Hydrodynamics and Fluctuations Outside of Local Equilibrium :

We derive hydrodynamic equations for systems not in local thermodynamic equilibrium, that is, where the local stationary measures are \non-Gibbsian" and do not satisfy detailed balance with respect to the microscopic dynamics. As a main example we consider the driven di usive systems(DDS), such as electrical conductors in an applied eld with di usion of charge carriers. In such systems, the hydrodynamical description is provided by a nonlinear drift-di usion equation, which we derive by a microscopic method of nonequilibrium distributions. The formal derivation yields a Green-Kubo formula for the bulk di usion matrix and microscopic prescriptions for the drift velocity and \nonequilibrium entropy" as functions of charge density. Properties of the hydrodynamical equations are established, including an \H-theorem" on increase of the thermodynamic potential, or \entropy," describing approach to the homogeneous steady-state. The results are shown to be consistent with the derivation of the linearized hydrodynamics for DDS by the Kadano -Martin correlation-function method and with rigorous results for particular models. We discuss also the internal noise in such systems, which we show to be governed by a generalized uctuation-dissipation relation(FDR), whose validity is not restricted to thermal equilibrium or to time-reversible systems. In the case of DDS, the FDR yields a version of a relation proposed some time ago by Price between the covariance matrix of electrical current noise and the bulk di usion matrix of charge density. Our derivation of the hydrodynamical laws is in a form|the so-called \Onsager forceux form" |which allows us to exploit the FDR to construct the Langevin description of the uctuations. In particular, we show that the probability of large uctuations in the hydrodynamical histories is governed by a version of the Onsager \principle of least dissipation," which estimates the probability of uctuations 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, we consider the relation of long-range spatial correlations in the steady-state of the DDS and the validity of ordinary hydrodynamical laws. We discuss also brie y the application of the general methods of this paper to other cases, such as reaction-di usion systems or magnetohydrodynamics of plasmas. 2 TABLE OF CONTENTS