Composition law of $κ$-entropy for statistically independent systems

Abstract: The intriguing and still open question concerning the composition law of $\kappa$-entropy $$S_{\kappa}(f)=\frac{1}{2\kappa}\sum_i (f_i<sup>{1-\kappa}-f_i<sup>{1+\kappa})$$ with $0&lt;\kappa&lt;1$ 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.