Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions

Abstract

In this paper we analyze metastability and nucleation in the context of the Kawasaki dynamics for the two-dimensional 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 $\beta >0$ 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 $\Delta \in (U_1,U_1+U_2)$, so that the empty (respectively full) configuration is a metastable (respectively stable) configuration. We investigate how the transition from empty to full takes place with particular attention to the critical configurations that asymptotically have to be crossed with probability 1. The derivation of some geometrical properties of the saddles allows us to identify the full geometry of the minimal gates and their boundaries for the nucleation in the strongly anisotropic case. We observe very different behaviors for this case with respect to the isotropic ($U_1=U_2$) and weakly anisotropic ($U_1<2U_2$) ones. Moreover, we derive mixing time, spectral gap and sharp estimates for the asymptotic transition time for the strongly anisotropic case.

51 pages and 14 figures. In this paper we consider, with the same authors, the same model as our previous work arXiv:2108.02017, but in a complementary range of parameter values. Thus sections 1, 2, 3 and 4.1 in the present paper are similar to sections 1, 2, 3.1 and 4.1 in the previous paper arXiv:2108.02017

References in corpus (4)

• Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States
• Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics
• Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions. Extended version
• Effect of energy degeneracy on the transition time for a series of metastable states: application to Probabilistic Cellular Automata

Cited by in corpus (5)

• Ising model on clustered networks: A model for opinion dynamics
• Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics
• Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions. Extended version
• 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