Deriving fractional Fokker-Planck equations from a generalised master equation

A generalised master equation is constructed from a non-homogeneous random walk scheme. It is shown how fractional Fokker-Planck equations for the description of anomalous diffusion in external fields, recently proposed in the literature, can be derived from this framework. Long-tailed waiting time distributions which cause slowly decaying memory effects, are demonstrated to give rise to a time-fractional Fokker-Planck equation that describes systems close to thermal equilibrium. An extension to include also Levy flights leads to a generalised Laplacian in the corresponding fractional Fokker-Planck equation.