Asymmetric attractive zero-range processes with particle destruction at the origin
Marielle Simon, Linjie Zhao, Clément Erignoux
TL;DR
The paper analyzes an asymmetric attractive zero-range process on ${\mathbb Z}$ with particle destruction at the origin at rate ${\alpha N^{\beta}}$, deriving its hydrodynamic limit across three regimes of ${\beta}$. Using entropy-methods, couplings, and second-class particles, it proves convergence to the entropy solution of a hyperbolic conservation law with boundary conditions that depend on ${\beta}$: no boundary condition for ${\beta<0}$, a Robin-type current constraint for ${\beta=0}$, and a Dirichlet-type mass-removal regime for ${\beta>0}$. The authors also provide a transparent linear-case analysis with an explicit macroscopic formula and a duality-based proof. The results clarify how microscopic destruction strength at the origin shapes macroscopic flux, mass balance, and shock behavior in hyperbolic limits, and extend prior open-system hydrodynamics for ZRPs by incorporating interior boundary destruction dynamics.
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 $N^β$, where $β\in \mathbb{R}$ 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 $β$. Namely, if $β\geqslant 0$, 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.