Droplet dynamics in a two-dimensional rarefied gas under Kawasaki dynamics

This is the second in a series of three papers in which we study a lattice gas subject to Kawasaki conservative dynamics at inverse temperature $\beta >0$ in a large finite box ${\Lambda }_{\beta }\subset Z^2$ whose volume depends on $\beta$. Each pair of neighboring particles has a negative binding energy $-U<0$, while each particle has a positive activation energy $\Delta >0$. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime $\Delta \in (U,2U)$ and in the limit as $\beta \to \infty$, how and when the system nucleates, i.e., grows a supercritical droplet somewhere in ${\Lambda }_{\beta }$. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the present paper we use the results in the first paper to analyze how subcritical droplets form and dissolve on multiple space–time scales when the volume is moderately large, namely, $\left|{\Lambda }_{\beta }\right|=e^{\Theta \beta }$ with $\Delta <\Theta <2\Delta -U$. In the third paper we consider the setting where the volume is very large, namely, $\left|{\Lambda }_{\beta }\right|=e^{\Theta \beta }$ with $\Delta <\Theta <\Gamma -(2\Delta -U)$, where $\Gamma$ is the energy of the critical droplet in the local model, i.e., when ${\Lambda }_{\beta }$ has a fixed volume not depending on $\beta$ and particles can be created and annihilated at the boundary, and use the results in the first two papers to identify the nucleation time. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. Since Kawasaki dynamics is conservative, i.e., particles move around and interact but are preserved, we need to control non-local effects in the way droplets are formed and dissolved. This is done via a deductive approach: the tube of typical trajectories leading to nucleation is described via a series of events, whose complements have negligible probability, on which the evolution of the gas can be captured by a coarse-grained Markov chain on a space of droplets, which we refer to as droplet dynamics.