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‘Return to Equilibrium’ for Weakly Coupled Quantum Systems: A Simple Polymer Expansion

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Abstract

Recently, several authors studied small quantum systems weakly coupled to free boson or fermion fields at positive temperature. All the rigorous approaches we are aware of employ complex deformations of Liouvillians or Mourre theory (the infinitesimal version of the former). We present an approach based on polymer expansions of statistical mechanics. Despite the fact that our approach is elementary, our results are slightly sharper than those contained in the literature up to now. We show that, whenever the small quantum system is known to admit a Markov approximation (Pauli master equation aka Lindblad equation) in the weak coupling limit, and the Markov approximation is exponentially mixing, then the weakly coupled system approaches a unique invariant state that is perturbatively close to its Markov approximation.

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Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 19, D-69120, Heidelberg, Germany

    W. De Roeck

  2. Department of Mathematics, University of Helsinki, P. O. Box 68, Helsinki, FIN-00014, Finland

    A. Kupiainen

Authors
  1. W. De Roeck
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  2. A. Kupiainen
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Correspondence to W. De Roeck.

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Communicated by I.M. Sigal

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De Roeck, W., Kupiainen, A. ‘Return to Equilibrium’ for Weakly Coupled Quantum Systems: A Simple Polymer Expansion. Commun. Math. Phys. 305, 797–826 (2011). https://doi.org/10.1007/s00220-011-1247-4

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  • DOI: https://doi.org/10.1007/s00220-011-1247-4

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