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Register a new userThe distinct flavors of Zipfs law in the rank-size and in the size-distribution representations, and its maximum-likelihood fitting
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In the last years, researchers have realized the difficulties of fitting power-law distributions properly. These difficulties are higher in Zipfs systems, due to the discreteness of the variables and to the existence of two representations for these systems, i.e., t
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We study the avalanche statistics observed in a minimal random growth model. The growth is governed by a reproduction rate obeying a probability distribution with finite mean a and variance va. These two control parameters determine if the avalanche size tends to a stationary distribution, (Finite Scale statistics with finite mean and variance or Power-Law tailed statistics with exponent in (1, 3]), or instead to a non-stationary regime with Log-Normal statistics. Numerical results and their statistical analysis are presented for a uniformly distributed growth rate, which are corroborated and generalized by analytical results. The latter show that the numerically observed avalanche regimes exist for a wide family of growth rate distributions and provide a precise definition of the boundaries between the three regimes.
This paper develops an analytical and rigorous formulation of the maximum entropy generation principle. The result is suggested as the Fourth Law of Thermodynamics.
The in situ measurement of the particle size distribution (PSD) of a suspension of particles presents huge challenges. Various effects from the process could introduce noise to the data from which the PSD is estimated. This in turn could lead to the occurrence of artificial peaks in the estimated PSD. Limitations in the models used in the PSD estimation could also lead to the occurrence of these artificial peaks. This could pose a significant challenge to in situ monitoring of particulate processes, as there will be no independent estimate of the PSD to allow a discrimination of the artificial peaks to be carried out. Here, we present an algorithm which is capable of discriminating between artificial and true peaks in PSD estimates based on fusion of multiple data streams. In this case, chord length distribution and laser diffraction data have been used. The data fusion is done by means of multi-objective optimisation using the weighted sum approach. The algorithm is applied to two different particle suspensions. The estimated PSDs from the algorithm are compared with offline estimates of PSD from the Malvern Mastersizer and Morphologi G3. The results show that the algorithm is capable of eliminating an artificial peak in a PSD estimate when this artificial peak is sufficiently displaced from the true peak. However, when the artificial peak is too close to the true peak, it is only suppressed but not completely eliminated.
We present the first world-wide inter-laboratory comparison of small-angle X-ray scattering (SAXS) for nanoparticle sizing. The measurands in this comparison are the mean particle radius, the width of the size distribution and the particle concentration. The investigated sample consists of dispersed silver nanoparticles, surrounded by a stabilizing polymeric shell of poly(acrylic acid). The silver cores dominate the X-ray scattering pattern, leading to the determination of their radii size distribution using: i) Glatters Indirect Fourier Transformation method, ii) classical model fitting using SASfit and iii) a Monte Carlo fitting approach using McSAS. The application of these three methods to the collected datasets produces consistent mean number- and volume-weighted core radii of R$_n$ = 2.76 nm and R$_v$ = 3.20 nm, respectively. The corresponding widths of the log-normal radii distribution of the particles were $sigma_n$ = 0.65 nm and $sigma_v$ = 0.71 nm. The particle concentration determined using this method was 3.00 $pm$ 0.38 g/L (4.20 $pm$ 0.73 $times$ 10$^{-6}$ mol/L). We show that the results are slightly biased by the choice of data evaluation procedure, but that no substantial differences were found between the results from data measured on a very wide range of instruments: the participating laboratories at synchrotron SAXS beamlines, commercial and home-made instruments were all able to provide data of high quality. Our results demonstrate that SAXS is a qualified method for revealing particle size distributions in the sub-20 nm region (at least), out of reach for most other analytical methods.
We present a maximum-likelihood method for parameter estimation in terahertz time-domain spectroscopy. We derive the likelihood function for a parameterized frequency response function, given a pair of time-domain waveforms with known time-dependent noise amplitudes. The method provides parameter estimates that are superior to other commonly-used methods, and provides a reliable measure of the goodness of fit. We also develop a simple noise model that is parameterized by three dominant sources, and derive the likelihood function for their amplitudes in terms of a set of repeated waveform measurements. We demonstrate the method with applications to material characterization.
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