Spatially Correlated Noise Induces Transitions from the Diffusive to Ballistic Regime in Fluids

TL;DR

This paper explores how spatially correlated thermal noise affects fluid dynamics, revealing that such correlations can induce a transition from diffusive to ballistic motion by altering momentum transport.

Contribution

It introduces a formulation of the Navier-Stokes equation with spatially correlated noise that preserves thermal equilibrium and demonstrates its impact on tracer diffusion regimes.

Findings

Increasing correlation length enhances MSD and promotes ballistic behavior.

Decreasing correlation strength also increases MSD, affecting the diffusion regime.

Nonlocal momentum transport resembles slow dynamics in disordered systems.

Abstract

We investigate the fluctuating incompressible Navier--Stokes equation driven by spatially correlated thermal noise characterized by a single length scale. This formulation is constructed to preserve thermal equilibrium through the fluctuation--dissipation relation (FDR), which enforces the same spatial correlation in the viscous diffusion term and therefore gives rise to nonlocal momentum transport. Numerical simulations of tracer diffusion in fluids governed by this formulation reveal that the mean-squared displacement (MSD) depends monotonically on the correlation length $\ell$ and the correlation strength $\beta$. Intuitively, increasing $\ell$ enhances MSD and induces the emergence of an early-time ballistic regime, as a larger correlation length slows momentum diffusion. Counterintuitively, decreasing $\beta$ also increases the MSD, since a weaker correlation strength also retards momentum diffusion, whereas smaller $\ell$ or larger $\beta$ suppresses the ballistic regime and leads to a diffusive behavior. The emergence or suppression of the ballistic regime stems from how spatial correlations, incorporated through the FDR to maintain equilibrium, alter the effective momentum transport across scales. Interestingly, we further show that the resulting nonlocal diffusion is reminiscent of the slow dynamics in glassy and other disordered systems.