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

Abstract

The intriguing and still open question concerning the composition law of $\kappa$-entropy $S_{\kappa }(f)=\frac{1}{2\kappa }{\sum }_i(f_i^{1-\kappa }-f_i^{1+\kappa })$ 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 $f_{ij}=p_i{q}_j$, the joint entropy $S_{\kappa }(pq)$ 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 $\kappa \to 0$ limit.

arXiv comment14 pages

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