  We analyse diffusion at low temperature by bringing the
fluctuation-dissipation theorem (FDT) to bear on a physically natural, viscous
response-function R(t). The resulting diffusion-law exhibits several distinct
regimes of time and temperature, each with its own characteristic rate of
spreading. As with earlier analyses, we find logarithmic spreading in the
quantum regime, indicating that this behavior is robust. A consistent R(t) must
satisfy the key physical requirements of Wightman positivity and passivity, and
we prove that ours does so. We also prove in general that these two conditions
are equivalent when the FDT holds. Given current technology, our diffusion law
can be tested in a laboratory with ultra cold atoms.
