Slow to Fast Infinitely Extended Reservoirs for the Symmetric Exclusion Process with Long Jumps

#### C. Bernardin, P. Goncalves, B. Jimenez-Oviedo

2019, v.25, Issue 2, 217-274

ABSTRACT

We consider an exclusion process with long jumps in the box $\Lambda_N=\{1, \ldots,N-1\}$, for $N \ge 2$, in contact with infinitely extended reservoirs on its left and on its right. The jump rate is described by a transition probability $p(\cdot)$ which is symmetric, with infinite support but with finite variance. The reservoirs add or remove particles with rate proportional to $\kappa N^{-\theta}$, where $\kappa>0$ and $\theta \in\bb R$. If $\theta>0$ (resp.\ $\theta<0$) the reservoirs slowly (resp.\ fastly) add and remove particles in the bulk. According to the value of $\theta$ we prove that the time evolution of the spatial density of particles is described by some reaction-diffusion equations with various boundary conditions.

%has a distinct behavior according to the range of the parameter $\theta$.The hydrodynamic limit is given by: 1) a reaction-diffusion equation with inhomogeneous Dirichlet boundary conditions when $\theta<1$, 2) a heat equation with Robin boundary conditions if $\theta = 1$ and 3) a heat equation with Neumann boundary conditions if $\theta>1$.

Keywords: hydrodynamic limit, reaction-diffusion equation, boundary conditions, exclusion with long jumps