Brief Review on the Connection between the Micro-Canonical Ensemble and the S-Canonical Probability Distribution

Non-standard thermostatistical formalisms derived from generalizations of the Boltzmann-Gibbs entropy have attracted considerable attention recently. Among the various proposals, the one that has been most intensively studied, and most successfully applied to concrete problems in physics and other areas, is the one associated with the Sq non-additive entropies. The Sq-based thermostatistics exhibits a number of peculiar features that distinguish it from other generalizations of the Boltzmann-Gibbs theory. In particular, there is a close connection between the Sq-canonical distributions and the micro-canonical ensemble. The connection, first pointed out in 1994, has been subsequently explored by several researchers, who elaborated this facet of the Sq-thermo-statistics in a number of interesting directions. In the present work, we provide a brief review of some highlights within this line of inquiry, focusing on micro-canonical scenarios leading to Sq-canonical distributions. We consider works on the micro-canonical ensemble, including historical ones, where the Sq-canonical distributions, although present, were not identified as such, and also more resent works by researchers who explicitly investigated the Sq-micro-canonical connection.