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Another Return of “Return to Equilibrium”

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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 interaction 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 the Araki-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).

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Author notes

  1. Marco Merkli

    Present address: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke W., Montreal, Quebec, Canada, H3A 2K6

Authors and Affiliations

  1. Theoretical Physics, ETH-Hönggerberg, 8093, Zürich, Switzerland

    Jürg Fröhlich & Marco Merkli

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  1. Jürg Fröhlich
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  2. Marco Merkli
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Correspondence to Jürg Fröhlich.

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Communicated by A. Kupiainen

Acknowledgements We thank V. Bach and I.M. Sigal for countless discussions on related problems and spectral methods without which this work would never have been done. M.M. is grateful to V. Jaksić for illuminating discussions. We have enjoyed the hospitality of IHES during the initial and final stages of this work.

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Fröhlich, J., Merkli, M. Another Return of “Return to Equilibrium”. Commun. Math. Phys. 251, 235–262 (2004). https://doi.org/10.1007/s00220-004-1176-6

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