Geometric Entropies of Mixing (EOM)

TL;DR

This paper introduces geometric entropies based on trigonometric functions, linking them to convex geometry and phase functions, revealing new inequalities and extremal properties for regular polygons and convex sets.

Contribution

It presents novel geometric entropies of mixing, connecting them to convex body theory and phase functions, and explores their extremal and inequality properties.

Findings

Maximum entropy occurs with inscribed regular n-gons.

Circumscribed regular n-gons minimize entropy change.

Entropies relate to convex set inequalities and phase functions.

Abstract

Trigonometric and trigonometric-algebraic entropies are introduced. Regularity increases the entropy and the maximal entropy is shown to result when a regular $n$-gon is inscribed in a circle. A regular $n$-gon circumscribing a circle gives the largest entropy reduction, or the smallest change in entropy from the state of maximum entropy which occurs in the asymptotic infinite $n$ limit. EOM are shown to correspond to minimum perimeter and maximum area in the theory of convex bodies, and can be used in the prediction of new inequalities for convex sets. These expressions are shown to be related to the phase functions obtained from the WKB approximation for Bessel and Hermite functions.