In this paper, we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let β denote the inverse temperature and let Λβ⊂ℤ2 be a square box with periodic boundary conditions such that limβ→∞|Λβ|=∞. We run the dynamics on Λβ, starting from a random initial configuration where all of the droplets (clusters of plus-spins and clusters of particles, respectively) are small. For large β and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single critical droplet somewhere in Λβ. Using potential-theoretic methods, we compute the average nucleation time (the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to 1 as β→∞. It turns out that this time grows as KeΓβ/|Λβ| for Glauber dynamics and as KβeΓβ/|Λβ| for Kawasaki dynamics, where Γ is the local canonical (resp. grand-canonical) energy, to create a critical droplet and K is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on |Λβ|). The fact that the average nucleation time is inversely proportional to |Λβ| is referred to as homogeneous nucleation because it says that the critical droplet for the transition appears essentially independently in small boxes that partition Λβ.
"Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures." Ann. Probab. 38 (2) 661 - 713, March 2010. https://doi.org/10.1214/09-AOP492