Entropies from coarse-graining: convex polytopes vs. ellipsoids

TL;DR

This paper explores how different phase-space coarse-graining methods influence the form of various entropy measures in Hamiltonian systems, highlighting the role of convex geometry and asymptotic estimates.

Contribution

It introduces a geometric perspective on entropy functional forms based on coarse-graining approaches, connecting convexity, symplectic capacities, and Dvoretzky's theorem.

Findings

Different coarse-graining approaches lead to distinct entropy forms.

Dvoretzky's theorem estimates the dimension where approaches converge.

Dualities may influence the relationship between coarse-graining methods.

Abstract

We examine the Boltzmann/Gibbs/Shannon $\mathcal{S}_{BGS}$ and the non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \ $\mathcal{S}_q$ \ and the Kaniadakis $\kappa$-entropy \ $\mathcal{S}_\kappa$ \ from the viewpoint of coarse-graining, symplectic capacities and convexity. We argue that the functional form of such entropies can be ascribed to a discordance in phase-space coarse-graining between two generally different approaches: the Euclidean/Riemannian metric one that reflects independence and picks cubes as the fundamental cells and the symplectic/canonical one that picks spheres/ellipsoids for this role. Our discussion is motivated by and confined to the behaviour of Hamiltonian systems of many degrees of freedom. We see that Dvoretzky's theorem provides asymptotic estimates for the minimal dimension beyond which these two approaches are close to each other. We state and speculate about the role that dualities may play in this viewpoint.