Ergodicity and central-limit theorem in systems with long-range interactions

Figueiredo, Annibal; Filho, T. M. Rocha; Amato, M. A.

doi:10.1209/0295-5075/83/30011

Cited by 30 publications

(49 citation statements)

References 42 publications

(76 reference statements)

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“…The distribution of Y i in the limit of large N is again controversial [49,50]. For the specific initial conditions cited above it seems to first approach a fat-tailed distribution, interpreted by some as a q-Gaussian, before it finally tends to an ordinary Gaussian.…”

Section: Logistic Map and Hmf Modelmentioning

confidence: 99%

Central limit theorems for correlated variables: some critical remarks

Hilhorst¹

2009

Braz. J. Phys.

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In this talk I first review at an elementary level a selection of central limit theorems, including some lesser known cases, for sums and maxima of uncorrelated and correlated random variables. I recall why several of them appear in physics. Next, I show that there is room for new versions of central limit theorems applicable to specific classes of problems. Finally, I argue that we have insufficient evidence that, as a consequence of such a theorem, q-Gaussians occupy a special place in statistical physics.

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Section: Logistic Map and Hmf Modelmentioning

confidence: 99%

Central limit theorems for correlated variables: some critical remarks

Hilhorst¹

2009

Braz. J. Phys.

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“…We will specially review some available results in finance [130,131], earthquakes [132], and biology [133], among others [80,81,134]. We will not include in the present occasion the discussion of long-range-interacting Hamiltonian classical systems [82,135,136,137,138,139,140,141,142,143] which surely deserve a detailed study by themselves (in the present context we do not refer to systems whose elements short-range-interact, but only to those whose elements long-range-interact).…”

Section: Inter-occurrence Times In Finance Earthquakes and Genomesmentioning

confidence: 99%

Inter-occurrence times and universal laws in finance, earthquakes and genomes

Tsallis

2016

Chaos, Solitons & Fractals

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A plethora of natural, artificial and social systems exist which do not belong to the Boltzmann-Gibbs (BG) statistical-mechanical world, based on the standard additive entropy S BG and its associated exponential BG factor. Frequent behaviors in such complex systems have been shown to be closely related to q-statistics instead, based on the nonadditive entropy S q (with S 1 = S BG ), and its associated q-exponential factor which generalizes the usual BG one. In fact, a wide range of phenomena of quite different nature exist which can be described and, in the simplest cases, understood through analytic (and explicit) functions and probability distributions which exhibit some universal features. Universality classes are concomitantly observed which can be characterized through indices such as q. We will exhibit here some such cases, namely concerning the distribution of inter-occurrence (or interevent) times in the areas of finance, earthquakes and genomes.

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“…Long-range interacting systems are characterized by an interaction potential decaying at long distances as r −α such that α ≤ d, with d being the space dimension and may lead to anomalous behavior as non-Gaussian Quasi-Stationary States (QSS), negative (microcanonical) heat capacity, ensemble inequivalence and non-ergodicity [1][2][3][4][5][6]. Examples of systems with long-range interactions include self-gravitating systems (stars in galaxies and globular clusters), non-neutral plasmas and two-dimensional vortices [7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Dynamics and physical interpretation of quasistationary states in systems with long-range interactions

et al. 2014

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Although the Vlasov equation is used as a good approximation for a sufficiently large N , Braun and Hepp have showed that the time evolution of the one particle distribution function of a N particle classical Hamiltonian system with long range interactions satisfies the Vlasov equation in the limit of infinite N . Here we rederive this result using a different approach allowing a discussion of the role of inter-particle correlations on the system dynamics. Otherwise for finite N collisional corrections must be introduced. This has allowed the a quite comprehensive study of the Quasi Stationary States (QSS) but many aspects of the physical interpretations of these states remain unclear. In this paper a proper definition of timescale for long time evolution is discussed and several numerical results are presented, for different values of N . Previous reports indicates that the lifetimes of the QSS scale as N 1.7 or even the system properties scales with exp(N ). However, preliminary results presented here shows indicates that time scale goes as N 2 for a different type of initial condition. We also discuss how the form of the inter-particle potential determines the convergence of the N -particle dynamics to the Vlasov equation. The results are obtained in the context of following models: the Hamiltonian Mean Field, the Self Gravitating Ring Model, and a 2-D Systems of Gravitating Particles. We have also provided information of the validity of the Vlasov equation for finite N , i. e. how the dynamics converges to the mean-field (Vlasov) description as N increases and how inter-particle correlations arise.

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Ergodicity and central-limit theorem in systems with long-range interactions

Cited by 30 publications

References 42 publications

Central limit theorems for correlated variables: some critical remarks

Central limit theorems for correlated variables: some critical remarks

Inter-occurrence times and universal laws in finance, earthquakes and genomes

Dynamics and physical interpretation of quasistationary states in systems with long-range interactions

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