From a generalized Chapman-Kolmogorov equation to the fractional Klein-Kramers equation

A non-Markovian generalization of the Chapman-Kolmogorov transition equatio n for continuous time random processes governed by a waiting time distribut ion is investigated. It is shown under which conditions a long-tailed waiti ng 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 equa tion are deduced. These equations are characterized by a slow, nonexponenti al relaxation of the modes toward the Gibbs-Boltzmann and the Maxwell therm al equilibrium distributions. The derivation sheds some light on the physic al origin of the generalized diffusion and friction constants appearing in the fractional Fokker-Planck equation.