Correlations in nonequilibrium diffusive systems

We study the behavior of stationary nonequilibrium two-body correlation functions for diffusive systems with equilibrium reference states (DSe). We describe a DSe at the mesoscopic level by M locally conserved continuum fields that evolve through coupled Langevin equations with white noises. The dynamic is designed such that the system may reach equilibrium states for a set of boundary conditions. In this form, we make the system driven to a nonequilibrium stationary state by changing the equilibrium boundary conditions. We decompose the correlations in a known local equilibrium part and another one that contains the nonequilibrium behavior and that we call correlation's excess C[over ¯](x,z). We formally derive the differential equations for C[over ¯]. To solve them order by order, we define a perturbative expansion around the equilibrium state. We show that the C[over ¯]'s first-order expansion, C[over ¯]^{(1)}, is always zero for the unique field case, M=1. Moreover, C[over ¯]^{(1)} is always long range or zero when M>1. We obtain the surprising result that their associated fluctuations, the space integrals of C[over ¯]^{(1)}, are always zero. Therefore, fluctuations are dominated by local equilibrium up to second order in the perturbative expansion around the equilibrium. We derive the behaviors of C[over ¯]^{(1)} in real space for dimensions d=1 and 2 explicitly. Finally, we derive the two first perturbative orders of the correlation's excess for a generic M=2 case and a hydrodynamic model.