Diffusive transport by thermal velocity fluctuations.

We study the contribution of advection by thermal velocity fluctuations to the effective diffusion coefficient in a mixture of two identical fluids. We find good agreement between a simple fluctuating hydrodynamics theory and particle and finite-volume simulations. The enhancement of the diffusive transport depends on the system size L and grows as lnðL=L0Þ in quasi-two-dimensional systems, while in three dimensions it scales as L -1 - L -1 , where L0 is a reference length. Our results demonstrate that 0 fluctuations play an important role in the hydrodynamics of small-scale systems. Thermal fluctuations in nonequilibrium systems in which a constant (temperature, concentration, velocity) gradient is imposed externally exhibit remarkable behavior compared to equilibrium systems [1]. The solution of the linearized equations of fluctuating hydrodynamics shows that concentration and density fluctuations exhibit longranged correlations in the presence of a macroscopic concentration gradient rc [1–3]. The enhancement of large-scale (small wave number) concentration fluctua­ tions is dramatic during the early stages of diffusive mixing between initially phase-separated fluids. These giant fluc­ tuations [4–6] during free diffusive mixing have been measured using light scattering and shadowgraphy tech­ niques [4,5,7], finding good but imperfect agreement with theoretical predictions. The giant fluctuation phenomenon arises because of the appearance of long-ranged correlations between concen­ tration and velocity fluctuations in the presence of a con­ centration gradient. It has been predicted that these correlations give rise to fluctuation-renormalized transport coefficients [3,8]; however, the predicted enhancement of transport at hydrodynamic scales has not yet been compu­ tationally observed. In particular, it is important to under­ stand how the effective transport coefficients depend on the length scale of observation. In this Letter we consider diffusion in a mixture of identical but labeled (as components 1 and 2) fluids [9] enclosed in a box of size Lx X Ly X Lz , in the absence of gravity. Periodic boundary conditions are applied in the x (horizontal) and z (depth) directions, while the top and bottom boundaries are impermeable constant-temperature walls. A concentration gradient rc p ¼ð cT - cBÞ=Ly is imposed along the y axes by enforcing a constant concen­ tration c T at the top wall and c B at the bottom wall. Because the fluids are identical, concentration is passively transported by thermal fluctuations. Since species are not changed in particle collisions, the diffusive transport of concentration c ¼ p1=p can only occur via advective motion of the particles, where p de­ notes the mass density. The mass flux for a given species is therefore equal to the momentum density for particles of that species. At steady state the particles of a given spe­ cies have a nonzero macroscopic momentum density p j1 ¼ p p1v p 1 ¼- p cÞ, where x is the mass diffusion px ðr p coefficient [10]. The local fluctuations around the macro­ scopic mean, p1 ¼ p p 1 þ op1 and v1 ¼ v p 1 þ ov1, can also make a nontrivial contribution to the average mass flux if they are correlated,