Nonadditive entropies and statistical mechanics - some open mathematical problems

Textbook statistical mechanics is grounded on the additive Boltzmann-Gibbs-von Neumann-Shannon entropic functional SBG = k PW i=1 pi ln p1i . The possibility of generalizing the entire BG theory was advanced in 1988 by generalizing SBG into nonadditive entropic functionals such as S q = k 1−PW i=1 pq i q−1 (q 2 R) and, later on, such as Sδ = k PW i=1 pihln p1i iδ (δ 2 R+). A whole saga followed through successful applications in physics, mathematics, chemistry, engineering, medicine, computational sciences, economics, linguistics, theory of networks, to name but a few. Along this evolution, more than fifty nonadditive entropic functionals have been introduced and studied in the literature. Very few, however, have proved to be neatly useful in natural sciences. The present status of the various entropic functionals that have been advanced for complex systems is briefly reviewed, a function generalizing the q-logarithmic one is defined, and some open mathematical problems are focused on