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Summary: The problem of return to equilibrium is phrased in terms of a $$C^*$$-algebra $$\mathcal A$$, and two one-parameter groups of automorphisms $$\tau, \tau^p$$ corresponding to the unperturbed and locally perturbed evolutions. The asymptotic evolution, under $$\tau$$, of $$\tau^p$$-invariant, and $$\tau^p$$-K.M.S., states is considered. This study is a generalization of scattering theory and results concerning the existence of limit states are obtained by techniques similar to those used to prove the existence, and intertwining properties, of wave-operators. Conditions of asymptotic abelianness of $$(\mathcal A, \tau)$$ provide the necessary dispersive properties for the return to equilibrium. It is demonstrated that the $$\tau^p$$-equilibrium states and their limit states are coupled by automorphisms with a quasi-local property; they are not necessarily normal with respect to one another. An application to the $$X-Y$$ model is given which extends previously known results and other applications, and examples, are given for the Fermi gas.
 46L05 General theory of $$C^*$$-algebras 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 46N55 Applications of functional analysis in statistical physics