is applied to the Fokker-Planck equation in underdamped conditions when the energy becomes the slow variable. The resulting transition rates describe the kinetic behaviour of the system much more accurately than the asymptotic formula derived by Kramers. It is shown that in the two opposite frictional regimes of the Fokker-Planck equation, the transition matrix assumes the limiting forms of the random walk between nearest-neighbour sites, and of the random jump among all sites as in the strong collision model.