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Three Lectures on Metastability Under Stochastic Dynamics

Part of the Lecture Notes in Mathematics book series (LNM,volume 1970)

Abstract

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. L. Alonso and R. Cerf, The three-dimensional polyominoes of minimal area, Electron. J. Combin. 3 (1996) Research Paper 27.

    Google Scholar 

  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. A. Bovier, Metastability: a potential theoretic approach, Proceedings of ICM 2006, Vol. 3, EMS Publising House, 2006, pp. 499–518.

    MATH  Google Scholar 

  4. A. Bovier, Metastability. In this volume. Springer, Berlin, 2007.

    Google Scholar 

  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.

    CrossRef  MathSciNet  Google Scholar 

  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.

    CrossRef  MathSciNet  Google Scholar 

  7. A. Bovier, F. den Hollander and D. Ioffe, work in progress.

    Google Scholar 

  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.

    CrossRef  MathSciNet  Google Scholar 

  9. A. Bovier, F. den Hollander and C. Spitoni, EURANDOM Report 2008–021.

    Google Scholar 

  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.

    CrossRef  MathSciNet  Google Scholar 

  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.

    CrossRef  MathSciNet  Google Scholar 

  12. P. Dehghanpour and R.H. Schonmann, Metropolis dynamics relaxation via nucleation and growth, Comm. Math. Phys. 188 (1997) 89–119.

    CrossRef  MathSciNet  Google Scholar 

  13. P. Dehghanpour and R.H. Schonmann, A nucleation-and-growth model, Probab. Theory Rel. Fields. 107 (1997) 123–135.

    CrossRef  MathSciNet  Google Scholar 

  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. M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, Springer, Berlin, 1984.

    CrossRef  Google Scholar 

  16. A. Gaudillière, F. den Hollander, F.R. Nardi, E. Olivieri and E. Scoppola, work in progress.

    Google Scholar 

  17. F. den Hollander, Metastability under stochastic dynamics, Stoch. Proc. Appl. 114 (2004) 1–26.

    CrossRef  MathSciNet  Google Scholar 

  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.

    CrossRef  MathSciNet  Google Scholar 

  19. F. den Hollander, E. Olivieri and E. Scoppola, Metastability and nucleation for conservative dynamics, J. Math. Phys. 41 (2000) 1424–1498.

    CrossRef  MathSciNet  Google Scholar 

  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.

    CrossRef  MathSciNet  Google Scholar 

  21. F. Manzo and E. Olivieri, Dynamical Blume-Capel model: competing metastable states at infinite volume, J. Stat. Phys. 104 (2001) 1029–1090.

    CrossRef  MathSciNet  Google Scholar 

  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.

    CrossRef  MathSciNet  Google Scholar 

  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. E. Olivieri and M.E. Vares, Large Deviations and Metastability, Cambridge University Press, Cambridge, 2004.

    MATH  Google Scholar 

  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. C.E. Pfister, Large deviations and phase separation in the two-dimensional Ising model, Helv. Phys. Acta. 64 (1991) 953–1054.

    MathSciNet  Google Scholar 

  27. A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Relat. Fields. 104 (1996) 427–466.

    CrossRef  MathSciNet  Google Scholar 

  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.

    CrossRef  Google Scholar 

  29. R.H. Schonmann, Metastability and the Ising model, Proceedings of ICM 1998, Doc. Math., Extra Vol. 3, pp. 173–181.

    Google Scholar 

  30. R.H. Schonmann, S.B. Shlosman, Wulff droplets and the metastable relaxation of kinetic Ising models, Comm. Math. Phys. 194 (1998) 389–462.

    CrossRef  MathSciNet  Google Scholar 

  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. M.E. Vares, Large deviations and metastability, in: Disordered Systems, Traveaux en Cours 53, Hermann, Paris, 1996, pp. 1–62.

    Google Scholar 

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Correspondence to Frank den Hollander .

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Hollander, F. (2009). Three Lectures on Metastability Under Stochastic Dynamics. In: Kotecký, R. (eds) Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics(), vol 1970. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92796-9_5

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