Entropies of Mixing and the Lorenz Order

TL;DR

This paper explores how entropies of mixing derived from extreme value theory relate to different entropy types, revealing their ordering and maximal properties in specific distributions.

Contribution

It establishes a connection between entropies of mixing and pseudo-additive and Shannon entropies, demonstrating their ordering via Lorenz order and linking to geometric properties.

Findings

Entropies of mixing correspond to pseudo-additive and Shannon entropies.

Processes with pseudo-additive entropies majorize those with Shannon entropy.

Maximal entropy of mixing relates to regular polygons in the arcsine distribution.

Abstract

Entropies of mixing can be derived directly from the parent distributions of extreme value theory. They correspond to pseudo-additive entropies in the case of Pareto and power function distributions, while to the Shannon entropy in the case of the exponential distribution.The former tend to the latter when their shape parameters tend to infinity and zero, respectively. Hence processes whose entropies of mixing are pseudo-additive entropies majorize, in the Lorenz order sense, those whose entropy is the Shannon entropy. In the case of the arcsine distribution, maximal properties of regular polygons correspond to maximum entropy of mixing.