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

Abstract

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.

arXiv comment17 pages including 3 figures. Introduction and references expanded

References in corpus (14)

• Non-Gaussian equilibrium in a long-range Hamiltonian system
• Prediction of anomalous diffusion and algebraic relaxations for long-range interacting systems, using classical statistical mechanics
• Fingerprints of nonextensive thermodynamics in a long-range Hamiltonian system
• Numerical indications of a q-generalised central limit theorem
• Lynden-Bell and Tsallis distributions for the HMF model
• Quasi-stationary states and incomplete violent relaxation in systems with long-range interactions
• Hamiltonian dynamics reveals the existence of quasi-stationary states for long-range systems in contact with a reservoir
• Incomplete equilibrium in long-range interacting systems
• On the meaning of Tsallis functional in astrophysics
• Temperature Fluctuations of the Cosmic Microwave Background Radiation: A Case of Nonextensivity?
• Deviation from Gaussianity in the cosmic microwave background temperature fluctuations
• Influence of global correlations on central limit theorems and entropic extensivity
• Algebraic Correlation Function and Anomalous Diffusion in the HMF model
• Occupancy of phase space, extensivity of Sq, and q-generalized central limit theorem

Cited by in corpus (27)

• Quantitative features of multifractal subtleties in time series
• Generalization of symmetric $\alpha$-stable Lévy distributions for $q>1$
• Nonergodicity and Central Limit Behavior for Long-range Hamiltonians
• A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model
• Strictly and asymptotically scale-invariant probabilistic models of $N$ correlated binary random variables having {\em q}--Gaussians as $N\to \infty$ limiting distributions
• Note on a q-modified central limit theorem
• The variance of identity-by-descent sharing in the Wright-Fisher model
• Nonadditive entropy and nonextensive statistical mechanics - Some central concepts and recent applications
• On q-Gaussians and Exchangeability
• Validity and Failure of the Boltzmann Weight
• Nonlinear statistical coupling
• Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions
• Analytic results for the Tsallis thermodynamic variables
• Stability criteria for q-expectation values
• Non-Gaussian distributions under scrutiny
• Escort entropies and divergences and related canonical distribution
• Long-range interactions, doubling measures and Tsallis entropy
• On the role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables
• Vanishing largest Lyapunov exponent and Tsallis entropy
• Open Problems within Nonextensive Statistical Mechanics
• Restricted random walk model as a new testing ground for the applicability of q-statistics
• On "Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions" by Figueiredo et al
• On the robustness of the $q$-Gaussian family
• Central limit theorem for a class of globally correlated random variables
• Stochastic dynamics of N correlated binary variables and non-extensive statistical mechanics
• Luminosity calibration by means of van-der-Meer scan for Q-Gaussian beams
• Reply to 'Comment on "Vortex distribution in a confining potential"