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.
Categories: cond-mat.stat-mech