Universite ́ de Cergy-Pontoise,

We consider a quantum system S which interacts in a successive way with el-ements Ek of a chain of independent quantum subsystems. We will consider two situations: either identical interactions or random ones. In both cases, we show that, under suitable assumptions, the system approaches a repeated interaction asymptotic state in the limit of large times. In the case of identical interactions, we also show that if the reference state is chosen so that S and E are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics. Our method is based on the analysis of products of effective operators modelling the effects of each in-teraction. In the random situation we obtain results on the infinite products of independant identically distributed random matrices. These results also apply to e.g. inhomogeneous Markov chains (products of random stochastic matrices)