Journal of Statistical Physics

, Volume 153, Issue 3, pp 412–441 | Cite as

Geometrical Excess Entropy Production in Nonequilibrium Quantum Systems

  • Tatsuro Yuge
  • Takahiro Sagawa
  • Ayumu Sugita
  • Hisao Hayakawa


For open systems described by the quantum Markovian master equation, we study a possible extension of the Clausius equality to quasistatic operations between nonequilibrium steady states (NESSs). We investigate the excess heat divided by temperature (i.e., excess entropy production) which is transferred into the system during the operations. We derive a geometrical expression for the excess entropy production, which is analogous to the Berry phase in unitary evolution. Our result implies that in general one cannot define a scalar potential whose difference coincides with the excess entropy production in a thermodynamic process, and that a vector potential plays a crucial role in the thermodynamics for NESSs. In the weakly nonequilibrium regime, we show that the geometrical expression reduces to the extended Clausius equality derived by Saito and Tasaki (J. Stat. Phys. 145:1275, 2011). As an example, we investigate a spinless electron system in quantum dots. We find that one can define a scalar potential when the parameters of only one of the reservoirs are modified in a non-interacting system, but this is no longer the case for an interacting system.


Nonequilibrium steady state Entropy production Clausius equality Geometrical phase Quantum Markovian master equation 



The authors thank Keiji Saito for his helpful advice. This work was supported by a JSPS Research Fellowship for Young Scientists (No. 24-1112), a Grant-in-Aid for Research Activity Start-up (KAKENHI 11025807), and a Grant-in-Aid (KAKENHI 25287098). A part of this study was performed when TY and TS were in the Yukawa Institute for Theoretical Physics.


  1. 1.
    Agarwal, G.S.: Open quantum Markovian systems and the microreversibility. Z. Phys. 258, 409–422 (1973) ADSCrossRefGoogle Scholar
  2. 2.
    Alicki, R.: On the detailed balance condition for non-Hamiltonian systems. Rep. Math. Phys. 10, 249–258 (1976) MathSciNetADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984) ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Towards a nonequilibrium thermodynamics: a self-contained macroscopic description of driven diffusive systems. J. Stat. Phys. 135, 857–872 (2009) MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Bertini, L., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Thermodynamic transformations of nonequilibrium states. J. Stat. Phys. 149, 773–802 (2012) MathSciNetADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Bertini, L., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Clausius inequality and optimality of quasistatic transformations for nonequilibrium stationary states. Phys. Rev. Lett. 110, 020601 (2013) ADSCrossRefGoogle Scholar
  7. 7.
    Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, London (2002) zbMATHGoogle Scholar
  8. 8.
    Brouwer, P.W.: Scattering approach to parametric pumping. Phys. Rev. B 58, R10135 (1998) ADSCrossRefGoogle Scholar
  9. 9.
    De Groot, S.R., Mazur, P.: Non-Equilibrium Thermodynamics. North-Holland, Amsterdam (1962). Dover, New York (1984) Google Scholar
  10. 10.
    Deffner, S., Lutz, E.: Generalized Clausius inequality for nonequilibrium quantum processes. Phys. Rev. Lett. 105, 170402 (2010) ADSCrossRefGoogle Scholar
  11. 11.
    Esposito, M., Harbola, U., Mukamel, S.: Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81, 1665 (2009) MathSciNetADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Gardiner, C.W., Zoller, P.: Quantum Noise. Springer, Berlin, Heidelberg (2004) zbMATHGoogle Scholar
  13. 13.
    Gaspard, P., Nagaoka, M.: Slippage of initial conditions for the Redfield master equation. J. Chem. Phys. 111, 5668 (1999) ADSCrossRefGoogle Scholar
  14. 14.
    Hatano, T., Sasa, S.: Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463 (2001) ADSCrossRefGoogle Scholar
  15. 15.
    Komatsu, T.S., Nakagawa, N., Sasa, S., Tasaki, H.: Steady state thermodynamics for heat conduction—microscopic derivation. Phys. Rev. Lett. 100, 230602 (2008) ADSCrossRefGoogle Scholar
  16. 16.
    Komatsu, T.S., Nakagawa, N., Sasa, S., Tasaki, H.: Entropy and nonlinear nonequilibrium thermodynamic relation for heat conducting steady states. J. Stat. Phys. 142, 127–153 (2010) MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Kouwenhoven, L.P., Johnson, A.T., van der Vaart, N.C., Harmans, C.J.P.M., Foxon, C.T.: Quantized current in a quantum-dot turnstile using oscillating tunnel barriers. Phys. Rev. Lett. 67, 1626 (1991) ADSCrossRefGoogle Scholar
  18. 18.
    Landauer, R.: dQ=TdS far from equilibrium. Phys. Rev. A 18, 255–266 (1978) ADSCrossRefGoogle Scholar
  19. 19.
    Nakagawa, N.: Work relation and the second law of thermodynamics in nonequilibrium steady states. Phys. Rev. E 85, 051115 (2012) ADSCrossRefGoogle Scholar
  20. 20.
    Oono, Y., Paniconi, M.: Steady state thermodynamics. Prog. Theor. Phys. Suppl. 130, 29–44 (1998) MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Prigogine, I.: Thermodynamics of Irreversible Processes. Wiley, New York (1967) Google Scholar
  22. 22.
    Ruelle, D.: Extending the definition of entropy to nonequilibrium steady states. Proc. Natl. Acad. Sci. USA 100, 3054–3058 (2003) MathSciNetADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Sagawa, T., Hayakawa, H.: Geometrical expression of excess entropy production. Phys. Rev. E 84, 051110 (2011) ADSCrossRefGoogle Scholar
  24. 24.
    Saito, K., Tasaki, H.: Extended Clausius relation and entropy for nonequilibrium steady states in heat conducting quantum systems. J. Stat. Phys. 145, 1275–1290 (2011) MathSciNetADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Sasa, S., Tasaki, H.: Steady state thermodynamics. J. Stat. Phys. 125, 125–224 (2006) MathSciNetADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Sinitsyn, N.A., Nemenman, I.: The Berry phase and the pump flux in stochastic chemical kinetics. Europhys. Lett. 77, 58001 (2007) MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Spohn, H., Lebowitz, J.L.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 38, 109 (1978) CrossRefGoogle Scholar
  28. 28.
    Switkes, M., Marcus, C.M., Campman, K., Gossard, A.C.: An adiabatic quantum electron pump. Science 283, 1905 (1999) ADSCrossRefGoogle Scholar
  29. 29.
    Takara, K., Hasegawa, H.-H., Driebe, D.J.: Generalization of the second law for a transition between nonequilibrium states. Phys. Lett. A 375, 88–92 (2010) MathSciNetADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Thouless, D.J.: Quantization of particle transport. Phys. Rev. B 27, 6083 (1983) MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Wichterich, H., Henrich, M.J., Breuer, H.P., Gemmer, J., Michel, M.: Modeling heat transport through completely positive maps. Phys. Rev. E 76, 031115 (2007) MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Yuge, T., Sagawa, T., Sugita, A., Hayakawa, H.: Geometrical pumping in quantum transport: quantum master equation approach. Phys. Rev. B 86, 235308 (2012) ADSCrossRefGoogle Scholar
  33. 33.
    Zubarev, D.N.: Nonequilibrium Statistical Thermodynamics. Consultants Bureau, New York (1974) Google Scholar
  34. 34.
    Zulkowski, P.R., Sivak, D.A., Crooks, G.E., DeWeese, M.R.: Geometry of thermodynamic control. Phys. Rev. E 86, 041148 (2012) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Tatsuro Yuge
    • 1
  • Takahiro Sagawa
    • 2
    • 3
    • 4
  • Ayumu Sugita
    • 5
  • Hisao Hayakawa
    • 3
  1. 1.Department of PhysicsOsaka UniversityToyonakaJapan
  2. 2.The Hakubi CenterKyoto UniversityKyotoJapan
  3. 3.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan
  4. 4.Department of Basic ScienceThe University of TokyoTokyoJapan
  5. 5.Department of Applied PhysicsOsaka City UniversityOsakaJapan

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