406 research outputs found

    Are galaxy distributions scale invariant? A perspective from dynamical systems theory

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    Unless there is evidence for fractal scaling with a single exponent over distances .1 <= r <= 100 h^-1 Mpc then the widely accepted notion of scale invariance of the correlation integral for .1 <= r <= 10 h^-1 Mpc must be questioned. The attempt to extract a scaling exponent \nu from the correlation integral n(r) by plotting log(n(r)) vs. log(r) is unreliable unless the underlying point set is approximately monofractal. The extraction of a spectrum of generalized dimensions \nu_q from a plot of the correlation integral generating function G_n(q) by a similar procedure is probably an indication that G_n(q) does not scale at all. We explain these assertions after defining the term multifractal, mutually--inconsistent definitions having been confused together in the cosmology literature. Part of this confusion is traced to a misleading speculation made earlier in the dynamical systems theory literature, while other errors follow from confusing together entirely different definitions of ``multifractal'' from two different schools of thought. Most important are serious errors in data analysis that follow from taking for granted a largest term approximation that is inevitably advertised in the literature on both fractals and dynamical systems theory.Comment: 39 pages, Latex with 17 eps-files, using epsf.sty and a4wide.sty (included) <[email protected]

    Zipf's law, 1/f noise, and fractal hierarchy

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    Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic events are typical ubiquitous general empirical observations across the individual sciences which cannot be understood within the set of references developed within the specific scientific domains. All these observations are associated with scaling laws and have caused a broad research interest in the scientific circle. However, the inherent relationships between these scaling phenomena are still pending questions remaining to be researched. In this paper, theoretical derivation and mathematical experiments are employed to reveal the analogy between fractal patterns, 1/f noise, and the Zipf distribution. First, the multifractal process follows the generalized Zipf's law empirically. Second, a 1/f spectrum is identical in mathematical form to Zipf's law. Third, both 1/f spectra and Zipf's law can be converted into a self-similar hierarchy. Fourth, fractals, 1/f spectra, Zipf's law, and the occurrence of large catastrophic events can be described with similar exponential laws and power laws. The self-similar hierarchy is a more general framework or structure which can be used to encompass or unify different scaling phenomena and rules in both physical and social systems such as cities, rivers, earthquakes, fractals, 1/f noise, and rank-size distributions. The mathematical laws on the hierarchical structure can provide us with a holistic perspective of looking at complexity such as self-organized criticality (SOC).Comment: 20 pages, 9 figures, 3 table

    Anomalous scaling and Lee-Yang zeroes in Self-Organized Criticality

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    We show that the generating functions of avalanche observables in SOC models exhibits a Lee-Yang phenomenon. This establishes a new link between the classical theory of critical phenomena and SOC. A scaling theory of the Lee-Yang zeroes is proposed including finite sampling effects.Comment: 33 pages, 19 figures, submitte

    Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality

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    In contrast to finite dimensions where disordered systems display multifractal statistics only at criticality, the tree geometry induces multifractal statistics for disordered systems also off criticality. For the Anderson tight-binding localization model defined on a tree of branching ratio K=2 with NN generations, we consider the Miller-Derrida scattering geometry [J. Stat. Phys. 75, 357 (1994)], where an incoming wire is attached to the root of the tree, and where KNK^{N} outcoming wires are attached to the leaves of the tree. In terms of the KNK^{N} transmission amplitudes tjt_j, the total Landauer transmission is T≡∑j∣tj∣2T \equiv \sum_j | t_j |^2, so that each channel jj is characterized by the weight wj=∣tj∣2/Tw_j=| t_j |^2/T. We numerically measure the typical multifractal singularity spectrum f(α)f(\alpha) of these weights as a function of the disorder strength WW and we obtain the following conclusions for its left-termination point α+(W)\alpha_+(W). In the delocalized phase W<WcW<W_c, α+(W)\alpha_+(W) is strictly positive α+(W)>0\alpha_+(W)>0 and is associated with a moment index q+(W)>1q_+(W)>1. At criticality, it vanishes α+(Wc)=0\alpha_+(W_c)=0 and is associated with the moment index q+(Wc)=1q_+(W_c)=1. In the localized phase W>WcW>W_c, α+(W)=0\alpha_+(W)=0 is associated with some moment index q+(W)<1q_+(W)<1. We discuss the similarities with the exact results concerning the multifractal properties of the Directed Polymer on the Cayley tree.Comment: v2=final version (16 pages

    Multi-scale magnetic field intermittence in the plasma sheet

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    This paper demonstrates that intermittent magnetic field fluctuations in the plasma sheet exhibit transitory, localized, and multi-scale features. We propose a multifractal based algorithm, which quantifies intermittence on the basis of the statistical distribution of the 'strength of burstiness', estimated within a sliding window. Interesting multi-scale phenomena observed by the Cluster spacecraft include large scale motion of the current sheet and bursty bulk flow associated turbulence, interpreted as a cross-scale coupling (CSC) process.Comment: 18 pages, 7 figure

    Scaling in the space climatology of the auroral indices: Is SOC the only possible explanation ?

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    The study of the robust features of the magnetosphere is motivated both by new "whole system" approaches, and by the idea of "space climate" as opposed to "space weather". We enumerate these features for the AE index, and discuss whether self-organised criticality (SOC) is the most natural explanation of the "stylised facts" so far known for AE. We identify and discuss some open questions, answers to which will clarify the extent to which AE's properties provide evidence for SOC. We then suggest an SOC-like reconnection-based scenario drawing on the result of Craig(2001) as an explanation of the very recent demonstration by Uritsky et al(2001b) of power laws in several properties of spatiotemporal features seen in auroral images.Comment: 24 pages including 7 figures. Based on an invited talk given at the IAGA meeting in Hanoi, Vietnam, August 2000. Retitled v2 has revisions, clearer statement of intent of paper i.e. part review/part critique/some new suggestions, and 1 new figure. In press, Nonlinear Processes in Geophysic
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