Metastability in a lattice gas with strong anisotropic interactions under Kawasaki dynamics

Abstract

In this paper we analyze metastability and nucleation in the context of a local version of the Kawasaki dynamics for the two-dimensional strongly anisotropic Ising lattice gas at very low temperature. Let $\Lambda \subset {\mathbb{Z}}^2$ be a finite box. Particles perform simple exclusion on $\Lambda$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $\Lambda$. 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. Thus, the boundary of $\Lambda$ plays the role of an infinite gas reservoir with density $\rho$. We consider the parameter regime $U_1>2U_2$ also known as the strongly anisotropic regime. We take $\Delta \in (U_1,U_1+U_2)$ and we prove that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We consider the asymptotic regime corresponding to finite volume in the limit of large inverse temperature $\beta$. We investigate how the transition from empty to full takes place. In particular, we estimate in probability, expectation and distribution the asymptotic transition time from the metastable configuration to the stable configuration. Moreover, we identify the size of the \emph{critical droplets}, as well as some of their properties. We observe very different behavior in the weakly and strongly anisotropic regimes. We find that the \emph{Wulff shape}, i.e., the shape minimizing the energy of a droplet at fixed volume, is not relevant for the nucleation pattern.

arXiv comment73 pages, 22 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
• Metastability and Nucleation for the Blume-Capel Model. Different mechanisms of transition
• 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
• Metastability and low lying spectra in reversible Markov chains
• Critical droplets in Metastable States of Probabilistic Cellular Automata
• 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 of hard-core dynamics on bipartite graphs
• Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus
• Asymptotic exponential law for the transition time to equilibrium of the metastable kinetic Ising model with vanishing magnetic field
• Kawasaki dynamics with two types of particles: critical droplets

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