Hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpi\'nski gasket

TL;DR

This paper establishes the hydrodynamic limit for Glauber-Kawasaki particle dynamics on the Sierpiński gasket, deriving a nonlinear reaction-diffusion equation that captures the macroscopic behavior on this fractal structure.

Contribution

It introduces a novel replacement lemma and estimates adapted to the fractal geometry, enabling analysis of particle systems on non-translationally invariant graphs.

Findings

Hydrodynamic limit proven for Glauber-Kawasaki dynamics on the Sierpiński gasket.

Derived a nonlinear reaction-diffusion equation for the particle density.

Developed new techniques for block estimates on fractal graphs.

Abstract

We prove the hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpi\'nski 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 Sierpi\'nski gasket. We establish this lemma by deriving 1-block and 2-blocks estimates on the Sierpi\'nski gasket graph, which require new arguments due to the absence of classical lattice structures.