The problem of return to equilibrium is phrased in terms of aC*-algebraU, and two one-parameter groups of automorphisms τ, τ P corresponding to the unperturbed and locally perturbed evolutions. The asymptotic evolution, under τ, of τ P -invariant, and τ 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 provide the necessary dispersive properties for the return to equilibrium. It is demonstrated that the τ 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 theX−Y model is given which extends previously known results and other applications, and examples, are given for the Fermi gas.
KeywordsNeural Network Statistical Physic Equilibrium State Complex System Nonlinear Dynamics
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