Quantum Hertz entropy increase in a quenched spin chain

  • Darshan G. Joshi
  • Michele CampisiEmail author
Regular Article


The classical Hertz entropy is the logarithm of the volume of phase space bounded by the constant energy surface; its quantum counterpart, the quantum Hertz entropy, is \(\hat S = k_B \ln \hat N\), where the quantum operator \(\hat N\) specifies the number of states with energy below a given energy eigenstate. It has been recently proved that, when an isolated quantum mechanical system is driven out of equilibrium by an external driving, the change in the expectation of its quantum Hertz entropy cannot be negative, and is null for adiabatic driving. This is in full agreement with the Clausius principle. Here, we test the behavior of the expectation of the quantum Hertz entropy in the case when two identical XY spin chains initially at different temperatures are quenched into a single XY chain. We observed no quantum Hertz entropy decrease. This finding further supports the statement that the quantum Hertz entropy is a proper entropy for isolated quantum systems. We further quantify how far the quenched chain is from thermal equilibrium and the temperature of the closest equilibrium.


Statistical and Nonlinear Physics 


  1. 1.
    M. Campisi, P. Hänggi, P. Talkner, Rev. Mod. Phys. 83, 771 (2011)ADSCrossRefGoogle Scholar
  2. 2.
    M. Campisi, P. Hänggi, P. Talkner, Rev. Mod. Phys. 83 1653(E) (2011)ADSGoogle Scholar
  3. 3.
    M. Esposito, U. Harbola, S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    E. Fermi, Thermodynamics (Dover Publ., New York, 1956)Google Scholar
  5. 5.
    A.E. Allahverdyan, T.M. Nieuwenhuizen, Physica A 305, 542 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)ADSCrossRefGoogle Scholar
  7. 7.
    H. Tasaki, arXiv:cond-mat/0009244 (2000)Google Scholar
  8. 8.
    J. Kurchan, arXiv:cond-mat/0007360 (2000)Google Scholar
  9. 9.
    P. Talkner, P. Hänggi, J. Phys. A 40, F569 (2007)ADSzbMATHCrossRefGoogle Scholar
  10. 10.
    A. Polkovnikov, K. Sengupta, A. Silva, M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    A. Polkovnikov, Ann. Phys. 326, 486 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    L.F. Santos, A. Polkovnikov, M. Rigol, Phys. Rev. Lett. 107, 040601 (2011)ADSCrossRefGoogle Scholar
  13. 13.
    H. Tasaki, arXiv:cond-mat/0009206 (2000)Google Scholar
  14. 14.
    M. Campisi, Stud. Hist. Phil. Mod. Phys. 39, 181 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    M. Campisi, Phys. Rev. E 78, 051123 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    P. Hertz, Ann. Phys. (Leipzig) 338, 225 (1910)ADSCrossRefGoogle Scholar
  17. 17.
    M. Campisi, Stud. Hist. Philos. Mod. Phys. 36, 275 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    M. Campisi, D.H. Kobe, Am. J. Phys. 78, 608 (2010)ADSCrossRefGoogle Scholar
  19. 19.
    A.V. Ponomarev, S. Denisov, P. Hänggi, Phys. Rev. Lett. 106, 010405 (2011)ADSCrossRefGoogle Scholar
  20. 20.
    A.V. Ponomarev, S. Denisov, P. Hänggi, J. Gemmer, Europhys. Lett. 98, 40011 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)ADSCrossRefGoogle Scholar
  22. 22.
    M. Campisi, D. Zueco, P. Talkner, Chem. Phys. 375, 187 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    R. Dorner, J. Goold, C. Cormick, M. Paternostro, V. Vedral, Phys. Rev. Lett. 109, 160601 (2012)ADSCrossRefGoogle Scholar
  24. 24.
    S. Hilbert, J. Dunkel, Phys. Rev. E 74, 011120 (2006)ADSCrossRefGoogle Scholar
  25. 25.
    J. Gibbs, Elementary Principles in Statistical Mechanics (Yale U.P., New Haven, 1902)Google Scholar
  26. 26.
    L. Landau, E. Lifschitz, Statistical Physics, 2nd edn. (Pergamon Press, Oxford, 1969)Google Scholar
  27. 27.
    K. Huang, Statistical Mechanics, 2nd edn. (Wiley, New York, 1987)Google Scholar
  28. 28.
    H.B. Callen, Thermodynamics: an introduction to the physical theories of equilibrium thermostatics and irreversible thermodynamics (Wiley, New York, 1960)Google Scholar
  29. 29.
    S. Braun, J.P. Ronzheimer, M. Schreiber, S.S. Hodgman, T. Rom, I. Bloch, U. Schneider, Science 339, 52 (2013)ADSCrossRefGoogle Scholar
  30. 30.
    F. Schlögl, Z. Phys. 191, 81 (1966)ADSCrossRefGoogle Scholar
  31. 31.
    G.N. Bochkov, Y.E. Kuzovlev, Physica A 106, 443 (1981)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    R. Kawai, J.M.R. Parrondo, C.V. den Broeck, Phys. Rev. Lett. 98, 080602 (2007)ADSCrossRefGoogle Scholar
  33. 33.
    S. Deffner, E. Lutz, Phys. Rev. Lett. 105, 170402 (2010)ADSCrossRefGoogle Scholar
  34. 34.
    G.E. Crooks, Phys. Rev. E 60, 2721 (1999)ADSCrossRefGoogle Scholar
  35. 35.
    R. Kubo, H. Ichimura, T. Usui, N. Hashitsume, Statistical Mechanics, 6th edn. (North Holland Publishing, Amsterdam, 1965)Google Scholar
  36. 36.
    N.F. Ramsey, Phys. Rev. 103, 20 (1956)ADSzbMATHCrossRefGoogle Scholar
  37. 37.
    E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961)MathSciNetADSzbMATHCrossRefGoogle Scholar
  38. 38.
    H.J. Mikeska, W. Pesch, Z. Phys. B 26, 351 (1977)ADSCrossRefGoogle Scholar
  39. 39.
    J. Dajka, J. Łuczka, P. Hänggi, Quantum Inf. Process. 10, 85 (2011)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität DresdenDresdenGermany
  2. 2.Institute of PhysicsUniversity of AugsburgAugsburgGermany

Personalised recommendations