Nonextensive Statistical Mechanics

348 BibliographyDiversity in Theory and Practice, eds. J.F. Grassle, G.P. Patil, W. Smith and C. Taillie (Int.Cooperat. Publ. House, Maryland, 1979), pages 3–27). Myself I rediscovered this form in 1985with the aim of generalizing Boltzmann-Gibbs statistical mechanics, but quote none of thepredecessors in my 1988 paper [39]. Indeed, I started knowing the whole story quite a fewyears later thanks to S.R.A. Salinas and R.N. Silver, who were the first to provide me with thecorresponding informations. Such rediscoveries can by no means be considered as particularlysurprising. Indeed, this happens in science more frequently than usually realized. This pointis lengthily and colorfully developed by S.M. Stigler, Statistics on the table – the history ofstatistical concepts and methods (Harvard University Press, Cambridge, MA, 1999). In page284, a most interesting example is described, namely that of the celebrated normal distribution.It was first introduced by Abraham De Moivre in 1733, then by Pierre-Simon Laplace in 1774,then by Robert Adrain in 1808, and finally by Carl Friedrich Gauss in 1809, nothing less than76 years after its first publication! This distribution is universally called Gaussian because ofthe remarkable insights of Gauss concerning the theory of errors, applicable in all experimentalsciences. A less glamorous illustration of the same phenomenon, but nevertheless interestingin the present context, is that of Renyi entropy [108]. According to I. Csiszar, Informationmeasures: A critical survey, inTransactions of the Seventh Prague Conference on InformationTheory, Statistical Decision Functions, Random Processes, and the European Meetingof Statisticians, 1974 (Reidel, Dordrecht, 1978), page 73, the Renyi entropy had alreadybeen essentially introduced by Paul-Marcel Schutzenberger, Contributions aux applicationsstatistiques de la theorie de l’ information, Publ. Inst. Statist. Univ. Paris 3, 3 (1954).108. A. Renyi, in Proceedings of the Fourth Berkeley Symposium, 1, 547 (University of CaliforniaPress, Berkeley, Los Angeles, 1961); A. Renyi, Probability theory (North-Holland, Amsterdam,1970), and references therein.109. G. Hardy, J.E. Littlewood and G. Polya, Inequalities (Cambridge University Press, Cambridge,1952).110. C. Tsallis and E. Brigatti, Nonextensive statistical mechanics: A brief introduction, inNonadditiveentropy and nonextensive statistical mechanics, ed. M. Sugiyama, Continuum Mechanicsand Thermodynamics 16, 223 (Springer-Verlag, Heidelberg, 2004).111. F. Jackson, Mess. Math. 38, 57 (1909); Quart. J. Pure Appl. Math. bf 41, 193 (1910).112. S. Abe, A note on the q-deformation theoretic aspect of the generalized entropies in nonextensivephysics,Phys.Lett.A224, 326 (1997).113. S. Abe, Stability of Tsallis entropy and instabilities of Renyi and normalized Tsallis entropies,Phys. Rev. E 66, 046134 (2002).114. C. Tsallis, P.W. Lamberti and D. Prato, A nonextensive critical phenomenon scenario forquantum entanglement, Physica A 295, 158 (2001).115. S. Abe, Axioms and uniqueness theorem for Tsallis entropy, Phys. Lett. A 271, 74 (2000).116. S. Abe, General pseudoadditivity of composable entropy prescribed by the existence of equilibrium,Phys. Rev. E 63, 061105 (2001).117. R.J.V. Santos, Generalization of Shannon’ s theorem for Tsallis entropy, J. Math. Phys. 38,4104 (1997).118. A.R. Plastino and A. Plastino, Generalized entropies,inCondensed Matter Theories,Vol.11,eds. E.V. Ludeña, P. Vashista and R.F. Bishop (Nova Science Publishers, New York, 1996),page 327.119. A.R. Plastino and A. Plastino, Tsallis entropy and Jaynes’ information theory formalism,Braz. J. Phys. 29, 50 (1999).120. E.M.F. Curado, General aspects of the thermodynamical formalism,inNonextensive StatisticalMechanics and Thermodynamics, eds. S.R.A. Salinas and C. Tsallis, Braz. J. Phys. 29,36(1999); E.M.F. Curado and F.D. Nobre, On the stability of analytic entropic forms, PhysicaA 335, 94 (2004).121. C. Anteneodo and A.R. Plastino, Maximum entropy approach to stretched exponential probabilitydistributions, J. Phys. A 32, 1089 (1999).122. P. Grassberger and M. Scheunert, Some more universal scaling laws for critical mappings,J.Stat. Phys. 26, 697 (1981).