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 $\Lambda \subset {\mathbb{H}}^2$ 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.

arXiv comment45 pages, 15 figures

References in corpus (19)

• Tunneling and Metastability of continuous time Markov chains
• Metastability in Glauber dynamics in the low-temperature limit: Beyond exponential asymptotics
• Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States
• Metastability for reversible probabilistic cellular automata with self--interaction
• Metastable states, quasi-stationary distributions and soft measures
• Metastability in the two-dimensional Ising model with free boundary conditions
• Metastability for general dynamics with rare transitions: escape time and critical configurations
• Nucleation and growth for the Ising model in $d$ dimensions at very low temperatures
• Hitting times asymptotics for hard-core interactions on grids
• Competitive nucleation in reversible Probabilistic Cellular Automata
• Sum of exit times in a series of two metastable states
• Kawasaki dynamics with two types of particles: stable/metastable configurations and communication heights
• Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics
• Low-temperature behavior of the multicomponent Widom-Rowlison model on finite square lattices
• Asymptotic exponential law for the transition time to equilibrium of the metastable kinetic Ising model with vanishing magnetic field
• Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions. Extended version
• Criticality of measures on 2-d Ising configurations: from square to hexagonal graphs
• Metastability of Ising and Potts models without external fields in large volumes at low temperatures
• Metastability of synchronous and asynchronous dynamics

Cited by in corpus (4)

• Ising model on clustered networks: A model for opinion dynamics
• Microscopic fluctuations in the spreading fronts of circular wetting liquid droplets
• Metastability for the degenerate Potts Model with positive external magnetic field under Glauber dynamics
• Critical configurations of the hard-core model on square grid graphs