The European Physical Journal Special Topics

, Volume 226, Issue 3, pp 455–466

Generalization of the possible algebraic basis of q-triplets

Regular Article

DOI: 10.1140/epjst/e2016-60159-x

Cite this article as:
Tsallis, C. Eur. Phys. J. Spec. Top. (2017) 226: 455. doi:10.1140/epjst/e2016-60159-x
Part of the following topical collections:
  1. Nonlinearity, Nonequilibrium and Complexity: Questions and Perspectives in Statistical Physics


The so called q-triplets were conjectured in 2004 [C. Tsallis, Physica A 340, 1 (2004)] and then found in nature in 2005 [L.F. Burlaga, A.F. Vinas, Physica A 356, 375 (2005)]. A relevant further step was achieved in 2005 [C. Tsallis, M. Gell-Mann, Y. Sato, PNAS 102, 15377 (2005)] when the possibility was advanced that they could reflect an entire infinite algebra based on combinations of the self-dual relations q → 2 - q (additive duality) and q → 1/q (multiplicative duality). The entire algebra collapses into the single fixed point q = 1, corresponding to the Boltzmann-Gibbs entropy and statistical mechanics. For q ≠ 1, an infinite set of indices q appears, corresponding in principle to an infinite number of physical properties of a given complex system describable in terms of the so called q-statistics. The basic idea that is put forward is that, for a given universality class of systems, a small number (typically one or two) of independent q indices exist, the infinite others being obtained from these few ones by simply using the relations of the algebra. The q-triplets appear to constitute a few central elements of the algebra. During the last decade, an impressive amount of q-triplets have been exhibited in analytical, computational, experimental and observational results in natural, artificial and social systems. Some of them do satisfy the available algebra constructed solely with the additive and multiplicative dualities, but some others seem to violate it. In the present work we generalize those two dualities with the hope that a wider set of systems can be handled within. The basis of the generalization is given by the selfdual relation qqa(q) ≡ ((a+2)−aq) / (a−(a−2)q) (a ∈ R). We verify that qa(1) = 1, and that q2(q) = 2 - q and q0(q) = 1/q. To physically motivate this generalization, we briefly review illustrative applications of q-statistics, in order to exhibit possible candidates where the present generalized algebras could be useful.

Copyright information

© EDP Sciences and Springer 2017

Authors and Affiliations

  1. 1.Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex SystemsRio de Janeiro-RJBrazil
  2. 2.Santa Fe InstituteSanta FeUSA