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 latticegas subject to Kawasaki conservative dynamics at inverse temperature $\beta>0$in a large finite box $\Lambda_\beta \subset\mathbb Z^2$ whose volume dependson $\beta$. Each pair of neighbouring particles has a negative binding energy$-U<0$, while each particle has a positive activation energy $\Delta>0$. Theinitial configuration is drawn from the grand-canonical ensemble restricted tothe set of configurations where all the droplets are subcritical. Our goal isto describe, in the metastable regime $\Delta\in(U,2U)$ and in the limit as$\beta\to\infty$, how and when the system nucleates. In the first paper we showed that subcritical droplets behave as quasi-randomwalks. In the present paper we use the results in the first paper to analysehow subcritical droplets form and dissolve on multiple space-time scales whenthe volume is moderately large, i.e., $|\Lambda_\beta|=\mathrm e^{\Theta\beta}$with $\Delta<\Theta<2\Delta-U$. In the third paper we consider the settingwhere the volume is very large, namely, $|\Lambda_\beta|=\mathrme^{\Theta\beta}$ with $\Delta<\Theta<\Gamma-(2\Delta-U)$, where $\Gamma$ is theenergy of the critical droplet in the local model with fixed volume, and usethe results in the first two papers to identify the nucleation time. We willsee that in a very large volume critical droplets appear more or lessindependently in boxes of moderate volume, a phenomenon referred to ashomogeneous nucleation. Since Kawasaki dynamics is conservative, i.e., particles are preserved, weneed 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 leadingto nucleation is described via a series of events on which the evolution of thegas consists of droplets wandering around on multiple space-time scales.