# Route from discreteness to the continuum for the non-logarithmic $q$-entropy

## TL;DR

This paper demonstrates that the discrete nonlogarithmic q-entropy converges to a well-defined continuous form, establishing its relevance for classical systems and linking it to known relative entropy measures.

## Contribution

It proves the continuum limit of the nonlogarithmic q-entropy and connects it to the Csiszár type q-relative entropy, resolving a key open problem.

## Key findings

- Discrete q-entropy converges to a continuous form.
- Negative q-entropy with continuous variables relates to q-relative entropy.
- No fundamental obstacle exists for applying q-entropy to continuous systems.

## Abstract

The existence and exact form of the continuum expression of the discrete nonlogarithmic $q$-entropy is an important open problem in generalized thermostatistics, since its possible lack implies that nonlogarithmic $q$-entropy is irrelevant for the continuous classical systems. In this work, we show how the discrete nonlogarithmic $q$-entropy in fact converges in the continuous limit and the negative of the $q$-entropy with continuous variables is demonstrated to lead to the (Csisz{\'a}r type) $q$-relative entropy just as the relation between the continuous Boltzmann-Gibbs expression and the Kullback-Leibler relative entropy. As a result, we conclude that there is no obstacle for the applicability of the $q$-entropy to the continuous classical physical systems.