  We describe the far from equilibrium non-local transport in a diffusive
superconducting wire with a Zeeman splitting, taking into account the different
spin relaxation mechanisms. We demonstrate that due to the Zeeman splitting an
injection of a current in a superconducting wire creates a spin accumulation
that can only relax via thermalization. In addition the Zeeman splitting also
causes a suppression of the spin-orbit and spin-flip scattering rates. These
two effects lead to long-range spin and charge accumulations detectable in the
non-local signal. Our model explains the main qualitative features of recent
experimental results in terms of realistic parameters and predicts a strong
dependence of the non-local signal on the orbital depairing effect from an
induced magnetic field.
