Deformed Mathematical Objects Stemming from the q-Logarithm Function

Reminiscences of Half a Century of Life in the World of Theoretical Physics

Selma Lagerlöf said that culture is what remains when one has forgotten everything we had learned. Without any warranty, through ongoing research tasks, that I will ever attain this high level of wisdom, I simply share here reminiscences that have played, during my life, an important role in my incursions in science, mainly in theoretical physics. I end by presenting some perspectives for future developments.

Senses along Which the Entropy Sq Is Unique

The Boltzmann-Gibbs-von Neumann-Shannon additive entropy SBG=-k∑ipilnpi as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988 into the nonextensive statistical mechanics-as currently referred to-grounded on the nonadditive entropy Sq=k1-∑ipiqq-1 as well as its corresponding continuous and quantum counterparts. In the literature, there exist nowadays over fifty mathematically well defined entropic functionals. Sq plays a special role among them. Indeed, it constitutes the pillar of a great variety of theoretical, experimental, observational and computational validations in the area of complexity-plectics, as Murray Gell-Mann used to call it. Then, a question emerges naturally, namely In what senses is entropy Sq unique? The present effort is dedicated to a-surely non exhaustive-mathematical answer to this basic question.

Enthusiasm and Skepticism: Two Pillars of Science—A Nonextensive Statistics Case

Science and its evolution are based on complex epistemological structures. Two of the pillars of such a construction definitively are enthusiasm and skepticism, both being ingredients without which solid knowledge is hardly achieved and certainly not guaranteed. Our friend and colleague Jean Willy André Cleymans (1944–2021), with his open personality, high and longstanding interest for innovation, and recognized leadership in high-energy physics, constitutes a beautiful example of the former. Recently, Joseph I. Kapusta has generously and laboriously offered an interesting illustration of the latter pillar, in the very same field of physics, concerning the very same theoretical frame, namely, nonextensive statistical mechanics and the nonadditive q-entropies on which it is based. I present here a detailed analysis, point by point, of Kapusta’s 19 May 2021 talk and, placing the discussion in a sensibly wider and updated perspective, I refute his bold conclusion that indices q have no physical foundation.