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q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations

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Thistleton W., Marsh J., Nelson K., Tsallis C.

Open Physics

2009

vol. 7

issue 3

387-394

We study a strictly scale-invariant probabilistic N-body model with symmetric, uniform, identically distributed random variables. Correlations are induced through a transformation of a multivariate Gaussian distribution with covariance matrix decaying out from the unit diagonal, as ρ/r α for r =1, 2, ..., N-1, where r indicates displacement from the diagonal and where 0 ⩽ ρ ⩽ 1 and α ⩾ 0. We show numerically that the sum of the N dependent random variables is well modeled by a compact support q-Gaussian distribution. In the particular case of α = 0 we obtain q = (1-5/3 ρ) / (1- ρ), a result validated analytically in a recent paper by Hilhorst and Schehr. Our present results with these q-Gaussian approximants precisely mimic the behavior expected in the frame of non-extensive statistical mechanics. The fact that the N → ∞ limiting distributions are not exactly, but only approximately, q-Gaussians suggests that the present system is not exactly, but only approximately, q-independent in the sense of the q-generalized central limit theorem of Umarov, Steinberg and Tsallis. Short range interaction (α > 1) and long range interactions (α < 1) are discussed. Fitted parameters are obtained via a Method of Moments approach. Simple mechanisms which lead to the production of q-Gaussians, such as mixing, are discussed.

The q-exponential family in statistical physics

88%

Naudts J.

Open Physics

2009

vol. 7

issue 3

405-413

The notion of a generalized exponential family is considered in the restricted context of non-extensive statistical physics. Examples are given of models belonging to this family. In particular, the q-Gaussians are discussed and it is shown that the configurational probability distributions of the micro-canonical ensemble belong to the q-exponential family.

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2 Open Physics

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1 Marsh J.

1 Naudts J.

1 Nelson K.

1 Thistleton W.

1 Tsallis C.

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q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations

The q-exponential family in statistical physics