A computational fluctuating hydrodynamics approach revealing how anomalous long-range correlations emerge hierarchically through mode-coupling between concentration and velocity fluctuations.
An experimentally well-established phenomenon without a known dynamical explanation.
When a macroscopic concentration gradient is imposed on a binary fluid mixture, anomalously large long-range correlations emerge in the concentration field. The structure factor diverges as S(q) ~ q-4, indicating power-law spatial correlations extending far beyond any molecular interaction range. While the steady-state properties are well understood, the dynamical mechanism by which these correlations emerge from an initially uncorrelated state remained an open problem.
Understanding the dynamical origin reveals the mechanism underlying the emergence of nonequilibrium order -- the process, not just the endpoint. It yields predictions for transient experiments where the gradient is suddenly imposed, and establishes the hierarchy of timescales governing the approach to the nonequilibrium steady state (NESS). This is item 5 in Maes' list of open problems in nonequilibrium statistical mechanics (arXiv:2601.16716, 2026).
Combined analytical and computational approach based on linearized fluctuating hydrodynamics.
Linearized FHD equations for concentration fluctuations coupled to velocity via the macroscopic gradient. Derive the transient structure factor as an exponential relaxation toward the NESS, with a q-dependent rate encoding the coupled dynamics.
Euler-Maruyama integration of the coupled concentration-velocity system in Fourier space on a 128x128 periodic lattice. 5000 time steps with adaptive dt ~ 9.27x10-4, covering ~0.33 tau_D of the relaxation.
Radially averaged power spectra, power-law fitting in log-log space, dynamical scaling collapse test, and real-space correlation function extraction via the Wiener-Khinchin theorem.
| Parameter | Symbol | Value | Description |
|---|
Watch the q-4 nonequilibrium enhancement build up hierarchically from high-q to low-q modes.
Select time snapshots to see how S(q) evolves from flat equilibrium toward the q-4 NESS envelope. High-q modes (short wavelengths) saturate first.
The fitted exponent alpha in S(q) ~ q^alpha evolves from ~0 (flat equilibrium) toward -4 (NESS). At the final simulation time (~0.33 tau_D), alpha ~ -1.96.
The characteristic spatial extent of concentration correlations grows diffusively as sqrt(Dt) at early times, saturating at 1/q_c.
Comparison of numerical (FHD) and analytical predictions. The green dashed reference shows the sqrt(Dt) diffusive scaling.
At early times (Dt much less than 1/q_c^2): xi(t) ~ sqrt(Dt)
At late times: xi approaches 1/q_c = 0.119
The normalized S_neq(q,t)/S_neq^ss(q) collapses onto a universal function of the scaling variable q^2 Dt.
Each point is one (q, t) pair from the simulation. The red curve is the universal function f(x) = 1 - e^(-2x).
This dynamical scaling collapse is a key prediction. The data follow this universal function with an RMS deviation of 0.27. Scatter is larger at small x (early times, small q) where stochastic fluctuations dominate.
Use the slider to filter data points by minimum scaling variable value, showing how the collapse improves at larger q^2 Dt.
The q-dependent rate controlling how fast each wavevector mode approaches its NESS value.
At large q, Gamma ~ Dq^2 (purely diffusive). Near q_c, buoyancy coupling slows the relaxation.
S_neq(q,t)/S_neq^ss(q) for selected wavevectors. High-q modes saturate first (hierarchical buildup).
How long-range concentration correlations emerge through mode-coupling.
The system starts at thermal equilibrium with short-range correlations only. A macroscopic concentration gradient is suddenly applied across the fluid layer.
The gradient couples concentration fluctuations to velocity fluctuations through the advective term. Each wavevector mode q begins developing nonequilibrium enhancement at rate Gamma(q).
Short-wavelength (large q) modes reach their NESS values first on diffusive timescales ~1/(Dq^2). The correlation length xi(t) ~ sqrt(Dt) grows as progressively longer-wavelength modes are populated.
All modes saturate. The full q^-4 spectrum is established. The correlation length reaches its maximum value 1/q_c, set by the balance of buoyancy and diffusion.
The fundamental mechanism is the time-dependent mode-coupling between concentration and velocity fluctuations, mediated by the macroscopic gradient. The long-range character of the correlations is inherited from the long-range nature of hydrodynamic interactions, encoded in the Stokes Green's function G(q) ~ 1/q^2 (the Oseen tensor).
The advective coupling (grad c_0) * v_z in the concentration equation transfers the 1/q^2 range of G(q) into the concentration fluctuations, producing the q^-4 enhancement in S(q). This transfer occurs progressively from fast (high-q) to slow (low-q) modes, creating the hierarchical buildup.
where xi_c and f_z are stochastic noise terms satisfying fluctuation-dissipation relations.
Four main contributions from this combined analytical and computational study.
S_neq(q, t) = S_neq^ss(q) [1 - exp(-2 Gamma(q) t)] provides the complete transient from equilibrium to NESS, governed by a q-dependent relaxation rate Gamma(q) encoding the coupled concentration-velocity dynamics.
Short-wavelength modes equilibrate on diffusive timescales ~1/(Dq^2) before long-wavelength modes. This is confirmed by SPDE simulations showing the power-law exponent evolving from ~0 to -1.96 (toward -4.0 at NESS).
The normalized structure factor S_neq(q,t)/S_neq^ss(q) collapses onto the universal function f(q^2 Dt) = 1 - exp(-2 q^2 Dt), verified by simulation with RMS deviation of 0.27.
The fundamental mechanism is the time-dependent mode-coupling between concentration and velocity fluctuations, mediated by the macroscopic gradient, which transfers the long-range character of hydrodynamic interactions (Oseen tensor G(q) ~ 1/q^2) into concentration correlations.
Detailed numerical results from the SPDE simulation and analytical predictions.
| Quantity | Value | Expected |
|---|
| t | xi_num | xi_ana | alpha |
|---|
| q | t=0.001 | t=0.65 | t=1.30 | t=1.95 | t=2.60 | t=3.25 | t=3.90 | t=4.64 | S_ss(ana) |
|---|