The Annals of Probability

Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures

Anton Bovier, Frank den Hollander, and Cristian Spitoni

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Abstract

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→∞β|=∞. 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.

Article information

Source
Ann. Probab., Volume 38, Number 2 (2010), 661-713.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1268143530

Digital Object Identifier
doi:10.1214/09-AOP492

Mathematical Reviews number (MathSciNet)
MR2642889

Zentralblatt MATH identifier
1193.60114

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C26: Dynamic and nonequilibrium phase transitions (general)

Keywords
Glauber dynamics Kawasaki dynamics critical droplet metastable transition time last-exit biased distribution Dirichlet principle Berman–Konsowa principle capacity flow cluster expansion

Citation

Bovier, Anton; den Hollander, Frank; Spitoni, Cristian. Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures. Ann. Probab. 38 (2010), no. 2, 661--713. doi:10.1214/09-AOP492. https://projecteuclid.org/euclid.aop/1268143530


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