Originally posted by Loren Booda
Statistical mechanics is a classical method which involves both qualitative and quantitative assessment of many similar particles interacting thermally. Atoms numbering on the order of Avogadro may be modelled in a closed system (say a box with sides of length x) toward the distribution of states (say quantum numbers). Although every possible state of a particle has an equal probability, the constraint of thermal equilibrium requires a very well defined overall statistical distribution of states in relation to, say, temperature. You may have heard of the Boltzmann distribution, which gives the population of classical particles (most atoms and molecules) as a function of energy or temperature.
Einstein showed in 1905 that Brownian motion, a process much like diffusion, is a quantum mechanical process. Basically, each particle in suspension has an uncertainty in position and velocity associated with itself that manifests as random motion. This movement overcomes the bonds present in the liquid, and diffusion procedes.

Thanks for that great explanation :)