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Physica A 340 (2004) 110–116www.elsevier.com/locate/physaGeneralized statistics: yet another generalizationPetr Jizba ∗ , Toshihico ArimitsuInstitute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, JapanAbstractWe provide a unifying axiomatics for R, enyi’s entropy and non-extensive entropy of Tsallis.It is shown that the resulting entropy coincides with Csisz, ar’s measure of directed divergenceknown from communication theory.c ? 2004 Elsevier B.V. All rights reserved.PACS: 05.90.+m; 65.40.Gr; 02.90.+pKeywords: R, enyi’s information entropy; Tsallis entropy; Non-extensive entropy1. IntroductionIt has been known already since Shannon’s seminal paper [1] that Shannon’sinformation measure (or entropy) represents mere idealized information appearing onlyin situations when the bu=er memory (or storage capacity) of a transmitting channelis in>nite. As the latter is not satis>ed in many practical situations, information theo-rists have invented various remedies to deal with such cases. This usually consists ofsubstituting Shannon’s information measure with information measures of other types.Particularly distinct among them is a one-parametric class of information measures dis-covered by A. R, enyi. It was later on realized by Linnik that these, so-called, R, enyientropies (REs) are associated to the decoding limit if the source is compressed toI q and the parameter q essentially tells how much the tail of a probability distribu-tion should count in the calculation of the R, enyi entropy. Recently, an operationalcharacterization of RE in terms of ?-cuto= rates was provided by Csisz, ar [2].On the other hand, pioneering works of E. Jaynes [3] in mid 1950s revealed thatthe Gibbs entropy of statistical physics represents the Shannon entropy whenever thesample space of Shannon’s entropy is identi>ed with the set of all (coarse-grained)∗ Corresponding author. Tel.: +81-298-53-4322; fax: +81-298-53-4492.E-mail addresses: petr@cm.ph.tsukuba.ac.jp (P. Jizba), arimitsu@cm.ph.tsukuba.ac.jp (T. Arimitsu).0378-4371/$-see front matter c ? 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2004.03.085