Hydrostatic Limit and Fick’s Law for the Symmetric Exclusion with Long Jumps

Abstract

Hydrostatic behavior and Fick’s law for the one dimensional exclusion process with long jumps in contact with infinite reservoirs at different densities are derived. The jump rate is described by a transition probability p which is proportional to $\vert \cdot \vert ^{-(\gamma +1)}$ for $\gamma >2$. The reservoirs add or remove particles with rate proportional to $\kappa N^{-\theta }$, where $\kappa >0$ and $\theta =2-\gamma$. The behavior of the solution of the hydrostatic equation is also studied.