Metastability for Kawasaki dynamics on the hexagonal lattice

TL;DR

This paper investigates the metastable behavior of the Ising model under Kawasaki dynamics on a hexagonal lattice at low temperatures, identifying stable states, transition times, and how lattice structure influences dynamics.

Contribution

It provides a detailed analysis of metastability for Kawasaki dynamics on the hexagonal lattice, including transition times, critical configurations, and lattice-specific effects.

Findings

Empty hexagon is the unique metastable state.

Full hexagon is the unique stable state.

Transition times and critical configurations depend on thermodynamical parameters.

Abstract

In this paper we analyze the metastable behavior for the Ising model that evolves under Kawasaki dynamics on the hexagonal lattice $\mathbb{H}^2$ in the limit of vanishing temperature. Let $\Lambda\subset\mathbb{H}^2$ a finite set which we assume to be arbitrarily large. Particles perform simple exclusion on $\Lambda$, but when they occupy neighboring sites they feel a binding energy $-U<0$. Along each bond touching the boundary of $\Lambda$ from the outside to the inside, particles are created with rate $\rho=e^{-\Delta\beta}$, while along each bond from the inside to the outside, particles are annihilated with rate 1, where $\beta$ is the inverse temperature and $\Delta>0$ is an activity parameter. For the choice $\Delta\in{(U,\frac{3}{2}U)}$ we prove that the empty (resp.\ full) hexagon is the unique metastable (resp.\ stable) state. We determine the asymptotic properties of the transition time from the metastable to the stable state and we give a description of the critical configurations. We show how not only their size but also their shape varies depending on the thermodynamical parameters. Moreover, we emphasize the role that the specific lattice plays in the analysis of the metastable Kawasaki dynamics by comparing the different behavior of this system with the corresponding system on the square lattice.