From Einstein to generalized diffusion
Jean Pierre Boon and James F. Lutsko, "From Einstein to generalized diffusion", AIP Conference Proceedings, 965, 157 (2007) http://jimlutsko.github.io/files/Boon_Lutsko_AIP_2007.pdf
We show that from a generalization of Einstein's master equation for the random walk one obtains a generalized equation for diffusion processes. The master equation is generalized by making the particle jump probability Pj(r) a functional of the particle distribution function f(r,t). If one demands that the resulting generalized diffusion equation admits of scaling solutions: f(r;t) = t−gamma/2phi(r/tgamma/2), a power law Pj(r)[proportional]f(r)alpha−1 (with alpha>1) follows, and the solutions exhibit q-exponential forms which are found to be in agreement with the results of Monte-Carlo simulations, providing a microscopic basis validating the nonlinear diffusion equation. We also show that the phenomenological porous media equation is an approximation to the generalized advection-diffusion equation.