Nonextensive Statistical Mechanics

306 8 Final Comments and Perspectives– the experiments with Hydra viridissima reported in 2001 [774] (the measuredvalue q = 1.5 ± 0.05 implies, through Eq. (4.16), γ = 1.33 ± 0.05, which isconsistent with the measured value γ = 1.24 ± 0.1; see Figs. 7.88 and 7.89);– the experiments in defect turbulence reported in 2004 [427] (the measured valueq ≃ 1.5 implies, through Eq. (4.16), γ ≃ 1.33, which is consistent with themeasured value γ = 1.16 − 1.50; see Figs. 7.7, 7.8 and 7.9);– the molecular dynamical simulations for the long-range classical inertial α-XYferromagnet reported in 2005 [41] (γ (3 − q)/2 = 1.0 ± 0.1; see also [820, 841],and Figs. 5.60 and 5.61);– the computational simulations for silo drainage reported in 2007 [451, 452](q ≃3/2 and γ ≃ 4/3; see Figs. 7.16 and 7.17);– and the experiments with dusty plasma reported in 2008 [462] ( ¯γ (3 − q)/2 =1.00 ± 0.016, where ¯γ is an averaged value; see Figs. 7.23, 7.24, 7.25 and 7.26).In all but the molecular dynamics approach, the value for q was determined fromthe index of the q-Gaussian distribution of velocities. In the molecular dynamicscase, q was determined from the time-relaxation of the velocity auto-correlationfunction. The precise relation (or even, perhaps, identity under some circumstances)of this q with that of the velocity distribution remains to be clarified.(b) q-Gaussian distributions of velocities of cold atoms in dissipative opticallatticesLutz predicted in 2003 [460] that the distribution of velocities of cold atoms indissipative optical lattices should be q-Gaussian with q = 1 + 44 E RU 0(Eq. (7.1)).The prediction was checked in 2006 [461] through quantum Monte Carlo calculations,as well as through experiments with Cs atoms: see Fig. 7.1. The MonteCarlo calculations neatly confirmed both the q-Gaussian shape of the distribution(with a correlation factor R 2 = 0.995, and Lutz formula (Eq. (7.1)) within the range50 ≤ U 0 /E R ≤ 240. The laboratory experiments provided a laser-frequency dependenceof q qualitatively the same as Lutz formula; the quantitative check would havedemanded the direct measure of E R and of U 0 , which was out of the scope of theexperiment. In what concerns the form of the distribution, the experiments verifiedthe predicted q-Gaussian shape with R 2 = 0.9985, and obtained (in the illustrationthat is presented in [461]) q = 1.38 ± 0.12 from the body of the distribution, andthe consistent value q = 1.396 ± 0.005 from the tail of the distribution.(c) Generalized central limit theorem leading to stable q-Gaussian distributionsThe possible generalization of the standard and the Levy–Gnedenko CentralLimit Theorems (CLT) was suggested in 2000 [826], and was then formally conjecturedin 2004 [191]. Its proof started in 2006 [246], and was finally published in2008 [247] (see also [252, 253]).(d) Existence of q, λ q and K q , and the identity K q = λ qIt was argued in 1997 [127] that, whenever the Lyapunov exponent λ 1 vanishes,(the upper bound of the) the sensitivity is given by ξ = e λ qsen tq sen, which determinesa special value of q, noted q sen . It was further argued that, at the edge of chaos,