Anomalous diffusion and anisotropic nonlinear Fokker-Planck equation

Abstract

We analyse a bidimensional nonlinear Fokker-Planck equation by considering an anisotropic case, whose diffusion coefficients are $D_x\propto |x{|}^{-\theta }$ and $D_y\propto |y{|}^{-\gamma }$ with $\theta ,\gamma \in \mathcal{R}$. In this context, we also investigate two situations with the drift force $\vec{F}(\vec{r},t)=(-k_{x}x,-k_{y}y)$. The first one is characterized by $k_x/k_y=(2+\gamma )/(2+\theta )$ and the second is given by $k_x=k$ and $k_y=0$. In these cases, we can verify an anomalous behavior induced in different directions by the drift force applied. The found results are exact and exhibit, in terms of the $q$-exponentials, functions which emerge from the Tsallis formalism. The generalization for the $D$-dimensional case is discussed.

6 pages, tex file