Correlated fluctuating hydrodynamics. I. Persistent transport from spatial correlations

Transport and reaction-diffusion processes in fluids are fundamentally influenced by fluctuations, which can modify the transport of momentum, particles, and other chemical species. In fluids at thermal equilibrium, these fluctuations arise from molecular thermal motion. These thermal fluctuations become particularly important for transport at mesoscopic scales, as they can be comparable with hydrodynamic transport mechanisms such as advection and viscous diffusion. While thermal fluctuations at mesoscopic scales are often assumed to be spatially uncorrelated in thermal equilibrium, the underlying fluid microstructure can generate finite-range spatial correlations. However, how such intrinsically correlated thermal fluctuations couple with hydrodynamics to influence mesoscopic transport remains poorly understood. To address this gap, we develop a fluctuating-hydrodynamic framework that incorporates spatially correlated thermal noise as a coarse-grained representation of its microscopic origin. To preserve thermal equilibrium, the fluctuation-dissipation relation requires the energy injected by the correlated thermal noise to remain balanced by viscous dissipation. Consequently, viscous dissipation inherits the same spatial correlations, thereby modifying momentum diffusion. Numerical simulations of particle diffusion show that increasing the correlation length prolongs transport persistence by nearly two orders of magnitude, thereby substantially delaying the crossover to normal diffusion relative to the white-noise limit. More generally, independently varying the correlation range and amplitude reveals that equilibrium spatial correlations can either enhance or suppress transport persistence relative to classical Brownian diffusion. These results demonstrate that intrinsic spatial correlations in equilibrium thermal fluctuations can qualitatively reshape mesoscopic transport.