Asymmetric attractive zero-range processes with particle destruction at the origin

Abstract

We investigate the macroscopic behavior of asymmetric attractive zero-range processes on $\mathbb{Z}$ where particles are destroyed at the origin at a rate of order $N^2$, where ${\hat{I}}^2\in \mathbb{R}$ and $N\in \mathbb{N}$ is the scaling parameter. We prove that the hydrodynamic limit of this particle system is described by the unique entropy solution of a hyperbolic conservation law, supplemented by a boundary condition depending on the range of ${\hat{I}}^2$. Namely, if ${\hat{I}}^2⩾0$, then the boundary condition prescribes the particle current through the origin, whereas if ${\hat{I}}^2<0$, the destruction of particles at the origin has no macroscopic effect on the system and no boundary condition is imposed at the hydrodynamic limit.