# Composition law of $\kappa$-entropy for statistically independent systems

## TL;DR

This paper derives a closed-form composition law for the $$-entropy, a generalization of Shannon entropy, for independent systems, revealing how joint entropy relates to individual entropies and extending classical additivity.

## Contribution

It provides the first explicit composition law for $$-entropy of independent systems, generalizing the classical additivity law and involving the $$-Napier number.

## Key findings

- Derived a closed-form composition law for $$-entropy.
- Showed the law reduces to Shannon entropy in the limit $ o 0$.
- Extended the additivity principle to $$-entropy for independent systems.

## Abstract

The intriguing and still open question concerning the composition law of $\kappa$-entropy $S_{\kappa}(f)=\frac{1}{2\kappa}\sum_i (f_i^{1-\kappa}-f_i^{1+\kappa})$ with $0<\kappa<1$ and $\sum_i f_i =1$ is here reconsidered and solved. It is shown that, for a statistical system described by the probability distribution $f=\{ f_{ij}\}$, made up of two statistically independent subsystems, described through the probability distributions $p=\{ p_i\}$ and $q=\{ q_j\}$, respectively, with $f_{ij}=p_iq_j$, the joint entropy $S_{\kappa}(p\,q)$ can be obtained starting from the $S_{\kappa}(p)$ and $S_{\kappa}(q)$ entropies, and additionally from the entropic functionals $S_{\kappa}(p/e_{\kappa})$ and $S_{\kappa}(q/e_{\kappa})$, $e_{\kappa}$ being the $\kappa$-Napier number. The composition law of the $\kappa$-entropy is given in closed form, and emerges as a one-parameter generalization of the ordinary additivity law of Boltzmann-Shannon entropy recovered in the $\kappa \rightarrow 0$ limit.