Emergence of long-range non-equilibrium correlations in free liquid diffusion

It is experimentally well-established that non-equilibrium long-range correlations of concentration fluctuations appear in free diffusion of a solute in a solvent, but it remains unknown how such correlations are established dynamically. We address this problem in a model of Donev, Fai, and Vanden-Eijnden (DFV), obtained from the high-Schmidt limit of the Landau-Lifshitz fluctuating hydrodynamic equations for a binary mixture. We consider an initial planar interface of the mean concentration field in an infinite space domain, idealizing prior experiments. Using methods borrowed from turbulence theory, we show both analytically and numerically that a quasi-steady regime with self-similar time decay of concentration correlations appears at long time. In addition to the expected "giant concentration fluctuations" with correlations proportional to r for r less than or similar to L(t) = (Dt)(1/2), with diffusivity D, a new regime with spatial decay proportional to 1/r appears for r greater than or similar to L(t). The quasi-steady regime arises from an initial stage of transient growth proportional to t, confirming the prediction of DFV for r greater than or similar to L(t) and discovering an analogous result for r less than or similar to L(t). Our results give new insight into the emergence of non-equilibrium long-range correlations and provide novel predictions that may be investigated experimentally.