Skip to main content
SpringerLink
Log in

Metastability: From Mean Field Models to SPDEs

  • Conference paper
  • First Online:
Probability in Complex Physical Systems

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 11))

Abstract

Kramer’s equation of a diffusion in a double well potential has been the pardigm for a metastable system since 1940. The theme of this note is to partially explain, why and in what sense this is a good model for metastable systems. In the process, I review recent progress in a variety of models, ranging from mean field spin systems to stochastic partial differential equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 149.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Note that I will not seriously enter the discussion of infinite volume dynamics and so I also do not enter the formalism of infinite volume Gibbs measures.

  2. 2.

    I am not very careful with time here. We may assume that our sets are big enough so that the optimal connecting paths do so in ε-independent time.

  3. 3.

    This may be general forms, see [3] for examples.

References

  1. Barret, F., Bovier, A., Méléard, S.: Uniform estimates for metastable transition times in a coupled bistable system. Elect. J. Probab. 15, 323–345 (2010)

    MATH  Google Scholar 

  2. Ben Arous, G., Cerf, R.: Metastability of the three-dimensional Ising model on a torus at very low temperatures. Electron. J. Probab. 1(10), p. 55 (1996)

    MathSciNet  Google Scholar 

  3. Berglund, N., Gentz, B.: Anomalous behavior of the Kramers rate at bifurcations in classical field theories. J. Phys. A 42(5), 052001, 9 (2009)

    Google Scholar 

  4. Berman, K.A., Konsowa, M.H.: Random paths and cuts, electrical networks, and reversible Markov chains. SIAM J. Discrete Math. 3(3), 311–319 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bianchi, A., Bovier, A., Ioffe, D.: Pointwise estimates and exponential laws in metastable systems via coupling methods. preprint, SFB 611, Bonn University (2009) (to appear in Ann. Probab)

    Google Scholar 

  6. Bianchi, A., Bovier, A., Ioffe, D.: Sharp asymptotics for metastability in the random field Curie-Weiss model. Electron. J. Probab. 14(53), 1541–1603 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Bovier, A.: Metastability: a potential theoretic approach. In: International Congress of Mathematicians. Vol. III, pp. 499–518. Eur. Math. Soc., Zürich (2006)

    Google Scholar 

  8. Bovier, A.: Metastability. In: Methods of contemporary mathematical statistical physics, volume 1970 of Lecture Notes in Math., pp. 177–221. Springer, Berlin (2009)

    Google Scholar 

  9. Bovier, A., den Hollander, F., Spitoni, C.: Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes and low temperature. Ann. Probab. 38, 661–713 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean-field models. Probab. Theor. Relat. Field. 119(1), 99–161 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low lying spectra in reversible Markov chains. Comm. Math. Phys. 228(2), 219–255 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. (JEMS) 6(4), 399–424 (2004)

    Google Scholar 

  13. Bovier, A., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. (JEMS) 7(1), 69–99 (2005)

    Google Scholar 

  14. Cassandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behavior of stochastic dynamics: a pathwise approach. J. Statist. Phys. 35(5–6), 603–634 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Catoni, O., Cerf, R.: The exit path of a Markov chain with rare transitions. ESAIM Probab. Stat. 1, 95–144 (1995/97)

    Google Scholar 

  16. Comets, F.: Nucleation for a long range magnetic model. Ann. Inst. H. Poincaré Probab. Statist. 23(2), 135–178 (1987)

    MathSciNet  MATH  Google Scholar 

  17. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions, volume 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  18. Dawson, D.A., Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20(4), 247–308 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dawson, D.A., Gärtner, J.: Long time behaviour of interacting diffusions. In Stochastic calculus in application (Cambridge, 1987), volume 197 of Pitman Res. Notes Math. Ser., pp. 29–54. Longman Sci. Tech., Harlow (1988)

    Google Scholar 

  20. den Hollander, F., Nardi, F.R., Olivieri, E., Scoppola, E.: Droplet growth for three-dimensional Kawasaki dynamics. Probab. Theor. Relat. Field. 125(2), 153–194 (2003)

    Article  MATH  Google Scholar 

  21. den Hollander, F., Olivieri, E., Scoppola, E.: Metastability and nucleation for conservative dynamics. J. Math. Phys. 41(3), 1424–1498 (2000); Probabilistic techniques in equilibrium and nonequilibrium statistical physics.

    Google Scholar 

  22. Eyring, H.: The activated complex in chemical reactions. J. Chem. Phys 3, 107 (1935)

    Article  Google Scholar 

  23. Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, New York (1984); Translated from the Russian by Joseph Szücs.

    Google Scholar 

  24. Gärtner, J.: On the McKean-Vlasov limit for interacting diffusions. Math. Nachr. 137, 197–248 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  26. Külske, C.: On the Gibbsian nature of the random field Kac model under block-averaging. J. Stat. Phys. 104(5–6), 991–1012 (2001)

    Article  MATH  Google Scholar 

  27. Levin, D.A., Luczak, M.J., Peres, Y.: Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Probab. Theor. Relat. Field. 146(1–2), 223–265 (2010)

    Article  MathSciNet  Google Scholar 

  28. Maier, R., Stein, D.: Droplet nucleation and domain wall motion in a bounded interval. Phys. Rev. Lett. 87, 270601–1–270601–4 (2001)

    Google Scholar 

  29. Martinelli, F., Olivieri, E., Scoppola, E.: Small random perturbations of finite- and infinite-dimensional dynamical systems: unpredictability of exit times. J. Stat. Phys. 55(3–4), 477–504 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  30. Neves, E.J., Schonmann, R.H.: Critical droplets and metastability for a Glauber dynamics at very low temperatures. Comm. Math. Phys. 137(2), 209–230 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  31. Neves, E.J., Schonmann, R.H.: Behavior of droplets for a class of Glauber dynamics at very low temperature. Probab. Theor. Relat. Field. 91(3–4), 331–354 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Olivieri, E., Scoppola, E.: Markov chains with exponentially small transition probabilities: first exit problem from a general domain. I. The reversible case. J. Statist. Phys. 79(3–4), 613–647 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Olivieri, E., Scoppola, E.: Markov chains with exponentially small transition probabilities: first exit problem from a general domain. II. The general case. J. Stat. Phys. 84(5–6), 987–1041 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Olivieri, E., Vares, M.E.: Large deviations and metastability, volume 100 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  35. Pollak, E., Talkner, P.: Reaction rate theory: What it was, where is it today, and where is it going? Chaos 15, 026116 (2005)

    Article  MathSciNet  Google Scholar 

  36. Schonmann, R.H., Shlosman, S.B.: Wulff droplets and the metastable relaxation of kinetic Ising models. Comm. Math. Phys. 194(2), 389–462 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ventcel′, A.D.: Formulas for eigenfunctions and eigenmeasures that are connected with a Markov process. Teor. Verojatnost. i Primenen. 18, 3–29 (1973)

    Google Scholar 

  38. Ventcel′, A.D.: The asymptotic behavior of the first eigenvalue of a second order differential operator with a small parameter multiplying the highest derivatives. Teor. Verojatnost. i Primenen. 20(3), 610–613 (1975)

    Google Scholar 

  39. Weidenmüller, H.A., Zhang, J.S.: Stationary diffusion over a multidimensional potential barrier: a generalization of Kramers’ formula. J. Stat. Phys. 34(1–2), 191–201 (1984)

    Article  Google Scholar 

Download references

Acknowledgements

This notes summarizes thoughts that have come up through extensive work on metastability with numerous people. The entire subject started with an intensive collaboration with Michael Eckhoff, Véronique Gayrard, and Markus Klein. These early works set the stage for the potential theoretic approach. More recently I collaborated with Florent Barret, Alessandra Bianchi, Alessandra Faggionato, Frank den Hollander, Dima Ioffe, Francesco Manzo, Sylvie Méléard, Francesca Nardi, and Cristian Spitoni, on various special issues and models. I thank all of them for sharing their thoughts and insights.

The huge project on metastability was possible also only due to the excellent working conditions I had from 1992 on at the WIAS. This was in in more than one way due to Jürgen Gärtner to whom I am deeply grateful.

One person deserves spatial thanks: Erwin Bolthausen handled our first paper [ 10] on metastability as Editor-in-Chief of PTRF, and through an extensive correspondence, that paper was finally published.

These notes were written while I was holding a Lady Davis Visiting Professorship at the Technion, Haifa. I thank the William Davidson Faculty of Industrial Engineering and Management and in particular Dmitry Ioffe for their kind hospitality. Much of the work reported on here was also supported by a grant from the German-Israeli Foundation (GIF).

Financial support from the German Research Council (DFG) through SFB 611 and the Hausdorff Center for Mathematics is gratefully acknoledged.

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität, Endenicher Allee 60, 53115, Bonn, Germany

    Anton Bovier

Authors
  1. Anton Bovier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anton Bovier .

Editor information

Editors and Affiliations

  1. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Jean-Dominique Deuschel

  2. Faculty of Mathematics, University of Bielefeld, P.O. Box 10 01 31, Bielefeld, 33501, Germany

    Barbara Gentz

  3. for Applied Analysis and Stochastics, Weierstrass Institute Berlin, Mohrenstr. 39, Berlin, 10117, Germany

    Wolfgang König

  4. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Max von Renesse

  5. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Michael Scheutzow

  6. Institute for Mathematical, Methods in Economics, Vienna University of Technology, Wiedner Hauptstraße 8-10/105-1, Vienna, 1040, Austria

    Uwe Schmock

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bovier, A. (2012). Metastability: From Mean Field Models to SPDEs. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_18

Download citation

Publish with us

Policies and ethics

Search

Navigation