Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative homogenization and hydrodynamic limit of non-gradient exclusion process

Tadahisa Funaki, Chenlin Gu, Han Wang

TL;DR

This work advances quantitative homogenization for non-gradient exclusion by marrying the renormalization framework of AKM with a novel coarse-grained lifting that embeds exclusion dynamics into a larger independent-particle space. A gradient coupling between Kawasaki dynamics and a Poissonized (grand canonical) ensemble, together with a density-free local corrector, yields density-uniform convergence of the conductivity and a quantitative hydrodynamic limit. The approach avoids reliance on closed-form characterizations and extends to models with bond disorder, providing a robust route to rate-based hydrodynamics in non-gradient systems. The results unify homogenization and entropy methods to deliver explicit decay rates and uniform-in-density estimates, advancing both theory and potential applications to disordered lattice gases. The framework promises applicability to broader non-gradient models and lays groundwork for future extensions to GSEP and multi-type exclusion processes.

Abstract

For the non-gradient exclusion process, we prove the quantitative homogenization of the diffusion matrix and the conductivity by local functions. The proof relies on the renormalization approach developed by Armstrong, Kuusi, Mourrat, and Smart, while the new challenge here is the hard core constraint of particle number on every site. Therefore, a coarse-grained method is proposed to lift the configuration to a larger space without exclusion, and a gradient coupling between two systems is applied to capture the spatial cancellation. We then strengthen the convergence rate to be uniform concerning the density, and integrate it into the work by Funaki, Uchiyama, and Yau [IMA Vol. Math. Appl., 77 (1996), pp. 1-40.] to yield a quantitative hydrodynamic limit. Our new approach avoids showing the characterization of closed forms and provides stronger results. The extension is discussed for the model in the presence of disorder on the bonds.

Quantitative homogenization and hydrodynamic limit of non-gradient exclusion process

TL;DR

This work advances quantitative homogenization for non-gradient exclusion by marrying the renormalization framework of AKM with a novel coarse-grained lifting that embeds exclusion dynamics into a larger independent-particle space. A gradient coupling between Kawasaki dynamics and a Poissonized (grand canonical) ensemble, together with a density-free local corrector, yields density-uniform convergence of the conductivity and a quantitative hydrodynamic limit. The approach avoids reliance on closed-form characterizations and extends to models with bond disorder, providing a robust route to rate-based hydrodynamics in non-gradient systems. The results unify homogenization and entropy methods to deliver explicit decay rates and uniform-in-density estimates, advancing both theory and potential applications to disordered lattice gases. The framework promises applicability to broader non-gradient models and lays groundwork for future extensions to GSEP and multi-type exclusion processes.

Abstract

For the non-gradient exclusion process, we prove the quantitative homogenization of the diffusion matrix and the conductivity by local functions. The proof relies on the renormalization approach developed by Armstrong, Kuusi, Mourrat, and Smart, while the new challenge here is the hard core constraint of particle number on every site. Therefore, a coarse-grained method is proposed to lift the configuration to a larger space without exclusion, and a gradient coupling between two systems is applied to capture the spatial cancellation. We then strengthen the convergence rate to be uniform concerning the density, and integrate it into the work by Funaki, Uchiyama, and Yau [IMA Vol. Math. Appl., 77 (1996), pp. 1-40.] to yield a quantitative hydrodynamic limit. Our new approach avoids showing the characterization of closed forms and provides stronger results. The extension is discussed for the model in the presence of disorder on the bonds.
Paper Structure (68 sections, 61 theorems, 615 equations, 3 figures)

This paper contains 68 sections, 61 theorems, 615 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy of the proof
  4. Homogenization via the renormalization approach
  5. Two analytic inequalities and coarse-grained lifting
  6. Regularity and uniform convergence
  7. Hydrodynamic Limit
  8. Perspectives
  9. Notations
  10. Geometry
  11. Probability spaces
  12. Function spaces
  13. Operators
  14. Constants
  15. Analytic tools
  16. ...and 53 more sections

Key Result

Theorem 1.3

Under Hypothesis hyp, there exists an exponent $\gamma(d,\lambda, \mathbf{r}) > 0$ and a positive constant $C(d, \lambda, \mathbf{r}) < \infty$, such that

Figures (3)

  • Figure 1: The outline of proof.
  • Figure 2:
  • Figure 3: An illustration for the decomposition in \ref{['eq.FreeDecom']}: the bonds in the term $\mathbf{I}$, $\mathbf{II}$ and $\mathbf{III}$ are respectively marked in blue, green and red.

Theorems & Definitions (124)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2: Efron--Stein inequality
  • proof
  • Corollary 2.3: Spectral inequality for Glauber dynamics
  • proof
  • ...and 114 more