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Register a new userGeneralized Huberman-Rudnick scaling law and robustness of $q$-Gaussian probability distributions
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\\\"Ozg\\\"ur Af\\c{s}ar
Publication date
2012
fields
Physics
and research's language is
English
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We generalize Huberman-Rudnick universal scaling law for all periodic windows of the logistic map and show the robustness of $q$-Gaussian probability distributions in the vicinity of chaos threshold. Our scaling relation is universal for the self-similar windows of the map which exhibit period-doubling subharmonic bifurcations. Using this generalized scaling argument, for all periodic windows, as chaos threshold is approached, a developing convergence to $q$-Gaussian is numerically obtained both in the central regions and tails of the probability distributions of sums of iterates.
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The stationary distributions of sums of positions of trajectories generated by the logistic map have been found to follow a basic renormalization group (RG) structure: a nontrivial fixed-point multi-scale distribution at the period-doubling onset of chaos and a Gaussian trivial fixed-point distribution for all chaotic attractors. Here we describe in detail the crossover distributions that can be generated at chaotic band-splitting points that mediate between the aforementioned fixed-point distributions. Self affinity in the chaotic region imprints scaling features to the crossover distributions along the sequence of band splitting points. The trajectories that give rise to these distributions are governed first by the sequential formation of phase-space gaps when, initially uniformly-distributed, sets of trajectories evolve towards the chaotic band attractors. Subsequently, the summation of positions of trajectories already within the chaotic bands closes those gaps. The possible shapes of the resultant distributions depend crucially on the disposal of sets of early positions in the sums and the stoppage of the number of terms retained in them.
In this paper, the scaling-law vector calculus, which is related to the connection between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The Gauss-Ostrogradsky-like theorem, Stokes-like theorem, Green-like theorem, and Green-like identities are considered in the sense of the scaling-law vector calculus. The Navier-Stokes-like equations are obtained in detail. The obtained result is as a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows.
We study the collision frequencies of particles in the weakly and highly ionized plasmas with the power-law q-distributions in nonextensive statistics. We derive the average collision frequencies of neutral-neutral particle, electron-neutral particle, ion-neutral particle, electron-electron, ion-ion and electron-ion, respectively, in the q-distributed plasmas. We show that the average collision frequencies depend strongly on the q-parameter in a complex form and thus their properties are significantly different from that in Maxwell-distributed plasmas. These new average collision frequencies are important for us to study accurately the transport property in the complex plasmas with non-Maxwell/power-law velocity distributions.
Weight at delivery is a standard cumulative measure of placental growth. But weight is a crude summary of other placental characteristics, such as the size and shape of the chorionic plate and the location of the umbilical cord insertion. Distributions of such measures across a cohort reveal information about the developmental history of the chorionic plate that is unavailable from an analysis based solely on the mean and standard deviation. Various measures were determined from digitized images of chorionic plates obtained from the Pregnancy, Infection, and Nutrition Study, a prospective cohort study of preterm birth in central North Carolina between 2002 and 2004. The centroids (the geometric centers) and umbilical cord insertions were taken directly from the images. The chorionic plate outlines were obtained from an interpolation based on a Fourier series, while eccentricity (of the best-fit ellipse), skewness, and kurtosis were determined from a shape analysis using the method of moments. The distribution of each variable was compared against the normal, lognormal, and Levy distributions. We found only a single measure (eccentricity) with a normal distribution. All other placental measures required lognormal or heavy-tailed distributions to account for moderate to extreme deviations from the mean, where relative likelihoods in the cohort far exceeded those of a normal distribution. Normal and lognormal distributions result from the accumulated effects of a large number of independent additive (normal) or multiplicative (lognormal) events. Thus, while most placentas appear to develop by a series of small, regular, and independent steps, the presence of heavy-tailed distributions suggests that many show shape features which are more consistent with a large number of correlated steps or fewer, but substantially larger, independent steps.
The Nearest Neighbour Spacing (NNS) distribution can be computed for generalized symmetric 2x2 matrices having different variances in the diagonal and in the off-diagonal elements. Tuning the relative value of the variances we show that the distributions of the level spacings exhibit a crossover from clustering to repulsion as in GOE. The analysis is extended to 3x3 matrices where distributions of NNS as well as Ratio of Nearest Neighbour Spacing (RNNS) show similar crossovers. We show that it is possible to calculate NNS distributions for Hermitian matrices (N=2, 3) where also crossovers take place between clustering and repulsion as in GUE. For large symmetric and Hermitian matrices we use interpolation between clustered and repulsive regimes and identify phase diagrams with respect to the variances.
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