8,766 research outputs found

    A Divergence Formula for Randomness and Dimension (Short Version)

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    If SS is an infinite sequence over a finite alphabet Σ\Sigma and β\beta is a probability measure on Σ\Sigma, then the {\it dimension} of S S with respect to β\beta, written dimβ(S)\dim^\beta(S), is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension dim(S)\dim(S) when β\beta is the uniform probability measure. This paper shows that dimβ(S)\dim^\beta(S) and its dual \Dim^\beta(S), the {\it strong dimension} of SS with respect to β\beta, can be used in conjunction with randomness to measure the similarity of two probability measures α\alpha and β\beta on Σ\Sigma. Specifically, we prove that the {\it divergence formula} \dim^\beta(R) = \Dim^\beta(R) =\CH(\alpha) / (\CH(\alpha) + \D(\alpha || \beta)) holds whenever α\alpha and β\beta are computable, positive probability measures on Σ\Sigma and RΣR \in \Sigma^\infty is random with respect to α\alpha. In this formula, \CH(\alpha) is the Shannon entropy of α\alpha, and \D(\alpha||\beta) is the Kullback-Leibler divergence between α\alpha and β\beta

    On the concept of complexity in random dynamical systems

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    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

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    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

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    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

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    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

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    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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