Correlated fluctuating hydrodynamics. III. Spatiotemporal correlations and mesoscopic transport

Many chemical, biological, and soft-matter phenomena involve the mesoscopic transport of particles through complex fluid environments. Such fluids often remain close to thermal equilibrium, where thermal fluctuations are intrinsically coupled to hydrodynamics and play an essential role in mesoscopic transport. In complex fluids, however, interactions within the underlying microstructure can correlate these fluctuations across space and time. How the resulting spatial and temporal correlations jointly influence particle transport remains poorly understood. To disentangle the roles of spatial and temporal correlations, we extend the correlated fluctuating-hydrodynamic framework developed in Papers I and II [Huang et al., J. Chem. Phys. 165, 064509 (2026) and J. Chem. Phys. 165, 064510 (2026)] of this series. The extended framework incorporates temporal correlations alongside spatial correlations while preserving thermal equilibrium through the fluctuation-dissipation relation. Allowing spatial and temporal correlations to be varied independently enables a systematic investigation of their individual and joint effects on mesoscopic particle transport. Numerical simulations of particle diffusion reveal that temporal correlations substantially enhance the persistence of particle motion at short times while producing a transient slowdown at intermediate times. The magnitude of this slowdown varies nonmonotonically with spatial correlation length. Analysis of the surrounding hydrodynamic flow reveals the origin of this behavior: temporal correlations produce oscillatory fluid relaxation, whereas spatial correlations reshape the oscillations, suppressing them at sufficiently large correlation lengths. Particle transport in correlated fluctuating fluids is therefore governed by the competition between spatial and temporal correlations rather than by a simple combination of their individual effects.