- fractional calculus modelling


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April 2001 - 010401
Recent paper on Fractional Fokker-Planck equation by Barkai

An interesting paper on Fractional Fokker-Planck equation is recentely appeared on a Physics prestigious journal:
E. Barkai, Fractional Fokker-Planck Equation, solution, and application
Physical Review E, Vol 63, pp 046118-1/17 (2001).(


Recently, Metzler et al. [Phys. Rev. Lett. 82, 3563 (1999)], introduced a fractional Fokker-Planck 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 Fokker-Planck equation onto the solution of the FFPE, and is based on Levy'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 force-free 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 one-dimensional fractional first passage time in an external linear field is investigated in detail. The FFPE is shown to be compatible with the Scher-Montroll 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.