  A model for diffusion in liquids that couples the dynamics of tracer
particles to a fluctuating Stokes equation for the fluid is investigated in the
limit of large Schmidt number. In this limit, the concentration of tracers is
shown to satisfy a closed-form stochastic advection-diffusion equation that is
used to investigate the collective diffusion of hydrodynamically-correlated
tracers through a combination of Eulerian and Lagrangian numerical methods.
This analysis indicates that transport in liquids is quite distinct from the
traditional Fickian picture of diffusion. While the ensemble-averaged
concentration follows Fick's law with a diffusion coefficient that obeys the
Stokes-Einstein relation, each instance of the diffusive mixing process
exhibits long-ranged giant fluctuations around its average behavior. We
construct a class of mesoscopic models for diffusion in liquids at different
observation scales in which the renormalized diffusion coefficient depends on
this scale. This indicates that the Fickian diffusion coefficient in liquids is
not a material constant, but rather, changes with the scale at which
experimental measurements are performed.
