Microscopic dynamics for the porous medium equation

Abstract

In this work, I present an interacting particle system whose dynamics conserves the total number of particles but with gradient transition rates that vanish for some configurations. As a consequence, the invariant pieces of the system, namely, the hyperplanes with a fixed number of particles can be decomposed into an irreducible set of configurations plus isolated configurations that do not evolve under the dynamics. By taking initial profiles smooth enough and bounded away from zero and one and for parabolic time scales, the macroscopic density profile evolves according to the porous medium equation. Perturbing slightly the microscopic dynamics in order to remove the degeneracy of the rates the same result can be obtained for more general initial profiles.