Journal of Statistical Physics

, Volume 75, Issue 3–4, pp 409–506 | Cite as

Shapes of growing droplets—A model of escape from a metastable phase

  • R. Kotecký
  • E. Olivieri


Nucleation from a metastable state is studied for an Ising ferromagnet with nearest and next nearest neighbor interaction and at very low temperatures. The typical escape path is shown to follow a sequence of configurations with a growing droplet of stable phase whose shape is determined by dynamical considerations and differs significantly from the equilibrium shape corresponding to the instantaneous volume.

Key Words

Stochastic dynamics Ising model next nearest neighbor interaction metastability crystal growth first excursion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. K. Burton, N. Cabrera, and F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces.Phil. Trans. Soc. 243A:40–359 (1951).Google Scholar
  2. 2.
    H. van Beijeren and I. Nolden,The Roughening Transition, W. Schommers and P. von Blackenhagen, eds. (Springer, Berlin, 1987), pp. 259–300.Google Scholar
  3. 3.
    M. Cassandro, A. Galves, E. Olivieri, and M. E. Vares, Metastable behaviour of stochastic dynamics: A pathwise approach,J. Stat. Phys. 35:603–634 (1984).CrossRefGoogle Scholar
  4. 4.
    R. L. Dobrushin, R. Kotecký, and S. Shlosman,The Wulff Construction: A Global Shape from Local Interactions (AMS, Providence, Rhode Island, 1992).Google Scholar
  5. 5.
    M. I. Freidlin and A. D. Wentzell,Random Perturbations of Dynamical Systems (Springer-Verlag, New York, 1979).Google Scholar
  6. 6.
    R. Holley, Possible rates of convergence in finite range, attractive spin systems,Contemp. Math. 41:215–234 (1985).Google Scholar
  7. 7.
    C. Herring, Some theorems on the free energies of crystal surfaces,Phys. Rev. 82:87–93 (1951).CrossRefGoogle Scholar
  8. 8.
    R. Kotecký and E. Olivieri, Droplet dynamics for asymmetric Ising model,J. Stat. Phys. 70:1121–1148 (1993).CrossRefGoogle Scholar
  9. 9.
    R. Kotecký and E. Olivieri, Stochastic models for nucleation and crystal growth, inProceedings of the International Workshop, Probabalistic Methods in Mathematical Physics (Siena, 1991), F. Guerra, M. I. Loffredo, and C. Marchioro, eds. (World Scientific, Singapore, 1992), pp. 264–275.Google Scholar
  10. 10.
    F. Martinelli, E. Olivieri, and E. Scoppola, Metastability and exponential approach to equilibrium for low-temperature stochastic models.J. Stat. Phys. 61:1105–1119 (1990).CrossRefGoogle Scholar
  11. 11.
    E. J. Neves and R. H. Schonmann, Critical droplets and metastability for a Glauber dynamics at very low temperatures,Commun. Math. Phys. 137:209–230 (1991).Google Scholar
  12. 12.
    E. J. Neves and R. H. Schonmann, Behavior of droplets for a class of Glauber dynamics at very low temperature,Prob. Theory Related Fields 91:331–354 (1992).CrossRefGoogle Scholar
  13. 13.
    C. Rottmann and M. Wortis, Statistical mechanics of equilibrium crystal shapes: Interfacial phase diagrams and phase transitions,Phys. Rep. 103:59–79 (1984).CrossRefGoogle Scholar
  14. 14.
    R. H. Schonmann, The pattern of escape from metastability of a stochastic Ising model,Commun. Math. Phys. 147:231–240 (1992).CrossRefGoogle Scholar
  15. 15.
    R. H. Schonmann, Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region, to be published (1993).Google Scholar
  16. 16.
    G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Krystallflachen,Z. Krystallogr. Mineral. 34:449 (1901).Google Scholar
  17. 17.
    R. K. P. Zia,Interfacial Problems in Statistical Physics, K. C. Bowler and A. J. McKane, eds.. (SUSSP, Edinburg, 1984).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • R. Kotecký
    • 1
    • 2
  • E. Olivieri
    • 3
  1. 1.Center for Theoretical StudyCharles UniversityPrague 3Czech Republic
  2. 2.Department of Theoretical PhysicsCharles UniversityPrague 8Czech Republic
  3. 3.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly

Personalised recommendations