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An Entropic Gradient Structure for Lindblad Equations and Couplings of Quantum Systems to Macroscopic Models

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Abstract

We show that all Lindblad operators (i.e., generators of quantum Markov semigroups) on a finite-dimensional Hilbert space satisfying the detailed balance condition with respect to the thermal equilibrium state can be written as a gradient system with respect to the relative entropy. We discuss also thermodynamically consistent couplings to macroscopic systems, either as damped Hamiltonian systems with constant temperature or as GENERIC systems.

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Acknowledgements

The research of M.M. was supported by ERC via AdG 267802 AnaMultiScale, and A.M. was partially supported by DFG via SFB 787 Nanophotonics (Subproject B4).

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Authors and Affiliations

  1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Markus Mittnenzweig & Alexander Mielke

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin-Adlershof, Germany

    Alexander Mielke

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  1. Markus Mittnenzweig
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  2. Alexander Mielke
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Correspondence to Alexander Mielke.

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Mittnenzweig, M., Mielke, A. An Entropic Gradient Structure for Lindblad Equations and Couplings of Quantum Systems to Macroscopic Models. J Stat Phys 167, 205–233 (2017). https://doi.org/10.1007/s10955-017-1756-4

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  • DOI: https://doi.org/10.1007/s10955-017-1756-4

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