Global and local relaxation of a spin-chain under exact Schroedinger and master-equation dynamics

We solve the Schroedinger equation for an interacting spin-chain locally coupled to a quantum environment with a specific degeneracy structure. The reduced dynamics of the whole spin-chain as well as of single spins is analyzed. We show, that the total spin-chain relaxes to a thermal equilibrium state independently of the internal interaction strength. In contrast, the asymptotic states of each individual spin are thermal for weak but non-thermal for stronger spin-spin coupling. The transition between both scenarios is found for couplings of the order of $0.1 \times \Delta E$, with $\Delta E$ denoting the Zeeman-splitting. We compare these results with a master equation treatment; when time averaged, both approaches lead to the same asymptotic state and finally with analytical results.