From Exclusion to Slow and Fast Diffusion

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.