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 $\ln (L/L_0)$ in quasi-two-dimensional systems, while in three dimensions it scales as $L_0^{-1}-L^{-1}$, where $L_0$ is a reference length. Our results demonstrate that fluctuations play an important role in the hydrodynamics of small-scale systems.

Figure 1

Discrete structure factor ${\mathcal{S}}_{{\rho }_1,v_{\parallel }^{(1)}}$ from quasi-two-dimensional DSMC runs with $L_x=64\lambda$, $L_y=512\lambda$, and $L_z=2\lambda$, for several wave numbers $k_x={\kappa }_x\times 2\pi /L_x$ (see legend), compared to the discrete equivalent of the continuum prediction (4) (solid lines of the same color). Note that for a fixed $k_x$ we expect the structure factor to decay as $k_y^{-4}$.

Figure 2

(a) The effective ${\chi }_{\mathrm{eff}}$ and the renormalized ${\chi }_0$ diffusion coefficients as a function of the width of the system $L_x$ in two dimensions. Numerical results for system $A$ (DSMC and SPDE) and system $B$ (DSMC) are shown with symbols (see legend). The error bars for all of the simulation data are comparable to or smaller than the size of the symbols. The theoretical predictions [12] are evaluated numerically and shown with lines. (b) Same as (a) but in three dimensions. The inset highlights the $L^{-1}$ behavior.