Entropy is Not Extensive

The Gibbs Paradox (concerning the entropy of mixing and entropic extensivity) was explored in depth by Edwin Jaynes (1992). We take up Jaynes’ treatment, considering the special cases for which entropy is (approximately) extensive, and the general case in which it is not. We also explore the Holographic Principle which (strictly speaking) excludes the extensivity of entropy. The formalism of Quantitative Geometrical Thermodynamics shows that, being isomorphic to energy, it is entropy production (not entropy) that is extensive. As a corollary, Shannon information is also not extensive, although information production is extensive.