Nonextensive Statistical Mechanics

Chapter 8Final Comments and PerspectivesI think it is safe to say that no one understands QuantumMechanicsRichard Feynman8.1 Falsifiable Predictions and Conjectures,and Their VerificationsAccording to the deep epistemological observations of Karl Raimund Popper, a scientifictheory cannot be considered as such if it is not capable of providing falsifiablepredictions. This is to say predictions that can in principle be checked to be true orfalse. And a successful theory is of course that one which accumulates predictionsthat have been verified to be correct, and whose basic hypothesis has not been provedto be violated within the restricted domain of conditions for which the theory isthought to be applicable.It is needless to say that nonextensive statistical mechanics cannot and mustnot escape to the necessity of satisfying such requirements. Although several suchillustrations have already been presented in the body of this book, let us brieflyand systematically list here some of the falsifiable predictions or conjectures of thetheory, as well as their verification in recent years. This list is not exhaustive: forsimplicity, I restrict here to those examples in which I have been, in one way oranother, personally involved.(a) The scaling relation γ = 23−q .Within the context of the nonlinear Fokker–Planck equation in the absence ofexternal forces, and its exact q-Gaussian solution for all space-time (x, t), it wasanalytically proved in 1996 [349] that x 2 scales like t γ with (Eq. (4.16)) γ = 23−q(hence, for instance, if 〈x 2 〉 is finite, it must be 〈x 2 〉∝t 23−q ). Through the perceptionof the crucial role that this equation plays in many complex systems addressed bynonextensive statistical mechanics, the rather generic applicability of this scalingrelation was conjectured, and also illustrated, in 2004 [881]. Five verifications areavailable at the present date, namely inC. Tsallis, Introduction to Nonextensive Statistical Mechanics,DOI 10.1007/978-0-387-85359-8 8, C○ Springer Science+Business Media, LLC 2009305