Non Equilibrium Current Fluctuations in Stochastic Lattice Gases

Abstract

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 space-time 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 Gallavotti-Cohen theorem for the entropy production.

Vortices in the Two-Dimensional Simple Exclusion Process

“…For more general diffusive systems the hydrodynamic large deviations are governed by functionals of the type (2.3) which depend on diffusion and conductivity matrices [2]. One could extend the previous discussion to these cases and the large deviation function F h (J) would vanish as soon as h < 1.…”

“…Recently, it has been shown how to compute the large deviation function of the current in one dimensional diffusive systems [2]- [7]. The hydrodynamic large deviation theory [2,19,12], yields explicit expressions for the large deviation function as well as the cumulants of the current fluctuations (under some stability condition [5,7]).…”

“…The hydrodynamic large deviation theory [2,19,12], yields explicit expressions for the large deviation function as well as the cumulants of the current fluctuations (under some stability condition [5,7]). The same hydrodynamical approach applies in principle also to currents in higher dimension.…”

“…In section 2, we recall the hydrodynamic large deviation theory [2,6] and show the asymptotics (1.6). Although the hydrodynamic large deviation theory does not predict the correct scaling of the current deviation, the analysis of section 2 suggests that local current fluctuations are dominated by vortices.…”

Scaling Limits of Additive Functionals of Interacting Particle Systems

“…sequence of mean zero and variance ρ(1 − ρ). Following [5], we call χ(ρ) = ρ(1 − ρ) the mobility of the system. In particular, for any t > 0 fixed, the S ′ (R)-valued random variable Y n t converges in distribution, as n → ∞ to a white noise of variance χ(ρ).…”