## The Annals of Applied Probability

### Large deviations of the empirical currents for a boundary-driven reaction diffusion model

#### Abstract

We derive a large deviation principle for the empirical currents of lattice gas dynamics which combine a fast stirring mechanism (Symmetric Simple Exclusion Process) and creation/annihilation mechanisms (Glauber dynamics). Previous results on the density large deviations can be recovered from this general large deviation principle. The contribution of external driving forces due to reservoirs at the boundary of the system is also taken into account.

#### Article information

Source
Ann. Appl. Probab., Volume 22, Number 6 (2012), 2282-2319.

Dates
First available in Project Euclid: 23 November 2012

https://projecteuclid.org/euclid.aoap/1353695954

Digital Object Identifier
doi:10.1214/11-AAP826

Mathematical Reviews number (MathSciNet)
MR3024969

Zentralblatt MATH identifier
1269.60029

#### Citation

Bodineau, Thierry; Lagouge, Maxime. Large deviations of the empirical currents for a boundary-driven reaction diffusion model. Ann. Appl. Probab. 22 (2012), no. 6, 2282--2319. doi:10.1214/11-AAP826. https://projecteuclid.org/euclid.aoap/1353695954

#### References

• [1] Bertin, E. (2006). An exactly solvable dissipative transport model. J. Phys. A 39 1539–1546.
• [2] Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G. and Landim, C. (2003). Large deviations for the boundary driven symmetric simple exclusion process. Math. Phys. Anal. Geom. 6 231–267.
• [3] Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G. and Landim, C. (2007). Large deviations of the empirical current in interacting particle systems. Theory Probab. Appl. 51 2–27.
• [4] Bertini, L., De Sole, A., Gabrielli, D., Jona Lasinio, G. and Landim, C. (2007). Stochastic interacting particle systems out of equilibrium. J. Stat. Mech. P07014.
• [5] Bertini, L., Landim, C. and Mourragui, M. (2009). Dynamical large deviations for the boundary driven weakly asymmetric exclusion process. Ann. Probab. 37 2357–2403.
• [6] Bodineau, T. and Derrida, B. (2007). Cumulants and large deviations of the current through non-equilibrium steady states. C. R. Physique 8 540–555.
• [7] Bodineau, T. and Lagouge, M. (2010). Current large deviations in a driven dissipative model. J. Stat. Phys. 139 201–218.
• [8] De Masi, A., Ferrari, P. A. and Lebowitz, J. L. (1985). Rigorous derivation of reaction-diffusion equations with fluctuations. Phys. Rev. Lett. 55 1947–1949.
• [9] De Masi, A., Ferrari, P. A. and Lebowitz, J. L. (1986). Reaction–diffusion equations for interacting particle systems. J. Stat. Phys. 44 589–644.
• [10] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin.
• [11] Derrida, B. (2007). Non-equilibrium steady states: Fluctuations and large deviations of the density and of the current. J. Stat. Mech. Theory Exp. 7 P07023, 45 pp. (electronic).
• [12] Donsker, M. D. and Varadhan, S. R. S. (1989). Large deviations from a hydrodynamic scaling limit. Comm. Pure Appl. Math. 42 243–270.
• [13] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
• [14] Evans, L. C. (1998). Partial Differential Equations. Graduate Studies in Mathematics 19. Amer. Math. Soc., Providence, RI.
• [15] Farfan Vargas, J. S., Landim, C. and Mourragui, M. (2011). Hydrodynamic behavior of boundary driven exclusion processes in dimension $d>1$. Stochastic Process. Appl. 121 725–758.
• [16] Gallavotti, G. (2007). Fluctuation relation, fluctuation theorem, thermostats and entropy creation in non equilibrium statistical physics. C. R. Acad. Sci. Ser. B 8 486–494.
• [17] Jona-Lasinio, G., Landim, C. and Vares, M. E. (1993). Large deviations for a reaction diffusion model. Probab. Theory Related Fields 97 339–361.
• [18] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin.
• [19] 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.
• [20] Landim, C., Mourragui, M. and Sellami, S. (2002). Hydrodynamic limit for a nongradient interacting particle system with stochastic reservoirs. Theory Probab. Appl. 45 604–623.
• [21] Levanony, D. and Levine, D. (2006). Correlation and response in a driven dissipative model. Phys. Rev. E 73 055102(R).
• [22] Quastel, J., Rezakhanlou, F. and Varadhan, S. R. S. (1999). Large deviations for the symmetric simple exclusion process in dimensions $d\geq 3$. Probab. Theory Related Fields 113 1–84.
• [23] Shokef, Y. and Levine, D. (2006). Energy distribution and effective temperatures in a driven dissipative model. Phys. Rev. E 74 051111.