A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model

doi:10.1016/j.physa.2008.01.112

Physica A: Statistical Mechanics and its Applications

Volume 387, Issue 13, 15 May 2008, Pages 3121-3128

https://doi.org/10.1016/j.physa.2008.01.112 Get rights and content

Abstract

We give a closer look at the Central Limit Theorem (CLT) behavior in quasi-stationary states of the Hamiltonian Mean Field model, a paradigmatic one for long-range-interacting classical many-body systems. We present new calculations which show that, following their time evolution, we can observe and classify three kinds of long-standing quasi-stationary states (QSS) with different correlations. The frequency of occurrence of each class depends on the size of the system. The different microscopic nature of the QSS leads to different dynamical correlations and therefore to different results for the observed CLT behavior.

Introduction

Very recently there has been a lot of interest in generalizations of the Central Limit Theorem (CLT) [1], [2], [3] and on their possible (strict or numerically approximate) application to systems with long-range correlations [4], [5], at the edge-of-chaos [6], nonlinear dynamical systems the maximal Lyapunov exponent of which is either exactly zero or tends to vanish in the thermodynamic limit (increasingly large systems) [7], hindering in this way mixing and thus the application of standard statistical mechanics. A possible application of nonextensive statistical mechanics [8], [9] has been advocated in these cases. Along this line we discuss in the present paper a detailed study of a paradigmatic toy model for long-range interacting Hamiltonian systems [10], [11], [12], [13], [14], [15], [16], i.e. the Hamiltonian Mean Field (HMF) model which has been intensively studied in the last years. In a recent article [17], we presented molecular dynamics numerical results for the HMF model showing three kinds of quasi-stationary states (QSS) starting from the same water-bag initial condition with unitary magnetization ( $M_{0} = 1$ ). In the following we present how the applicability of the standard or $q$ -generalized CLT is influenced by the different microscopic dynamics observed in the three classes of QSS. In general, averaging over the three classes can be misleading. Indeed, the frequency of appearance of each of these classes depends on the size of the system under investigation, and there is no clear evidence that a predominant class exists.

Section snippets

Quasi-stationary behavior in the HMF model

The HMF model consists of $N$ fully-coupled classical inertial XY spins (rotors) $\vec{s_{i}} = (cos θ_{i}, sin θ_{i}), i = 1, \dots, N$ , with unitary module and mass [10]. One can also think of these spins as rotating particles on the unit circle. The Hamiltonian is given by $H = \sum_{i = 1}^{N} \frac{{p_{i}}^{2}}{2} + \frac{1}{2 N} \sum_{i, j = 1}^{N} [1 - cos (θ_{i} - θ_{j})],$ where $θ_{i}$ ( $0 < θ_{i} \leq 2 π$ ) is the angle and $p_{i}$ the conjugate variable representing the rotational velocity of spin $i$ .

At equilibrium the model can be solved exactly and the solution predicts a second-order phase transition

Discussion of numerical results for the CLT

In this section, following the prescription of the CLT and the procedure adopted in Refs. [16], [18], [19], we construct probability density functions (pdf) of quantities expressed as a finite sum of stochastic variables and we select these variables along the deterministic time evolutions of the $N$ rotors. More formally, we study the pdf of the quantity $y$ defined as $y_{i} = \frac{1}{\sqrt{n}} \sum_{k = 1}^{n} p_{i} (k) for i = 1, 2, \dots, N,$ where $p_{i} (k)$ , with $k = 1, 2, \dots, n$ , are the rotational velocities of the $i th$ rotor taken at fixed intervals

Conclusions

On the basis of the new calculations presented here, one should distinguish among different classes of QSS for a given size of the HMF system. Our tests do confirm that correlations can be different for different dynamical realizations of the same system, starting from the same class of initial conditions, and therefore also the Central Limit Theorem behavior can change. According to the class considered we can have a Gaussian pdf, a $q$ -Gaussian one or a mixture between the two. In this respect,

Acknowledgments

The present calculations were done within the Trigrid project and we thank M. Iacono Manno for technical help. A.P. and A.R. acknowledge financial support from the PRIN05-MIUR project “Dynamics and Thermodynamics of Systems with Long-Range Interactions”. C.T. acknowledges partial financial support from the Brazilian Agencies CNPq and Faperj.

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Citation Excerpt :
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A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model

Abstract

Introduction

Section snippets

Quasi-stationary behavior in the HMF model

Discussion of numerical results for the CLT

Conclusions

Acknowledgments

Phys. Rev. Lett.

J. Phys. B

Milan J. Math.

AIP Conf. Proc.

Phys. Rev. Lett.

Europhys. Lett.

Physica A

AIP Conf. Proc.

AIP Conf. Proc.

J. Stat. Mech.

Phys. Rev. E

J. Stat. Phys.

Europhys. News

Proc. Natl. Acad. Sci. USA