Nonextensive Statistical Mechanics

166 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanicsa=1.7a=1.8a=1.9σ 2 = 0.018610 0 σ 2 = 0.1248σ 2 = 0.0613N=2x10 6n ini=1x10 6ρ y(y)10 –110 –210 –3–2.0 –1.5 –1.0 –0.5 0.0 0.5 1.0 1.5 2.0yFig. 5.14 Probability density of rescaled sums of iterates of the logistic map for a = 1.7, 1.8, 1.9and N = 2 · 10 6 , n ini = 10 6 .Thesolid lines show Gaussians e −y2 /(2σ 2) / √ 2πσ 2 with varianceparameter σ 2 (from [370]).d∑i=1λ (i)1= 0 . (5.29)If, in addition to that, it is symplectic, d is an even integer, and we can thereforeconveniently define d = 2N (N = 1, 2,...). Furthermore, the Lyapunov exponentsare in pairs which differ only in the sign. Obviously, two-dimensional conservativemaps are necessarily symplectic.Entropic properties in low-dimensional maps have already been addressed ford = 2 ( [85, 138, 356–358], among others) and d = 4 ( [356, 357], among others).The review of some of their peculiarities will pave the understanding of many-bodyHamiltonian systems, the primary object of study in statistical mechanics.5.2.1 Strongly Chaotic Two-Dimensional Conservative MapsIn order to illustrate relevant properties, we shall focus here on three paradigmatic(strongly chaotic, area-preserving, and transforming the unit square into itself), twodimensionalconservative maps, first the so-called baker map, second the generalizedcat map [85], and third the standard map [85, 356].The baker map is defined as follows [138]: