  We study long-time spin diffusion of harmonically trapped lattice fermions in
one dimension. Combining thermodynamic Bethe ansatz approach and local density
approximation, we calculate spin current and spin diffusion coefficient driven
by the population imbalance. We find spin current is driven by susceptibility
effects rather than typical diffusion where magnetization would transport from
regions of high magnetization to low. As expected, spin transport is zero
through insulating regions and are only present in the metallic regions. In the
weak coupling limit, the local spin diffusion coefficient shows maxima at all
the insulating regions. Further, we estimate damping rate of diffusion modes in
the weak coupling limit within the lower metallic portion of the cloud. The
predicted spin current pattern can be probed via currently available
experimental techniques.
