Spin propagation in systems of one-dimensional interacting fermions at finite temperature is intrinsically diffusive. The spreading rate of a spin packet is controlled by a transport coefficient termed spin drag relaxation time $tau_{rm sd}$. In this paper we present both numerical and analytical calculations of $tau_{rm sd}$ for a two-component spin-polarized cold Fermi gas trapped inside a tight atomic waveguide. At low temperatures we find an activation law for $tau_{rm sd}$, in agreement with earlier calculations of Coulomb drag between slightly asymmetric quantum wires, but with a different and much stronger temperature dependence of the prefactor. Our results provide a fundamental input for microscopic time-dependent spin-density functional theory calculations of spin transport in 1D inhomogeneous systems of interacting fermions.
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