  We investigate transport in several translationally invariant spin-1/2 chains
in the limit of high temperatures. We concretely consider spin transport in the
anisotropic Heisenberg chain, the pure Heisenberg chain within an alternating
field, and energy transport in an Ising chain which is exposed to a tilted
field. Our approach is essentially based on a connection between the evolution
of the variance of an inhomogeneous non-equilibrium density and the current
auto-correlation function at finite times. Although this relationship is not
restricted to the case of diffusive transport, it allows to extract a
quantitative value for the diffusion constant in that case. By means of
numerically exact diagonalization we indeed observe diffusive behavior in the
considered spin chains for a range of model parameters and confirm the
diffusion coefficients which were obtained for these systems from
non-equilibrium bath scenarios.
