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


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.


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



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).


  1. 1.
    Aschbacher, W., Jakšić, V., Pautrat, Y., Pillet, C.-A.: Topics in non-equilibrium quantum statistical mechanics. In: Attal S., Joye A., Pillet C.-A. (edS.) Open Quantum Systems III. Lecture Notes Mathematics, vol. 1882, pp. 1–116. Springer (2006)Google Scholar
  2. 2.
    Alicki, R.: On the detailed balance condition for non-Hamiltonian systems. Rep. Math. Phys. 10(2), 249–258 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Albert, V.V., Jiang, L.: Symmetries and conserved quantities in Lindblad master equations. Phys. Rev. A 59, 022118 (2014)ADSCrossRefGoogle Scholar
  4. 4.
    Baumgartner, B., Narnhofer, H.: Analysis of quantum semigroups with GKS-Lindblad generators. II. General. J. Phys. A 41, 395303 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baumgartner, B., Narnhofer, H.: The structures of state space concerning quantum dynamical semigroups. Rev. Math. Phys. 24(2), 1250001 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baumgartner, B., Narnhofer, H., Thirring, W.: Analysis of quantum semigroups with GKS-Lindblad generators. I. Simple generators. J. Phys. A 41(6), 065201 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carlen, E.A., Maas, J.: An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy. Commun. Math. Phys. 331(3), 887–926 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carlen, E.A., Maas, J.: Gradient flow and entropy inequalities for quantum markov semigroups with detailed balance. arXiv:1609.01254 (2016)
  9. 9.
    Chow, S.-N., Huang, W., Li, Y., Zhou, H.: Fokker-Planck equations for a free energy functional or Markov process on a graph. Arch. Ration. Mech. Anal. 203(3), 969–1008 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Davies, E.B.: Markovian master equations. Commun. Math. Phys. 39, 91–110 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Duong, M.H., Peletier, M.A., Zimmer, J.: GENERIC formalism of a Vlasov-Fokker-Planck equation and connection to large-deviation principles. Nonlinearity 26(11), 2951–2971 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dumas, É.: Global existence for Maxwell-Bloch systems. J. Differ. Equ. 219(2), 484–509 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Erbar, M., Maas, J.: Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206(3), 997–1038 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grmela, M., Öttinger, H.C.: Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E 56(6), 6620–6632 (1997)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Jordan, R., Kinderlehrer, D., Otto, F.: Free energy and the fokker-planck equation. Phys. D 107(2–4), 265–271 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Joly, J.-L., Metivier, G., Rauch, J.: Transparent nonlinear geometric optics and Maxwell-Bloch equations. J. Differ. Equ. 166(1), 175–250 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jakšić, V., Pillet, C.-A., Westrich, M.: Entropic fluctuations of quantum dynamical semigroups. J. Stat. Phys. 154(1–2), 153–187 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kossakowski, A., Frigerio, A., Gorini, V., Verri, M.: Quantum detailed balance and KMS condition. Commun. Math. Phys. 57(2), 97–110 (1977). Erratum. CMP 60, 96 (1978)Google Scholar
  20. 20.
    Kubo, R.: Some aspects of the statistical-mechanical theory of irreversible processes. In: Brittin, W.E., Dunham, L.G. (eds.) Lectures in Theoretical Physics. Interscience Publishers, New York (1959)Google Scholar
  21. 21.
    Lüdge, K., Malić, E., Schöll, E.: The role of decoupled electron and hole dynamics in the turn-on behavior of semiconductor quantum-dot lasers. In: Caldas, M.J., Studart, N. (eds.) 29th Conference on the Physics of Semiconductors (2009)Google Scholar
  22. 22.
    Maas, J.: Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261, 2250–2292 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mielke, A.: Formulation of thermoelastic dissipative material behavior using GENERIC. Contin. Mech. Thermodyn. 23(3), 233–256 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mielke, A.: A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24, 1329–1346 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mielke, A.: Dissipative quantum mechanics using GENERIC. In: Johann, A., Kruse, H.-P., Rupp, F., Schmitz, S. (eds.) Recent Trends in Dynamical Systems, pp. 555–586. Springer Verlag (2013). Proceedings of a Conference in Honor of Jürgen ScheurleGoogle Scholar
  26. 26.
    Mielke, A.: Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial Differ. Equ. 48(1), 1–31 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mielke, A.: On thermodynamical couplings of quantum mechanics and macroscopic systems. In: Exner, P., önig, W.K., Neidhardt, H. (eds.) Mathematical Results in Quantum Mechanics, pp. 331–348, Singapore (2015). World Scientific. Proceedings of the QMath12 ConferenceGoogle Scholar
  28. 28.
    Mielke, A., Mittnenzweig, M., Rotundo, N.: On a thermodynamically consistent coupling of quantum systems to reaction-rate equation. In preparation (2017)Google Scholar
  29. 29.
    Mielke, A., Thomas, M.: GENERIC—A powerful tool for thermomechanical modeling. In preparation (2016)Google Scholar
  30. 30.
    Morrison, P.J.: Bracket formulation for irreversible classical fields. Phys. Lett. A 100(8), 423–427 (1984)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Morrison, P.J.: A paradigm for joined Hamiltonian and dissipative systems. Phys. D 18(1–3), 410–419 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Morrison, P.J.: Thoughts on brackets and dissipation: old and new. J. Phys. 169, 012006 (2009)Google Scholar
  33. 33.
    Mielke, A., Peletier, M.A., Renger, D.R.M.: On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion. Potential Anal. 41(4), 1293–1327 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Onsager, L.: Reciprocal relations in irreversible processes, I+II. Phys. Rev., 37, 405–426 (1931). (part II, 38:2265–2279)Google Scholar
  35. 35.
    Öttinger, H.C., Grmela, M.: Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys. Rev. E 56(6), 6633–6655 (1997)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Öttinger, H.C.: The nonlinear thermodynamic quantum master equation. Phys. Rev. A 82, 052119 (2010)ADSCrossRefGoogle Scholar
  37. 37.
    Öttinger, H.C.: The geometry and thermodynamics of dissipative quantum systems. Europhys. Lett. 94, 10006 (2011)CrossRefGoogle Scholar
  38. 38.
    Ritter, S., Gartner, P., Gies, C., Jahnke, F.: Emission properties and photon statistics of a single quantum dot laser. Opt. Express 18(10), 9909–9921 (2010)ADSCrossRefGoogle Scholar
  39. 39.
    Spohn, H.: Entropy production for quantum dynamical semigroups. J. Math. Phys. 19(5), 1227–1230 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Wilcox, R.M.: Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8(4), 962–982 (1967)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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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