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

Thistleton, William J.; Marsh, John A.; Nelson, Kenric P.; Tsallis, Constantino

doi:10.2478/s11534-009-0054-4

Cited by 25 publications

(42 citation statements)

References 11 publications

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“…An important goal along these lines is therefore to describe in simple terms the basic physical assumptions behind the mathematical requirement of q-independence. Two types of simple models have been recently introduced [11,12] in order to provide this insight. They are hereafter referred to as the MTG and the TMNT models respectively.…”

Section: Introductionmentioning

confidence: 99%

Strictly and asymptotically scale invariant probabilistic models ofNcorrelated binary random variables havingq-Gaussians asN→ ∞ limiting distributions

2008

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The celebrated Leibnitz triangle has a remarkable property, namely that each of its elements equals the sum of its South-West and South-East neighbors. In probabilistic terms, this corresponds to a specific form of correlation of N equally probable binary variables which satisfy scaleinvariance. Indeed, the marginal probabilities of the N -system precisely coincide with the joint probabilities of the (N − 1)-system. On the other hand, the nonadditive entropy, which grounds nonextensive statistical mechanics, is, under appropriate constraints, extremized by the). These distributions also result, as attractors, from a generalized central limit theorem for random variables which have a finite generalized variance, and are correlated in a specific way called q-independence. In order to physically enlighten this concept, we introduce here three types of asymptotically scale-invariant probabilistic models with binary random variables, namely (i) a family, characterized by an index ν = 1, 2, 3, . . . , unifying the Leibnitz triangle (ν = 1) and the case of independent variables (ν → ∞); (ii) two slightly different discretizations of q-Gaussians; (iii) a special family, characterized by the parameter χ, which generalizes the usual case of independent variables (recovered for χ = 1/2). Models (i) and (iii) are in fact strictly scale-invariant. For models (i), we analytically show that the N → ∞ probability distribution is a q-Gaussian with q = (ν − 2)/(ν − 1). Models (ii) approach q-Gaussians by construction, and we numerically show that they do so with asymptotic scale-invariance. Models (iii), like two other strictly scale-invariant models recently discussed by Hilhorst and Schehr (2007), approach instead limiting distributions which are not q-Gaussians. The scenario which emerges is that asymptotic (or even strict) scale-invariance is not sufficient but it might be necessary for having strict (or asymptotic) q-independence, which, in turn, mandates q-Gaussian attractors.

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Section: Introductionmentioning

confidence: 99%

Strictly and asymptotically scale invariant probabilistic models ofNcorrelated binary random variables havingq-Gaussians asN→ ∞ limiting distributions

2008

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“…The compact-support distributions are representative of the gains in information due to reductions in fluctuations which eliminate the probability of rare states. These distributions can also arise in finite domain systems in which correlations or losses of information increase the probability of states near the boundary [5,6,36]. For negative values of Q, the decoupled states and decreased coupled-surprisal weakens the rate of decay, resulting in a heavy-tail distribution.…”

Section: An Interpretation Of Q-statistics Using Nonlinear Couplingmentioning

confidence: 99%

Nonlinear statistical coupling

Nelson

Umarov

2010

Physica A: Statistical Mechanics and its Applications

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-By considering a nonlinear combination of the probabilities of a system, a physical interpretation of Tsallis statistics as representing the nonlinear coupling or decoupling of statistical states is proposed. The escort probability is interpreted as the coupled probability, with Gaussian distributions. This conjugate relationship has been used to extend the generalized Fourier transform to the compact-support domain and to define a scale-invariant correlation structure with heavy-tail limit distribution. In the present paper, we show that the conjugate is a mapping between the source of nonlinearity in non-stationary stochastic processes and the nonlinear coupling which defines the coupled-Gaussian limit distribution. The effects of additive and multiplicative noise are shown to be separable into the coupled-variance and the coupling parameter Q, providing further evidence of the importance of the generalized moments.

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“…The celebrated Leibnitz triangle is one such example. Both models introduced in Thistleton et al, 2009] also are nontrivially scale-invariant. But they are not q-independent.…”

Section: Q-generalized Central Limit Theoremsmentioning

confidence: 99%

“…Possible relation between q-independence and scale-invariance Two probabilistic models, and [Thistleton et al, 2009] respectively, involving N equally distributed random variables were introduced some time ago. Their numerical discussion suggested that, in the N → ∞ limit, q-Gaussians emerged with q ≤ 1, after appropriate centering and scaling.…”

Section: Q-generalized Central Limit Theoremsmentioning

confidence: 99%