WIAS Preprint No. 2320, (2016)

An entropic gradient structure for Lindblad equations and GENERIC for quantum systems coupled to macroscopic models


  • Mittnenzweig, Markus
    ORCID: 0000-0002-8502-1702
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 82B10 82C10 47D07 49S05


  • Quantum Markov semigroups, open quantum systems, Lindblad operator, detailed-balance condition, relative entropy, Onsager, operators, general equations for non-equilibrium reversible irreversible coupling, GENERIC




We show that all Lindblad operators (i.e. generators of quantum 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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