Frank den Hollander, Enzo Olivieri, Elisabetta Scoppola
Metastability and nucleation for conservative dynamics
(1285K, PS gzipped uuencoded)
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.