406 research outputs found
Are galaxy distributions scale invariant? A perspective from dynamical systems theory
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
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
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
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 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 outcoming wires are attached to the leaves of
the tree. In terms of the transmission amplitudes , the total
Landauer transmission is , so that each channel
is characterized by the weight . We numerically measure the
typical multifractal singularity spectrum of these weights as a
function of the disorder strength and we obtain the following conclusions
for its left-termination point . In the delocalized phase ,
is strictly positive and is associated with a
moment index . At criticality, it vanishes and is
associated with the moment index . In the localized phase ,
is associated with some moment index . 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
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 ?
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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