Long-range correlations for conservative dynamics

We investigate the origin of long-range spatial correlations in certain anisotropic translation-invariant stationary nonequilibrium states of systems with conservative dynamics. We consider both lattice-gas models with anisotropic but reflection-invariant stochastic dynamics and driven diffusive systems described by a Ginzburg-Landau equation with an electric field E. Carrying out perturbation expansions about an equilibrium state with short-range correlations we find that, in general, the spatial correlations in the stationary state decay only via a power law; the spatial decay thus reflects the well-known diffusive decay in time for systems with conservative dynamics. The typical spatial decay of the pair correlation behaves like the electrostatic potential produced by a quadrupole charge density at the origin. Exponential decay of spatial correlations, so familiar from equilibrium, appears here as the exception; it occurs generically only when there are special constraints on the dynamics, such as detailed balance and possibly spatial symmetry. The paradigm for this generic long-range behavior or self-organized criticality is found in the solutions of the linear Langevin equation describing the behavior of fluctuations of a conserved macroscopic variable. The fluctuating hydrodynamics underlying such a description is justified here rigorously for the macroscopic scaling limit of certain lattice-gas models.