Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solution
Abstract
The nonlinear diffusion equation ∂ρ/∂t=DΔ~ρν is analyzed here, where Δ~≡(1/rd-1)(∂/∂r)rd-1- θ∂/∂r, and d, θ, and ν are real parameters. This equation unifies the anomalous diffusion equation on fractals (ν=1) and the spherical anomalous diffusion for porous media (θ=0). An exact point-source solution is obtained, enabling us to describe a large class of subdiffusion [θ>(1-ν)d], ``normal'' diffusion [θ=(1-ν)d] and superdiffusion [θ<(1-ν)d]. Furthermore, a thermostatistical basis for this solution is given from the maximum entropic principle applied to the Tsallis entropy.
- Publication:
-
Physical Review E
- Pub Date:
- March 2001
- DOI:
- 10.1103/PhysRevE.63.030101
- arXiv:
- arXiv:cond-mat/0010142
- Bibcode:
- 2001PhRvE..63c0101M
- Keywords:
-
- 05.20.-y;
- 05.40.Fb;
- 05.40.Jc;
- Classical statistical mechanics;
- Random walks and Levy flights;
- Brownian motion;
- Condensed Matter - Statistical Mechanics
- E-Print:
- 3 pages, 2 eps figures