Stresses in non-equilibrium fluids: Exact formulation and coarse grained theory

Starting from the stochastic equation for the density operator, we formulate the exact (instantaneous) stress tensor for interacting Brownian particles, whose average value agrees with expressions derived previously. We analyze the relation between the stress tensor and forces on external potentials, and observe that, out of equilibrium, particle currents give rise to extra forces. Next, we derive the stress tensor for a Landau-Ginzburg theory in non-equilibrium situations, finding an expression analogous to that of the exact microscopic stress tensor, and discuss the computation of out-of-equilibrium (classical) Casimir forces. We use these relations to study the spatio-temporal correlations of the stress tensor in a Brownian fluid, which we derive exactly to leading order in the interaction potential strength. We observe that, after integration over time, the spatial correlations generally decay as power laws in space. These are expected to be of importance for driven confined systems. We also show that divergence-free parts of the stress tensor do not contribute in the Green-Kubo relation for the viscosity.