A note on q-Gaussians and non-Gaussians in statistical mechanics

The sum of $N$ sufficiently strongly correlated random variables will not in general be Gaussian distributed in the limit N\to\infty. We revisit examples of sums x that have recently been put forward as instances of variables obeying a q-Gaussian law, that is, one of type (cst)\times[1-(1-q)x^2]^{1/(1-q)}. We show by explicit calculation that the probability distributions in the examples are actually analytically different from q-Gaussians, in spite of numerically resembling them very closely. Although q-Gaussians exhibit many interesting properties, the examples investigated do not support the idea that they play a special role as limit distributions of correlated sums.Comment: 17 pages including 3 figures. Introduction and references expande

Some Open Points in Nonextensive Statistical Mechanics

We present and discuss a list of some interesting points that are currently open in nonextensive statistical mechanics. Their analytical, numerical, experimental or observational advancement would naturally be very welcome.Comment: 30 pages including 6 figures. Invited paper to appear in the International Journal of Bifurcation and Chao

Strictly and asymptotically scale-invariant probabilistic models of NN correlated binary random variables having {\em q}--Gaussians as N→∞N\to \infty limiting distributions

In order to physically enlighten the relationship between {\it $q$--independence} and {\it scale-invariance}, we introduce three types of asymptotically scale-invariant probabilistic models with binary random variables, namely (i) a family, characterized by an index $\nu=1,2,3,...$, unifying the Leibnitz triangle ($\nu=1$) and the case of independent variables ($\nu\to\infty$); (ii) two slightly different discretizations of $q$--Gaussians; (iii) a special family, characterized by the parameter $\chi$, which generalizes the usual case of independent variables (recovered for $\chi=1/2$). Models (i) and (iii) are in fact strictly scale-invariant. For models (i), we analytically show that the $N \to\infty$ probability distribution is a $q$--Gaussian with $q=(\nu -2)/(\nu-1)$. Models (ii) approach $q$--Gaussians by construction, and we numerically show that they do so with asymptotic scale-invariance. Models (iii), like two other strictly scale-invariant models recently discussed by Hilhorst and Schehr (2007), approach instead limiting distributions which are {\it not} $q$--Gaussians. The scenario which emerges is that asymptotic (or even strict) scale-invariance is not sufficient but it might be necessary for having strict (or asymptotic) $q$--independence, which, in turn, mandates $q$--Gaussian attractors.Comment: The present version is accepted for publication in JSTA