Jim Lutsko

Jim Lutsko

My interests include Statistical Physics; Crystallization; Density Functional Theory, Simulation methods - particularly rare event techniques, Protein physics, Granular physics and Crystal Growth. I am based in the Physics of Complex Systems and Statistical Mechanics group at the Universite Libre de Bruxelles.

[064] Nonlinear diffusion from Einstein’s master equation

J. P. Boon and J. F. Lutsko, "Nonlinear diffusion from Einstein's master equation", EuroPhys. Lett., 80, 60006 (2007) http://jimlutsko.github.io/files/Boon_Lutsko_EPL_2007.pdf

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 Pj(r)[?]f(r)a-1 (with a>1) follows from the requirement that the generalized equation admits scaling solutions (f(r;t)=t[?]gph(r/tg)). 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.