The Stokes-Einstein equation
It is a beautiful and simple example of a fluctuation-dissipation relation.
But suppose now we think about one of the individual atoms or molecules in the fluid. It also undergoes Brownian motion and one can define a self-diffusion constant.
It is amazing to me that the Stokes-Einstein relation still holds for a wide range of liquids, temperatures, and pressures with r being of the order of the molecular radius.
The figure and table below are taken from this paper.
Can this relation be derived from microscopic theory?
Zwanzig gave a heuristic justification here.
Rah and Eu gave a derivation from stat. mech. here.
The Stokes-Einstein relation does break down as one approaches the glass temperature in a supercooled liquid, as for example shown here. The origin of that breakdown is controversial, as is many phenomena involving glasses.