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Return to equilibrium. (English) Zbl 0257.46091
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.

MSC:
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
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