Journal of Statistical Physics

, Volume 11, Issue 4, pp 343–357 | Cite as

Gibbs vs. Shannon entropies

  • Richard L. Liboff


The Gibbs neg-entropy -ηG=∫ II ln II is compared to the Shannon negentropy ηs=∑p Inp. The coarse-grained density is II, whilep is a probability sequence. Both objects are defined over partitions of the energy shell within a set-theoretic framework. The dissimilarity of these functionals is exhibited throughηG vs.GηS curves. A positive information interpretation of ηG is given referring it to the maximum information contained in the solution to the Liouville equation. The physical relevance ofηG over ηS in classical physics is argued. In quantum mechanics, the fine-grained Shannon entropy remains relevant to the uncertainty principle, while the coarsegrained densities maintain their properties as in the classical case.

Key words

Coarse-grained density density operator energy shell ensemble points fine-grained density Gibbs-Ehrenfest theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Katz, Principles of Statistical Mechanics, W. H. Freeman, San Francisco (1967); E. A. Jackson, Equilibrium Statistical Mechanics, Prentice Hall, Englewood Cliffs, New Jersey (1968); L. Brillouin, Science and Information Theory, Academic Press, New York (1956); M. Tribus, Thermostatics and Thermodynamics, Von Nostrand, Princeton, New Jersey (1961); D. ter Haar, Elements of Statistical Mechanics, Holt, Rinehart and Winston, New York (1954); E. T. Jaynes,Phys. Rev. 106:620 (1957).Google Scholar
  2. 2.
    R. L. Liboff,Introduction to the Theory of Kinetic Equations, John Wiley, New York (1969).Google Scholar
  3. 3.
    J. W. Gibbs, Elementary Principles in Statistical Mechanics (Vol. II of his collected works), Chapter XII, New Haven (1948).Google Scholar
  4. 4.
    C. E. Shannon,Bell. Syst. Tech. J. 27(I):379 (1948);Proc. IRE 37:10 (1949).Google Scholar
  5. 5.
    H. R. Raemer,Statistical Communication Theory and Applications, Prentice Hall, Englewood Cliffs, New Jersey (1969).Google Scholar
  6. 6.
    P. Ehrenfest and T. Ehrenfest, in Encyklopadie der Mathematischen Wissenshaften IV, part 32, Leipzig-Berlin (1911);Physik Z. 8:311 (1907).Google Scholar
  7. 7.
    M. Kac,Probability and Related Topics in Physical Sciences, Interscience, New York (1959).Google Scholar
  8. 8.
    D. ter Haar,Rev. Mod. Phys. 27:289 (1955).Google Scholar
  9. 9.
    V. Fano,Rev. Mod. Phys. 29:74 (1957).Google Scholar
  10. 10.
    W. Pauli, in Sommerfeld Festschrift, Leipzig (1928).Google Scholar
  11. 11.
    R. C. Tolman, The Principles of Statistical Mechanics, Oxford, New York (1938).Google Scholar
  12. 12.
    O. Klein,Z. Physik 72:767 (1931).Google Scholar
  13. 13.
    K. Gottfried,Quantum Mechanics, Vol. 1, Chapters IV and V, Benjamin, New York (1966).Google Scholar

Copyright information

© Plenum Publishing Corporation 1974

Authors and Affiliations

  • Richard L. Liboff
    • 1
  1. 1.Department of Applied Physics and School of Electrical EngineeringCornell UniversityIthaca

Personalised recommendations