Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach

We introduce a fractional Fokker-Planck equation describing the stochastic evolution of a particle under the combined influence of an external, nonlin ear force and a thermal heat bath. For the force-free case, a subdiffusive behavior is recovered. The equation is shown to obey generalized Einstein r elations, and its stationary solution is the Boltzmann distribution. The re laxation of single modes is shown to follow a Mittag-Leffler decay. We disc uss the example of a particle in a harmonic potential. [S0031-9007(99)09071 -7].