  We investigate transport properties of one-dimensional fermionic tight
binding models featuring nearest and next-nearest neighbor hopping, where the
fermions are additionally subject to a weak short range mutual interaction. To
this end we employ a pertinent approach which allows for a mapping of the
underlying Schr\"odinger dynamics onto an adequate linear quantum Boltzmann
equation. This approach is based on a suitable projection operator method. From
this Boltzmann equation we are able to numerically obtain diffusion
coefficients in the case of non-vanishing next-nearest neighbor hopping, i.e.,
the non-integrable case, whereas the diffusion coefficient diverges without
next-nearest neighbor hopping. For the latter case we analytically investigate
the decay behavior of the current with the result that arbitrarily small parts
of the current relax arbitrarily slowly which suggests anomalous diffusive
transport behavior within the scope of our approach.
