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

Abstract

We investigate the macroscopic behavior of asymmetric attractive zero-range processes on Z where particles are destroyed at the origin at a rate of order Nβ, where β∈R and N∈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 β. Namely, if β⩾0, then the boundary condition prescribes the particle current through the origin, whereas if β<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.