Composition law of κ-entropy for statistically independent systems.

Abstract

The intriguing and still open question concerning the composition law of κ-entropy S_{κ}(f)=1/2κ∑_{i}(f_{i}^{1-κ}-f_{i}^{1+κ}) with 0<κ<1 and ∑_{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_{κ}(pq) 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→0 limit.