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Hydrodynamic limit for an active-passive exclusion process

Deyue Li

TL;DR

The paper rigorously derives the hydrodynamic limit for a two-type active-passive lattice gas with exclusion and Brownian angular diffusion, yielding coupled PDEs for orientation densities on $ ext{T}^2 imes ext{S}$ that couple diffusion, drift, and rotation. The authors generalize the non-gradient method to accommodate continuous orientations and mixed particle types by introducing measure-valued local equilibria, a broadened function class, and new replacement lemmas, supported by a detailed Hilbert-space framework and spectral-gap estimates. The main contributions are the derivation of the macroscopic equations, the development of non-gradient decomposition and replacement techniques for active-passive mixtures, and the establishment of a robust spectral-gap infrastructure to control fluctuations. This work provides a rigorous bridge from microscopic active-passive exclusion dynamics to macroscopic, orientation-resolved diffusion-drift equations, with potential implications for understanding MIPS-like phenomena and cross-diffusion in active matter systems.

Abstract

The collective non-equilibrium dynamics of multi-component mixtures of interacting active (self-propelled) and passive (diffusive) particles have garnered great interest in the physics community. However, the mathematical understanding of these systems remains partial. In this work, we consider a lattice gas model of active-passive particle mixtures with exclusion, where the self-propulsion orientations of active particles undergo Brownian motion on a torus. We derive the hydrodynamic equations governing the particle densities. Due to the presence of two types of particles with continuous-valued orientations, further generalizations of non-gradient decomposition and spectral gap estimation developed for the pure active case are necessitated, which entails novel challenges and new proofs.

Hydrodynamic limit for an active-passive exclusion process

TL;DR

The paper rigorously derives the hydrodynamic limit for a two-type active-passive lattice gas with exclusion and Brownian angular diffusion, yielding coupled PDEs for orientation densities on that couple diffusion, drift, and rotation. The authors generalize the non-gradient method to accommodate continuous orientations and mixed particle types by introducing measure-valued local equilibria, a broadened function class, and new replacement lemmas, supported by a detailed Hilbert-space framework and spectral-gap estimates. The main contributions are the derivation of the macroscopic equations, the development of non-gradient decomposition and replacement techniques for active-passive mixtures, and the establishment of a robust spectral-gap infrastructure to control fluctuations. This work provides a rigorous bridge from microscopic active-passive exclusion dynamics to macroscopic, orientation-resolved diffusion-drift equations, with potential implications for understanding MIPS-like phenomena and cross-diffusion in active matter systems.

Abstract

The collective non-equilibrium dynamics of multi-component mixtures of interacting active (self-propelled) and passive (diffusive) particles have garnered great interest in the physics community. However, the mathematical understanding of these systems remains partial. In this work, we consider a lattice gas model of active-passive particle mixtures with exclusion, where the self-propulsion orientations of active particles undergo Brownian motion on a torus. We derive the hydrodynamic equations governing the particle densities. Due to the presence of two types of particles with continuous-valued orientations, further generalizations of non-gradient decomposition and spectral gap estimation developed for the pure active case are necessitated, which entails novel challenges and new proofs.
Paper Structure (34 sections, 49 theorems, 399 equations)

This paper contains 34 sections, 49 theorems, 399 equations.

Table of Contents

  1. Introduction
  2. Previous works and our contributions
  3. Structure of the article
  4. Notations and results
  5. Particle configurations
  6. Active-passive exclusion process
  7. Hydrodynamic limit
  8. Proof of Theorem \ref{['thm:main']}
  9. Reduction to weaker solutions
  10. Dynkin's formula and instantaneous currents
  11. Non-gradient decomposition and replacement lemma
  12. Relative compactness and regularity of the densities
  13. The second integration by parts
  14. Conversion to function of empirical measures
  15. Proof of Theorem \ref{['thm:main2']}
  16. ...and 19 more sections

Key Result

Theorem 2.3

The sequence $(Q^N )_{n\in \mathbb{N}}$ is weakly relatively compact, and any of its limit points $Q^*$ are concentrated on trajectories $\bm{\pi}^{[0,T]}=(\pi^{a},\pi^{p}) \in \mathcal{M}^{[0,T]}\times \mathcal{M}^{[0,T]}$, which are weak solutions of eq:pde in the sense of Definition def:weak.

Theorems & Definitions (91)

  • Remark 2.1
  • Definition 2.2: Weak solution to \ref{['eq:pde']}
  • Theorem 2.3
  • Remark 2.4
  • Definition 3.1: Weaker solution of \ref{['eq:pde']}
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.3
  • Lemma 3.4
  • proof : Proof of Lemma \ref{['lemma:nongradreplacement']}
  • ...and 81 more