Current Fluctuations in the One-Dimensional Symmetric Exclusion Process with Open Boundaries

Derrida, Bernard; Douçot, Benoît; Roche, Philippe-Emmanuel

doi:10.1023/b:joss.0000022379.95508.b2

Cited by 125 publications

(224 citation statements)

References 40 publications

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“…Instead of trying to check an expression like (A.1) for the large deviation function we look for a symmetry at the level of its Legendre transform. As in [11], one knows that This is invariant under the symmetry λ → −2 log 2 − λ, implying that (A.2) is satisfied.…”

Section: Psfrag Replacementsmentioning

confidence: 96%

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Vortices in the Two-Dimensional Simple Exclusion Process

Bodineau

Derrida²,

Lebowitz

2008

J Stat Phys

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Abstract. We show that the fluctuations of the partial current in two dimensional diffusive systems are dominated by vortices leading to a different scaling from the one predicted by the hydrodynamic large deviation theory. This is supported by exact computations of the variance of partial current fluctuations for the symmetric simple exclusion process on general graphs. On a two-dimensional torus, our exact expressions are compared to the results of numerical simulations. They confirm the logarithmic dependence on the system size of the fluctuations of the partial flux. The impact of the vortices on the validity of the fluctuation relation for partial currents is also discussed in an Appendix.

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Section: Psfrag Replacementsmentioning

confidence: 96%

“…The procedure to derive (3.11) is similar to what was done in [11]. One considers all the possible moves occurring during an infinitesimal time interval dτ and their contributions to exp λQ A (τ ) .…”

Section: Current Fluctuations On a General Graphmentioning

confidence: 99%

Vortices in the Two-Dimensional Simple Exclusion Process

Bodineau

Derrida²,

Lebowitz

2008

J Stat Phys

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“…We begin with examining the simple lattice random walk case. We continue with an interacting lattice gas, namely the one dimensional exclusion process with periodic boundary conditions, for which our analytic results are somewhat less extensive, but that has in the recent past [11,12,8,29] served as a testbench for many of the ideas discussed in this introduction. In the case of the symmetric exclusion process we found that, though there is no first order dynamical phase transition, the event-counting observable mentioned above shows signs of a second order dynamical phase transition.…”

Section: Motivations and Outlinementioning

confidence: 99%

Thermodynamic Formalism for Systems with Markov Dynamics

2007

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The thermodynamic formalism allows one to access the chaotic properties of equilibrium and out-of-equilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not suitable for continuous time Markov dynamics. Here we propose another interpretation of the definition that allows us to apply the thermodynamic formalism to continuous time.We also generalize the formalism -a dynamical Gibbs ensemble constructionto a whole family of observables and their associated large deviation functions. This allows us to make the connection between the thermodynamic formalism and the observable involved in the much-studied fluctuation theorem.We illustrate our approach on various physical systems: random walks, exclusion processes, an Ising model and the contact process. In the latter cases, we identify a signature of the occurrence of dynamical phase transitions. We show that this signature can already be unraveled using the simplest dynamical ensemble one could define, based on the number of configuration changes a system has undergone over an asymptotically large time window.

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“…It is one of the few examples for which it has been possible to determine the distribution P (Q t ) both by microscopic and macroscopic approaches [4,5,8,13]. This distribution reveals that, for a long chain of length L connected at its two ends to two reservoirs at densities ρ a = 1 and ρ b = 0, the Fano factor (the ratio of the first two cumulants) is given, in the long time limit, by…”

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confidence: 99%

Universal current fluctuations in the symmetric exclusion process and other diffusive systems

Akkermans¹,

Bodineau²,

Derrida³

et al. 2013

EPL

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-We show, using the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, that the statistics of the current of the symmetric simple exclusion process (SSEP) connected to two reservoirs are the same on an arbitrary large finite domain in dimension d as in the one dimensional case. Numerical results on squares support this claim while results on cubes exhibit some discrepancy. We argue that the results of the macroscopic fluctuation theory should be recovered by increasing the size of the contacts. The generalization to other diffusive systems is straightforward.Introduction. -When a system is connected for a long time to two heat baths at unequal temperatures or to two reservoirs of particles at different densities, it reaches a nonequilibrium steady state, with a non vanishing current of heat or of particles. This current fluctuates with time and the study of its fluctuations and of its large deviations has become a central aspect in the theory of non equilibrium systems [4,5,12,16,19,20,23,24,26,27,29,34,35]. A quantity which characterizes these fluctuations is the probability P (Q t ) of observing an energy or a number of particles Q t flowing through the system during a time window t. A notorious property of these fluctuations is known as the fluctuation theorem [15,17,18,28] which establishes a general relation between P (Q t ) and P (−Q t ), starting from some time reversal symmetry of the microscopic dynamics. Apart from simple models, however, it is usually difficult to predict the whole shape of the distribution P (Q t ).

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Current Fluctuations in the One-Dimensional Symmetric Exclusion Process with Open Boundaries

Cited by 125 publications

References 40 publications

Vortices in the Two-Dimensional Simple Exclusion Process

Vortices in the Two-Dimensional Simple Exclusion Process

Thermodynamic Formalism for Systems with Markov Dynamics

Universal current fluctuations in the symmetric exclusion process and other diffusive systems

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