Nonlinear anomalous diffusion equation and fractal dimension: Exact generalized gaussian solution

Details Nonlinear anomalous diffusion equation and fractal dimension: Exact generalized gaussian solution Nonlinear anomalous diffusion equation and fractal dimension: Exact generalized gaussian solution

2002-02-14

arxiv.org/abs/cond-mat/0202233v1

Phys. Rev. E 65, 041108 (2002)

Physics

Condensed Matter

Statistical Mechanics

Latex, 6 pages. To appear in Phys. Rev. E

Scientific paper

10.1103/PhysRevE.65.041108

In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the $N$-dimensional nonlinear diffusion equation $\partial\rho /\partial{t}={\bf \nabla} \cdot (K{\bf \nabla} \rho^{\nu})-{\bf \nabla}\cdot(\mu{\bf F} \rho)-\alpha \rho ,$ where $K=D r^{-\theta}$, $\nu$, $\theta$, $\mu$ and $D$ are real parameters and $\alpha$ is a time-dependent source. This equation unifies the O'Shaugnessy-Procaccia anomalous diffusion equation on fractals ($\nu =1$) and the spherical anomalous diffusion for porous media ($\theta=0$). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.

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