The European Physical Journal Special Topics

, Volume 226, Issue 4, pp 705–723 | Cite as

Comparison of Boltzmann and Gibbs entropies for the analysis of single-chain phase transitions

  • T. Shakirov
  • S. Zablotskiy
  • A. Böker
  • V. Ivanov
  • W. PaulEmail author
Regular Article
Part of the following topical collections:
  1. Recent Advances in Phase Transitions and Critical Phenomena


In the last 10 years, flat histogram Monte Carlo simulations have contributed strongly to our understanding of the phase behavior of simple generic models of polymers. These simulations result in an estimate for the density of states of a model system. To connect this result with thermodynamics, one has to relate the density of states to the microcanonical entropy. In a series of publications, Dunkel, Hilbert and Hänggi argued that it would lead to a more consistent thermodynamic description of small systems, when one uses the Gibbs definition of entropy instead of the Boltzmann one. The latter is the logarithm of the density of states at a certain energy, the former is the logarithm of the integral of the density of states over all energies smaller than or equal to this energy. We will compare the predictions using these two definitions for two polymer models, a coarse-grained model of a flexible-semiflexible multiblock copolymer and a coarse-grained model of the protein poly-alanine. Additionally, it is important to note that while Monte Carlo techniques are normally concerned with the configurational energy only, the microcanonical ensemble is defined for the complete energy. We will show how taking the kinetic energy into account alters the predictions from the analysis. Finally, the microcanonical ensemble is supposed to represent a closed mechanical N-particle system. But due to Galilei invariance such a system has two additional conservation laws, in general: momentum and angular momentum. We will also show, how taking these conservation laws into account alters the results.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B.A. Berg, Fields Inst. Comm. 26, 1 (2000)Google Scholar
  2. 2.
    B.A. Berg, Comp. Phys. Comm. 147, 52 (2002)ADSCrossRefGoogle Scholar
  3. 3.
    W. Janke, Physica A 254, 164 (1998)ADSCrossRefGoogle Scholar
  4. 4.
    W. Janke, Lect. Notes Phys. 739, 79 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    F. Wang, D.P. Landau, Phys. Rev. Lett. 86, 2050 (2001)ADSCrossRefGoogle Scholar
  6. 6.
    F. Wang, D.P. Landau, Phys. Rev. E 64, 056101 (2001)ADSCrossRefGoogle Scholar
  7. 7.
    T. Wuest, Y.W. Li, D.P. Landau, J. Stat. Phys. 144, 638 (2011)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    F. Liang, J. Stat. Phys. 122, 511 (2006)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    F. Liang, C. Liu, R.J. Carroll, J. Amer. Stat. Ass. 102, 305 (2007)CrossRefGoogle Scholar
  10. 10.
    F. Liang, Statist. Prob. Lett. 79, 581 (2009)CrossRefGoogle Scholar
  11. 11.
    D.H.E. Gross, Microcanonical Thermodynamics: Phase Transitions in “Small” Systems (World Scientific, Singapore, 2001)Google Scholar
  12. 12.
    C. Junghans, M. Bachmann, W. Janke, Phys. Rev. Lett. 97, 218103 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    W. Janke, Nucl. Phys. B (Proc. Suppl.) 63A-C, 631 (1998)ADSCrossRefGoogle Scholar
  14. 14.
    W. Paul, F. Rampf, T. Strauch, K. Binder, Comp. Phys. Commun. 178, 17 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    W. Janke, W. Paul, Soft Matter 12, 642 (2016)ADSCrossRefGoogle Scholar
  16. 16.
    R.S. Ellis, Entropy, Large Deviations and Statistical Mechanics (Springer, Berlin, 2006)Google Scholar
  17. 17.
    H. Touchette, Phys. Rep. 478, 1 (2009)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    P. Hertz, Ann. Phys. 33, 225 (1910)CrossRefGoogle Scholar
  19. 19.
    P. Hertz, Ann. Phys. 33, 537 (1910)CrossRefGoogle Scholar
  20. 20.
    J.W. Gibbs, Elementary principles of statistical mechanics (Charles Scribner's Sons, New York, 1902)Google Scholar
  21. 21.
    J. Dunkel, S. Hilbert, Physica A 370, 390 (2006)ADSCrossRefGoogle Scholar
  22. 22.
    S. Hilbert, J. Dunkel, Phys. Rev. E 74, 011120 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    J. Dunkel, S. Hilbert, Nature Phys. 10, 67 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    S. Hilbert, P. Hänggi, J. Dunkel, Phys. Rev. E 90, 062116 (2014)ADSCrossRefGoogle Scholar
  25. 25.
    P. Hänggi, S. Hilbert, J. Dunkel, Phil. Trans. Roy. Soc. A 374, 20150039 (2016)CrossRefGoogle Scholar
  26. 26.
    R.H. Swendsen, J.S. Wang, Phys. Rev. E 92, 020103 (2015)ADSCrossRefGoogle Scholar
  27. 27.
    D. Frenkel, P. Warren, Am. J. Phys. 83, 163 (2015)ADSCrossRefGoogle Scholar
  28. 28.
    P. Ehrenfest, Phil. Mag. 23, 500 (1917)CrossRefGoogle Scholar
  29. 29.
    T. Levi-Civita, Abhandl. Mathem. Semin. Hamburg 6, 323 (1928)CrossRefGoogle Scholar
  30. 30.
    A.I. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover Publications, New York, 1949)Google Scholar
  31. 31.
    V.L. Berdichevskii, J. Appl. Math. Mech. 52, 738 (1988)MathSciNetCrossRefGoogle Scholar
  32. 32.
    P. Schierz, J. Zierenberg, W. Janke, J. Chem. Phys. 143, 134114 (2015)ADSCrossRefGoogle Scholar
  33. 33.
    P. Schierz, J. Zierenberg, W. Janke, Phys. Rev E 94, 021301(R) (2016)ADSCrossRefGoogle Scholar
  34. 34.
    S.V. Zablotskiy, J.A. Martemyanova, V.A. Ivanov, W. Paul, J. Chem. Phys. 144, 244903 (2016)ADSCrossRefGoogle Scholar
  35. 35.
    S.V. Zablotskiy, V.A. Ivanov, W. Paul, Phys. Rev. E 93, 063303 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    S.V. Zablotskiy, J.A. Martemyanova, V.A. Ivanov, W. Paul, Polym. Sci. Ser. A 58, 899 (2016)CrossRefGoogle Scholar
  37. 37.
    M. Cheon, I. Chang, C.K. Hall, Proteins: Struct. Funct. Bioinf. 78, 2950 (2010)CrossRefGoogle Scholar
  38. 38.
    A. Voegler Smith, C.K. Hall, Proteins: Struct. Funct. Bioinf. 44, 344 (2001)CrossRefGoogle Scholar
  39. 39.
    B. Werlich, M.P. Taylor, W. Paul, Phys. Proc. 57, 82 (2014)ADSCrossRefGoogle Scholar
  40. 40.
    B. Werlich, T. Shakirov, M.P. Taylor, W. Paul, Comp. Phys. Commun. 186, 65 (2015)ADSCrossRefGoogle Scholar
  41. 41.
    S. Schnabel, D.T. Seaton, D.P. Landau, M. Bachmann, Phys. Rev. E 84, 011127 (2011)ADSCrossRefGoogle Scholar
  42. 42.
    J. Zierenberg, W. Janke, Phys. Rev. E 92, 012134 (2015)ADSCrossRefGoogle Scholar
  43. 43.
    M.P. Taylor, K. Isik, J. Luettmer-Strathmann, Phys. Rev. E 78, 051805 (2008)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    V. Sadovnichy, A. Tikhonravov, V. Voevodin, V. Opanasenko, in “Contemporary High Performance Computing: From Petascale toward Exascale” (Chapman & Hall/CRC Computational Science, Boca Raton, USA, CRC Press, 2013), pp. 283–307Google Scholar

Copyright information

© EDP Sciences and Springer 2017

Authors and Affiliations

  • T. Shakirov
    • 1
  • S. Zablotskiy
    • 2
  • A. Böker
    • 1
  • V. Ivanov
    • 2
  • W. Paul
    • 1
    Email author
  1. 1.Institut für Physik, Martin-Luther-Universität Halle-WittenbergHalle (Saale)Germany
  2. 2.Faculty of Physics, Moscow State UniversityMoscowRussia

Personalised recommendations