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Fluctuations of the Boundary-Driven Symmetric Zero-Range Process from the NESS

Patrícia Gonçalves, Adriana Neumann, Maria Chiara Ricciuti

TL;DR

The paper analyzes stationary fluctuations of a boundary-driven symmetric zero-range process on a finite interval with reservoirs, establishing that under diffusive scaling and bounded jump rate $g$, the fluctuations converge to a generalized Ornstein–Uhlenbeck process with boundary conditions chosen according to the reservoir strength parameter $\theta$, ranging from Dirichlet to Robin to Neumann. The authors develop a robust framework combining Dynkin martingales, a tailored Boltzmann–Gibbs principle, and boundary replacement lemmas, adapted to non-reversible and non-translation-invariant stationary measures, to prove tightness and identify the limit. They construct a hierarchy of test-function spaces $\mathcal{S}_{\theta}$ (and an extended $\widetilde{S}$ for $\theta\ge1$) to formulate well-posed martingale problems whose unique stationary solutions yield the limiting fluctuations, including new Robin–Neumann formulations. The results unify the fluctuation theory for boundary-driven ZRP across all reservoir strengths, complementing the hydrodynamic limit and offering tools potentially applicable to related open systems and to weakly asymmetric regimes.

Abstract

We study the non-equilibrium stationary fluctuations of a symmetric zero-range process on the discrete interval $\{1, \ldots, N-1\}$ coupled to reservoirs at sites $1$ and $N-1$, which inject and remove particles at rates proportional to $N^{-θ}$ for any value of $θ\in\mathbb{R}$. We prove that, if the jump rate is bounded and under diffusive scaling, the fluctuations converge to the solution of a generalised Ornstein-Uhlenbeck equation with characteristic operators that depend on the stationary density profile. The limiting equation is supplemented with boundary conditions of Dirichlet, Robin, or Neumann type, depending on the strength of the reservoirs. We also introduce two notions of solutions to the corresponding martingale problems, which differ according to the choice of test functions.

Fluctuations of the Boundary-Driven Symmetric Zero-Range Process from the NESS

TL;DR

The paper analyzes stationary fluctuations of a boundary-driven symmetric zero-range process on a finite interval with reservoirs, establishing that under diffusive scaling and bounded jump rate , the fluctuations converge to a generalized Ornstein–Uhlenbeck process with boundary conditions chosen according to the reservoir strength parameter , ranging from Dirichlet to Robin to Neumann. The authors develop a robust framework combining Dynkin martingales, a tailored Boltzmann–Gibbs principle, and boundary replacement lemmas, adapted to non-reversible and non-translation-invariant stationary measures, to prove tightness and identify the limit. They construct a hierarchy of test-function spaces (and an extended for ) to formulate well-posed martingale problems whose unique stationary solutions yield the limiting fluctuations, including new Robin–Neumann formulations. The results unify the fluctuation theory for boundary-driven ZRP across all reservoir strengths, complementing the hydrodynamic limit and offering tools potentially applicable to related open systems and to weakly asymmetric regimes.

Abstract

We study the non-equilibrium stationary fluctuations of a symmetric zero-range process on the discrete interval coupled to reservoirs at sites and , which inject and remove particles at rates proportional to for any value of . We prove that, if the jump rate is bounded and under diffusive scaling, the fluctuations converge to the solution of a generalised Ornstein-Uhlenbeck equation with characteristic operators that depend on the stationary density profile. The limiting equation is supplemented with boundary conditions of Dirichlet, Robin, or Neumann type, depending on the strength of the reservoirs. We also introduce two notions of solutions to the corresponding martingale problems, which differ according to the choice of test functions.
Paper Structure (25 sections, 39 theorems, 225 equations, 3 figures)

This paper contains 25 sections, 39 theorems, 225 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Framework and Results
  3. The Model
  4. The Invariant Measure
  5. Hydrodynamic Limit
  6. The Space of Test Functions
  7. The Martingale Problems
  8. Convergence of Stationary Fluctuations
  9. Proof of Theorem \ref{['thm:fluctuations']}
  10. Dynkin's Martingales
  11. The Boltzmann-Gibbs Principle
  12. $\boldsymbol{L^2}$ Bounds and Dirichlet Forms
  13. Proof of Theorems \ref{['thm:boltzmann_gibbs']} and \ref{['thm:local_boltzmann_gibbs']}
  14. Replacement Lemma at the Boundary
  15. Tightness
  16. ...and 10 more sections

Key Result

Lemma 2.1

Define the map $\bar{\varphi}_N:I_N\to[0, \infty)$ as For $\alpha, \beta, \lambda, \delta\ge0$, $\theta\in\mathbb{R}$ and $N\in\mathbb{N}$ satisfying the measure eq:nu with fugacity profile $\bar{\varphi}_N$ is the unique invariant measure of $\{\eta_t^N, t\ge0\}$.

Figures (3)

  • Figure 1: Microscopic dynamics of the boundary-driven symmetric zero-range process.
  • Figure 2: Plot of the asymptotic fugacity profile $\bar{\varphi}_\theta$ for the choice $\alpha=2, \beta=1, \lambda=\frac{3}{2}$ and $\delta=\frac{1}{2}$.
  • Figure 3: Plot of the functions $\psi_{\alpha,\beta}$ and $\psi_\varepsilon$ for some values of the parameters $\alpha, \beta, \varepsilon$.

Theorems & Definitions (76)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 66 more