Results 1  10
of
42
Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process
, 2003
"... Dedicated to Michael E. Fisher on the occasion of his seventieth birthday. 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 ρa and ρb. As ρa and ρb are varied, the typical mac ..."
Abstract

Cited by 20 (7 self)
 Add to MetaCart
Dedicated to Michael E. Fisher on the occasion of his seventieth birthday. 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 ρa and ρb. As ρa and ρb are varied, the typical macroscopic steady state density profile ¯ρ(x), x ∈ [a,b], obtained in the limit N = L(b − a) → ∞, exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile ρ(x): PN({ρ(x)}) ∼ exp[−LF [a,b]({ρ(x)});ρa,ρb], 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 ρ(x). Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which F({ρ(x)}) is not convex and others for which F({ρ(x)}) has discontinuities in its second derivatives at ρ(x) = ¯ρ(x); the fluctuations near ¯ρ(x) are then nonGaussian and cannot be calculated from the large deviation function. 1
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 ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
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.
Large deviation approach to non equilibrium processes in stochastic lattice gases
, 2006
"... 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 ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
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.
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) ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
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.
Current large deviations for asymmetric exclusion processes with open boundaries
, 2006
"... Abstract. 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 JensenVaradhan ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
Abstract. 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 JensenVaradhan functional. Conjectures for generalizing the JensenVaradhan functional to open systems are also stated. 02.50.r, 05.40.a, 05.70 Ln, 82.20w 1.
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 ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
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.
TimeReversal and Entropy
, 2002
"... There is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of timereversal symmetry, and the entropy production in such processes. We explain on an elementary mathematical level the relations between entropy production, phasespace contraction and timer ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
There is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of timereversal symmetry, and the entropy production in such processes. We explain on an elementary mathematical level the relations between entropy production, phasespace contraction and timereversal starting from a deterministic dynamics. Both closed and open systems, in the transient and in the steady regime, are considered. The main result identifies under general conditions the statistical mechanical entropy production as the source term of timereversal breaking in the path space measure for the evolution of reduced variables. This provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful (in)equalities. We also discuss the Markov approximation. Important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior.
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 ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
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.
From Dynamic to Static Large Deviations in Boundary Driven Exclusion Particle Systems
, 2002
"... We consider the large deviations for the stationary measures associated to a boundary driven symmetric simple exclusion process. Starting from the large deviations for the hydrodynamics and following the Freidlin and Wentzell's strategy, we prove that the rate function is given by the quasipote ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
We consider the large deviations for the stationary measures associated to a boundary driven symmetric simple exclusion process. Starting from the large deviations for the hydrodynamics and following the Freidlin and Wentzell's strategy, we prove that the rate function is given by the quasipotential of the Freidlin and Wentzell theory.