Nonextensive Statistical Mechanics

308 8 Final Comments and Perspectives(h) Existence of a q-triplet with q sen < 1,q rel > 1, and q stat > 1It was conjectured in 2004 [880] that complex systems (of the nonextensive type)would exist exhibiting q-exponential behavior for the time dependence of the sensitivityto the initial conditions (with index q sen < 1), for the time dependenceof the relaxation of relevant physical quantities towards the final stationary state(with index q rel > 1), and for the energy distribution at the stationary state (withindex q stat > 1). In the Boltzmannian, thermal equilibrium, limit (correspondingto full mixing, and ergodicity) one would expect the collapse of this q-triplet (orq-triangle, as sometimes called) into q sen = q rel = q stat = 1. This conjecture wasindeed verified by Burlaga and Vinas in 2005 [361], through processing data sentto NASA by the Voyager 1, in the solar wind at the distant heliosphere, and also,more recently, in the heliosheath [362–364] (see also [365–368]). The Voyager 1spacecraft was launched in 1977, over 30 years ago. It is therefore unreasonable toexpect high precision results. This said, the values advanced by Burlaga and Vinas in2005 [361] were (q sen , q rel , q stat ) = (−0.6±0.2, 3.8±0.3, 1.75±0.06). Since onlyone of them is expected to be independent, one expects a priori two relations to existbetween these three indices. Such relations were heuristically advanced in [199].The outcome that was found is (q sen , q rel , q stat ) = (−1/2, 4, 7/4), which, withinthe error bars, is consistent with the NASA results.More recently, another q-triplet was completed, namely at the edge of chaos ofthe logistic map: (q sen , q rel , q stat ) = (0.24448..., 2.24978..., 1.65 ± 0.05) (see [370,371] and references therein). Although far from transparent, we have assumed herethat the value of q corresponding to the q-Gaussian attractor (summing successiveiterates) is to be identified with q stat .These and the NASA results together seem to indicate that perhaps the generalscheme for the q-triplet is q sen ≤ 1 ≤ q stat ≤ q rel . A proof or clarifications wouldbe welcome.(i) Degree distributions of the q-exponential type for scale-free networksThe so-called scale-free networks (which are in fact only asymptotically scalefree)exhibit very frequently a degree distribution of the form k δ e −k/k 0q (q > 1; k 0 >0), with an exponent δ than can be either zero or positive, or negative. This was firstnoticed in 2004 [803] with δ = 0. The scale-invariance being a basic ingredientof nonextensive statistics (in particular in relation to the q-CLT), it was a kind ofnatural to expect that this q-exponential degree distribution would be somethingubiquitous. Indeed, it has been so verified since 2005 in many models [49, 50, 52–54, 790]. However, it is yet elusive what motivates δ to be zero or nonzero. Even itssign is presently an open question.8.2 Frequently Asked QuestionsAs the history of sciences profusely shows to us, every possible substantial progressin the foundations of any science is accompanied by doubts and controversies. This