Q-Gaussian Approximants Mimic Non-Extensive Statistical-Mechanical Expectation for Many-Body Probabilistic Model with Long-Range Correlations

Cent. Eur. J. Phys. • 7(3) • 2009 • 387-394 DOI: 10.2478/s11534-009-0054-4 Central European Journal of Physics q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations Research Article William J. Thistleton1∗, John A. Marsh2† , Kenric P. Nelson3‡ , Constantino Tsallis45§ 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 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 Received 3 November 2008; accepted 25 March 2009 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/rα for r =1, 2, …, N-1, where r indicates displacement from the diagonal and where 0 6 ρ 6 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. PACS (2008): 02.50.-r; 02.60.-x; 02.60.Cb; 02.70.-c; 05.10.-a Keywords: q-Gaussian • non-extensive statistical mechanics • correlated systems © Versita Warsaw and Springer-Verlag Berlin Heidelberg. 1. Introduction ∗ E-mail: [email protected] † E-mail: [email protected] Central Limit Theorems illuminate the mechanisms which ‡ E-mail: [email protected] § lead to the ubiquitous natural occurrence of certain prob- E-mail: [email protected] ability distributions. The most celebrated of these theorems, most commonly known as The Central Limit The- 387 q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations orem, is taught in introductory statistics courses and, in ples of size N drawn from the dependent uniform variables its simplest form, describes how the sum of a sequence and examines their asymptotic (large N) behavior. Section of independent, identically distributed random variables 6 further explores dependent uniform random variables by with finite first and second moments converges to a Gaus- examining the effect of power-law decay in the under- sian distribution [1]. A generalization of the traditional lying autocorrelation function. Finally, in Section 7 we Gaussian distribution which maximizes, under certain con- conclude with a discussion of the relevance of this result ditions, the non-additive entropy Sq [2] of a system has and suggestions for further study. been introduced and shown to model a wide variety of naturally occurring systems, especially those with heavy tails [3]. These q-Gaussian distributions are known to be 2. Dependent uniform random vari- attractors under summation for systems exhibiting the type ables of correlation known as q-Independence [4]. In this paper, we demonstrate another mechanism for the The basic objects of our study are uniform random vari- relevance of the q-Gaussian distribution. We show nu- ables exhibiting global correlations. To obtain such a merically that compact support q-Gaussian distributions system we first construct a multivariate Gaussian random can serve approximately as limiting distributions of spe- variable X with mean µX =0 and whose covariance matrix cially dependent systems of uniformly distributed random ΣX exhibits a particularly simple form of global correla- variables. The uniform distribution is important because tion, of the simplicity of the form and because it is the max- imum entropy distribution under the constraint of compact support. The generalized Central Limit Theorem of ρ ··· ρ et al 1 Umarov .[4] requires variables to have a special de- ρ ··· ρ q- 1 pendency structure, referred to as independent. In this x : Σ = . (1) paper, we demonstrate numerically that, for uniformly dis- . .. tributed random variables with global correlation simply ρ ρ ··· 1 equal throughout the system, the limiting distribution is approximately a q-Gaussian. This provides a simplified analytical tool for examining an important system, namely We transform each of these Gaussian distributions to pro- one with uniform random variables influenced by uniform duce uniformly distributed random variables via the Prob- global dependency. ability Integral Transformation. To do so, use the elemen- We also generalize the above through the study of a tary fact that if Y is a random variable obtained under strictly scale-invariant (power law decay in covariance an invertible transformation of random variable X as Y matrix) probabilistic N-body model with symmetric, uni- = g(X), then the cumulative distribution functions of X − form, identically distributed random variables where cor- and Y are related as FY (·) = FX (g 1(·)). In this way a relations are induced through a transformation of the mul- random variable X may be mapped to a uniform random tivariate normal distribution with covariance matrix decay- variable U by applying to X its own cumulative distribu- α ing out from the unit diagonal as ρ/r , where r indicates tion function. That is, setting g(·) = FX (·) yields FU (u) = r N α −1 displacement from the diagonal, =1, 2, …, -1, > 0 FX (FX (u))=u, the cumulative distribution function of the characterizes the range of the correlations, and 0 6 ρ 6 1 uniform distribution. Using this idea we apply a transfor- characterizes the strength of the correlation. We show mation involving the error function to each component of numerically that the non-Gaussian sum of the N depen- the multivariate Gaussian distribution X, yielding a mul- dent random variables is well modeled by a distribution tivariate random variable U of globally dependent, uni- which mimics a compact support q-Gaussian distribution formly distributed components. The detailed construction with q(ρ, α) 6 1. is given immediately below. This paper is organized as follows. The next section pro- Proceed by first defining N+1 independent, identically Z Z ∼ N vides the defining relationships and demonstrates the ef- distributed normal random variables 0,… N (0,1) fect of the induced dependency structure for small sample with zero mean, µ=0, and unit variance, σ 2=1. Then de- q- X sizes. Section 3 reviews Gaussian random variables, fine a new multivariate√ random vector with components X ≡ ρZ p − ρZ i N which are shown to approximate the distributions of the defined as i 0 + 1 i for =1,…, . The real mean for the system under study. Section 4 discusses parameter ρ introduces global correlations amongst the q- Z parameter fitting in Gaussian distributions. Section 5 Xi through dependence on the common term 0. The new discusses the probability distribution of the mean of sam- distribution X can then be written as 388 William J. Thistleton, John A. Marsh, Kenric P. Nelson, Constantino Tsallis We now transform the random vector X into a multivariate uniform distribution U by defining √ p Z0 ρ 1 − ρ 0 ··· 0 √ p Z Xi ρ − ρ ··· 1 U ≡ X − 1 1 √ : 0 1 0 Z i Φ( i) = erf (4) X 2 BZ: 2 2 = . = 2 . .. . √ p . ρ 0 0 ··· 1 − ρ i n x ZN For =1,…, where Φ( ) is the standard normal cumu- Ui (2) lative distribution function. Each is then distributed − 1 ; 1 Thus expressed, X is seen to be a linear transformation of uniformly on the interval 2 2 and inherits correlations a multivariate normal random vector Z, hence also multi- from the underlying multivariate normal distribution. With variate normal. It is straightforward to verify that X has our system so constructed we proceed to demonstrate a zero mean, and covariance matrix few basic properties. First consider the dependency structure inherited by the T Σx = BΣZ B = [ρ + (1 − ρ)δij ] (3) uniform random variables. Since each of the uniform random variables has mean E[Ui]=0, the covariance is seen reproducing the form given in Eq. (1) above. to be σ[Ui,Uj ] = E[UiUj ] where Z ∞ Z ∞ xi xj 1 1 1 2 2 σ UiUj √ √ − xi − ρxixj xj dxidxj ; [ ] = erf erf p exp − ρ2 2 + (5) −∞ −∞ 2 2 2 2π 1 − ρ2 2(1 ) with correlation coefficient ρ[Ui,Uj ] = 12 σ[Ui,Uj ]. We present the correlation of the uniform random variables as a function of the correlation of the underlying normal random variables in Fig.1 , where data corresponding to the resulting normal variables are transformed to uniformly − 1 ; 1 distributed random variables on 2 2 . The linear correlation coefficient (Pearson’s ρ) of the bivariate uniform marginal data is presented as a function of the normal correlation coefficient of the underlying normal system in Fig.1 and is obtained by numerical integration of Eq. (5) and also by direct calculation from the transformed uniform data. Induced correlations in the uniform random variables are slightly less than those of the normal random variables.

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