Interface Fluctuations in Non Equilibrium Stationary States: The SOS Approximation

  • Anna De MasiEmail author
  • Immacolata Merola
  • Stefano Olla


We study the 2d stationary fluctuations of the interface in the SOS approximation of the non equilibrium stationary state found in De Masi et al. (J Stat Phys 175:203–221, 2019). We prove that the interface fluctuations are of order \(N^{1/4}\), N the size of the system. We also prove that the scaling limit is a stationary Ornstein–Uhlenbeck process.


Non equilibrium stationary states Interfaces SOS model 



We thank S. Shlosman for helpful discussions. A.DM thanks very warm hospitality at the University of Paris-Dauphine where part of this work was performed. This work was partially supported by ANR-15-CE40-0020-01 grant LSD.


  1. 1.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctiation theory. Rev. Mod. Phys. 87, 593 (2015)ADSCrossRefGoogle Scholar
  2. 2.
    Bernardin, C., Olla, S.: Fourier law for a microscopic model of heat conduction. J. Stat. Phys. 121, 271–289 (2005)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)zbMATHGoogle Scholar
  4. 4.
    De Masi, A., Olla, S., Presutti, E.: A note on Fick’s law with phase transitions. J. Stat. Phys. 175, 203–21 (2019)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Derrida, B., Lebowitz, J.L., Speer, E.R.: Free energy functional for nonequilibrium systems: an exactly solvable case. Phys. Rev. Lett. 87, 150601 (2001)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    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(3/4), 599–634 (2002)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Gallavotti, G.: The phase separation line in the two-dimensional Ising model. Commun. Math. Phys. 27, 103–136 (1972)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model Journ. Stat. Phys. 27, 65–74 (1982)ADSCrossRefGoogle Scholar
  9. 9.
    Ioffe, D., Shlosman, S., Velenik, Y.: An invariance principle to Ferrari–Spohn diffusions. Commun. Math. Phys. 336, 905–932 (2015)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hanche-Olsen, Harald, Holden, Helge: The Kolmogorov–Riesz compactness theorem. Exp. Math. 28(4), 385–394 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Ann. Math. Soc. Transl. 26, 128 (1950)MathSciNetGoogle Scholar
  12. 12.
    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics, vol. 123. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversità degli studi dell’AquilaL’AquilaItaly
  2. 2.CNRS, CEREMADE, Université Paris-Dauphine, PSL Research UniversityParisFrance

Personalised recommendations