Mathematical Physics Another Return of “Return to Equilibrium”

Abstract: The property of “return to equilibrium ” is established for a class of quantum-mechanical models describing interactions of a (toy) atom with black-body radiation, or of a spin with a heat bath of scalar bosons, under the assumption that the inter-action strength is sufficiently weak. For models describing the first class of systems, our upper bound on the interaction strength is independent of the temperature T, (with 0 < T ≤ T0 < ∞), while, for the spin-boson model, it tends to zero logarithmically, as T → 0. Our result holds for interaction form factors with physically realistic infrared behaviour. Three key ingredients of our analysis are: a suitable concrete form of theAraki-Woods representation of the radiation field, Mourre’s positive commutator method combined with a recent virial theorem, and a norm bound on the difference between the equilibrium states of the interacting and the non-interacting system (which, for the system of an atom coupled to black-body radiation, is valid for all temperatures T ≥ 0, assuming only that the interaction strength is sufficiently weak). 1