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Three Lectures on Metastability Under Stochastic Dynamics

  • Frank den Hollander
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1970)

Metastability is a phenomenon where a physical, chemical or biological system, under the influence of a noisy dynamics, moves between different regions of its state space on different time scales. On short time scales the system is in a quasi-equilibrium within a single region, while on long time scales it undergoes rapid transitions between quasiequilibria in different regions (see Fig. 1).

Examples of metastability can be found in:
  • biology: folding of proteins;

  • climatology: effects of global warming;

  • economics: crashes of financial markets;

  • materials science: anomalous relaxation in disordered media;

  • physics: freezing of supercooled liquids.

The task of mathematics is to formulate microscopic models of the relevant underlying dynamics, to prove the occurrence of metastable behavior in these models on macroscopic space-time scales, and to identify the key mechanisms behind the experimentally observed universality in the metastable behavior of whole classes of systems. This is a challenging program!

Keywords

Ising Model Free Particle Simple Random Walk Metastable Behavior Glauber Dynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    L. Alonso and R. Cerf, The three-dimensional polyominoes of minimal area, Electron. J. Combin. 3 (1996) Research Paper 27.Google Scholar
  2. 2.
    G. Ben Arous and R. Cerf, Metastability of the three-dimensional Ising model on a torus at very low temperature, Electron. J. Probab. 1 (1996) Research Paper 10.Google Scholar
  3. 3.
    A. Bovier, Metastability: a potential theoretic approach, Proceedings of ICM 2006, Vol. 3, EMS Publising House, 2006, pp. 499–518.Google Scholar
  4. 4.
    A. Bovier, Metastability. In this volume. Springer, Berlin, 2007.Google Scholar
  5. 5.
    A. Bovier, M. Eckhoff, V. Gayrard and M. Klein, Metastability in stochastic dynamics of disordered mean-field models, Probab. Theory Rel. Fields. 119 (2001) 99–161.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Bovier, M. Eckhoff, V. Gayrard and M. Klein, Metastability and low lying spectra in reversible Markov chains, Commun. Math. Phys. 228 (2002) 219–255.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Bovier, F. den Hollander and D. Ioffe, work in progress.Google Scholar
  8. 8.
    A. Bovier, F. den Hollander and F.R. Nardi, Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary, Probab. Theory Relat. Fields. 135 (2006) 265–310.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Bovier, F. den Hollander and C. Spitoni, EURANDOM Report 2008–021.Google Scholar
  10. 10.
    A. Bovier and F. Manzo, Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics, J. Stat. Phys. 107 (2002) 757–779.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    M. Cassandro, A. Galves, E. Olivieri and M.E. Vares, Metastable behavior of stochastic dynamics: a pathwise approach, J. Stat. Phys. 35 (1984) 603–634.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    P. Dehghanpour and R.H. Schonmann, Metropolis dynamics relaxation via nucleation and growth, Comm. Math. Phys. 188 (1997) 89–119.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    P. Dehghanpour and R.H. Schonmann, A nucleation-and-growth model, Probab. Theory Rel. Fields. 107 (1997) 123–135.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    R.L. Dobrushin, R. Kotecký, S.B. Shlosman, Wulff construction: a global shape from local interaction, AMS translations series, Providence, RI, Am. Math. Soc. 1992.Google Scholar
  15. 15.
    M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, Springer, Berlin, 1984.Google Scholar
  16. 16.
    A. Gaudillière, F. den Hollander, F.R. Nardi, E. Olivieri and E. Scoppola, work in progress.Google Scholar
  17. 17.
    F. den Hollander, Metastability under stochastic dynamics, Stoch. Proc. Appl. 114 (2004) 1–26.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    F. den Hollander, F.R. Nardi, E. Olivieri and E. Scoppola, Droplet growth for three-dimensional Kawasaki dynamics, Probab. Theory Relat. Fields. 125 (2003) 153–194.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    F. den Hollander, E. Olivieri and E. Scoppola, Metastability and nucleation for conservative dynamics, J. Math. Phys. 41 (2000) 1424–1498.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    D. Ioffe, Exact deviation bounds up to T c for the Ising model in two dimensions, Probab. Theory Relat. Fields. 102 (1995) 313–330.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    F. Manzo and E. Olivieri, Dynamical Blume-Capel model: competing metastable states at infinite volume, J. Stat. Phys. 104 (2001) 1029–1090.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    E.J. Neves and R.H. Schonmann, Critical droplets and metastability for Glauber dynamics at very low temperature, Comm. Math. Phys. 137 (1991) 209–230.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    E. Olivieri and E. Scoppola, Metastability and typical exit paths in stochastic dynamics, Proceedings of ECM 1996, Progress in Mathematics 169, Birkhäuser, Basel, 1998, pp. 124–150.Google Scholar
  24. 24.
    E. Olivieri and M.E. Vares, Large Deviations and Metastability, Cambridge University Press, Cambridge, 2004.Google Scholar
  25. 25.
    O. Penrose and J.L. Lebowitz, Towards a rigorous molecular theory of metastability, in: Fluctuation Phenomena, 2nd. ed. (E.W. Montroll and J.L. Lebowitz, eds.), North-Holland, Amsterdam, 1987.Google Scholar
  26. 26.
    C.E. Pfister, Large deviations and phase separation in the two-dimensional Ising model, Helv. Phys. Acta. 64 (1991) 953–1054.MathSciNetGoogle Scholar
  27. 27.
    A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Relat. Fields. 104 (1996) 427–466.zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    R.H. Schonmann, Theorems and conjectures on the droplet-driven relaxation of stochastic Ising models, in: Probability and Phase Transitions (G. Grimmett ed.), NATO ASI Series, Kluwer Academic Publishers, Dordrecht, 1994, pp. 265–301.Google Scholar
  29. 29.
    R.H. Schonmann, Metastability and the Ising model, Proceedings of ICM 1998, Doc. Math., Extra Vol. 3, pp. 173–181.Google Scholar
  30. 30.
    R.H. Schonmann, S.B. Shlosman, Wulff droplets and the metastable relaxation of kinetic Ising models, Comm. Math. Phys. 194 (1998) 389–462.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    E. Scoppola, Metastability for Markov chains, in: Probability and Phase Transitions (G. Grimmett ed.), NATO ASI Series, Kluwer Academic Publishers, Dordrecht, 1994, pp. 303–322.Google Scholar
  32. 32.
    M.E. Vares, Large deviations and metastability, in: Disordered Systems, Traveaux en Cours 53, Hermann, Paris, 1996, pp. 1–62.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Mathematical InstituteLeiden UniversityThe Netherlands
  2. 2.EURANDOMThe Netherlands

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