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We establish the existence of two weak coupling regime effective dynamics foran open quantum system of repeated interactions (vanishing strength andindividual interaction duration, respectively). This generalizes known resultsin that the von Neumann algebras describing the system and the chain elementmay not be of finite type. Then (but now assuming that the small system is offinite type), we prove that both effective dynamics capture the long-termbehaviour of the system: existence of a unique asymptotic state for themimplies the same property for the respective exact dynamics--provided that theperturbation parameter is sufficiently small. The zero-th order term in a powerseries expansion in the perturbation parameter of such an asymptotic state isgiven by the asymptotic state of the effective dynamics. We conclude by workingout the case in which the small system and the chain element are spins.

Repeated Interaction Quantum Systems: van Hove Limits and Asymptotic States