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Journal of Statistical Physics

, Volume 167, Issue 2, pp 205–233 | Cite as

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

  • Markus Mittnenzweig
  • Alexander MielkeEmail author
Article

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.

Keywords

Quantum Markov semigroups Relative entropy Gradient structure General equations for non-equilibrium reversible irreversible coupling 

Notes

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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Weierstraß-Institut für Angewandte Analysis und StochastikBerlinGermany
  2. 2.Institut für MathematikHumboldt-Universität zu BerlinBerlin-AdlershofGermany

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