Composition law of $κ$-entropy for statistically independent systems
The intriguing and still open question concerning the composition law of $κ$-entropy $S_κ(f)=\frac{1}{2κ}\sum_i (f_i^{1-κ}-f_i^{1+κ})$ with $0<κ<1$ 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 $S_κ(p\,q)$ can be obtained starting from the $S_κ(p)$ and $S_κ(q)$ entropies, and additionally from the entropic functionals $S_κ(p/e_κ)$ and $S_κ(q/e_κ)$, $e_κ$ being the $κ$-Napier number. The composition law of the $κ$-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 $κ\rightarrow 0$ limit.