Nonextensive Statistical Mechanics

252 7 Thermodynamical and Nonthermodynamical Applications–1–2–3–4– – ––Fig. 7.41 Probability distribution of the avalanche size differences (returns) x(t) = S(t + 1) − S(t)for the OFC model on a small-world topology (critical state, open circles) and on a regular lattice(noncritical state, full circles). For comparison, a Gaussian and a q-Gaussian (with q = 2) areindicated as well. All the curves have been normalized so as to have unit area. For further detailssee [855].small-world lattice (referred to as the critical case) and a regular lattice (referredto as the noncritical case). The conclusion is highly interesting: at criticality q-Gaussian-like distributions are obtained, whereas something close to a Gaussian ontop of another (larger) Gaussian is obtained out of criticality. 3The (analytic) connection between the various qs that have been presented herefor earthquakes remains an open worthwhile question. 43 This fact is quite suggestive on quite different experimental grounds. Indeed, the velocity distributionof cold atoms in dissipative optical lattices has been measured by at least two differentgroups, namely in [857] and in [461]. The latter obtained a q-Gaussian velocity distribution (seeFig. 2(a) in [461]). The former, however, obtained a double-Gaussian distribution (see Fig. 11(a)in [857]). The reason for such a discrepancy is, to the best of our knowledge, not yet understood.A possibility could be that in the latter experiment, the apparatus is at “criticality,” whereas in theformer experiment it might be slightly out of it. The point surely is worthy of further clarification.4 Along this line, some hint might be obtained from the following observation. Series correspondingto thirteen earthquakes have been analyzed in [291]. It is claimed that the cumulative distributionof the distances between the epicenters of successive events is well fitted by a q s -exponential(where s stands for spatial); analogously, the cumulative distribution of the time intervals betweensuccessive events was also well fitted by a q t -exponential (where t stands for temporal).From the data corresponding to the set of 13 earthquakes (see Table 3 of [291]), we can calculateq s = 0.73 ± 0.09, q t = 1.32 ± 0.08, and q s + q t = 2.05 ± 0.07. If the distances and times betweensuccessive events were independent, we should obtain, for the standard deviation of q s +q t , roughly0.08+0.07 ≃ 0.17. Since the data yield 0.07 instead of 0.17, correlation is present, which suggestsq s + q t ≃ 2 for each earthquake.