Nonextensive Statistical Mechanics

5.5 The q-Triplet 19310 010 –1N = 20,000 - U = 0.690,8δ = 100 n = 2000 transient time = 2000,60,40,2event 1event 2event 3Gaussianq-Gaussian (q = 1.5 β = 1.8)PDF0,0–2 –10 1 210 –210 –3–6 –4 –2 0 24 6y/σFig. 5.42 We present for each class of the QSS found, the different central limit theorem behaviorobserved. A Gaussian (dashed curve) with unitary variance and a q-Gaussian p(x) = Ae q (−βx 2 )with A = 0.66, q = 1.5, and β = 1.8 (full curve) are also reported for comparison. In the inset, amagnification of the central part in linear scale is plotted (from [46]).is given byy = e axq . (5.55)These expressions, respectively, generalize expressions (5.49) and (5.50). As before,we may think of them in three different physical manners, related respectivelyto the sensitivity to the initial conditions, to the relaxation in phase-space, and, ifthe system is Hamiltonian, to the distribution of energies at a stationary state. In thefirst interpretation we reproduce Eq. (5.8). In the second interpretation, we typicallyexpect(t) = e −t/τ q relq rel, (5.56)where τ qrel is the relaxation time. Finally, in the third interpretation, we haveEq. (3.207) (with Eq. (3.208)), i.e.,Z qstat p i = e −β qstat E i, (5.57)