Large deviations of the empirical currents for a boundary-driven reaction diffusion model. (English) Zbl 1269.60029

Bodineau, Thierry; Lagouge, Maxime

Ann. Appl. Probab. 22, No. 6, 2282-2319 (2012).

The authors consider a model that describes reaction diffusion equations by combining the symmetric simple exclusion process to a Glauber dynamics which models the annihilation and creation of particles. A well-known result on density hydrodynamic large deviations [G. Jona-Lasinio, C. Landim and M. E. Vares, Probab. Theory Relat. Fields 97, No. 3, 339–361 (1993; Zbl 0792.60096)] is generalized, by deriving the joint large deviations of the density and of the (conservative and nonconservative) currents flowing in the system.

Reviewer: Nasir N. Ganikhodjaev (Kuantan)

MSC:

60F10	Large deviations
82C22	Interacting particle systems in time-dependent statistical mechanics

Keywords:

large deviations; interacting particle systems

Citations:

Zbl 0792.60096

References:

[1]	Bertin, E. (2006). An exactly solvable dissipative transport model. J. Phys. A 39 1539-1546. · Zbl 1088.82023
[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. · Zbl 1031.82039
[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. · Zbl 1190.82027
[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. · Zbl 1190.82027
[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. · Zbl 1187.82083
[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. · Zbl 1191.82029
[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. · Zbl 0629.60107
[10]	Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38 . Springer, Berlin. · Zbl 1177.60035
[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. · Zbl 0780.60027
[13]	Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes : Characterization and Convergence . Wiley, New York. · Zbl 0592.60049
[14]	Evans, L. C. (1998). Partial Differential Equations. Graduate Studies in Mathematics 19 . Amer. Math. Soc., Providence, RI. · Zbl 0902.35002
[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. · Zbl 1219.82115
[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. · Zbl 0792.60096
[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. · Zbl 0927.60002
[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. · Zbl 0644.76001
[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. · Zbl 1008.60106
[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. · Zbl 0928.60087
[23]	Shokef, Y. and Levine, D. (2006). Energy distribution and effective temperatures in a driven dissipative model. Phys. Rev. E 74 051111.