Fractional FokkerPlanck equation, solution, and application
Abstract
Recently, Metzler et al. [Phys. Rev. Lett. 82, 3563 (1999)], introduced a fractional FokkerPlanck equation (FFPE) describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field, and a Boltzmann thermal heat bath. In this paper we present the solution of the FFPE in terms of an integral transformation. The transformation maps the solution of ordinary FokkerPlanck equation onto the solution of the FFPE, and is based on Lévy's generalized central limit theorem. The meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk (CTRW). We investigate in detail (i) a forcefree particle, (ii) a particle in a uniform field, and (iii) a particle in a harmonic field. We also find an exact solution of the CTRW, and compare the CTRW result with the corresponding solution of the FFPE. The relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given. An example of the onedimensional fractional first passage time in an external linear field is investigated in detail. The FFPE is shown to be compatible with the ScherMontroll approach for dispersive transport, and thus is applicable in a large variety of disordered systems. The simple FFPE approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena.
 Publication:

Physical Review E
 Pub Date:
 April 2001
 DOI:
 10.1103/PhysRevE.63.046118
 Bibcode:
 2001PhRvE..63d6118B
 Keywords:

 02.50.r;
 05.40.Fb;
 05.30.Pr;
 05.45.Df;
 Probability theory stochastic processes and statistics;
 Random walks and Levy flights;
 Fractional statistics systems;
 Fractals