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Coupling hydrodynamics of several Facilitated Exclusion Processes with closed boundaries

Hugo Da Cunha, Lu Xu

TL;DR

The paper proves the hydrodynamic limit for ergodic Facilitated Exclusion Processes with closed boundaries across symmetric, asymmetric, and weakly asymmetric regimes by constructing a microscopic mapping to the Simple Exclusion Process. It then transfers the SEP hydrodynamic limits to FEP via a one-to-one macroscopic mapping, yielding PDEs that describe the macroscopic density under different regimes and boundary types, with Neumann, Robin, or Otto-type conditions depending on the regime parameter $\kappa$. The authors rigorously relate weak and entropy solutions between FEP and SEP, ensuring that the macroscopic mapping preserves solution classes and boundary behavior. This work extends hydrodynamic limit results to finite domains with impermeable walls, clarifying how phase structure and constrained dynamics manifest in continuum limits and providing a robust framework for analyzing kinetically constrained models via well-understood exclusion dynamics.

Abstract

In this paper, we prove the hydrodynamic limit for the ergodic dynamics of the Facilitated Exclusion Process with closed boundaries in the symmetric, asymmetric and weakly asymmetric regimes. For this, we couple it with a Simple Exclusion Process by constructing a mapping that transforms the facilitated dynamics into the simple one. As the hydrodynamic behaviour of the simple exclusion process with closed boundaries has been extensively studied, we can deduce the corresponding hydrodynamics for the facilitated exclusion process.

Coupling hydrodynamics of several Facilitated Exclusion Processes with closed boundaries

TL;DR

The paper proves the hydrodynamic limit for ergodic Facilitated Exclusion Processes with closed boundaries across symmetric, asymmetric, and weakly asymmetric regimes by constructing a microscopic mapping to the Simple Exclusion Process. It then transfers the SEP hydrodynamic limits to FEP via a one-to-one macroscopic mapping, yielding PDEs that describe the macroscopic density under different regimes and boundary types, with Neumann, Robin, or Otto-type conditions depending on the regime parameter . The authors rigorously relate weak and entropy solutions between FEP and SEP, ensuring that the macroscopic mapping preserves solution classes and boundary behavior. This work extends hydrodynamic limit results to finite domains with impermeable walls, clarifying how phase structure and constrained dynamics manifest in continuum limits and providing a robust framework for analyzing kinetically constrained models via well-understood exclusion dynamics.

Abstract

In this paper, we prove the hydrodynamic limit for the ergodic dynamics of the Facilitated Exclusion Process with closed boundaries in the symmetric, asymmetric and weakly asymmetric regimes. For this, we couple it with a Simple Exclusion Process by constructing a mapping that transforms the facilitated dynamics into the simple one. As the hydrodynamic behaviour of the simple exclusion process with closed boundaries has been extensively studied, we can deduce the corresponding hydrodynamics for the facilitated exclusion process.

Paper Structure

This paper contains 16 sections, 8 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and notations
  3. Facilitated Exclusion Processes
  4. Ergodicity property
  5. Main results
  6. Coupling with Simple Exclusion Process
  7. Hydrodynamic limit
  8. Proof of Theorem \ref{['thm:HLFEPs']}
  9. Macroscopic mapping
  10. Hydrodynamic limit for SEP
  11. Mapping of solutions
  12. For weak solutions
  13. For entropy solutions
  14. Notions of solution
  15. Weak solutions
  16. ...and 1 more sections

Key Result

theorem 1

Let $(\eta (t))_{t\ge 0}$ be the Markov process driven by the generator ${\mathcal{L}}_N$ defined in def:genFEP, and starting from an ergodic configuration $\eta (0)\in{\mathcal{E}}_N^k$. Then, $\xi(t):=\phi(\eta(t))$ is a Markov process evolving on $\Omega_M^\ell$ with $(M,\ell)=(k+2,2k-N+2)$, and where for $f:\Omega_M^\ell\longrightarrow\mathbb R$, ${\mathbb L}_M^\mathrm{S}$ and ${\mathbb L}_M^

Figures (1)

  • Figure 1: Illustration of the mapping $\phi$ on a configuration $\eta\in{\mathcal{E}}_{15}^9$. The hatched particles represent the fictive ones added at both ends of the configuration.

Theorems & Definitions (20)

  • theorem 1: Microscopic mapping
  • remark 1
  • theorem 2
  • proposition 1
  • remark 2
  • proof
  • proposition 2: Hydrodynamic limits for the SEPs
  • remark 3
  • proof
  • lemma 1: Mapping of weak solutions
  • ...and 10 more