Transforming Gaussian diffusion into fractional a generalized law of large numbers approach

Abstract

The fractional Fokker-Planck equation (FFPE) [R. Metzler, E. Barkai, J. Klafter, Phys. Rev. Lett., 82, 3563 (1999)] describes an anomalous sub diffusive behavior of a particle in an external force field. In this paper we present the solution of the FFPE in terms of an integral transformation. The transformation maps the solution of ordinary Fokker-Planck equation onto the solution of the FFPE. We investigate in detail the force free particle and the particle in uniform and harmonic fields. The meaning of the transformation is explained based on the asymptotic solution of the continuous time random walk (CTRW). We also find an exact solution of the CTRW and compare the CTRW result with the integral solution of the FFPE for the force free case.