Logarithmic and power-law entropies from convexity

TL;DR

This paper explores the geometric foundations of entropy functions, contrasting logarithmic and power-law forms, and discusses their implications for statistical mechanics and complex systems.

Contribution

It introduces a geometric perspective using convexity, Johnson-Lindenstrauss lemma, and Dvoretzky's theorem to analyze the origins of entropy functionals and suggests power-law entropies for certain systems.

Findings

Convexity-based geometric approach to entropy

Implications of Johnson-Lindenstrauss lemma and Dvoretzky's theorem

Power-law entropies may be suitable for some systems

Abstract

In an attempt to understand the origin and robustness of the Boltzmann/Gibbs/Shannon entropic functional, we adopt a geometric approach and discuss the implications of the Johnson-Lindenstrauss lemma and of Dvoretzky's theorem on convex bodies for the choice of this functional form. We contrast these results with a more recent result on flowers of balls, which may be interpreted as suggesting the use of power-law entropies for some systems.