Correlations in non-equilibrium diffusive systems

TL;DR

This paper investigates non-equilibrium stationary correlations in diffusive systems, revealing that local equilibrium dominates fluctuations up to second order and providing explicit solutions for specific cases.

Contribution

It introduces a perturbative framework to analyze non-equilibrium correlations and shows that the first-order excess correlations are either zero or long-range, with fluctuations dominated by local equilibrium.

Findings

First-order correlation excess is zero for M=1.

For M>1, the first-order excess is either zero or long-range.

Fluctuations are dominated by local equilibrium up to second order.

Abstract

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.