Nonextensive Statistical Mechanics

Chapter 7Thermodynamical and NonthermodynamicalApplicationsNothing is more practical than a good theory. 1The nature of the present chapter is quite different from all the others of the book.In all its other Chapters we have privileged the presentation and understanding ofnonextensive statistical mechanics itself, and of some of its delicate and unusualconcepts. In the present chapter, we focus on the concrete and typical applicationsthat are available in the literature, as well as on some connections that have emergedalong time with other areas such as quantum chaos, quantum entanglement, randommatrices, theory of networks.The present list is not an exhaustive one. It is aimed mainly to illustrate thespecific types of systems that have been handled in one way or another within thenonextensive framework. Some of them are genuine applications of the theory, othersare just possible explanations and connections. Whenever the microscopic, or atleast the mesoscopic, dynamics of the system is unknown, it is of course impossibleto determine q otherwise than through fitting (as astronomers determine the ellipticeccentricities of the orbits of the planets). This extra difficulty does not exist inBG thermostatistics, since the corresponding value trivially is just q = 1. In themore complex systems addressed in this chapter, all types of situations occur. Sometimesthe experimental measurements, or observations, or computational results existthrough very many numerical decades with satisfactory precision. In these cases,the correctness of the fitness constitutes already a strong argument favoring the applicabilityof the nonextensive theory, with its predictions and concepts. Sometimes,we have at our disposal only a few numerical decades and/or not very high precision.It might then be disputable whether the system under focus really belongs to thepresent frame, or to a somewhat different one. Sometimes, it becomes possible tomake precise falsifiable predictions, sometimes not. Sometimes the applications justconsist in improved algorithms for optimization, signal analysis, image processing,and similar techniques. In these cases, the quality of the improvement speaks byitself. In all cases, we do achieve a better understanding of the phenomenon, or atleast develop some intuition on it.1 Attributed to Lenin.C. Tsallis, Introduction to Nonextensive Statistical Mechanics,DOI 10.1007/978-0-387-85359-8 7, C○ Springer Science+Business Media, LLC 2009221