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Interface motion from Glauber-Kawasaki dynamics of non-gradient type

Tadahisa Funaki

TL;DR

This work analyzes Glauber-Kawasaki dynamics on a large d-dimensional torus with a diverging Glauber speed, establishing a quantitative hydrodynamic limit to a reaction-diffusion equation with nonlinear diffusion and a large reaction term. By using a relative-entropy framework with a second-order local-equilibrium correction, the authors derive precise entropy bounds and develop a non-gradient gradient-replacement machinery that yields sharp convergence rates. In the bistable regime, they show that microscopic phase separation yields a macroscopic two-phase profile whose interface evolves under anisotropic curvature flow, with the interface motion linked to a limiting sharp-interface PDE studied in a companion paper FP. The work extends non-gradient Glauber-Kawasaki analysis to quantitative limits, providing robust error control via gradient-replacement, CLT-variance estimates, and Schauder PDE techniques, and discusses the technical range for the Glauber scaling $K(N)$. The results have implications for understanding phase separation dynamics and interface evolution in non-gradient interacting particle systems with competing creation-annihilation dynamics.

Abstract

We consider the Glauber-Kawasaki dynamics on a $d$-dimensional periodic lattice of size $N$, that is, a stochastic time evolution of particles performing random walks with interaction subject to the exclusion rule (Kawasaki part), in general, of non-gradient type, together with the effect of the creation and annihilation of particles (Glauber part) whose rates are set to favor two levels of particle density, called sparse and dense. We then study the limit of our dynamics under the hydrodynamic space-time scaling, that is, $1/N$ in space and a diffusive scaling $N^2$ for the Kawasaki part and another scaling $K=K(N)$, which diverges slower, for the Glauber part in time. In the limit as $N\to\infty$, we show that the particles autonomously make phase separation into sparse or dense phases at the microscopic level, and an interface separating two regions is formed at the macroscopic level and evolves under an anisotropic curvature flow. In the present article, we show that the particle density at the macroscopic level is well approximated by a solution of a reaction-diffusion equation with a nonlinear diffusion term of divergence form and a large reaction term. Furthermore, by applying the results of Funaki, Gu and Wang [arXiv:2404.12234] for the convergence rate of the diffusion matrix approximated by local functions, we obtain a quantitative hydrodynamic limit as well as the upper bound for the allowed diverging speed of $K=K(N)$. The above result for the derivation of the interface motion is proved by combining our result with that in a companion paper by Funaki and Park [arXiv:2403.01732], in which we analyzed the asymptotic behavior of the solution of the reaction-diffusion equation obtained in the present article and derived an anisotropic curvature flow in the situation where the macroscopic reaction term determined from the Glauber part is bistable and balanced.

Interface motion from Glauber-Kawasaki dynamics of non-gradient type

TL;DR

This work analyzes Glauber-Kawasaki dynamics on a large d-dimensional torus with a diverging Glauber speed, establishing a quantitative hydrodynamic limit to a reaction-diffusion equation with nonlinear diffusion and a large reaction term. By using a relative-entropy framework with a second-order local-equilibrium correction, the authors derive precise entropy bounds and develop a non-gradient gradient-replacement machinery that yields sharp convergence rates. In the bistable regime, they show that microscopic phase separation yields a macroscopic two-phase profile whose interface evolves under anisotropic curvature flow, with the interface motion linked to a limiting sharp-interface PDE studied in a companion paper FP. The work extends non-gradient Glauber-Kawasaki analysis to quantitative limits, providing robust error control via gradient-replacement, CLT-variance estimates, and Schauder PDE techniques, and discusses the technical range for the Glauber scaling . The results have implications for understanding phase separation dynamics and interface evolution in non-gradient interacting particle systems with competing creation-annihilation dynamics.

Abstract

We consider the Glauber-Kawasaki dynamics on a -dimensional periodic lattice of size , that is, a stochastic time evolution of particles performing random walks with interaction subject to the exclusion rule (Kawasaki part), in general, of non-gradient type, together with the effect of the creation and annihilation of particles (Glauber part) whose rates are set to favor two levels of particle density, called sparse and dense. We then study the limit of our dynamics under the hydrodynamic space-time scaling, that is, in space and a diffusive scaling for the Kawasaki part and another scaling , which diverges slower, for the Glauber part in time. In the limit as , we show that the particles autonomously make phase separation into sparse or dense phases at the microscopic level, and an interface separating two regions is formed at the macroscopic level and evolves under an anisotropic curvature flow. In the present article, we show that the particle density at the macroscopic level is well approximated by a solution of a reaction-diffusion equation with a nonlinear diffusion term of divergence form and a large reaction term. Furthermore, by applying the results of Funaki, Gu and Wang [arXiv:2404.12234] for the convergence rate of the diffusion matrix approximated by local functions, we obtain a quantitative hydrodynamic limit as well as the upper bound for the allowed diverging speed of . The above result for the derivation of the interface motion is proved by combining our result with that in a companion paper by Funaki and Park [arXiv:2403.01732], in which we analyzed the asymptotic behavior of the solution of the reaction-diffusion equation obtained in the present article and derived an anisotropic curvature flow in the situation where the macroscopic reaction term determined from the Glauber part is bistable and balanced.
Paper Structure (46 sections, 31 theorems, 362 equations)

This paper contains 46 sections, 31 theorems, 362 equations.

Table of Contents

  1. Introduction --- model and results
  2. Model at microscopic level
  3. Diffusion matrix and reaction term at macroscopic level
  4. Quantitative and non-gradient hydrodynamic limit
  5. Interface motion from nonlinear Allen-Cahn equation
  6. Main result
  7. Outline of the article
  8. Discussion
  9. Proof of Theorem \ref{['Theorem 1.1']}
  10. Formulation of Theorem \ref{['Corollary 2.1']}: estimate on the relative entropy
  11. Proof of Theorem \ref{['Theorem 1.1']}
  12. Toward the proof of Theorem \ref{['Corollary 2.1']}
  13. Time derivative of $h_N(t)$ and one-block estimate
  14. Time derivative of the relative entropy $h_N(t)$
  15. Refined one-block estimate
  16. ...and 31 more sections

Key Result

Theorem 1.1

For each $K\ge 1$, let $\rho_K(t,v)$ be the solution of the equation 1.6 with an initial value $\rho_0(v)$ that satisfies the condition eq:1.12. We assume that $f_0$ and $\psi_{0,0}$ defined above satisfy $H(f_0| \psi_{0,0}) \le C_1N^{d-\kappa_1}$ for some $C_1>0$ and $\kappa_1>0$. Then, there exist for every $\alpha>d/2$, and some $C=C_{T,\alpha}>0$ and $\kappa=\kappa_{\frak{c},\alpha}>0$, where

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 42 more