Nonequilibrium Green-Kubo relations for hydrodynamic transport from an equilibrium-like fluctuation-response equality

Near equilibrium, Green-Kubo relations provide microscopic expressions for macroscopic transport coefficients in terms of equilibrium correlation functions. At their core, they are based on the intimate relationship between response and fluctuations as embodied by the equilibrium fluctuation-dissipation theorem, a connection generically broken far from equilibrium. In this paper, we identify a class of perturbations whose response around far-from-equilibrium steady states is linked to steady-state correlation functions via an equilibrium-like fluctuation-response equality. We then utilize this prediction to substantiate linearized hydrodynamic transport equations that describe how spatial inhomogeneities in macroscopic nonequilibrium systems relax. As a consequence, we derive nonequilibrium Green-Kubo relations for the transport coefficients of two types of hydrodynamic variables: local conserved densities and broken-symmetry modes. A by-product of this work is to provide a theoretical foundation for the validity of Onsager's regression hypothesis around nonequilibrium steady states. Our predictions are analytically and numerically corroborated for two model systems: density diffusion in a fluid of soft, spherical active Brownian particles and phase diffusion in the noisy Kuramoto model on a square lattice.