Brownian motion at absolute zero
  We derive a general quantum formula giving the mean-square displacement of a diffusing particle as a function of time. Near {\bf 0 K} we find a universal logarithmic behavior (valid for times longer than the relaxation time), and deviations from classical behavior can also be significant at larger values of time and temperature. Our derivation depends neither on the specific composition of the heat bath nor on the strength of the coupling between the bath and the particle. An experimental regime of microseconds and microdegrees Kelvin would elicit the pure logarithmic diffusion. 
Supurna Sinha and Rafael D. Sorkin