Entropy and its discontents: A note on definitions

TL;DR

This paper identifies inconsistencies in standard entropy definitions and proposes modifications to make them coherent and compatible through renormalization, with slight property differences.

Contribution

It introduces revised entropy measures that resolve incoherencies and establish a coherent relationship via renormalization.

Findings

Proposed modified entropy definitions are more consistent.

Modified entropies smoothly transition in the appropriate limits.

New properties slightly differ from traditional entropy measures.

Abstract

The routine definitions of both entropy, and differential entropy show inconsistencies that make them not reciprocally coherent. We propose a few possible modifications of these quantities so that 1) they no longer show incongruities, 2) they go one into the other in a suitable limit as the result of a renormalization. The properties of the new quantities would slightly differ from that of the usual entropies in a few other respects