Mapping hydrodynamics for the facilitated exclusion and zero-range processes

TL;DR

This paper derives the hydrodynamic limits for facilitated exclusion and zero-range lattice gases, revealing diffusive and hyperbolic Stefan problems, and introduces new results for asymmetric cases with simplified proofs.

Contribution

It provides the first hydrodynamic limit derivations for asymmetric facilitated exclusion and zero-range processes, using novel mappings and simplified proofs.

Findings

Hydrodynamic limits characterized by Stefan problems

New results for asymmetric processes due to degeneracy

Simplified proof for symmetric facilitated exclusion process

Abstract

We derive the hydrodynamic limit for two degenerate lattice gases, the \emph{facilitated exclusion process} (FEP) and the \emph{facilitated zero-range process} (FZRP), both in the symmetric and the asymmetric case. For both processes, the hydrodynamic limit in the symmetric case takes the form of a diffusive Stefan problem, whereas the asymmetric case is characterized by a hyperbolic Stefan problem. Although the FZRP is attractive, a property that we extensively use to derive its hydrodynamic limits in both cases, the FEP is not. To derive the hydrodynamic limit for the latter, we exploit that of the zero-range process, together with a classical mapping between exclusion and zero-range processes, both at the microscopic and macroscopic level. Due to the degeneracy of both processes, the asymmetric case is a new result, but our work also provides a simpler proof than the one that was previously proposed for the FEP in the symmetric case in \cite{blondel2021stefan}.