Reply to the comment on "Route from discreteness to the continuum for the Tsallis $q$-entropy" by Congjie Ou and Sumiyoshi Abe

TL;DR

This paper discusses the convergence of discrete $q$-entropy to the continuous case and addresses concerns about its expandability, showing that it depends on the origin of the convergence factor.

Contribution

It clarifies the conditions under which the discrete $q$-entropy converges and satisfies the expandability axiom, responding to previous critiques.

Findings

Discrete $q$-entropy can converge to the continuous case with proper factors.

Expandability of the discrete $q$-entropy depends on the convergence factor origin.

An example demonstrates when $ ilde{S}_q^{(n)}$ is expandable.

Abstract

It has been known for some time that the usual $q$-entropy $S_q^{(n)}$ cannot be shown to converge to the continuous case. In [Phys. Rev. E 97 (2018) 012104], we have shown that the discrete $q$-entropy $\widetilde{S}_q^{(n)}$ converges to the continuous case when the total number of states are properly taken into account in terms of a convergence factor. Ou and Abe [Phys. Rev. E 97, (2018) 066101, arXiv:1801.03035] noted that this form of the discrete $q$-entropy does not conform to the Shannon-Khinchin expandability axiom. As a reply, we note that the fulfillment or not of the expandability property by the discrete $q$-entropy strongly depends on the origin of the convergence factor, presenting an example in which $\widetilde{S}_q^{(n)}$ is expandable.