Nonlinear diffusion from Einstein's master equationJ. P. Boon and J. F. Lutsko
Physics Department, Université Libre de Bruxelles - CP 231 B-1050 Bruxelles, Belgium
received 8 September 2007; accepted in final form 19 October 2007; published December 2007
published online 13 November 2007
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 (with > 1) follows from the requirement that the generalized equation admits scaling solutions . 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.
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