Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process

By Bernard Derrida, J. Lebowitz, E. Speer. 2001

217 citations recorded by the source corpus.

AbstractWe 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[−N $$F$$ ({ρ})]; $$F$$ is the large deviation functional. In contrast to equilibrium systems, for which $$F$$ eq({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 non-local behavior of $$F$$ in general SNS, where the long range correlations have been observed experimentally.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9ec99178d3d8645977a47976e7b352ccdb494d4a

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Citations in the corpus

Non Equilibrium Current Fluctuations in Stochastic Lattice Gases (2005, background): “ar X iv :c on d- m at /0 50 66 64 Keywords and phrases: Stationary non equilibrium states, Stochastic lattice gases, Current fluctuations, Gallavotti-Cohen symmetry.”
Non-equilibrium steady states: fluctuations and large deviations of the density and of the current (2007, methodology): “Derivation of (71,72): in the original derivation of (71,72) from the matrix ansatz [25, 26] the idea was to decompose the chain into L/l boxes of l sites and to sum the weights given by the matrix ansatz (48,50) over all the microscopic configurations for which the number of particles is lρ1 in the”
The large deviation approach to statistical mechanics (2008, background): “For many of these models, large deviation principles have been derived at the level of the empirical density or density field [223–227], as well as at the level of the current [228–230], whichmeasures the average number of particles moving on the lattice.”
Macroscopic fluctuation theory (2014, methodology): “In particular, the result in (Derrida et al., 2001, 2002b) for the boundary driven symmetric simple exclusion process has been obtained, in a rather straightforward way, from the Hamilton-Jacobi equation for quasi-potential in (Bertini et al., 2002).”
Stochastic interacting particle systems out of equilibrium (2007, unknown): “This has been proven in [19, 20] by combinatorial methods and in [3, 4] by the dynamical/variational approach presented here.”
Fluctuation-dissipation: Response theory in statistical physics (2008, background): “[272,167] has a nonlocal nature, as easily understood in the works of Derrida and collaborators [273,166], who independently worked out the explicit solutions of what can now be seen as specific examples of the theory of [272,167].”