Journal of Statistical Physics

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

Gibbs vs. Shannon entropies

  • Richard L. Liboff
Articles

Abstract

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 

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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

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