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 ␤Ͼ0 be the inverse temperature and let ⌳ ʚ⌳ ␤ ʚZ 2 be two finite boxes. Particles perform independent random walks on ⌳ ␤ ‫⌳گ‬ and inside ⌳ feel exclusion as well as a binding energy UϾ0 with particles at neighboring sites, i.e., inside ⌳ the dynamics follows a Metropolis algorithm with an attractive lattice gas Hamiltonian. The initial configuration is chosen such that ⌳ is empty, while a total of ͉⌳ ␤ ͉ particles is distributed randomly over ⌳ ␤ ‫⌳گ‬ with no exclusion. That is to say, initially the system is in equilibrium with particle density conditioned on ⌳ being empty. For large ␤, the system in equilibrium has ⌳ fully occupied because of the binding energy. We consider the case where ϭe Ϫ⌬␤ for some ⌬ (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 ␤→ϱ for fixed ⌳ and lim ␤→ϱ (1/␤) log͉⌳ ␤ ͉ϭϱ. In addition, we obtain some information on the typical trajectory of the system prior to the creation of the critical droplet. The choice ⌬ (U,2U) corresponds to the situation where the critical droplet has side length l c (1,ϱ), i.e., the system is metastable. The side length of ⌳ must be much larger than l c and independent of ␤, 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 ⌳ ␤ ‫⌳گ‬ can be treated as a reservoir that creates particles with rate at sites on the interior boundary of ⌳ and annihilates particles with rate 1 at sites on the exterior boundary of ⌳ . Once this approximation has been achieved, the problem reduces to understanding the local metastable behavior inside ⌳ in the presence of a nonconservative boundary. The dynamics inside ⌳ 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 time͒ without being attracted by a square.

Droplet growth for three-dimensional Kawasaki dynamics

“…We consider the "local version" of the model, where particles live on a finite box and are created respectively annihilated at the boundary of this box in a way that reflects an infinite gas reservoir. Our main results generalize part of those obtained in den Hollander, Olivieri and Scoppola [5], [6], where the two-dimensional version of the same model was considered. In particular, we identify the size and shape of the critical droplet and the time of its creation in the limit of low temperature and low density.…”

“…Show that the same results apply when the creation and annihilation of particles at the boundary of occurs from an infinite gas reservoir surrounding rather than from a boundary mimicking this reservoir. This issue was settled in den Hollander, Olivieri and Scoppola [5] for the two-dimensional version of the model (for the case where outside particles do not interact). The proof relies on delicate coupling arguments, but probably carries over because it is largely independent of dimension.…”

“…The equivalent of C * for the model in two dimensions, as described in [5] and having ∈ (U, 2U) for its metastable regime, is simpler but still not trivial. Here too there is motion along the border of the droplet, this time at maximal cost U .…”

Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary

“…As was pointed out in den Hollander, Olivieri, and Scoppola [9], Section 1.2.3, the energy E( ) of an × droplet in − equals (recall (1.1.3) and see Fig. 1) 3.2) which is maximal at = U/(2U − ):…”

Three Lectures on Metastability Under Stochastic Dynamics

“…Part (a), together with a partial description of C * and the crude estimate lim β→∞ (1/β) log E (τ ) = Γ , were proved in [19]. Parts (b)-(e) were proved in [8].…”