Metastability and nucleation for conservative dynamics

Abstract

In this paper we study metastability and nucleation for a local version of the two-dimensional lattice gas with Kawasaki dynamics at low temperature and low density. Let β>0 be the inverse temperature and let Λ̄⊂Λβ⊂Z2 be two finite boxes. Particles perform independent random walks on Λβ\Λ̄ and inside Λ̄ feel exclusion as well as a binding energy U>0 with particles at neighboring sites, i.e., inside Λ̄ the dynamics follows a Metropolis algorithm with an attractive lattice gas Hamiltonian. The initial configuration is chosen such that Λ̄ is empty, while a total of ρ|Λβ| particles is distributed randomly over Λβ\Λ̄ with no exclusion. That is to say, initially the system is in equilibrium with particle density ρ conditioned on Λ̄ being empty. For large β, the system in equilibrium has Λ̄ fully occupied because of the binding energy. We consider the case where ρ=e−Δβ for some Δ∈(U,2U) and investigate how the transition from empty to full takes place under the dynamics. In particular, we identify the size and shape of the critical droplet and the time of its creation in the limit as β→∞ for fixed Λ and limβ→∞(1/β) log|Λβ|=∞. In addition, we obtain some information on the typical trajectory of the system prior to the creation of the critical droplet. The choice Δ∈(U,2U) corresponds to the situation where the critical droplet has side length lc∈(1,∞), i.e., the system is metastable. The side length of Λ̄ must be much larger than lc and independent of β, but is otherwise arbitrary. Because particles are conserved under Kawasaki dynamics, the analysis of metastability and nucleation is more difficult than for Ising spins under Glauber dynamics. The key point is to show that at low density the gas in Λβ\Λ̄ can be treated as a reservoir that creates particles with rate ρ at sites on the interior boundary of Λ̄ and annihilates particles with rate 1 at sites on the exterior boundary of Λ̄. Once this approximation has been achieved, the problem reduces to understanding the local metastable behavior inside Λ̄ in the presence of a nonconservative boundary. The dynamics inside Λ̄ is still conservative and this difficulty has to be handled via local geometric arguments. Here it turns out that the Kawasaki dynamics has its own peculiarities. For instance, rectangular droplets tend to become square through a movement of particles along the border of the droplet. This is different from the behavior under the Glauber dynamics, where subcritical rectangular droplets are attracted by the maximal square contained in the interior, while supercritical rectangular droplets tend to grow uniformly in all directions (at least for not too long a time) without being attracted by a square.