Issue
EPL
Volume 80, Number 6, December 2007
Article Number 60006
Number of page(s) 4
Section General
DOI http://dx.doi.org/10.1209/0295-5075/80/60006
Published online 13 November 2007
EPL, 80 (2007) 60006
DOI: 10.1209/0295-5075/80/60006

Nonlinear diffusion from Einstein's master equation

J. P. Boon and J. F. Lutsko

Physics Department, Université Libre de Bruxelles - CP 231 B-1050 Bruxelles, Belgium

jpboon@ulb.ac.be
jlutsko@ulb.ac.be

received 8 September 2007; accepted in final form 19 October 2007; published December 2007
published online 13 November 2007

Abstract
We generalize Einstein's master equation for random-walk processes by considering that the probability for a particle at position r to make a jump of length j lattice sites, Pj(r), is a functional of the particle distribution function f(r, t). By multiscale expansion, we obtain a generalized advection-diffusion equation. We show that the power law $P_{j}(r)\propto f(r)^{\alpha -1}$ (with $\alpha$ > 1) follows from the requirement that the generalized equation admits scaling solutions $(f(r;t)=t^{- \gamma}\phi (r/t^{\gamma }))$. The solutions have a q-exponential form and are found to be in agreement with the results of Monte Carlo simulations, so providing a microscopic basis validating the nonlinear diffusion equation. Although its hydrodynamic limit is equivalent to the phenomenological porous media equation, there are extra terms which, in general, cannot be neglected as evidenced by the Monte Carlo computations.

PACS
05.40.Fb - Random walks and Levy flights.
05.60.-k - Transport processes.
05.10.Gg - Stochastic analysis methods (Fokker-Planck, Langevin, etc.).

© EPLA 2007