99-319 Frank den Hollander, Enzo Olivieri, Elisabetta Scoppola

Metastability and nucleation for conservative dynamics (1285K, PS gzipped uuencoded) Aug 31, 99

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Abstract. In this paper we study metastability and nucleation for the two-dimen\-sional lattice gas with Kawasaki dynamics at low temperature and low density. Let $\b>0$ be the inverse temperature and let $\bar\L \subset \L_\b \subset \Z^2$ be two finite boxes. Particles perform independent random walks on $\L_\b \setminus \bar\L$ and inside $\bar\L$ feel exclusion as well as a binding energy $U>0$ with particles at neighboring sites, i.e., inside $\bar\L$ the dynamics follows a Metroplis algorithm with an attractive lattice gas Hamiltonian. The initial configuration is chosen such that $\bar\L$ is empty, while a total of $\rho |\L_\b|$ particles is distributed randomly over $\L_\b\setminus\bar\L$. That is to say, {\it initially the system is in equilibrium with particle density $\rho$ \it conditioned on $\bar\L$ being empty}. For large $\beta$, the system in equilibrium has $\bar\L$ fully occupied because of the binding energy. We consider the case where $\rho = e^{-\Delta\beta}$ for some $\Delta \in (U,2U)$ and investigate how the transition from empty to full takes place under the dynamics. In particular, we identify the size and shape of the {\it critical droplet\/} and the time of its creation in the limit as $\beta \to \infty$ for {\it fixed\/} $\bar\L$ and $\lim_{\b\to\infty}\frac{1}{\b} \log|\L_\b|=\infty$. In addition, we obtain some information on the typical trajectory of the system prior to the creation of the critical droplet. The choice $\Delta \in (U,2U)$ corresponds to the situation where the critical droplet has side length $\ell_c \in (1,\infty)$, i.e., the system is metastable. The side length of $\bar\L$ must be much larger than $\ell_c$, but is otherwise arbitrary.\par Because particles are {\it 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 $\L_\b\setminus\bar\L$ can be treated as a reservoir that creates particles with rate $\rho$ at sites on the interior boundary of $\bar\L$ and annihilates particles with rate 1 at sites on the exterior boundary of $\bar\L$. Once this approximation has been achieved, the problem reduces to understanding the {\it local metastable behavior\/} inside $\bar\L$, and standard techniques from non-conservative dynamics can be applied. Even so, the dynamics inside $\bar\L$ is still conservative, and this difficulty has to be handled via {\it 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 {\it 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 large times) without being attracted by a square.

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