Metastability for Kawasaki Dynamics on the Hexagonal Lattice

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 varLambda subset {mathbb {H}}^2 a finite set which we assume to be arbitrarily large. Particles perform simple exclusion on varLambda , but when they occupy neighboring sites they feel a binding energy -U<0. Along each bond touching the boundary of varLambda from the outside to the inside, particles are created with rate rho =e^{-varDelta beta }, while along each bond from the inside to the outside, particles are annihilated with rate 1, where beta is the inverse temperature and varDelta >0 is an activity parameter. For the choice varDelta 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.