The Annals of Applied Probability

Large deviation principles for nongradient weakly asymmetric stochastic lattice gases

Lorenzo Bertini, Alessandra Faggionato, and Davide Gabrielli

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We consider a lattice gas on the discrete $d$-dimensional torus $(\mathbb{Z}/N\mathbb{Z})^d$ with a generic translation invariant, finite range interaction satisfying a uniform strong mixing condition. The lattice gas performs a Kawasaki dynamics in the presence of a weak external field $E/N$. We show that, under diffusive rescaling, the hydrodynamic behavior of the lattice gas is described by a nonlinear driven diffusion equation. We then prove the associated dynamical large deviation principle. Under suitable assumptions on the external field (e.g., $E$ constant), we finally analyze the variational problem defining the quasi-potential and characterize the optimal exit trajectory. From these results we deduce the asymptotic behavior of the stationary measures of the stochastic lattice gas, which are not explicitly known. In particular, when the external field $E$ is constant, we prove a stationary large deviation principle for the empirical density and show that the rate function does not depend on $E$.

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Ann. Appl. Probab., Volume 23, Number 1 (2013), 1-65.

First available in Project Euclid: 25 January 2013

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C05: Classical dynamic and nonequilibrium statistical mechanics (general)
Secondary: 60F10: Large deviations 82C22: Interacting particle systems [See also 60K35]

Stochastic lattice gases stationary nonequilibrium states large deviations


Bertini, Lorenzo; Faggionato, Alessandra; Gabrielli, Davide. Large deviation principles for nongradient weakly asymmetric stochastic lattice gases. Ann. Appl. Probab. 23 (2013), no. 1, 1--65. doi:10.1214/11-AAP805.

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  • [1] Asselah, A. (1998). Nonreversible stationary measures for exchange processes. Ann. Appl. Probab. 8 1303–1311.
  • [2] Bahadoran, C. (2010). A quasi-potential for conservation laws with boundary conditions. Preprint. Available at arXiv:1010.3624.
  • [3] Bernardin, C. (2002). Regularity of the diffusion coefficient for lattice gas reversible under Bernoulli measures. Stochastic Process. Appl. 101 43–68.
  • [4] Bertini, L., Cirillo, E. N. M. and Olivieri, E. (1999). Renormalization-group transformations under strong mixing conditions: Gibbsianness and convergence of renormalized interactions. J. Stat. Phys. 97 831–915.
  • [5] Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G. and Landim, C. (2002). Macroscopic fluctuation theory for stationary non-equilibrium states. J. Stat. Phys. 107 635–675.
  • [6] Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G. and Landim, C. (2007). Stochastic interacting particle systems out of equilibrium. J. Stat. Mech. Theory Exp. 7 P07014, 35.
  • [7] Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G. and Landim, C. (2011). Action functional and quasi-potential for the Burgers equation in a bounded interval. Comm. Pure Appl. Math. 64 649–696.
  • [8] Bertini, L., Gabrielli, D. and Landim, C. (2009). Strong asymmetric limit of the quasi-potential of the boundary driven weakly asymmetric exclusion process. Comm. Math. Phys. 289 311–334.
  • [9] Bertini, L., Landim, C. and Mourragui, M. (2009). Dynamical large deviations for the boundary driven weakly asymmetric exclusion process. Ann. Probab. 37 2357–2403.
  • [10] Bodineau, T. and Giacomin, G. (2004). From dynamic to static large deviations in boundary driven exclusion particle systems. Stochastic Process. Appl. 110 67–81.
  • [11] Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.
  • [12] Cancrini, N. and Martinelli, F. (2000). On the spectral gap of Kawasaki dynamics under a mixing condition revisited. J. Math. Phys. 41 1391–1423.
  • [13] Enaud, C. and Derrida, B. (2004). Large deviation functional of the weakly asymmetric exclusion process. J. Stat. Phys. 114 537–562.
  • [14] Eyink, G. L., Lebowitz, J. L. and Spohn, H. (1996). Hydrodynamics and fluctuations outside of local equilibrium: Driven diffusive systems. J. Stat. Phys. 83 385–472.
  • [15] Farfan, J. (2009). Stationary large deviations of boundary driven exclusion processes. Preprint. Available at arXiv:0908.1798.
  • [16] Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical Systems, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, New York.
  • [17] Garrido, P. L., Lebowitz, J. L., Maes, C. and Spohn, H. (1990). Long-range correlations for conservative dynamics. Phys. Rev. A (3) 42 1954–1968.
  • [18] Jensen, L. H. (2000). Large deviations of the asymmetric simple exclusion process in one dimension. Ph.D. thesis, Courant Institute, New York Univ.
  • [19] Katz, S., Lebowitz, J. L. and Spohn, H. (1984). Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors. J. Stat. Phys. 34 497–537.
  • [20] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin.
  • [21] Kipnis, C., Olla, S. and Varadhan, S. R. S. (1989). Hydrodynamics and large deviation for simple exclusion processes. Comm. Pure Appl. Math. 42 115–137.
  • [22] Lu, S. L. and Yau, H.-T. (1993). Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys. 156 399–433.
  • [23] Presutti, E. (2009). Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Springer, Berlin.
  • [24] Protter, M. H. and Weinberger, H. F. (1984). Maximum Principles in Differential Equations. Springer, New York.
  • [25] Quastel, J. (1995). Large deviations from a hydrodynamic scaling limit for a nongradient system. Ann. Probab. 23 724–742.
  • [26] Quastel, J., Rezakhanlou, F. and Varadhan, S. R. S. (1999). Large deviations for the symmetric simple exclusion process in dimensions $d\geq3$. Probab. Theory Related Fields 113 1–84.
  • [27] Schmittmann, B. and Zia, R. K. P. (1995). Statistical mechanics of driven diffusive systems. In Phase Transitions and Critical Phenomena, Vol. 17 (C. Domb and J. L. Lebowitz, eds.). Academic Press, New York.
  • [28] Schonmann, R. H. and Shlosman, S. B. (1995). Complete analyticity for 2D Ising completed. Comm. Math. Phys. 170 453–482.
  • [29] Spohn, H. (1991). Large Scale Dynamics of Interacting Particles. Springer, Berlin.
  • [30] Spohn, H. and Yau, H. T. (1995). Bulk diffusivity of lattice gases close to criticality. J. Stat. Phys. 79 231–241.
  • [31] Varadhan, S. R. S. (1984). Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics 46. SIAM, Philadelphia, PA.
  • [32] Varadhan, S. R. S. (1994). Nonlinear diffusion limit for a system with nearest neighbor interactions. II. Pitman Res. Notes Math. 283 75–128.
  • [33] Varadhan, S. R. S. and Yau, H.-T. (1997). Diffusive limit of lattice gas with mixing conditions. Asian J. Math. 1 623–678.
  • [34] Yau, H.-T. (1996). Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phys. 181 367–408.