  We study diffusive mixing in the presence of thermal fluctuations under the
assumption of large Schmidt number. In this regime we obtain a limiting
equation that contains a diffusive thermal drift term with diffusion
coefficient obeying a Stokes-Einstein relation, in addition to the expected
advection by a random velocity. The overdamped limit correctly reproduces both
the enhanced diffusion in the ensemble-averaged mean and the long-range
correlated giant fluctuations in individual realizations of the mixing process,
and is amenable to efficient numerical solution. Through a combination of
Eulerian and Lagrangian numerical methods we demonstrate that diffusion in
liquids is not most fundamentally described by Fick's irreversible law; rather,
diffusion is better modeled as reversible random advection by thermal velocity
fluctuations. We find that the diffusion coefficient is effectively
renormalized to a value that depends on the scale of observation. Our work
reveals somewhat unexpected connections between flows at small scales,
dominated by thermal fluctuations, and flows at large scales, dominated by
turbulent fluctuations.
