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78
Stochastic interacting particle systems out of equilibrium
 J. Stat. Mech.
, 2007
"... Abstract This paper provides an introduction to some stochastic models of lattice gases out of equilibrium and a discussion of results of various kinds obtained in recent years. Although these models are different in their microscopic features, a unified picture is emerging at the macroscopic level ..."
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Cited by 37 (5 self)
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Abstract This paper provides an introduction to some stochastic models of lattice gases out of equilibrium and a discussion of results of various kinds obtained in recent years. Although these models are different in their microscopic features, a unified picture is emerging at the macroscopic level, applicable, in our view, to real phenomena where diffusion is the dominating physical mechanism. We rely mainly on an approach developed by the authors based on the study of dynamical large fluctuations in stationary states of open systems. The outcome of this approach is a theory connecting the non equilibrium thermodynamics to the transport coefficients via a variational principle. This leads ultimately to a functional derivative equation of HamiltonJacobi type for the non equilibrium free energy in which local thermodynamic variables are the independent arguments. In the first part of the paper we give a detailed introduction to the microscopic dynamics considered, while the second part, devoted to the macroscopic properties, illustrates many consequences of the HamiltonJacobi equation. In both parts several novelties are included.
Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process.
 J. Statist. Phys.
, 2003
"... We consider the asymmetric exclusion process (ASEP) in one dimension on sites i=1,..., N, in contact at sites i=1 and i=N with infinite particle reservoirs at densities r a and r b . As r a and r b are varied, the typical macroscopic steady state density profile r(x), x ¥ [a, b], obtained in the li ..."
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Cited by 34 (10 self)
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We consider the asymmetric exclusion process (ASEP) in one dimension on sites i=1,..., N, in contact at sites i=1 and i=N with infinite particle reservoirs at densities r a and r b . As r a and r b are varied, the typical macroscopic steady state density profile r(x), x ¥ [a, b], obtained in the limit N=L(b − a) Q ., exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile r(x): [a, b] ({r(x)}; r a , r b )], so that F is the large deviation functional, a quantity similar to the free energy of equilibrium systems. We find, as in the symmetric, purely diffusive case q=1 (treated in an earlier work), that F is in general a nonlocal functional of r(x). Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which F({r(x)}) is not convex and others for which F({r(x)}) has discontinuities in its second derivatives at r(x)=r(x). In the latter ranges the fluctuations of order 1/`N in the density profile near r(x) are then nonGaussian and cannot be calculated from the large deviation function. KEY WORDS: Large deviations; asymmetric simple exclusion process; open system; stationary nonequilibrium state.
Large deviation of the density profile in the steady state of the open symmetric simple exclusion process
 J. Statist. Phys
, 2002
"... Abstract We consider an open one dimensional lattice gas on sites i = 1,...,N, with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoir ..."
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Cited by 33 (7 self)
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Abstract We consider an open one dimensional lattice gas on sites i = 1,...,N, with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when N → ∞. The probability of microscopic configurations corresponding to some other profile ρ(x), x = i/N, has the asymptotic form exp[−NF({ρ})]; F is the large deviation functional. In contrast to equilibrium systems, for which Feq({ρ}) is just the integral of the appropriately normalized local free energy density, the F we find here for the nonequilibrium system is a nonlocal function of ρ. This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar nonlocal behavior of F in general SNS, where the long range correlations have been observed experimentally. Key words: Large deviations, symmetric simple exclusion process, open system, stationary nonequilibrium state.
Non equilibrium current fluctuations in stochastic lattice gases
 J. STAT. PHYS
, 2006
"... We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles. More precisely, we establish a large deviation principle for a spacetime fluctuation j of the empirical current with a rate functional I(j) ..."
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Cited by 31 (6 self)
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We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles. More precisely, we establish a large deviation principle for a spacetime fluctuation j of the empirical current with a rate functional I(j). We then estimate the probability of a fluctuation of the average current over a large time interval; this probability can be obtained by solving a variational problem for the functional I. We discuss several possible scenarios, interpreted as dynamical phase transitions, for this variational problem. They actually occur in specific models. We finally discuss the time reversal properties of I and derive a fluctuation relationship akin to the GallavottiCohen theorem for the entropy production.
Large deviation approach to non equilibrium processes in stochastic lattice gases.
 Bull. Braz. Math. Soc., New Series
, 2006
"... Abstract. We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of c ..."
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Cited by 30 (3 self)
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Abstract. We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.
FluctuationDissipation: Response Theory in Statistical Physics
 PHYSICS REPORTS 461 (2008) 111195
, 2008
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Large deviations for the boundary driven symmetric simple exclusion process
, 2003
"... The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in non–equilibrium, namely for non reversible systems. In this paper we consider a simple example of a non–equilibrium situation, the symmetric simple exclusion ..."
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Cited by 25 (3 self)
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The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in non–equilibrium, namely for non reversible systems. In this paper we consider a simple example of a non–equilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the probability of fluctuations from the solutions of the hydrodynamic equation. The so called quasi potential, which measures the cost of a fluctuation from the stationary state, is then defined by a variational problem for the dynamical large deviation rate function. By characterizing the optimal path, we prove that the quasi potential can also be obtained from a static variational problem introduced by Derrida, Lebowitz, and Speer.
Current fluctuations of the one dimensional symmetric simple exclusion process with step initial condition
 J. Stat. Phys
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Large deviation functional of the weakly asymmetric exclusion process
"... Abstract We obtain the large deviation functional of a density profile for the asymmetric exclusion process of L sites with open boundary conditions when the asymmetry scales like 1 L. We recover as limiting cases the expressions derived recently for the symmetric (SSEP) and the asymmetric (ASEP) ca ..."
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Cited by 24 (8 self)
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Abstract We obtain the large deviation functional of a density profile for the asymmetric exclusion process of L sites with open boundary conditions when the asymmetry scales like 1 L. We recover as limiting cases the expressions derived recently for the symmetric (SSEP) and the asymmetric (ASEP) cases. In the ASEP limit, the non linear differential equation one needs to solve can be analysed by a method which resembles the WKB method. Key words: Large deviations, asymmetric simple exclusion process, open system, stationary nonequilibrium state.