From Exclusion to Slow and Fast Diffusion

TL;DR

This paper introduces a particle system model that transitions smoothly from slow to fast diffusion regimes, with a hydrodynamic limit described by a nonlinear parabolic PDE, unifying several known diffusion models.

Contribution

It constructs a microscopic particle system that interpolates between porous medium and fast diffusion models, deriving their hydrodynamic limits using the entropy method.

Findings

Unified microscopic model for slow and fast diffusion

Hydrodynamic limit described by a nonlinear PDE with variable diffusion coefficient

Established the hydrodynamic limit for the local particle density on the torus

Abstract

We construct a nearest-neighbour interacting particle system of exclusion type, which illustrates a transition from slow to fast diffusion. More precisely, the hydrodynamic limit of this microscopic system in the diffusive space-time scaling is the parabolic equation $\partial_t\rho=\nabla (D(\rho)\nabla \rho)$, with diffusion coefficient $ D(\rho)=m\rho^{m-1} $ where $ m\in(0,2] $, including therefore the fast diffusion regime in the range $ m\in(0,1) $, and the porous {medium} equation for $ m\in(1,2) $. The construction of the model is based on the generalized binomial theorem, and interpolates continuously in $ m $ the already known microscopic porous medium model with parameter $ m=2 $, the symmetric simple exclusion process with $ m=1 $, going down to a fast diffusion model up to any $ m>0$. The derivation of the hydrodynamic limit for the local density of particles on the one-dimensional torus is achieved via the entropy method -- with additional technical difficulties depending on the regime (slow or fast diffusion) and where new properties of the porous medium model need to be derived.