We study the large deviation functional of the current for the Weakly Asymmetric Simple Exclusion Process in contact with two reservoirs. We compare this functional in the large drift limit to the one of the Totally Asymmetric Simple Exclusion Process, in particular to the Jensen-Varadhan functional. Conjectures for generalizing the Jensen-Varadhan functional to open systems are also stated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Bahadoran, Hydrodynamics and hydrostatics for a class of asymmetric particle systems with open boundaries. preprint (2005).
C. Bardos, A. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions. Comm. Partial Differential Equations 4(9):1017–1034 (1979).
O. Benois, R. Esposito, R. Marra, and M. Mourragui, Hydrodynamics of a driven lattice gas with open boundaries: the asymmetric simple exclusion. Markov Process Relat. Fields 10(1):89–112 (2004).
L. Bertini, A. De Sole, D. Gabrielli, G. Jona–Lasinio, and C. Landim, Macroscopic fluctuation theory for stationary non equilibrium states. J. Stat. Phys. 107:635–675 (2002).
L. Bertini, A. De Sole, D. Gabrielli, G. Jona–Lasinio, and C. Landim, Large deviations for the boundary driven symmetric simple exclusion process. Math. Phys. Anal. Geometry 6:231–267 (2003).
L. Bertini, A. De Sole, D. Gabrielli, G. Jona–Lasinio, and C. Landim, Current fluctuations in stochastic lattice gases. Phys. Rev. Lett. 94:030601 (2005).
L. Bertini, A. De Sole, D. Gabrielli, G. Jona–Lasinio, and C. Landim, Non equilibrium current fluctuations in stochastic lattice gases. preprint, cond-mat/0506664 (2005).
T. Bodineau and B. Derrida, Current fluctuations in non-equilibrium diffusive systems: an additivity principle. Phys. Rev. Lett. 92:180601 (2004).
T. Bodineau and B. Derrida, Distribution of current in non-equilibrium diffusive systems and phase transitions. Phys. Rev. E 72: 0661100 (2005).
B. Derrida and C. Appert, Universal large deviation function of the Kardar-Parisi-Zhang equation in one dimension. J. Stat. Phys. 94:1–30 (1999).
B. Derrida, B. Douçot, and P.-E. Roche, Current fluctuations in the one dimensional symmetric exclusion process with open boundaries. J. Stat. Phys. 115:717–748 (2004).
B. Derrida, C. Enaud, C. Landim, and S. Olla, Fluctuations in the weakly asymmetric exclusion process with open boundary conditions. J. Stat. Phys. 118(5–6):795–811 (2005).
B. Derrida, C. Enaud, and J. L. Lebowitz, The asymmetric exclusion process and Brownian excursions. J. Stat. Phys. 115:365–382 (2004).
B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a 1d asymmetric exclusion model using a matrix formulation. J. Phys. A 26:1493–1517 (1993).
B. Derrida and J. L. Lebowitz, Exact large deviation function in the asymmetric exclusion process. Phys. Rev. Lett. 80:209–213 (1998).
B. Derrida, J. L. Lebowitz, and E. R. Speer, Exact free energy functional for a driven diffusive open stationary nonequilibrium system. Phys. Rev. Lett. 89:030601 (2002).
B. Derrida, J. L. Lebowitz, and E. R. Speer, Exact large deviation functional of a stationary open driven diffusive system: The asymmetric exclusion process. J. Stat. Phys. 110, 775–810 (2003).
M. Depken and R. Stinchcombe, Exact joint density-current probability function for the asymmetric exclusion process. Phys. Rev. Lett. 93:040602 (2004).
M. Depken and R. Stinchcombe, Exact probability function for bulk density and current in the asymmetric exclusion process. Phys. Rev. E 71:036120 (2005).
C. Enaud and B. Derrida, Large deviation functional of the weakly asymmetric exclusion process. J. Stat. Phys. 114:537–562 (2004).
G. Eyink, J. L. Lebowitz, and H. Spohn, Hydrodynamics of stationary nonequilibrium states for some stochastic lattice gas models. Comm. Math. Phys. 132(1):253–283 (1990).
G. Eyink, J. L. Lebowitz, and H. Spohn, Lattice gas models in contact with stochastic reservoirs: Local equilibrium and relaxation to the steady state. Comm. Math. Phys. 140(1):119–131 (1991).
M. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, 1998.
L. Jensen, The asymmetric exclusion process in one dimension, Ph.D. dissertation, New York Univ., New York, 2000; S. R. Varadhan, Large deviations for the asymmetric simple exclusion process. Stochastic Analysis on Large Scale Interacting Systems, pp. 1–27, Adv. Stud. Pure Math., Vol. 39, Math. Soc. Japan, Tokyo, 2004.
A. N. Jordan, E. V. Sukhorukov, and S. Pilgram, Fluctuation statistics in networks: A stochastic path integral approach. J. Math. Phys. 45:4386–4417 (2004).
C. Kipnis, S. Olla, and C. Landim, Macroscopic properties of a stationary non-equilibrium distribution for a non-gradient interacting particle system, Ann. Inst. H. Poincaré Probab. Statist. 31(1):191–221 (1995).
C. Kipnis, S. Olla, and S. R. Varadhan, Hydrodynamics and large deviations for simple exclusion processes. Commun. Pure Appl. Math. 42:115–137 (1989).
C. Kipnis and C. Landim, Scaling limits of interacting particle systems. Grund. fur Math. Wissen. 320 (Springer 1999).
J. Krug, Boundary-induced phase-transitions in driven diffusive systems. Phys. Rev. Lett. 67:1882–1885 (1991).
C. Landim, M. Mourragui, and S. Sellami, Hydrodynamic limit for a nongradient interacting particle system with stochastic reservoirs. Theory Probab. Appl. 45(4):604–623 (2002).
T. Liggett, Stochastic interacting systems: Contact, voter and exclusion processes. Grund. fur Math. Wissen. 324 (Springer 1999).
S. Pilgram, A. N. Jordan, E. V. Sukhorukov, and M. Büttiker, Stochastic path integral formulation of full counting statistics Phys. Rev. Lett. 90, 206801 (2003).
F. Rezakhanlou, Hydrodynamic limit for attractive particle systems on Z d. Comm. Math. Phys. 140(3):417–448 (1991).
G. Schütz, Exactly solvable models for many-body systems far from equilibri. In: C. Domb and J. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 19, pp. 1–251, Academic Press, London, 2000.
G. Schütz and E. Domany, Phase transitions in an exactly soluble one-dimensional asymmetric exclusion model. J. Stat. Phys. 72:277–296 (1993).
D. Serre, Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, Cambridge, 1999.
D. Serre, Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-Boundary Value Problems, Cambridge University Press, Cambridge, 2000.
H. Spohn, Large Scale Dynamics of Interacting Particles, Springer-Verlag, Berlin, 1991.
Author information
Authors and Affiliations
Corresponding author
Additional information
PACS: 02.50.-r, 05.40.-a, 05.70 Ln, 82.20-w
Rights and permissions
About this article
Cite this article
Bodineau, T., Derrida, B. Current Large Deviations for Asymmetric Exclusion Processes with Open Boundaries. J Stat Phys 123, 277–300 (2006). https://doi.org/10.1007/s10955-006-9048-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-006-9048-4