Metastability and nucleation for conservative dynamics

In this paper we study metastability and nucleation for a local version of the two-dimensional lattice gas with Kawasaki dynamics at low temperature a nd low density. Let beta > 0 be the inverse temperature and let <(Lambda)ov er bar>subset of Lambda(beta)subset of Z(2) be two finite boxes. Particles perform independent random walks on Lambda(beta)\<(Lambda)over bar> and ins ide feel exclusion as well as a binding energy U > 0 with particles at neig hboring sites, i.e., inside <(Lambda)over bar> the dynamics follows a Metro polis algorithm with an attractive lattice gas Hamiltonian. The initial con figuration is chosen such that <(Lambda)over bar> is empty, while a total o f rho\Lambda(beta\) particles is distributed randomly over Lambda(beta)\<(L ambda)over bar> with no exclusion. That is to say, initially the system is in equilibrium with particle density rho conditioned on <(Lambda)over bar> being empty. For large beta, the system in equilibrium has <(Lambda)over ba r> fully occupied because of the binding energy. We consider the case where rho=e(-Delta beta) for some Delta is an element of(U,2U) and investigate h ow the transition from empty to full takes place under the dynamics. In par ticular, we identify the size and shape of the critical droplet and the tim e of its creation in the limit as beta -->infinity for fixed Lambda and lim (beta -->infinity)(1/beta) log\Lambda(beta\)=infinity. In addition, we obta in some information on the typical trajectory of the system prior to the cr eation of the critical droplet. The choice Delta is an element of(U,2U) cor responds to the situation where the critical droplet has side length l(c)is an element of(1,infinity), i.e., the system is metastable. The side length of <(Lambda)over bar> must be much larger than l(c) and independent of bet a, but is otherwise arbitrary. Because particles are conserved under Kawasa ki 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 Lambda(beta)\<(Lambda)over bar> can be treated a s a reservoir that creates particles with rate rho at sites on the interior boundary of Lambda<(Lambda)over bar> and annihilates particles with rate 1 at sites on the exterior boundary of <(Lambda)over bar>. Once this approxi mation has been achieved, the problem reduces to understanding the local me tastable behavior inside <(Lambda)over bar> in the presence of a nonconserv ative boundary. The dynamics inside <(Lambda)over bar> is still conservativ e and this difficulty has to be handled via local geometric arguments. Here it turns out that the Kawasaki dynamics has its own peculiarities. For ins tance, rectangular droplets tend to become square through a movement of par ticles along the border of the droplet. This is different from the behavior under the Glauber dynamics, where subcritical rectangular droplets are att racted by the maximal square contained in the interior, while supercritical rectangular droplets tend to grow uniformly in all directions (at least fo r not too long a time) without being attracted by a square. (C) 2000 Americ an Institute of Physics. [S0022-2488(00)01503-6].