Extensivity of entropy and modern form of Gibbs paradox


The extensivity property of entropy is clarified in the light of a critical examination of the entropy formula based on quantum statistics and the relevant thermodynamic requirement. The modern form of the Gibbs paradox, related to the discontinuous jump in entropy due to identity or non-identity of particles, is critically investigated. Qualitative framework of a new resolution of this paradox, which analyses the general effect of distinction mark on the Hamiltonian of a system of identical particles, is outlined.

This is a preview of subscription content, log in to check access.


  1. Bridgman P W 1961 The Nature of thermodynamics (University Press: Cambridge, Mass p. 169

    Google Scholar 

  2. Casper B M and Freier S 1973 Am. J. Phys. 41 509

    Article  ADS  Google Scholar 

  3. Ehrenfest O P and Trkal V 1921 Ann. Physik 65 609

    Article  ADS  Google Scholar 

  4. Fong P 1963 Foundations of thermodynamics (New York: Oxford University Press)

    Google Scholar 

  5. Fowler R H 1966 Statistical mechanics (London: Cambridge University Press) pp. 205–206

    Google Scholar 

  6. Gibbs J W 1931 The collected works (London: Longman, Green) Part I, pp. 187–207

    Google Scholar 

  7. Huang K 1963 Statistical mechanics (New York: John Wiley) pp. 153–154

    Google Scholar 

  8. Jackson E A 1968 Equilibrium statistical mechanics (Englewood Cliffs, N J: Prentice Hall, pp. 229–231

    Google Scholar 

  9. Klein M J 1958 Am. J. Phys. 26 80

    MATH  Article  ADS  Google Scholar 

  10. Lande A 1960 From dualism to unity in quantum physics (London: Cambridge University Press)

    Google Scholar 

  11. Landsberg P T 1961 Thermodynamics with quantum statistical illustrations (New York: Interscience Publishers) pp. 128–142

    Google Scholar 

  12. Landsberg P T and Tranah D 1978 Am. J. Phys. 46 228

    Article  ADS  Google Scholar 

  13. Landsberg P T and Tranah D 1980 Collect. phenom. 3 73

    MathSciNet  Google Scholar 

  14. Münster A 1969 Statistical thermodynamics (Berlin-Springer-Verlag) Vol. 1, pp. 212–256

    Google Scholar 

  15. Penrose O 1970 Foundations of statistical mechanics (New York: Pergamon Press) pp. 171–172

    Google Scholar 

  16. Reif F 1965 Fundamentals of statistical and thermal physics (New York: McGraw-Hill), pp. 243–244

    Google Scholar 

  17. Schrödinger E 1967 Statistical thermodynamics (London: Cambridge University Press)

    Google Scholar 

  18. Sommerfeld A 1965 Thermodynamics and statistical mechanics (New York: Academic Press)

    Google Scholar 

  19. Sudarshan E C G and Mehra J 1970 Int. J. Theor. Phys. 3 245

    Article  Google Scholar 

  20. Von Neumann J 1955 Mathematical foundations of quantum mechanics (Princeton, N.J: Princeton University Press), pp. 370–379

    Google Scholar 

  21. Wright P G 1970 Proc. R. Soc. London A317 487

    ADS  Google Scholar 

  22. Yourgrau W, Merwe A and Raw G 1966 A treatise on irreversible and statistical thermophysics (New York: Macmillan), pp. 235–241

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to D Home.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Home, D., Sengupta, S. Extensivity of entropy and modern form of Gibbs paradox. Pramana - J. Phys 17, 509–514 (1981). https://doi.org/10.1007/BF02848159

Download citation


  • Extensivity of entropy
  • Gibbs paradox
  • distinguishability
  • identical particles
  • Hamiltonian