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.

17 pages including 3 figures. Introduction and references expanded

References in corpus (7)

• Numerical indications of a q-generalised central limit theorem
• Lynden-Bell and Tsallis distributions for the HMF model
• 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
• 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

Cited by in corpus (6)

• A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model
• Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions
• Stability criteria for q-expectation values
• Non-Gaussian distributions under scrutiny
• On the role of ergodicity and mixing in the central limit theorem for Casati-Prosen triangle map variables
• On "Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions" by Figueiredo et al