Nonlinear diffusion from Einstein's master equation

TL;DR

This paper extends Einstein's master equation to nonlinear diffusion processes where jump probabilities depend on the particle distribution, deriving a generalized equation with solutions matching Monte Carlo simulations and connecting to porous media equations.

Contribution

It introduces a nonlinear dependence of jump probabilities on the distribution, deriving a generalized advection-diffusion equation with scaling solutions and validating it through simulations.

Findings

Derivation of a nonlinear diffusion equation with power-law jump probabilities.

Solutions exhibit q-exponential form consistent with Monte Carlo results.

Extra terms in the hydrodynamic limit distinguish it from standard porous media equations.

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, $P_j(r)$ 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 of 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.}