Homogeneous nucleation for two-dimensional Kawasaki dynamics

TL;DR

The paper advances the metastability analysis of a two-dimensional Kawasaki lattice gas in very large volumes, proving that nucleation is homogeneous: a critical droplet appears roughly independently in disjoint moderate boxes. It identifies the nucleation time scale as exponential in the energy barrier, with a leading order given by e^{Γβ}/|Λ_β|, and it characterizes the path to nucleation through a controlled sequence of quasi-squares (the tube of typical trajectories). The authors combine lower/upper bounds, recurrence and tube arguments, and a novel near-independence construction across boxes to describe both the timing and geometry of nucleation, while leveraging results from prior work on subcritical droplets. The work thus completes the metastability description for Kawasaki dynamics in large volumes and highlights a mechanism by which homogeneous nucleation emerges in sparse, low-density regimes.

Abstract

This is the third in a series of three papers in which we study a lattice gas subject to Kawasaki dynamics at inverse temperature $β>0$ in a large finite box $Λ_β\subset \mathbb{Z}^2$ whose volume depends on $β$. Each pair of neighbouring particles has a negative binding energy $-U<0$, while each particle has a positive activation energy $Δ>0$. 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 $Δ\in (U,2U)$ and in the limit as $β\to\infty$, how and when the system nucleates, i.e., creates a critical droplet somewhere in $Λ_β$ that subsequently grows by absorbing particles from the surrounding gas. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the second paper we used the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large, namely, $|Λ_β| = \mathrm{e}^{θβ}$ with $Δ< θ< 2Δ-U$. In the present paper we consider the setting where the volume is very large, namely, $|Λ_β| = \mathrm{e}^{Θβ}$ with $Θ< Γ-(2Δ-U)$, where $Γ$ is the energy of the critical droplet in the local model with fixed volume, and use the results in the first two papers to identify the nucleation time and the tube of typical trajectories towards nucleation. 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. One of the key ingredients in the proof is an estimate showing that no information can travel between these boxes on relevant time scales.

Figures (5)

• Figure 1: A critical droplet in a finite volume $\Lambda$: a protocritical droplet, consisting of an $(\ell_c-1) \times \ell_c$ quasi-square with a single protuberance attached to one of the longest sides, and a free particle nearby. When the free particle attaches itself to the protuberance, the droplet becomes supercritical.
• Figure 2: Cost of adding or removing a row of length $\ell$ in a finite volume.
• Figure 3: Each particle is represented by a unit square. Particles 1--5 and 16 are free, particles 6--9, 10, 11--15 are not free. All other particles are clusterised.
• Figure 4: Construction of the clouds $(C_i)_{i<3}$ at time $t=0$. Note that $\mathrm{dist}(C_{i_1},C_{i_2})=\sqrt{T_{\theta}\mathrm e^{-\kappa\beta}}$, with $\kappa>0$ small enough, for $i_1,i_2<3$ and $i_1\neq i_2$.
• Figure 5: We depict the construction of the times $(\tau^*_{j,i})_{j < 4}$ for $i<N(j)$, where $N(0)=N(1)=3$ and $N(2)=N(3)=5$. In particular, each cross represents a return time $\bar{\tau}_{j,i}$, where we highlight with a circle the times $\tau^*_{j,i}$ and with a square the times $\tau_j^*$.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Definition 2.1
• Definition 2.2
• Remark 5.1
• Remark 5.2
• Proposition 5.3