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 throughη G vs.G η S 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 ofη G 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.
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This research was supported by the Physics Branch of the Office of Naval Research under contract N 00014-67-A-0077-0015.
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Liboff, R.L. Gibbs vs. Shannon entropies. J Stat Phys 11, 343–357 (1974). https://doi.org/10.1007/BF01009793
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DOI: https://doi.org/10.1007/BF01009793