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

TL;DR

This paper studies the large-scale behavior of asymmetric zero-range particle systems with destruction at the origin, revealing how destruction rate influences boundary conditions in the resulting hyperbolic conservation law.

Contribution

It characterizes the hydrodynamic limit of the process with destruction at the origin, showing the boundary condition depends on the destruction rate parameter.

Findings

Hydrodynamic limit described by a hyperbolic conservation law

Boundary condition depends on destruction rate parameter

Particle destruction has no macroscopic effect when destruction rate is negative

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 $N^\beta$, where $\beta \in \mathbb{R}$ 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 $\beta$. Namely, if $\beta \geqslant 0$, then the boundary condition prescribes the particle current through the origin, whereas if $\beta<0$, the destruction of particles at the origin has no macroscopic effect on the system and no boundary condition is imposed at the hydrodynamic limit.