Non-Gaussian statistics of concentration fluctuations in free liquid diffusion

Abstract

We show that the three-point skewness of concentration fluctuations is non-vanishing in free liquid diffusion, even in the limit of vanishingly small mean concentration gradients. We exploit a high-Schmidt reduction of nonlinear Landau-Lifshitz hydrodynamics for a binary fluid, both analytically and by a massively parallel Lagrangian Monte Carlo simulation. Non-Gaussian statistics result from nonlinear coupling of concentration fluctuations to thermal velocity fluctuations, analogous to the turbulent advection of a passive scalar. Concentration fluctuations obey no central limit theorem, counter to the predictions of macroscopic fluctuation theory for generic diffusive systems.

Figures (4)

• Figure 1: Triple cumulants for the vertical equilateral triangle with side lengths $r^*=50$ and for initial mean concentration profiles with $\tau^*=10^4$, $10^6$, $10^8$ and $10^{10}$ going from left to right, top to bottom, respectively. Solid lines with symbols and shades show the numerical results and corresponding errors from the Lagrangian Monte Carlo computation. Dashed lines correspond to the analytical expression in SM, § A, valid asymptotically for short times, as highlighted by the insets. For corresponding results on the combined second cumulant $({\mathcal{C}}_{13}{\mathcal{C}}_{23}{\mathcal{C}}_{12})^{1/2}$ used to calculate the three-point skewness, see End Matter, Fig.\ref{['C_2 r=50']}.
• Figure 2: Three-point skewness for the vertical equilateral triangle with side lengths $r^*=50$ and small concentration gradients (colors and symbols are coded as in Fig. \ref{['Fig1']} for $\tau^*=10^6$ - cyan squares (${\boldsymbol \square}$), $10^8$ - green circles (${\boldsymbol \circ}$), $10^{10}$ - yellow triangles (${\boldsymbol \triangle}$). The result for $\tau^*=10^4$ can be found in SM, § C.) Solid lines with symbols and shades represent the numerical Lagrangian results and Monte Carlo errors, while dashed lines show the analytical prediction. The times $t$ in seconds on the upper axis are calculated with reference values $\sigma=10$ nm and and $D=2.2\times 10^{-5}\,{\rm mm}^2/{\rm sec}$ for which the maximum skewness is achieved in about $4$ ms. Short-time behavior is zoomed in the inset, with the red line (${\boldsymbol -}\!\!{\boldsymbol -}$) showing the short-time asymptotic prediction for vanishing concentration gradient.
• Figure 3: Illustration of the triangle formed by vertices ${\bf x}_1,$${\bf x}_2,$${\bf x}_3,$ in reference position in the horizontal plane.
• Figure 4: Combined second cumulants $({\mathcal{C}}_{13}{\mathcal{C}}_{23}{\mathcal{C}}_{12})^{1/2}$ for the vertical equilateral triangle with side lengths $r^*=50$ and for initial mean concentration profiles with $\tau^*=10^4$, $10^6$, $10^8$ and $10^{10}$ going from left to right, top to bottom, respectively. Solid lines with symbols and shades represent numerical results and dashed lines the analytical prediction. Monte Carlo error is too small to be visible at this scale.