From a Generalized Chapman−Kolmogorov Equation to the Fractional Klein−Kramers Equation

A non-Markovian generalization of the Chapman−Kolmogorov transition equation for continuous time random processes governed by a waiting time distribution is investigated. It is shown under which conditions a long-tailed waiting time distribution with a diverging characteristic waiting time leads to a fractional generalization of the Klein−Kramers equation. From the latter equation a fractional Rayleigh equation and a fractional Fokker−Planck equation are deduced. These equations are characterized by a slow, nonexponential relaxation of the modes toward the Gibbs−Boltzmann and the Maxwell thermal equilibrium distributions. The derivation sheds some light on the physical origin of the generalized diffusion and friction constants appearing in the fractional Fokker−Planck equation.