Nonextensive Statistical Mechanics

5.2 Low-Dimensional Conservative Maps 175where sgnx =±1 is the sign of x, and α, β are two parameters (n = 0, 1....).This map is linearly unstable. For rational values of α, β the system is in principleintegrable, as the dynamics is confined on invariant curves. If β = 0 and α is irrational,the dynamics is ergodic but the phase-space is filled very slowly, while forincommensurate irrational values of α, β the dynamics is ergodic and mixing withdynamical correlation function decaying as t −3/2 (i.e., q rel = 5/3, according to anotation that will be discussed later on). This map does not have any secondary timescales, and the exploration of the phase-space by a given orbit is arbitrarily close tothat of a random model.For the sake of definiteness, in the following we will fix (see [358]) the parametervalues α = [ 1 2 (√ 5 − 1) − e −1 ]/2, β = [ 1 2 (√ 5 − 1) + e −1 ]/2 although it should benoticed that qualitatively identical results are obtained for other irrational parametervalues. Figure 5.25 shows the mixing process of an ensemble of points initiallylocalized inside a small square. The action of the map (5.34) initially divides thearea covered by the ensemble into different unconnected portions, each essentiallystretched along a straight line. After a certain amount of time, these portions overlapuntil a slow relaxation process to a complete mixing is observed. We can verify in(a)1n = 0(b)1n = 100.5initial ensemble0.5y0y0–0.5–0.5(c)–1 –11–0.50 0.5 1xn = 100(d)–1 –11–0.50 0.5 1xn = 10 60.50.5y0y0–0.5–0.5–1 –1–0.50 0.5 1x–1–1 –0.5 0 0.5 1xFig. 5.25 Time evolution of an ensemble of points in phase-space. (a) The ensemble is initiallylocated inside a single cell. (b, c, andd) Phase-space distribution after n = 10, 10 2 , 10 6 mapiterations (from [358]).