Journal of Statistical Physics

, Volume 171, Issue 4, pp 599–631 | Cite as

Stationary States of Boundary Driven Exclusion Processes with Nonreversible Boundary Dynamics

  • C. Erignoux
  • C. Landim
  • T. Xu


We prove a law of large numbers for the empirical density of one-dimensional, boundary driven, symmetric exclusion processes with different types of non-reversible dynamics at the boundary. The proofs rely on duality techniques.


Nonequilibrium stationary states Boundary driven interacting particle systems Hydrostatics 



We thank H. Spohn for suggesting the problem and S. Grosskinsky for fruitful discussions. C. Landim has been partially supported by FAPERJ CNE E-26/201.207/2014, by CNPq Bolsa de Produtividade em Pesquisa PQ 303538/2014-7, and by ANR-15-CE40-0020-01 LSD of the French National Research Agency.


  1. 1.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory for stationary non-equilibrium states. J. Stat. Phys. 107, 635 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory. Rev. Mod. Phys. 87, 593–636 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Derrida, B.: Non-equilibrium steady states: Fluctuations and large deviations of the density and of the current. J. Stat. Mech. Theory Exp. P07023 (2007)Google Scholar
  4. 4.
    Derrida, B., Lebowitz, J.L., Speer, E.R.: Large deviation of the density profile in the steady state of the open symmetric simple exclusion process. J. Stat. Phys. 107, 599 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Erignoux, C.: Hydrodynamic limit of boundary exclusion processes with nonreversible boundary dynamics. preprint arXiv:1712.04877 (2017)
  6. 6.
    Eyink, G., Lebowitz, J., Spohn, H.: Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models. Commun. Math. Phys. 132, 253–283 (1990)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Friedman, A.: Stochastic Differential Equations and Applications, vol. 1. Academic Press, New York (1975)zbMATHGoogle Scholar
  8. 8.
    Kipnis, C., Landim, C., Olla, S.: Macroscopic properties of a stationary nonequilibrium distribution for a nongradient interacting particle system. Ann. Inst. H. Poincaré, Prob. et Stat. 31, 191221 (1995)zbMATHGoogle Scholar
  9. 9.
    Landim, C., Olla, S., Volchan, S.: Driven tracer particle in one-dimensional symmetric simple exclusion. Commun. Math. Phys. 192, 287–307 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lawler, G.E.: Intersections of Random Walks. Modern Birkhäuser Classics. Birkhäuser, Basel (1991)CrossRefGoogle Scholar
  11. 11.
    Onsager, L.: Reciprocal relations in irreversible processes. I, II. Phys. Rev. 37, 405 (1931)ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91, 1505 (1953)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sonigo, N.: Semi-infinite TASEP with a complex boundary mechanism. J. Stat. Phys. 136, 1069–1094 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil
  2. 2.CNRS UMR 6085, Université de RouenMont-Saint-AignanFrance

Personalised recommendations