Advertisement

Quenched convergence and strong local equilibrium for asymmetric zero-range process with site disorder

  • C. BahadoranEmail author
  • T. Mountford
  • K. Ravishankar
  • E. Saada
Article
  • 18 Downloads

Abstract

We study asymmetric zero-range processes on \(\mathbb {Z}\) with nearest-neighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. We prove quenched strong local equilibrium at subcritical and critical hydrodynamic densities, and dynamic local loss of mass at supercritical hydrodynamic densities. Our results do not assume starting from local Gibbs states. As byproducts of these results, we prove convergence of the process from given initial configurations with an asymptotic density of particles to the left of the origin. In particular, we relax the weak convexity assumption of Bahadoran et al. (Braz J Probab Stat 29(2):313–335, 2015; Ann Inst Henri Poincaré Probab Stat 53(2):766–801, 2017) for the escape of mass property.

Keywords

Asymmetric zero-range process Site disorder Phase transition Condensation Hydrodynamic limit Strong local equilibrium Large-time convergence 

Mathematics Subject Classification

60K35 82C22 

Notes

Acknowledgements

We thank Gunter Schütz for many interesting discussions. This work was partially supported by laboratoire MAP5, Grants ANR-15-CE40-0020-02 and ANR-14-CE25-0011, LabEx CARMIN (ANR-10-LABX-59-01), Simons Foundation Collaboration Grant 281207 awarded to K. Ravishankar. We thank Universités Clermont Auvergne and Paris Descartes for hospitality. This work was partially carried out during C.B.’s 2017–2018 délégation CNRS, whose support is acknowledged. Part of it was done during the authors’ stay at the Institut Henri Poincaré (UMS 5208 CNRS-Sorbonne Université) - Centre Emile Borel for the trimester “Stochastic Dynamics Out of Equilibrium”, and during the authors’ stay at NYU Shanghai. The authors thank these institutions for hospitality and support.

References

  1. 1.
    Andjel, E.D.: Invariant measures for the zero-range process. Ann. Probab. 10, 525–547 (1982)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andjel, E.D., Kipnis, C.: Derivation of the hydrodynamical equation for the zero-range interaction process. Ann. Probab. 12(2), 325–334 (1984)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Andjel, E.D., Vares, M.E.: Hydrodynamic equations for attractive particle systems on \({\mathbb{Z}}\). J. Stat. Phys. 47(1–2), 265–288 (1987)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Andjel, E.D., Vares, M.E.: Correction to: Hydrodynamic equations for attractive particle systems on \({\mathbb{Z}}\). J. Stat. Phys. 113(1–2), 379–380 (2003)Google Scholar
  5. 5.
    Andjel, E., Ferrari, P.A., Guiol, H., Landim, C.: Convergence to the maximal invariant measure for a zero-range process with random rates. Stoch. Process. Appl. 90, 67–81 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Constructive Euler hydrodynamics for one-dimensional attractive particle systems. Hal 01447200. In: Sidoravicius, V. (ed.) Sojourns in Probability and Statistical Physics, Springer (2019). (to appear). arXiv:1701.07994
  7. 7.
    Bahadoran, C., Mountford, T.S.: Convergence and local equilibrium for the one-dimensional nonzero mean exclusion process. Probab. Theory Relat. Fields 136(3), 341–362 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Supercriticality conditions for the asymmetric zero-range process with sitewise disorder. Braz. J. Probab. Stat. 29(2), 313–335 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Supercritical behavior of zero-range process with sitewise disorder. Ann. Inst. Henri Poincaré Probab. Stat. 53(2), 766–801 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Hydrodynamics in a condensation regime: the disordered asymmetric zero-range process. Ann. Probab. (to appear). arXiv:1801.01654
  11. 11.
    Benjamini, I., Ferrari, P.A., Landim, C.: Asymmetric conservative processes with random rates. Stoch. Process. Appl. 61(2), 181–204 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Evans, M.R.: Bose–Einstein condensation in disordered exclusion models and relation to traffic flow. Europhys. Lett. 36(1), 13 (1996)Google Scholar
  13. 13.
    Ferrari, P., Krug, J.: Phase transitions in driven diffusive systems with random rates. J. Phys. A. 29, L:465–471 (1996)zbMATHGoogle Scholar
  14. 14.
    Harris, T.E.: Nearest-neighbour Markov interaction processes on multidimensional lattices. Adv. Math. 9, 66–89 (1972)zbMATHGoogle Scholar
  15. 15.
    Kipnis, C., Landim, C.: Scaling Limits for Interacting Particle Systems. Springer, Berlin (1999)zbMATHGoogle Scholar
  16. 16.
    Kosygina, E.: The behaviour of specific entropy in the hydrodynamic scaling limit. Ann. Probab. 29(3), 1086–1110 (2001)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Landim, C.: Conservation of local equilibrium for attractive particle systems on \({\mathbb{Z}}\). Ann. Probab. 21(4), 1782–1808 (1993)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Landim, C.: Hydrodynamical limit for space inhomogeneous one-dimensional totally asymmetric zero-range processes. Ann. Probab. 24, 599–638 (1996)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Liggett, T.M.: Interacting Particle Systems. Reprint of the 1985 Original. Classics in Mathematics. Springer, Berlin (2005)zbMATHGoogle Scholar
  20. 20.
    Pardoux, E.: Processus de Markov et Applications. Dunod, Paris (2007)zbMATHGoogle Scholar
  21. 21.
    Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on \({\mathbb{Z}}\). Commun. Math. Phys. 140(3), 417–448 (1991)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Seppäläinen, T.: Existence of hydrodynamics for the totally asymmetric simple \(K\)-exclusion process. Ann. Probab. 27(1), 361–415 (1999)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Seppäläinen, T., Krug, J.: Hydrodynamics and platoon formation for a totally asymmetric exclusion process with particlewise disorder. J. Stat. Phys. 95, 525–567 (1999)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Blaise PascalUniversité Clermont AuvergneAubièreFrance
  2. 2.Institut de MathématiquesÉcole Polytechnique FédéraleLausanneSwitzerland
  3. 3.NYU-ECNU Institute of Mathematical Sciences at NYU ShanghaiShanghaiChina
  4. 4.CNRS, UMR 8145, MAP5Université Paris DescartesParis cedex 06France

Personalised recommendations