Abstract
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.
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References
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctiation theory. Rev. Mod. Phys. 87, 593 (2015)
Bernardin, C., Olla, S.: Fourier law for a microscopic model of heat conduction. J. Stat. Phys. 121, 271–289 (2005)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
De Masi, A., Olla, S., Presutti, E.: A note on Fick’s law with phase transitions. J. Stat. Phys. 175, 203–21 (2019)
Derrida, B., Lebowitz, J.L., Speer, E.R.: Free energy functional for nonequilibrium systems: an exactly solvable case. Phys. Rev. Lett. 87, 150601 (2001)
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)
Gallavotti, G.: The phase separation line in the two-dimensional Ising model. Commun. Math. Phys. 27, 103–136 (1972)
Kipnis, C., Marchioro, C., Presutti, E.: Heat flow in an exactly solvable model Journ. Stat. Phys. 27, 65–74 (1982)
Ioffe, D., Shlosman, S., Velenik, Y.: An invariance principle to Ferrari–Spohn diffusions. Commun. Math. Phys. 336, 905–932 (2015)
Hanche-Olsen, Harald, Holden, Helge: The Kolmogorov–Riesz compactness theorem. Exp. Math. 28(4), 385–394 (2010). https://doi.org/10.1016/j.exmath.2010.03.001
Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Ann. Math. Soc. Transl. 26, 128 (1950)
Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics, vol. 123. Cambridge University Press, Cambridge (2010)
Acknowledgements
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.
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Communicated by Michael Aizenman.
Dedicated to Joel for his important contributions to the theory of phase transition and interfaces.
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De Masi, A., Merola, I. & Olla, S. Interface Fluctuations in Non Equilibrium Stationary States: The SOS Approximation. J Stat Phys 180, 414–426 (2020). https://doi.org/10.1007/s10955-019-02450-w
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DOI: https://doi.org/10.1007/s10955-019-02450-w