Giant concentration fluctuations far-from-equilibrium and reduction to a soluble model of turbulent advection

Giant concentration fluctuations are a manifestation of nonequilibrium long-range correlations in diffusive mixing of a solute in a quiescent fluid. Donev et al. in 2014 pointed out unexpected links of liquid diffusion with turbulent advection of a passive scalar, showing that power-law structure functions arise by a cascade process. In the asymptotic limit of high Schmidt number, typical of liquid mixtures, they found that the Landau-Lifschitz fluctuating hydrodynamic equations for a binary fluid reduce to a version of the soluble Kraichnan model of turbulent advection. However, their numerical simulations of the model did not obviously reproduce the experimentally observed k^{-4} structure function. We resolve this discrepancy by solving the model analytically. Surprisingly, although the theory accounts for nonlinear advection, it yields precisely the same static and dynamic structure functions predicted by linearized fluctuating hydrodynamics. We argue that this is a simple example of anomaly non-renormalization. In addition, however, the model predicts non-Gaussian higher-order statistics of concentration fluctuations and applies also to difficult problems with large concentration gradients, high concentrations, and transient processes beyond the scope of linearized theory.