Hydrodynamic Limit of the Symmetric Zero-Range Process with Slow Boundary

TL;DR

This paper analyzes how the symmetric zero-range process behaves on a finite interval with slow boundary reservoirs, showing that the empirical density follows a nonlinear heat equation with boundary conditions depending on reservoir strength.

Contribution

It establishes the hydrodynamic limit of the symmetric zero-range process with slow boundary reservoirs, deriving the nonlinear heat equation with boundary conditions for this setting.

Findings

Empirical density evolves according to a nonlinear heat equation.

Boundary conditions depend on the reservoir strength parameter.

Results hold under mild assumptions on jump rates and initial measures.

Abstract

We study the hydrodynamic behaviour of the symmetric zero-range process on the finite interval $\{1, \ldots, N-1\}$ in contact with slow reservoirs at the boundary. Particles are injected and removed at sites $1$ and $N-1$ at rates that scale like $N^{-\theta}$ with $\theta\ge1$. Under mild assumptions on the jump rate and the sequence of initial measures, we show that the empirical density evolves on the diffusive scale according to a nonlinear heat equation, with boundary conditions reflecting the strength of the reservoirs.