From Exclusion to Slow and Fast Diffusion

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.