A recent paper by T. Dauxois entitled "Non-Gaussian distributions under scrutiny" is submitted to scrutiny. Several comments on its content are made, which constitute, at the same time, a brief state-of-the-art review of nonextensive statistical mechanics, a current generalization of the Boltzmann-Gibbs theory. Some inadvertences and misleading sentences are pointed out as well.
Deep Dive into T. Dauxois "Non-Gaussian distributions under scrutiny" under scrutiny.
A recent paper by T. Dauxois entitled “Non-Gaussian distributions under scrutiny” is submitted to scrutiny. Several comments on its content are made, which constitute, at the same time, a brief state-of-the-art review of nonextensive statistical mechanics, a current generalization of the Boltzmann-Gibbs theory. Some inadvertences and misleading sentences are pointed out as well.
T. Dauxois has recently commented [1] an interesting paper by Hilhorst and Schehr [2].
In the latter paper, two specific probabilistic models, namely a discrete one [3] (from now on referred to as the MTG model) and a continuous one [4] (from now on referred to as the TMNT model), are analytically discussed. Both models consist in N correlated random variables (respectively discrete and continuous), and the point under study is what is the limiting distribution when N → ∞. Numerical indications 1 have been found that quite strongly suggest that these limiting distributions could be of the q-Gaussian class (Eq. (1) in [1]). Hilhorst and Schehr have shown [2] that they are not, even if they are numerically intriguingly close. The relations between the entropic indices q (of the possible q-Gaussian limiting distributions) and the parameters of the models have been addressed as well. It was analytically confirmed in [2] the correctness of the relations conjectured respectively in [3] and in [4], if these analytically exact non q-Gaussian distributions were to be approached by q-Gaussians! Why may all this have some general interest? The reason lies on the fact that q-Gaussian distributions play a special role in nonextensive statistical mechanics [5,6,7], a current generalization (based on the entropy (2) of [1]) of the celebrated BG theory. Recent reviews of q-statistics and a vast set of experimental, observational and computational applications and/or verifications can be seen in [8,9,10,11,12]. Very specifically, it has been recently proved [13] a q-generalization of the Central Limit Theorem for q ≥ 1. More precisely, if we have N random variables that are strongly correlated in a special manner (called q-independence; see [13] for details), it can be rigorously proved that their sum approaches, when appropriately centered and scaled, a q-Gaussian distribution. The standard CLT is recovered as the q = 1 particular instance. The proof is under progress for q < 1 (which are distributions with a compact support, even in the N → ∞ limit). Nevertheless, preliminary studies suggest that the q-CLT should remain applicable even for q < 1. Since the MTG and TMNT models have a compact support (i.e., of the q < 1 type), one expects q-Gaussians whenever the correlation is of the q-independent type. Given the results in [1] See the title of the present Ref. [3].
[2], one is naturally led to argue that the strong correlations in those two models are not q-independent but only almost q-independent. An interesting question remains then open.
What physical ingredient have the MTG and TMNT models failed to incorporate? In other words, there is something which is still missing in those models in order for the correlation to be exactly q-independent. What is it? Progress is presently being achieved along this line (deeply related to asymptotic scale-invariance), but this remains outside the present scope, which primarily is the careful scrutiny of the paper [1]. Let us address now some of the weaknesses of that paper.
(i) In contrast with what one reads in [1], the T MNT model is not at all described nor presented in its Ref. [3]. In fact, it is not even mentioned there, and has never been published! 2 . Dauxois visibly confuses with [14], whose content has absolutely nothing to do with the model discussed by Hilhorst and Schehr. Had the Author of [1] paid more attention to his own Ref. [3], or at least to its title, the confusion would have been avoided.
(ii) The statement “could be the basis for a generalized central limit theorem” (below Eq.
(1) of [1]) reveals that the Author is possibly unaware of the proof existing since already some time in [13] (further extensions are possible along the lines of [15,16]).
(iii) The statement “The basis for this suggestion …” (above Eq. ( 2) of [1]) reveals that the Author is unaware that the crucial point is not the isolated fact that q-Gaussians optimize S q (something which, in contrast with what is referred in [1], is not even vaguely mentioned in Ref. [2] of [1]!), but rather the remarkable fact that q-Gaussians also happen to be [17,18] exact stable solutions of the nonlinear Fokker-Planck equation (since many decades called Porous Medium Equation). The mere fact that q-Gaussians would optimize [2] The details of the T M N T model, as well as all our numerical and graphical results, were transmitted privately by me to Hilhorst, who had heard, in Natal-Brazil in March 2007, an oral presentation of mine including that subject some specific entropic functional would certainly not be of any particular significance. The entire rationale of what was at the time a conjecture (and is now a proved theorem) can be seen in [19].
(iv) The qualification “in the absence of firm grounds…” reveals that the Author of [1] is possibly unaware of the numerous rigorous results which precisely provide a firm mathematical basis for the entropy S q and the associated nonextensive statistical
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