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

TL;DR

The paper investigates how intrinsic spatial correlations in thermal noise modify diffusion in fluids at thermal equilibrium by formulating a spatially correlated fluctuating incompressible Navier–Stokes equation that preserves the fluctuation-dissipation relation. They introduce two Lorentzian-type correlation functions C1(r) and C2(r) that allow independent control of correlation length ell and strength beta, and implement a Fourier-Galerkin 2D simulation with passive tracers to study diffusion. The results show that the mean-squared displacement (MSD) increases with ell and with decreasing beta, and a ballistic regime emerges at early times due to slowed momentum diffusion encoded in a nonlocal viscosity nu_eff(r). This nonlocal diffusion exhibits slow, glass-like dynamics and suggests experimental tests in quasi-2D fluids.

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 $β$. 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 $β$ also increases the MSD, since a weaker correlation strength also retards momentum diffusion, whereas smaller $\ell$ or larger $β$ 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.

Figures (6)

• Figure 1: FDR requires a scale-by-scale balance between noise-energy injection and viscous dissipation. (a) For uncorrelated noise, both noise-energy injection and viscous dissipation scale as $k^2$, corresponding to the Laplacian, and thus balance each other. The material derivative is $D_t=\partial_t + \bm{u}\cdot\bm{\nabla}$. Insets show instantaneous noise and velocity fields, both spatially random, consistent with thermal equilibrium. The black line in the velocity field represents a sample particle trajectory. (b) Introducing spatial correlations with length scale $\ell$ redistributes the noise-energy injection across scales. If the viscous term remains unmodified, the balance between energy injection and dissipation is lost. (c) Incorporating the same spatial correlation into the viscous term restores the scale-by-scale energy balance. In the insets, the correlated noise field develops large-scale structures, whereas the velocity field remains random, confirming thermal equilibrium. The particle trajectory exhibits enhanced diffusion compared with (a).
• Figure 2: Enhancement of particle diffusion as the noise correlation length $\ell$ increases for the correlation function $C_1$. Particle radius $a=\qty{0.05}{\um}$. (a) Representative instantaneous snapshots of the correlated noise (upper halves) and the resulting fluid velocity (lower halves) for $\ell=0.05,0.1$ and $\qty{0.2}{\um}$ (from left to right). Only the components in the $x$-direction are shown ($\mathcal{Z}_{xx}$ and $u_x$) thanks to spatial isotropy. Scale bar: 1. (b) Representative tracer trajectories over the same time span. Scale bar: 0.02. Simulation details are described in Sec. II of the Supplemental Material supplemental.
• Figure 3: Particle VACF and MSD exhibit a monotonic dependence on the correlation length $\ell$ for $C_1$. (a) Normalized VACF, and (b) MSD $\Delta X^2$. Insets in (a) show the absolute value of the VACF on a log--log scale, and in (b) the local slope $\alpha$ of $\Delta X^2$. As $\ell$ increases, the MSD rises, whereas the VACF exhibits an extended ballistic regime. Simulation parameters are the same as \ref{['fig:2']}.
• Figure 4: Enhancement of particle diffusion with increasing noise correlation length $\ell$ increases or decreasing correlation strength $\beta$ for the correlation function $C_2$. Particle radius $a=\qty{0.05}{\um}$. Representative instantaneous snapshots of the correlated noise (upper halves) and the resulting fluid velocity (lower halves) for (a) $\beta=1,0.5$ and $0.1$ (left to right) at $\ell=\qty{0.2}{\um}$, and (b) $\ell=0.1,0.2$ and 0.5 at $\beta=1$. Only the $x$-components ($\mathcal{Z}_{xx}$ and $u_x$) are shown, owing to spatial isotropy. Scale bars: 1. (c,d) Representative tracer trajectories over the same time span. Scale bars: $2\times10^{-3}~\unit{\um}$ in (c), 0.02 in (d), $4\times10^{-4}~\unit{\um}$ in the inset of (c), and $2\times10^{-3}~\unit{\um}$ in the inset of (d). Simulation methods and parameters are described in Sec. II of the Supplemental Material supplemental.
• Figure 5: Particle VACF and MSD exhibit a monotonic dependence on the correlation length $\ell$ and correlation strength $\beta$ for $C_2$. (a) Normalized VACF, and (b) MSD $\Delta X^2$ at fixed $\ell=\qty{0.2}{\um}$. Insets in (a) show the absolute value of the VACF on a log--log scale, and in (b) the local slope $\alpha$ of $\Delta X^2$. As $\beta$ decreases, the MSD rises, whereas the VACF enters a diffusive regime as $\beta$ increases. (c) Normalized VACF, and (d) MSD $\Delta X^2$ at fixed $\beta=1$. As $\ell$ increases, the MSD rises, and the VACF transitions away from the diffusive regime. Simulation parameters are the same as \ref{['fig:4']}.