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Hydrodynamic limits of collisions and fluxes in the exclusion process

Mario Ayala, D. R. Michiel Renger

TL;DR

This work extends the hydrodynamic theory of the symmetric exclusion process by incorporating collision events and unidirectional flux as additional macroscopic observables, enabling a quantitative decomposition of transport and dissipation. By leveraging distribution-valued Markov process methods, self-duality of SEP, and a flexible scaling framework between lattice spacing $\epsilon$ and particle number $n$, the authors derive deterministic hydrodynamic limits for density and net flux, and regime-dependent limits for unidirectional flux, unidirectional collisions, and net collisions, including a space-time white noise SPDE in certain regimes. The main results reveal how microscopic blocking manifests at macroscopic and fluctuation levels: collisions contribute a quadratic reaction-like term to the unidirectional flux in dense regimes, while in sparse regimes this impact vanishes, and stochastic fluctuations emerge in the net collision observable. These findings provide a principled framework to quantify dissipative effects in exclusion dynamics and pave the way for extensions to more complex interacting particle systems and fluctuating hydrodynamics.

Abstract

We extend the usual hydrodynamic description of the symmetric exclusion process by keeping track of collision events corresponding to jumps into already occupied sites, thereby quantifying the dissipated part of the microscopic activity that is otherwise discarded by the empirical density in the macroscopic limit. In addition to the empirical density and net current, we study unidirectional fluxes and collision counts under flexible joint scalings of the lattice spacing and particle number. These collision and flux observables have regime dependent hydrodynamic limits, with deterministic unidirectional behaviour and a stochastic space time white noise limit for the net collision count. Our results provide a quantitative decomposition of exclusion dynamics into transport and collision effects and clarify how microscopic blocking manifests at the macroscopic and fluctuation levels.

Hydrodynamic limits of collisions and fluxes in the exclusion process

TL;DR

This work extends the hydrodynamic theory of the symmetric exclusion process by incorporating collision events and unidirectional flux as additional macroscopic observables, enabling a quantitative decomposition of transport and dissipation. By leveraging distribution-valued Markov process methods, self-duality of SEP, and a flexible scaling framework between lattice spacing and particle number , the authors derive deterministic hydrodynamic limits for density and net flux, and regime-dependent limits for unidirectional flux, unidirectional collisions, and net collisions, including a space-time white noise SPDE in certain regimes. The main results reveal how microscopic blocking manifests at macroscopic and fluctuation levels: collisions contribute a quadratic reaction-like term to the unidirectional flux in dense regimes, while in sparse regimes this impact vanishes, and stochastic fluctuations emerge in the net collision observable. These findings provide a principled framework to quantify dissipative effects in exclusion dynamics and pave the way for extensions to more complex interacting particle systems and fluctuating hydrodynamics.

Abstract

We extend the usual hydrodynamic description of the symmetric exclusion process by keeping track of collision events corresponding to jumps into already occupied sites, thereby quantifying the dissipated part of the microscopic activity that is otherwise discarded by the empirical density in the macroscopic limit. In addition to the empirical density and net current, we study unidirectional fluxes and collision counts under flexible joint scalings of the lattice spacing and particle number. These collision and flux observables have regime dependent hydrodynamic limits, with deterministic unidirectional behaviour and a stochastic space time white noise limit for the net collision count. Our results provide a quantitative decomposition of exclusion dynamics into transport and collision effects and clarify how microscopic blocking manifests at the macroscopic and fluctuation levels.
Paper Structure (36 sections, 23 theorems, 89 equations, 1 figure)

This paper contains 36 sections, 23 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline.
  3. Model and notation
  4. Notation and spaces
  5. Dynamics
  6. Initial and path measure
  7. Scaling regimes
  8. Metatheorems
  9. Deterministic hydrodynamic limits
  10. Stochastic hydrodynamic limits
  11. Tightness criteria
  12. Correlations
  13. Convergence of correlations
  14. Estimates and convergence of time-integrated correlations
  15. Tightness
  16. ...and 21 more sections

Key Result

Proposition 2.3

Assume $n\sim 1/\epsilon^d$. Then $\rho(x,t)\leq\frac{\lVert\rho_0\rVert_{L^1(\mathbb{T}^d)}}{\alpha}$ at all $x\in\mathbb{T}^d, t\geq0$.

Figures (1)

  • Figure 1: In dimension $d=2$: all nearest neighbours $x\sim y$ of $x$ (left), and all nearest neighbours in positive direction $x\leadsto y$ (right).

Theorems & Definitions (52)

  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 3.1
  • Theorem 3.2: KipnisLandim1999
  • proof
  • Theorem 3.3: KipnisLandim1999
  • Remark 3.4
  • Remark 3.5
  • proof
  • ...and 42 more