  We consider the increase of the spatial variance of some inhomogeneous,
non-equilibrium density (particles, energy, etc.) in a periodic quantum system
of condensed matter-type. This is done for a certain class of initial quantum
states which is supported by static linear response and typicality arguments.
We directly relate the broadening to some current auto-correlation function at
finite times. Our result is not limited to diffusive behavior, however, in that
case it yields a generalized Einstein relation. These findings facilitate the
approximation of diffusion constants/conductivities on the basis of current
auto-correlation functions at finite times for finite systems. Pursuing this,
we quantitatively confirm the magnetization diffusion constant in a spin chain
which was recently found from non-equilibrium bath scenarios.
