Cumulants and large deviations of the current through non-equilibrium steady states

Abstract Using a generalisation of detailed balance for systems maintained out of equilibrium by contact with 2 reservoirs at unequal temperatures or at unequal densities, one can recover the fluctuation theorem for the large deviation function of the current. For large diffusive systems, we show how the large deviation function of the current can be computed using a simple additivity principle. The validity of this additivity principle and the occurrence of phase transitions are discussed in the framework of the macroscopic fluctuation theory. To cite this article: T. Bodineau, B. Derrida, C. R. Physique 8 (2007).

[1]  Evans,et al.  Probability of second law violations in shearing steady states. , 1993, Physical review letters.

[2]  B. Derrida,et al.  Exact solution of a 1d asymmetric exclusion model using a matrix formulation , 1993 .

[3]  Robin Stinchcombe,et al.  Exact joint density-current probability function for the asymmetric exclusion process. , 2004, Physical review letters.

[4]  B Derrida,et al.  Exact free energy functional for a driven diffusive open stationary nonequilibrium system. , 2002, Physical review letters.

[5]  C. Landim,et al.  Scaling Limits of Interacting Particle Systems , 1998 .

[6]  E. R. Speer,et al.  Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process , 2002 .

[7]  Large Deviations for the Boundary Driven Symmetric Simple Exclusion Process , 2003, cond-mat/0307280.

[8]  J. Lebowitz,et al.  Fourier's Law: a Challenge for Theorists , 2000, math-ph/0002052.

[9]  Exact probability function for bulk density and current in the asymmetric exclusion process. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Debra J. Searles,et al.  The Fluctuation Theorem , 2002 .

[11]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[12]  C. Landim,et al.  Fluctuations in stationary nonequilibrium states of irreversible processes. , 2001, Physical review letters.

[13]  C. Landim,et al.  Macroscopic Fluctuation Theory for Stationary Non-Equilibrium States , 2001, cond-mat/0108040.

[14]  Jorge Kurchan,et al.  Fluctuation theorem for stochastic dynamics , 1998 .

[15]  J. L. Lebowitz,et al.  Exact Large Deviation Function in the Asymmetric Exclusion Process , 1998 .

[16]  David Ruelle,et al.  Conversations on nonequilibrium physics with an extraterrestrial , 2004 .

[17]  Stefano Olla,et al.  Hydrodynamics and large deviation for simple exclusion processes , 1989 .

[18]  Lee,et al.  Universal statistics of transport in disordered conductors. , 1994, Physical review. B, Condensed matter.

[19]  G. Gallavotti.  Entropy production and thermodynamics of nonequilibrium stationary states: a point of view. , 2004, Chaos.

[20]  D. Ruelle.  Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics , 1998, chao-dyn/9812032.

[21]  B. Derrida,et al.  Large Deviation of the Density Profile in the Symmetric Simple Exclusion Process , 2001 .

[22]  C. Maes.  On the origin and the use of fluctuation relations for the entropy , 2003 .

[23]  B. Derrida,et al.  Free energy functional for nonequilibrium systems: an exactly solvable case. , 2001, Physical review letters.

[24]  J. Eckmann,et al.  Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures , 1998, chao-dyn/9804001.

[25]  Current Fluctuations in the One-Dimensional Symmetric Exclusion Process with Open Boundaries , 2003, cond-mat/0310453.

[26]  J. Farago.  Power fluctuations in stochastic models of dissipative systems , 2004, cond-mat/0409645.

[27]  B. Derrida,et al.  Current Large Deviations for Asymmetric Exclusion Processes with Open Boundaries , 2005, cond-mat/0509179.

[28]  B. Derrida,et al.  Current fluctuations in nonequilibrium diffusive systems: an additivity principle. , 2004, Physical review letters.

[29]  A. Jordan,et al.  Stochastic path integral formulation of full counting statistics. , 2002, Physical review letters.

[30]  Thermal conduction in classical low-dimensional lattices , 2001, cond-mat/0112193.

[31]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[32]  E. Cohen,et al.  Dynamical ensembles in stationary states , 1995, chao-dyn/9501015.

[33]  Lorenzo Bertini,et al.  Large Deviations for a Stochastic Model of Heat Flow , 2005 .

[34]  B Derrida,et al.  Distribution of current in nonequilibrium diffusive systems and phase transitions. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[36]  E. Domany,et al.  Phase transitions in an exactly soluble one-dimensional exclusion process , 1993, cond-mat/9303038.

[37]  Breakdown of Gallavotti-Cohen symmetry for stochastic dynamics , 2005, cond-mat/0512159.

[38]  Large Deviations in Weakly Interacting Boundary Driven Lattice Gases , 2004, cond-mat/0404358.

[39]  Y. Blanter,et al.  Shot noise in mesoscopic conductors , 1999, cond-mat/9910158.

[40]  L. Bertini,et al.  Non Equilibrium Current Fluctuations in Stochastic Lattice Gases , 2006 .

[41]  Fluctuation statistics in networks: A stochastic path integral approach , 2004, cond-mat/0401650.

[42]  Bernard Derrida,et al.  Universal Large-Deviation Function of the Kardar–Parisi–Zhang Equation in One Dimension , 1999 .

[43]  T. Liggett,et al.  Stochastic Interacting Systems: Contact, Voter and Exclusion Processes , 1999 .

[44]  The Fluctuation Theorem as a Gibbs Property , 1998, math-ph/9812015.

[45]  L Bertini,et al.  Current fluctuations in stochastic lattice gases. , 2005, Physical review letters.

[46]  Giovanni Gallavotti,et al.  Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem , 1995, chao-dyn/9506006.

[47]  P. Visco.  Work fluctuations for a Brownian particle between two thermostats , 2006, cond-mat/0605069.

[48]  P. Ferrari,et al.  A remark on the hydrodynamics of the zero-range processes , 1984 .

[49]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

[50]  Herbert Spohn,et al.  Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors , 1984 .

[51]  J. Lebowitz,et al.  A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics , 1998, cond-mat/9811220.

[52]  Current fluctuations in the zero-range process with open boundaries , 2005, cond-mat/0504711.