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.
Highlights
Metastability is a dynamical phenomenon that occurs when a system is close to first order phase transition, i.e., a crossover that involves a jump in some intrinsic physical parameter such as the energy density or the magnetization
The phenomenon of metastability occurs when a system is trapped for a long time in a state different from the equilibrium state for specific values of the thermodynamical parameters, and subsequently at some random time the system undergoes a sudden transition from the metastable to the stable state
In this paper we study the metastable behavior of the two-dimensional strongly anisotropic Ising lattice gas that evolves according to Kawasaki dynamics, i.e., a discrete time Markov chain defined by the Metropolis algorithm with transition probabilities given precisely later in (2.9)
Summary
Metastability is a dynamical phenomenon that occurs when a system is close to first order phase transition, i.e., a crossover that involves a jump in some intrinsic physical parameter such as the energy density or the magnetization. A special feature of Kawasaki dynamics is that in the metastable regime (see (2.36)) particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. If the evolution is according to Kawasaki dynamics, it turns out that particles can move along the border of a droplet more rapidly than they can arrive from the boundary of the container. For any anisotropic Glauber dynamics the critical configurations are not Wulff-shaped and the tube of typical paths crosses the Wulff shape only during the supercritical growth (see [42]). In our regime the energy of the Wulff shape is different from Γ and this holds in the case in which ∆ → U1 + U2 that corresponds to ε → 0
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