The boundary driven zero-range process

TL;DR

This paper analyzes the asymptotic behavior of a symmetric zero-range process with slow boundary conditions, deriving the invariant measure, hydrostatic limit, and hydrodynamic equations depending on a parameter.

Contribution

It provides the invariant measure, hydrostatic limit, and hydrodynamic equations for the zero-range process with slow boundary, highlighting the dependence on the parameter .

Findings

Invariant measure explicitly characterized

Hydrostatic limit obtained for the model

Hydrodynamic equation with boundary conditions depending on 

Abstract

We study the asymptotic behaviour of the symmetric zero-range process in the finite lattice $\{1,\ldots, N-1\}$ with slow boundary, in which particles are created at site $1$ or annihilated at site $N\!-\!1$ with a rate proportional to $N^{-\theta}$, for $\theta\geq 1$. We present the invariant measure for this model and obtain the hydrostatic limit. In order to understand the asymptotic behaviour of the spatial-temporal evolution of this model under the diffusive scaling, we start to analyze the hydrodynamic limit, exploiting attractiveness as an essential ingredient. We obtain that the hydrodynamic equation has boundary conditions that depend on the value of $\theta$.