Derivation of a Fokker–Planck equation for generalized Langevin dynamics

A Fokker–Planck equation describing the statistical properties of Brownian particles acted upon by long-range stochastic forces with power-law correlations is derived. In contrast with previous approaches (Wang, Phys. Rev. A 45 (1992) 2), it is shown that the distribution of Brownian particles after release from a point source is broader than Gaussian and described by a Fox function. Transport is shown to be ballistic at short times and either sub-diffusive or super-diffusive at large times. The imposition of occasional trapping events onto the Brownian dynamics can result in confined diffusion (d/dt〈x2〉→0) at long times when the mean trapping time is divergent. It is suggested that such dynamics describe protein motions in cell membranes.