Nonextensive Statistical Mechanics

Chapter 1Historical Background and Physical MotivationsBeauty is the first test:there is no permanent place in the world for ugly mathematics.G.H. Hardy(A Mathematician’s Apology, 1941)1.1 IntroductionLet us consider the free surface of a glass covering a table. And let us idealize it asbeing planar. What is its volume? Clearly zero since it has no height. An uninterestinganswer to an uninteresting question. What is its length? Clearly infinity. Onemore uninteresting answer to another uninteresting question. Now, if we ask what isits area, we will have a meaningful answer, say 2 m 2 .Afinite answer. Not zero, notinfinity – correct but poorly informative features. A finite answer for a measurablequantity, as expected from good theoretical physics, good experimental physics, andgood mathematics. Who “told” us that the interesting question for this problem wasthe area? The system did! Its planar geometrical nature did. If we were focusing ona fractal, the interesting question would of course be its measure in d f dimensions,d f being the corresponding fractal or Hausdorff dimension. Its measure in any dimensiond larger than d f is zero, and in any dimension smaller than d f is infinity.Only the measure at precisely d f dimensions yields a finite number. For instance,if we consider an ideal 10 cm long straight segment, and we proceed through thecelebrated Cantor-set construction (i.e., eliminate the central third of the segment,and then also eliminate the central third of each of the two remaining thirds, andhypothetically continue doing this for ever) we will ultimately arrive to a remarkablegeometrical set – the triadic Cantor set – which is embedded in a one-dimensionalspace but whose Lebesgue measure is zero. The fractal dimension of this set isd f = ln 2/ ln 3 = 0.6309 ... Therefore, the interesting information about ourpresent hypothetical system is that its measure is (10 cm) 0.6309... ≃ 4.275 cm 0.6309 .And, interestingly enough, the nature of this valuable geometric information wasdictated by the system itself!This entire book is written within precisely this philosophy: it is the natural(or artificial or social) system itself which, through its geometrical-dynamicalC. Tsallis, Introduction to Nonextensive Statistical Mechanics,DOI 10.1007/978-0-387-85359-8 1, C○ Springer Science+Business Media, LLC 20093