Fractional Fokker–Planck equation on heterogeneous fractal structures in external force fields and its solutions

We introduce a heterogeneous fractional Fokker–Planck equation (HFFPE) involving external force fields describing systems on heterogeneous fractal structure medium. The HFFPE is shown to obey generalized Einstein relation, and its stationary solution is the generalized Boltzmann distribution. It is proved that the asymptotic shape of its solution is a stretched Gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for generic potentials. Furthermore, it is shown that the solution of FFPE with parameter has the same properties as that ones of HFFPE.