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Correlations in non-equilibrium diffusive systems

We study the behavior of stationary non-equilibrium 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 non-equilibrium stationary state by changing the equilibrium boundary conditions. We decompose the correlations in a known local equilibrium part and another one that contains the non-equilibrium behavior and that we call {\it correlation's excess} $\bar C(x,z)$. We formally derive the differential equations for $\bar C$. To solve them order by order, we define a perturbative expansion around the equilibrium state. We show that the $\bar C$'s first-order expansion, $\bar C^{(1)}$, is always zero for the unique field case, $M=1$. Moreover $\bar C^{(1)}$ is always long-range or zero when $M>1$. Surprisingly we show that their associated fluctuations, the space integrals of $\bar C^{(1)}$, 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 $\bar C^{(1)}$ 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.