q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations |
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Authors: | William J Thistleton John A Marsh Kenric P Nelson Constantino Tsallis |
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Institution: | (1) Department of Mathematics, SUNY Institute of Technology, Utica, NY 13504, USA;(2) Department of Computer and Information Sciences, SUNY Institute of Technology, Utica, NY 13504, USA;(3) Raytheon Integrated Defense Systems, Principal Systems Engineer, San Diego, CA, USA;(4) Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro - RJ, Brazil;(5) Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA |
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Abstract: | 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.
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Keywords: | q-Gaussian non-extensive statistical mechanics correlated systems |
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