Gibbs vs. Shannon entropies

The Gibbs neg-entropy -η_G=∫ II ln II is compared to the Shannon negentropy η_s=∑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 η_G is given referring it to the maximum information contained in the solution to the Liouville equation. The physical relevance ofη _G over η_S 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.