8,766 research outputs found
A Divergence Formula for Randomness and Dimension (Short Version)
If is an infinite sequence over a finite alphabet and is
a probability measure on , then the {\it dimension} of with
respect to , written , is a constructive version of
Billingsley dimension that coincides with the (constructive Hausdorff)
dimension when is the uniform probability measure. This paper
shows that and its dual \Dim^\beta(S), the {\it strong
dimension} of with respect to , can be used in conjunction with
randomness to measure the similarity of two probability measures and
on . Specifically, we prove that the {\it divergence formula}
\dim^\beta(R) = \Dim^\beta(R) =\CH(\alpha) / (\CH(\alpha) + \D(\alpha ||
\beta)) holds whenever and are computable, positive
probability measures on and is random with
respect to . In this formula, \CH(\alpha) is the Shannon entropy of
, and \D(\alpha||\beta) is the Kullback-Leibler divergence between
and
On the concept of complexity in random dynamical systems
We introduce a measure of complexity in terms of the average number of bits
per time unit necessary to specify the sequence generated by the system. In
random dynamical system, this indicator coincides with the rate K of divergence
of nearby trajectories evolving under two different noise realizations.
The meaning of K is discussed in the context of the information theory, and
it is shown that it can be determined from real experimental data. In presence
of strong dynamical intermittency, the value of K is very different from the
standard Lyapunov exponent computed considering two nearby trajectories
evolving under the same randomness. However, the former is much more relevant
than the latter from a physical point of view as illustrated by some numerical
computations for noisy maps and sandpile models.Comment: 35 pages, LaTe
Analysis of Daily Streamflow Complexity by Kolmogorov Measures and Lyapunov Exponent
Analysis of daily streamflow variability in space and time is important for
water resources planning, development, and management. The natural variability
of streamflow is being complicated by anthropogenic influences and climate
change, which may introduce additional complexity into the phenomenological
records. To address this question for daily discharge data recorded during the
period 1989-2016 at twelve gauging stations on Brazos River in Texas (USA), we
use a set of novel quantitative tools: Kolmogorov complexity (KC) with its
derivative associated measures to assess complexity, and Lyapunov time (LT) to
assess predictability. We find that all daily discharge series exhibit long
memory with an increasing downflow tendency, while the randomness of the series
at individual sites cannot be definitively concluded. All Kolmogorov complexity
measures have relatively small values with the exception of the USGS (United
States Geological Survey) 08088610 station at Graford, Texas, which exhibits
the highest values of these complexity measures. This finding may be attributed
to the elevated effect of human activities at Graford, and proportionally
lesser effect at other stations. In addition, complexity tends to decrease
downflow, meaning that larger catchments are generally less influenced by
anthropogenic activity. The correction on randomness of Lyapunov time
(quantifying predictability) is found to be inversely proportional to the
Kolmogorov complexity, which strengthens our conclusion regarding the effect of
anthropogenic activities, considering that KC and LT are distinct measures,
based on rather different techniques
Local time in diffusive media and applications to imaging
Local time is the measure of how much time a random walk has visited a given
position. In multiple scattering media, where waves are diffuse, local time
measures the sensitivity of the waves to the local medium's properties. Local
variations of absorption, velocity and scattering between two measurements
yield variations in the wave field. These variations are proportionnal to the
local time of the volume where the change happened and the amplitude of
variation. The wave field variations are measured using correlations and can be
used as input in a inversion algorithm to produce variation maps. The present
article gives the expression of the local time in dimensions one, two and three
and an expression of its fluctuations, in order to perform such inversions and
estimate their accuracy.Comment: 10 pages, 2 figures and 3 table
Quantum Griffiths effects and smeared phase transitions in metals: theory and experiment
In this paper, we review theoretical and experimental research on rare region
effects at quantum phase transitions in disordered itinerant electron systems.
After summarizing a few basic concepts about phase transitions in the presence
of quenched randomness, we introduce the idea of rare regions and discuss their
importance. We then analyze in detail the different phenomena that can arise at
magnetic quantum phase transitions in disordered metals, including quantum
Griffiths singularities, smeared phase transitions, and cluster-glass
formation. For each scenario, we discuss the resulting phase diagram and
summarize the behavior of various observables. We then review several recent
experiments that provide examples of these rare region phenomena. We conclude
by discussing limitations of current approaches and open questions.Comment: 31 pages, 7 eps figures included, v2: discussion of the dissipative
Ising chain fixed, references added, v3: final version as publishe
A mechanical model of normal and anomalous diffusion
The overdamped dynamics of a charged particle driven by an uniform electric
field through a random sequence of scatterers in one dimension is investigated.
Analytic expressions of the mean velocity and of the velocity power spectrum
are presented. These show that above a threshold value of the field normal
diffusion is superimposed to ballistic motion. The diffusion constant can be
given explicitly. At the threshold field the transition between conduction and
localization is accompanied by an anomalous diffusion. Our results exemplify
that, even in the absence of time-dependent stochastic forces, a purely
mechanical model equipped with a quenched disorder can exhibit normal as well
as anomalous diffusion, the latter emerging as a critical property.Comment: 16 pages, no figure
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