Skip to main content
SpringerLink
Log in

Metastability of Non-reversible, Mean-Field Potts Model with Three Spins

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

Abstract

We examine a non-reversible, mean-field Potts model with three spins on a set with \(N\uparrow \infty \) points. Without an external field, there are three critical temperatures and five different metastable regimes. The analysis can be extended by a perturbative argument to the case of small external fields, and it can be carried out in the case where the external field is in the direction or in the opposite direction to one of the values of the spins. Numerical computations permit to identify other phenomena which are not present in the previous situations.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

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.

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  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Berglund, N.: Kramers’ law: validity, derivations and generalisations. Markov Process. Relat. Fields 19, 459–490 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Bouchet, F., Reygner, J.: Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes. Preprint (2015). arXiv:1507.02104

  6. Bovier, A., den Hollander, F.: Metastability: A Potential-Theoretic Approach. Grundlehren der mathematischen Wissenschaften, vol. 351. Springer, Berlin (2015)

  7. 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 

  8. 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  ADS  MathSciNet  MATH  Google Scholar 

  9. Cuff, P., Ding, J., Louidor, O., Lubetzky, E., Peres, Y., Sly, A.: Glauber dynamics for the mean-field Potts model. J. Stat. Phys. 149, 432–477 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Gaudillière, A., Landim, C.: A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probab. Theory Relat. Fields 158, 55–89 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Landim, C.: Metastability for a non-reversible dynamics: the evolution of the condensate in totally asymmetric zero range processes. Commun. Math. Phys. 330, 1–32 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Landim, C.: A topology for limits of Markov chains. Stoch. Proc. Appl. 125, 1058–1098 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Landim, C., Seo, I.: Metastability of non-reversible random walks in a potential field, the Eyring–Kramers transition rate formula (2016, submitted). arXiv:1605.01009

  14. Landim, C., Misturini, R., Tsunoda, K.: Metastability of reversible random walks in potential field. J. Stat. Phys. 160, 1449–1482 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Misturini, R.: Evolution of the ABC model among the segregated configurations in the zero temperature limit. Ann. Inst. H. Poincaré Probab. Stat. 52, 669–702 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Potts, R.B.: Mathematical investigation of some cooperative phenomena. Ph.D. Thesis, University of Oxford (1950)

  17. Slowik, M.: A note on variational representations of capacities for reversible and nonreversible Markov chains. Unpublished, Technische Universität Berlin (2012)

  18. Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54, 235–268 (1982)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    C. Landim

  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. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012-1185, USA

    I. Seo

Authors
  1. C. Landim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Seo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Seo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Landim, C., Seo, I. Metastability of Non-reversible, Mean-Field Potts Model with Three Spins. J Stat Phys 165, 693–726 (2016). https://doi.org/10.1007/s10955-016-1638-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-016-1638-1

Keywords

Search

Navigation