95-168 Eyink, G. L., Lebowitz, J. L., Spohn, H.

Hydrodynamics and Fluctuations Outside of Local Equilibrium: Driven Diffusive Systems (224K, LaTex) Mar 28, 95

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Abstract. 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 {\em driven diffusive systems}(DDS), such as electrical conductors in an applied field with diffusion of charge carriers. In such systems, the hydrodynamical description is provided by a nonlinear drift-diffusion equation, which we derive by a microscopic method of {\em nonequilibrium distributions}. The formal derivation yields a Green-Kubo formula for the bulk diffusion 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 Kadanoff-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 {\em fluctuation-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 diffusion matrix of charge density. Our derivation of the hydrodynamical laws is in a form---the so-called ``Onsager force-flux form'' ---which allows us to exploit the FDR to construct the Langevin description of the fluctuations. In particular, we show that the probability of large fluctuations in the hydrodynamical histories is governed by a version of the Onsager ``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, 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 briefly the application of the general methods of this paper to other cases, such as reaction-diffusion systems or magnetohydrodynamics of plasmas.

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