Diffusion in fluctuating hydrodynamics with spatially-correlated noise

Understanding diffusion in fluctuating hydrodynamic environments is crucial for molecular transport in biophysical systems. Diffusion facilitates the movement of nutrients and signaling molecules within cells, ensuring efficient transport and supporting essential functions. However, in such complex fluid environments, hydrodynamic fluctuations likely involve more than thermal noise, including collective motions driven by cellular structures. In this study, we investigate the fluctuating Navier-Stokes equations with spatially correlated noise, maintaining detailed balance to ensure thermal equilibrium. The correlated noise models large-scale motions potentially induced by large cellular structures. Previous studies have primarily used Langevin equations with correlated noise to study diffusion, rather than the full fluctuating Navier-Stokes dynamics. We explore how variations in the spatial correlation length scale affect the effective diffusion coefficient of passive particles. Using a pseudo-spectral method to solve the fluctuating Navier-Stokes equations and Lagrangian particle tracking for simulating particle motions, we demonstrate that spatial correlations in noise significantly alter diffusion compared to standard thermal fluctuations. Notably, the effective diffusion depends on the correlation length scale relative to particle size, offering deeper insights into diffusive transport at mesoscopic scales.