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Hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket

Patrick van Meurs, Kenkichi Tsunoda

TL;DR

The paper proves a hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket, yielding a nonlinear reaction-diffusion equation as the macroscopic limit. It extends previous lattice/fractal results by incorporating locally dependent Glauber birth-death rates, which introduce a nonlinear reaction term $\Phi(\rho)$ in the limit. A central technical advance is a replacement lemma tailored to the gasket's self-similar geometry, built from 1-block and 2-block estimates and a fractal moving-particle lemma. Depending on the boundary scaling parameter $b$, the limiting boundary condition is Dirichlet for $b<\tfrac{5}{3}$ and Robin/Neumann for $b\ge\tfrac{5}{3}$, enabling a comprehensive view of bulk and boundary interactions on fractal media. The results pave the way for studying fluctuations and large deviations for similar systems on fractals.

Abstract

We prove the hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket, a prototypical fractal graph that lacks translational invariance. The main novelty lies in incorporating Glauber dynamics, allowing for particle creation and annihilation with birth-death rates depending locally on the particle configuration. In the macroscopic limit, the particle density evolves according to a nonlinear reaction--diffusion equation, where the reaction term is explicitly determined by the microscopic rates. The key new ingredient is a replacement lemma adapted to the fractal geometry of the Sierpiński gasket. We establish this lemma by deriving 1-block and 2-blocks estimates on the Sierpiński gasket graph, which require new arguments due to the absence of classical lattice structures.

Hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket

TL;DR

The paper proves a hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket, yielding a nonlinear reaction-diffusion equation as the macroscopic limit. It extends previous lattice/fractal results by incorporating locally dependent Glauber birth-death rates, which introduce a nonlinear reaction term in the limit. A central technical advance is a replacement lemma tailored to the gasket's self-similar geometry, built from 1-block and 2-block estimates and a fractal moving-particle lemma. Depending on the boundary scaling parameter , the limiting boundary condition is Dirichlet for and Robin/Neumann for , enabling a comprehensive view of bulk and boundary interactions on fractal media. The results pave the way for studying fluctuations and large deviations for similar systems on fractals.

Abstract

We prove the hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpiński gasket, a prototypical fractal graph that lacks translational invariance. The main novelty lies in incorporating Glauber dynamics, allowing for particle creation and annihilation with birth-death rates depending locally on the particle configuration. In the macroscopic limit, the particle density evolves according to a nonlinear reaction--diffusion equation, where the reaction term is explicitly determined by the microscopic rates. The key new ingredient is a replacement lemma adapted to the fractal geometry of the Sierpiński gasket. We establish this lemma by deriving 1-block and 2-blocks estimates on the Sierpiński gasket graph, which require new arguments due to the absence of classical lattice structures.
Paper Structure (28 sections, 7 theorems, 174 equations, 2 figures)

This paper contains 28 sections, 7 theorems, 174 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and main result
  3. The Sierpiński gasket $K$ and its discretization
  4. Glauber-Kawasaki dynamics on $\mathcal{G}_N$
  5. Discrete and Continuum Laplacians on the Sierpiński gasket
  6. The reaction--diffusion equation with Dirichlet boundary condition
  7. The hydrodynamic limit for $b < \frac{5}{3}$
  8. Proof of Theorem \ref{['t']}
  9. Preparation: lower bounds on Dirichlet forms
  10. The choice of the reference density $\varrho$
  11. The associated martingale
  12. Heuristically passing to $N \to \infty$
  13. Tightness of $\mathbb Q_N$
  14. A priori properties of the limit path measure $\mathbb Q$
  15. The limiting density $\rho$ is a weak solution
  16. ...and 13 more sections

Key Result

Theorem 2.4

Weak solutions of HDL are unique for any $T, \rho_B, \rho_\circ, \Phi$ in the setting above.

Figures (2)

  • Figure 1: Discretizations of the Sierpiński gasket at depths $N=0,1,2$. $\varphi_w(V_0)$ is highlighted for several words $w$.
  • Figure 2: All shapes $\Lambda \in \mathcal{S}$ for $L_0 = 1$ and $L_0 = 2$. Instead of the sites, the edges (all of unit length) connecting them are shown. The dot indicates the origin in $\mathbb R^2$. Each illustrated shape appears 3 times in $\mathcal{S}$ under the rotations by $0$, $120$, and $240$ degrees.

Theorems & Definitions (13)

  • Example 2.2
  • Definition 2.3: Weak solution
  • Theorem 2.4
  • Theorem 2.5: Hydrodynamic limit
  • Proposition 3.1
  • proof
  • proof : Proof of Theorem \ref{['t:HDL:un']}
  • Lemma 4.1: Main replacement
  • Lemma 4.2: Moving Particle
  • Definition 5.1: Weak solution for $b \geq \frac{5}{3}$
  • ...and 3 more