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We prove two statements about the long time dynamics of integrableHamiltonian systems. In classical mechanics, we prove the microcanonicalversion of the Generalized Gibbs Ensemble (GGE) by mapping it to a knowntheorem and then extend it to the limit of infinite number of degrees offreedom. In quantum mechanics, we prove GGE for maximal Hamiltonians - a classof models stemming from a rigorous notion of quantum integrability understoodas the existence of conserved charges with prescribed dependence on a systemparameter, e.g. Hubbard U, anisotropy in the XXZ model etc. In analogy withclassical integrability, the defining property of these models is that theyhave the maximum number of independent integrals. We contrast their dynamicsinduced by quenching the parameter to that of random matrix Hamiltonians.

Generalized microcanonical and Gibbs ensembles in classical and quantum integrable dynamics