Unitary Proof of the Second Law of Thermodynamics

Part of the Springer Theses book series (Springer Theses)


In this chapter, we review how to derive the second law of thermodynamics for systems that obey quantum mechanics at the microscopic level. Starting with the statement of the second law, we derive it based on quantum statistical mechanics [1, 2, 3, 4, 5]. We formulate the theory such that the total system of the thermodynamic system and the heat baths obey the unitary evolution, and assume that the initial states of the heat baths are in the canonical distribution. Mathematically, our derivation is based on the Klein inequality (or equivalently, the positivity of the quantum relative entropy). The reason why the second law can be derived from the reversible unitary evolution is due to the fact that we select the canonical distributions as the initial states.


Heat Bath External Parameter Helmholtz Free Energy Unitary Evolution Thermodynamic System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    J. Kurchan, arXiv:cond-mat/0007360 (2000).Google Scholar
  2. 2.
    H. Tasaki, arXiv:cond-mat/0009244 (2000).Google Scholar
  3. 3.
    C. Jarzynski, D.K. Wójcik, Phys. Rev. Lett. 92, 230602 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    T. Sagawa, Prog. Theor. Phys. 127, 1 (2012)ADSMATHCrossRefGoogle Scholar
  5. 5.
    T. Sagawa, Second law-Like inequalities with quantum relative entropy: an introduction. In: Lectures on Quantum Computing, Thermodynamics and Statistical Physics, Kinki University Series on Quantum Computing (World Scientific, Hackensack, 2012) e-print: arXiv:1202.0983 (To appear).Google Scholar
  6. 6.
    S. Carnot, Réflexions sur la pussance motrice du feu et sur les machines propresà développer atte puissance, (Bachelier, 1824).Google Scholar
  7. 7.
    L. Tisza, P.M. Quay, Annal. Phys. 25, 48 (1963)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    E.H. Lieb, J. Yngvason, Phys. Rept. 310, 1 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd edn. (Wiley, New York, 1985).Google Scholar
  10. 10.
    H. Tasaki, Thermodynanmics–From a Modern Point of View (Baifu-kan, Tokyo, 2000) (in Japanese).Google Scholar
  11. 11.
    S. Sasa, Introduction to Thermodynamics (Kyoritsu, 2000), (in Japanese).Google Scholar
  12. 12.
    A. Shimizu, Principles of Thermodynamics (Univ, Tokyo Press, 2007). (in Japanese)Google Scholar
  13. 13.
    M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. Lett. 102, 210401 (2009)ADSCrossRefGoogle Scholar

Copyright information

© Springer Japan 2012

Authors and Affiliations

  1. 1.Kyoto UniversityKyotoJapan

Personalised recommendations