Nonextensive Statistical Mechanics, Anomalous Diffusion and Central Limit Theorems

Abstract

We briefly review Boltzmann-Gibbs and nonextensive statistical mechanics as well as their connections with Fokker-Planck equations and with existing central limit theorems. We then provide some hints that might pave the road to the proof of a new central limit theorem, which would play a fundamental role in the foundations and ubiquity of nonextensive statistical mechanics. The basic novelty introduced within this conjectural theorem is the generalization of the hypothesis of independence of the N random variables being summed. In addition to this, we also advance some nonlinear dynamical (possibly exact) relations which generalize the concepts of Lyapunov exponents, entropy production per unit time, and their interconnection as first proved by Pesin for chaotic systems.

Axiomatic nonextensive statistics at NICA energies

“…In a recent work [3], we have shown that the blind implementation of Tsallis algebra (replacing exponential and logarithmic functions with their Tsallis counter parts) [4,5,6] in the partition-function of the hadron resonance gas (HRG) model, which is -per definition -constructed from summing up independent contributions from different hadron resonances, fails to assure full incorporation of nonextensivity (due to correlations or interactions, among others) in such statistical thermal models. The resulting HRG is no longer able to reproduce the lattice QCD calculations, and furthermore results in very low temperatures even from the statistical fitting with the transverse momentum spectra (p T ), which are exceptionally (but seem-ingly unjustly) celebrated as the best implication for the Tsallis statistics in high-energy physics [7,8].…”

“…q differing from unity characterizes Tsallis statistics, k = 1/(1−q) and q-exponential function exp q (−x) [4,5,6]. For an arbitrary b, a more general form for the quantum statistical distribution should be obtained.…”

Central limit theorems for correlated variables: some critical remarks

“…Considerations from a q-generalized statistical mechanics [25][26][27] have led Tsallis [28] to surmise that in the limit N → ∞ the sum of N correlated random variables becomes, under appropriate conditions, q-Gaussian distributed; that is, on this hypothesis q-Gaussians are attractors in a similar sense as ordinary Gaussians. Now, variables can be correlated in very many ways.…”

q-Deformed Loewner Evolution

“…In particular, a form of q-calculus is involved in the description of non-extensive statistical mechanics [34], whose scope of application ranges from phenomena where the Loewner approach has already proved useful, like the Ising model [35] and viscous fingering [36], to broader and more general subjects, as generalizations of the α-stable distributions [37] and anomalous diffusion [38]. We do not pursue these connections in this paper, but they constitute the physical motivations for a systematic approach to q-deformed equations in statistical mechanics.…”