Room P3.10, Mathematics Building

Gabriel Nahum, Instituto Superior Técnico
An interacting particle system interpolating the Symmetric Simple Exclusion Process and the Porous Media Model

The Porous Media Model (PMM) is an interacting particle system of exclusion-type (two particles cannot occupy the same site) belonging to the class of kinetically constrained lattice gases (KCLG), which have gained a lot of attention as simple models for the liquid/glass transition. In particular, the PMM was first introduced to derive the Porous Media Equation (PME), which can be seen as a generalisation of the heat equation with a non-linear diffusion coefficient, $D(u)=mu^{m-1}$, with $m\gt 1$.

As it was defined, the PMM allows one to derive the PME when the diffusion is a positive integer power of the solution by constraining nearest neighbour jumps with occupation conditions on a neighbourhood of where the jumps take place. This model has not been directly extended yet to encompass the case where the diffusion is a non-integer power. In this talk we present a direct extension of the PMM for $m\in (1,2)$ via the use of the generalised binomial coefficients. The resulting dynamics is very simple and interpolates continuously the Symmetric Simple Exclusion Process, which allows one to derive the Heat Equation, and the PMM with $m=2$, while being of gradient and exclusion types.

If there is still time, we also discuss the difficulties of deriving the Fast Diffusion Equation ($m\lt 1$) and the extension to $m\gt 2$.