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Mapping hydrodynamics for the facilitated exclusion and zero-range processes

Clément Erignoux, Marielle Simon, Linjie Zhao

TL;DR

This work derives the hydrodynamic limits for two degenerate lattice gases, the facilitated exclusion process (FEP) and the facilitated zero-range process (FZRP), in both symmetric (diffusive) and asymmetric (hyperbolic) regimes. The authors establish a robust microscopic EX↔ZR mapping and show that the ZR limits can be transferred to EX to obtain Stefan-type macroscopic equations: a diffusive Stefan problem for the symmetric case and a hyperbolic Stefan problem for the asymmetric case, with precise weak/entropy solution formulations. A key novelty is handling non-product initial measures for the ZRP and constructing a macroscopic mapping between weak solutions of the Stefan problems, using smoothing of nonlinear fluxes and careful one-/two-block analyses fortified by attractiveness. The results include a new proof for the symmetric FEP, and, notably, the first derivation of the asymmetric FEP hydrodynamics, offering a comprehensive framework for degenerate interacting particle systems and their phase-boundary dynamics with potential applications to transport in constrained media.

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}.

Mapping hydrodynamics for the facilitated exclusion and zero-range processes

TL;DR

This work derives the hydrodynamic limits for two degenerate lattice gases, the facilitated exclusion process (FEP) and the facilitated zero-range process (FZRP), in both symmetric (diffusive) and asymmetric (hyperbolic) regimes. The authors establish a robust microscopic EX↔ZR mapping and show that the ZR limits can be transferred to EX to obtain Stefan-type macroscopic equations: a diffusive Stefan problem for the symmetric case and a hyperbolic Stefan problem for the asymmetric case, with precise weak/entropy solution formulations. A key novelty is handling non-product initial measures for the ZRP and constructing a macroscopic mapping between weak solutions of the Stefan problems, using smoothing of nonlinear fluxes and careful one-/two-block analyses fortified by attractiveness. The results include a new proof for the symmetric FEP, and, notably, the first derivation of the asymmetric FEP hydrodynamics, offering a comprehensive framework for degenerate interacting particle systems and their phase-boundary dynamics with potential applications to transport in constrained media.

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}.
Paper Structure (28 sections, 22 theorems, 238 equations, 4 figures)

This paper contains 28 sections, 22 theorems, 238 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. General notations
  4. Main results
  5. Hydrodynamic limit for the facilitated exclusion process
  6. Hydrodynamic limit for the facilitated zero-range process
  7. Strategy of the proof
  8. The symmetric case
  9. Mapping EX $\mapsto$ ZR
  10. Macroscopic mapping
  11. Construction
  12. Smoothing out solutions to the Stefan problems
  13. Mapping between weak solutions to \ref{['eq:PDEstrong']} and \ref{['eq:PDEstrongZR']}
  14. Proof of Theorem \ref{['thm:ex']} in the symmetric case
  15. The asymmetric case
  16. ...and 13 more sections

Key Result

Theorem 1

Let us assume that the initial profile $\rho^{\mathrm{ini}}: {\mathbb L} \rightarrow [0,1]$ is Riemann integrable on $\mathbb{L}$ and bounded away from 1, namely: $\rho^{\mathrm{ini}}(u)\leqslant \rho_\star < 1$ for any $u \in {\mathbb T}$. both with initial condition $\rho_0=\rho^{\mathrm{ini}}$. Then, for any $\varepsilon>0$, any test function $\varphi \in C^2(\mathbb{L})$ with compact support,

Figures (4)

  • Figure 1: Example of configurations belonging to the ergodic, frozen and transient sets, with $N=8$ sites in a periodic setting (lattice for the symmetric case).
  • Figure 2: Example of configurations belonging to the ergodic, frozen and transient sets, with $M=4$ sites in a periodic setting (lattice for the symmetric case).
  • Figure 3: An example of $\eta$ and its corresponding $\widehat{\omega}^\eta$ in a periodic setting with $N=8$.
  • Figure 4: Representation of the event $A(t)$ for two configurations $\omega$ (white) and $\zeta$ (red), under the basic coupling. The particles with the red-white gradient are the coupled ones, the arrows represent couplings that occur before time $t+T$.

Theorems & Definitions (50)

  • Remark 1
  • Definition 1: Parabolic Stefan problem for exclusion
  • Definition 2: Hyperbolic Stefan problem for exclusion
  • Remark 2: Uniqueness of solutions
  • Theorem 1: Hydrodynamic limit for the facilitated exclusion process
  • Remark 3
  • Remark 4
  • Theorem 2: Hydrodynamic limit for the facilitated zero-range process
  • Remark 5
  • Lemma 1: Mapping
  • ...and 40 more