Skip to main content
SpringerLink
Log in

Metastability of Reversible Random Walks in Potential Fields

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Let \(\Xi \) be an open and bounded subset of \({\mathbb {R}}^d\), and let \(F:\Xi \rightarrow {\mathbb {R}}\) be a twice continuously differentiable function. Denote by \(\Xi _N\) the discretization of \(\Xi \), \(\Xi _N = \Xi \cap (N^{-1} {\mathbb {Z}}^d)\), and denote by \(X_N(t)\) the continuous-time, nearest-neighbor, random walk on \(\Xi _N\) which jumps from \({\varvec{x}}\) to \({\varvec{y}}\) at rate \( e^{-(1/2) N [F({\varvec{y}}) - F({\varvec{x}})]}\). We examine in this article the metastable behavior of \(X_N(t)\) among the wells of the potential F.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys. 140, 1065–1114 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains II. J. Stat. Phys. 149, 598–618 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Beltrán, J., Landim, C.: A Martingale approach to metastability. Probab. Theory Relat. Fields 161(1—-2), 267–307 (2015)

    Article  MATH  Google Scholar 

  4. Landim, C.: A topology for limits of Markov chains. Stoch. Process. Appl. 125, 1058–1088 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems, 2nd edn. Translated from the 1979 Russian original by Joseph Szücs. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260. Springer, New York (1998)

  7. Galves, A., Olivieri, E., Vares, M.E.: Metastability for a class of dynamical systems subject to small random perturbations. Ann. Probab. 15, 1288–1305 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean field models. Probab. Theory Relat. Fields 119, 99–161 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. 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. 6, 399–424 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Cameron, M., Vanden-Eijnden, E.: Flows in complex networks: theory, algorithms, and application to LennardJones cluster rearrangement. J. Stat. Phys. 156, 427–454 (2014)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Noé, F., Wu, H., Prinz, J.H., Plattner, N.: Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules (2013). arxiv:1309.3220v1

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Weinan, E., Vanden-Eijnden, E.: Towards a theory of transition paths. J. Stat. Phys. 123, 503–523 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Metzner, P., Schütte, Ch., Vanden-Eijnden, E.: Transition path theory for Markov jump processes. SIAM Multiscale Model. Simul. 7, 1192–1219 (2009)

    Article  MATH  Google Scholar 

  17. Avena, L., Gaudillière, A.: On some random forests with determinantal roots (2013). arXiv:1310.1723v3

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Bianchi, A., Bovier, A., Ioffe, D.: Pointwise estimates and exponential laws in metastable systems via coupling methods. Ann. Probab. 40, 339–371 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gaudillière, A.: Condenser physics applied to Markov chains: a brief introduction to potential theory (2009). arXiv:0901.3053

Download references

Acknowledgments

R. Misturini was supported by CAPES and CNPq-Brazil during the preparation of this work, and K. Tsunoda was supported by the Program for Leading Graduate Course for Frontiers of Mathematical Sciences and Physics.

Author information

Authors and Affiliations

  1. IMPA, Estrada Dona Castorina 110, Rio de Janeiro, CEP 22460, Brazil

    C. Landim & R. Misturini

  2. CNRS UMR 6085, Université de Rouen, Avenue de l’Université, BP 12, Technopôle du Madrillet, 76801, Saint-Étienne-du-Rouvray, France

    C. Landim

  3. Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    K. Tsunoda

Authors
  1. C. Landim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Misturini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. Tsunoda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Landim.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Landim, C., Misturini, R. & Tsunoda, K. Metastability of Reversible Random Walks in Potential Fields. J Stat Phys 160, 1449–1482 (2015). https://doi.org/10.1007/s10955-015-1298-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-015-1298-6

Keywords

Search

Navigation