HYDRODYNAMICS AND FLUCTUATIONS OUTSIDE OF LOCAL EQUILIBRIUM - DRIVEN DIFFUSIVE SYSTEMS

We derive hydrodynamic equations for systems not in local thermodynami c equilibrium, that is, where the local stationary measures are ''non- Gibbsian'' and do not satisfy detailed balance with respect to the mic roscopic dynamics. As a main example we consider the driven diffusive systems (DDS), such as electrical conductors in an applied field with diffusion of charge carriers. In such systems, the hydrodynamic descri ption is provided by a nonlinear drift-diffusion equation, which we de rive by a microscopic method of nonequilibrium distributions. The form al derivation yields a Green-Kubo formula for the bulk diffusion matri x and microscopic prescriptions for the drift velocity and ''nonequili brium entropy'' as functions of charge density. Properties of the hydr odynamic equations are established, including an ''H-theorem'' on incr ease of the thermodynamic potential, or ''entropy,'' describing approa ch to the homogeneous steady state. The results are shown to be consis tent with the derivation of the linearized hydrodynamics for DDS by th e Kadanoff-Martin correlation-function method and with rigorous result s for particular models. We discuss also the internal noise in such sy stems, which we show to be governed by a generalized fluctuation-dissi pation relation (FDR), whose validity is not restricted to thermal equ ilibrium or to time-reversible systems. In the case of DDS, the FDR yi elds a version of a relation proposed some time ago by Price between t he covariance matrix of electrical current noise and the bulk diffusio n matrix of charge density. Our derivation of the hydrodynamic laws is in a form-the so-called ''Onsager force-flux form'' which allows us t o exploit the FDR to construct the Langevin description of the fluctua tions. In particular, we show that the probability of large fluctuatio ns in the hydrodynamic histories is governed by a version of the Onsag er ''principle of least dissipation,'' which estimates the probability of fluctuations in terms of the Ohmic dissipation required to produce them and provides a variational characterization of the most probable behavior as that associated to least (excess) dissipation. Finally, w e consider the relation of long-range spatial correlations in the stea dy slate of the DDS and the validity of ordinary hydrodynamic laws. We also discuss briefly the application of the general methods of this p aper to other cases, such as reaction-diffusion systems or magnetohydr odynamics of plasmas.