Communications in Mathematical Physics

, Volume 330, Issue 1, pp 1–32 | Cite as

Metastability for a Non-reversible Dynamics: The Evolution of the Condensate in Totally Asymmetric Zero Range Processes

  • C. LandimEmail author


It has been observed (Evans in Braz J Phys 30:42–57, 2000; Jeon et al. in Ann Probab 28:1162–1194, 2000) that some zero-range processes exhibit condensation, a macroscopic fraction of particles concentrates on one single site. We examined in (Beltrán and Landim in Probab Theory Relat Fields 152:781–807, 2012) the asymptotic evolution of the condensate in the case where the dynamics is reversible, the number of sites is fixed, and the total number of particles diverges. We proved in that paper that in an appropriate time-scale the condensate evolves according to a symmetric random walk whose transition rates are proportional to the capacities of the underlying random walk. In this article, we extend this result to the condensing totally asymmetric zero-range process, a non-reversible dynamics.


Random Walk Sector Condition Dirichlet Form Metastable Behavior Jump Rate 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Armendáriz I., Loulakis M.: Thermodynamic limit for the invariant measures in supercritical zero range processes. Probab. Theory Relat. Fields 145, 175–188 (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Armendáriz I., Loulakis M.: Conditional distribution of heavy tailed random variables on large deviations of their sum. Stoch. Proc. Appl. 121, 1138–1147 (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Armendáriz I., Großkinsky S., Loulakis M.: Zero range condensation at criticality. Stoch. Process. Appl. 123, 346–3496 (2013)Google Scholar
  4. 4.
    Beltrán J., Landim C.: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys. 140, 1065–1114 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Beltrán J., Landim C.: Metastability of reversible condensed zero range processes on a finite set. Probab. Theory Relat. Fields 152, 781–807 (2012)CrossRefzbMATHGoogle Scholar
  6. 6.
    Beltrán J., Landim C.: Metastability of reversible finite state Markov processes. Stoch. Proc. Appl. 121, 1633–1677 (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Beltrán, J., Landim, C.: Tunneling of the Kawasaki dynamics at low temperatures in two dimensions. To appear in Ann. Inst. H. Poincaré, Probab. Statist. (2014)Google Scholar
  8. 8.
    Beltrán J., Landim C.: Tunneling and metastability of continuous time Markov chains II, the nonreversible case. J. Stat. Phys. 149, 598–618 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Beltrán, J., Landim, C.: A martingale approach to metastability. To appear in Probab. Theory Related Fields (2014)Google Scholar
  10. 10.
    Bianchi, A., Gaudillière, A.: Metastable states, quasi-stationary and soft measures, mixing time asymptotics via variational principles. arXiv:1103.1143 (2011)
  11. 11.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Bovier A., Eckhoff M., Gayrard V., Klein M.: Metastability and low lying spectra in reversible Markov chains. Commun. Math. Phys. 228, 219–255 (2002)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Cassandro M., Galves A., Olivieri E., Vares M.E.: Metastable behavior of stochastic dynamics: a pathwise approach. J. Stat. Phys. 35, 603–634 (1984)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Doyle, P.: Energy for Markov Chains. Preprint (1994)
  15. 15.
    Evans M.R.: Phase transitions in one-dimensional nonequilibrium systems. Braz. J. Phys. 30, 42–57 (2000)ADSCrossRefGoogle Scholar
  16. 16.
    Evans M.R., Hanney T.: Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A 38(19), R195–R240 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Ferrari P.A., Landim C., Sisko V.V.: Condensation for a fixed number of independent random variables. J. Stat. Phys. 128, 1153–1158 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Gaudillière, A.: Condenser physics applied to Markov chains: A brief introduction to potential theory. Online
  19. 19.
    Gaudillière A., Landim C.: A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probab. Theory Relat. Fields 158, 55–89 (2014)CrossRefzbMATHGoogle Scholar
  20. 20.
    Gois, B., Landim, C.: Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus. To appear in Ann. Probab. (2014)Google Scholar
  21. 21.
    Godrèche C., Luck J.M.: Dynamics of the condensate in zero-range processes. J. Phys. A 38, 7215–7237 (2005)ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    Großkinsky S., Schütz G.M., Spohn H.: Condensation in the zero range process: stationary and dynamical properties. J. Stat. Phys. 113, 389–410 (2003)CrossRefzbMATHGoogle Scholar
  23. 23.
    Jara M., Landim C., Teixeira A.: Quenched scaling limits of trap models. Ann. Probab. 39, 176–223 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Jara, M., Landim, C., Teixeira, A.: Universality of trap models in the ergodic time scale. To appear in Annals of Probability (2014)Google Scholar
  25. 25.
    Jeon I., March P., Pittel B.: Size of the largest cluster under zero-range invariant measures. Ann. Probab. 28, 1162–1194 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov processes. Time symmetry and martingale approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 345. Springer, Heidelberg, (2012)Google Scholar
  27. 27.
    Lacoin, H., Teixeira, A.: A Mathematical Perspective on Metastable Wetting. arXiv:1312.7732 (2013)
  28. 28.
    Landim, C.: A Topology for Limits of Markov Chains. arXiv:1310.3646 (2013)
  29. 29.
    Olivieri, E., Vares, M.E.: Large Deviations and Metastability. Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge (2005)Google Scholar
  30. 30.
    Slowik, M.: A Note on Variational Representations of Capacities for Reversible and Non-reversible Markov Chains. Preprint (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil
  2. 2.CNRS UMR 6085Université de RouenSaint-Étienne-du-RouvrayFrance

Personalised recommendations