Nonextensive Statistical Mechanics

256 7 Thermodynamical and Nonthermodynamical ApplicationsO(t)10 0 δ = .00030slope = −1/τ 210 −1−1q = 3.3−210 −2 t 2 1 2 3 ×10 610 0 10 1 10 2 10 3tln q[O(t)]O(t)10 0 δ = .010slope = −1/τ 210 −1−2q = 4.25−410 −2 t 2 2000 400010 0 10 1 10 2tln q[O(t)]Fig. 7.45 Overlap vs. time for an initial angular momentum coherent state located at the borderbetween regular and chaotic zones of the QKT of spin 240 and α = 3. This region, the edgeof quantum chaos, shows the expected power law decrease in overlap. The top figure is for aperturbation strength in the weak perturbation regime, δ = 0.0003 and the bottom figure is for aperturbation strength of δ = 0.01, within the FGR regime. On the log–log plot the power law decayregion, from about 600–2500 in the weak perturbation regime and 20–70 in the FGR regime, islinear. We can fit the decrease in overlap with the expression [1+(q rel −1)(t/τ qrel ) 2 ] 1/(1−qrel) where,in the weak perturbation regime, the entropic index q rel = 3.3 andτ qrel = 1300 and in the FGRregime q rel = 4.25 and τ qrel = 34. The insets of both figures show ln qrel O ≡ (O 1−qrel −1)/(1−q rel )vs. t 2 ; since ln q x is the inverse function of eq x 1≡ [1 + (1 − q) x]1−q , this produces a straight linewith a slope −1/τ 2 (also plotted) (from [547]).54.54q rel3.5310002.5τ q100210.001 .011.5δ0 0.005 0.01 0.015δFig. 7.46 Values of q rel and τ q (inset) forJ = 120 (x), 240 (circles), 360 (diamonds), and 480(stars). q rel remains constant for perturbation strengths below the critical perturbation and abovethe saturation perturbation. In between q rel increases with a rate dependent on J. The values ofq c rel , qs rel , δ c,andδ s can be seen in the figure. In addition the rate of growth of q rel with increasedperturbation strength can be seen. The inset shows a log–log plot of the value of τ q vs. δ for theabove values of J. The data can be fit with a lines of slope −1.06, −1.03, −1.07, and −1.08 forJ = 120, 240, 360, and 480 (top to bottom) (from [547]).