Nonextensive Statistical Mechanics

178 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanicsa special value of q below unity exists such that S q is extensive. In other words, S qasymptotically increases linearly with N, whereas S BG does not.The Moore map we shall study is a paradigmatic one belonging to the generalizedshift family of maps proposed by Moore [134]. This class of dynamical systemsposes some sort of undecidability, as compared with other low-dimensional chaoticsystems [134, 135]. It is equivalent to the piecewise linear map shown in Fig. 5.27.When this map is recurrently applied, the area in phase-space is conserved, while thecorresponding shape keeps changing in time, becoming increasingly complicated.This map appears to be ergodic, possibly exhibits a Lyapunov exponent λ 1 = 0, and,presumably, the divergence of close initial conditions follows a power-law behavior[137]. When we consider a partition of W equal cells and select N random initialconditions inside one random cell, the points spread much slower than they do onthe baker map. More precisely, they spread, through a slow relaxation process, allover the phase-space, each orbit appearing to gradually fill up the entire square. SeeFigs. 5.28, 5.29, and 5.30.300250MOORE MAPN = 10Winitial condition: (1,1) cellln 0.1 W200W = 20 x 20precision:18precision:100precision:40S 0.1150precision:16100precision:100ln 0.1 W50W = 10 x 10precision:1600 3000 6000 9000 12000tFig. 5.29 Numerical study of the Moore map I. Top figures: Evolution of occupancy in phasespace.Bottom figure: Evolution of S 0.1 .