WIAS Preprint No. 1710, (2012)

Dissipative quantum mechanics using GENERIC


  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 34D20 37N20 47N50 80A99 81Q05 81V19


  • Quantum mechanics, density matrices, Hamiltonian systems, gradient systems, Onsager systems, GENERIC, von Neumann entropy, canonical correlation


Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework (General Equations for Non-Equilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition. One of our models couples a quantum system to a finite number of heat baths each of which is described by a time-dependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the Kubo-Mori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative double-bracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finite-dimensional and isothermal case we provide a general existence result and discuss sufficient conditions that guarantee that all solutions converge to the unique thermal equilibrium state. Finally, we compare of our gradient flow formulation for quantum systems with the Wasserstein gradient flow formulation for the Fokker-Planck equation and the entropy gradient flow formulation for reversible Markov chains.

Appeared in

  • Recent Trends in Dynamical Systems, Proceedings of a Conference in Honor of Jürgen Scheurle, A. Johann, H.-P. Kruse, S. Schmitz, eds., vol. 35 of Proceedings in Mathematics and Statistics, Springer, Heidelberg, 2013, pp. 555--585

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