Lognormal Law Research Articles

Behaviors of the bubble, cloud, and emulsion phases in a fluidized bed

AbstractAn intrusive bioptical probe provides radial profiles of the PDF (probability density function) of the bubble, the cloud, and the emulsion phases behaviors in a fluidized bed of Geldart‐A particles. The fluidizing velocity ranges between 0.20 and 0.80 m/s. The bed porosity, the overall averaged bubble chords and velocities are well‐predicted by correlations of the literature. The cloud thickness is closely related to the bubble chord. The amount of gas traveling through the bubble phase never exceeds 15–20% of the total gas flow rate. The averaged porosity of the bubble, the cloud and the emulsion phases is of 88–92%, 65–70%, and 50–55%, respectively. The reduced bubble porosity distribution obeys a constant normal law over the fluidizing velocity range. Each distribution of the reduced bubble chord and velocity, and of the reduced cloud thickness obey a constant log normal law over the fluidizing velocity range. © 2007 American Institute of Chemical Engineers AIChE J, 2008

Analysis of floc size distributions in a mixing tank

The aim of this work is to analyse the relation between floc size distribution and hydrodynamics in a mixing tank. The objectives are to answer the two following questions: (1) Does a steady state exist in flocculation? (2) Do the floc size distributions depend on the impeller? Flocculation experiments are realised for fixed physico-chemical conditions (pH 3.5) in a standardized tank with two different impellers at the same dissipated power: an axial impeller (Lightnin A310) and a radial impeller (Rushton turbine). To analyse the sensitivity of flocculation to hydrodynamics, a hydrodynamic sequencing is performed. Whatever the impeller, a steady state, characterised by a balance between agglomeration and rupture phenomena, is rapidly reached. The characteristic size of the floc is shown to be related to the Kolmogorov scale and thus, to the dissipated power. In the case of the Rushton turbine, the floc size distributions are monomodal and can be fitted by a log-normal law. In the case of the Lightnin A310, floc size distributions are rather bi-modal and can be fitted by a weighted combination of two log-normal distributions. Given a dissipated power, different impellers, with different power numbers, rotate at different velocities; the mixing efficiency of these different impellers is shown to generate different floc size distributions. Thus the dissipated power cannot be the single parameter sufficient to model aggregation.

On particle size distribution in rocks under failure

Based on the Weibull theory, with allowance for experimental data on rock failure, a formula is derived for calculating the output of particles less than 100 µm, that includes the relationship between the strength and the size of a forming particle. The obtained outputs of the particles give an evidence that the examined size distribution deviate greatly from the lognormal law and the Weibull law when small sizes are considered. The established size distribution law for the rock failure products is more universal in nature as compared with the laws by Weibull or Rosen-Rammler.

Closed-form and numerical solutions for the probability distribution function of fracture diameters

After summarizing the available methods intended to estimate the fracture size distribution from a given trace length distribution, Santalò closed-form integral solution is used to derive a closed-form expression of the fracture size distribution for each of the most common trace length distributions, i.e. uniform, exponential, gamma, and power law. Numerical integration is used for the lognormal distribution. Expressions are given for the mean fracture diameter as a function of the mean trace length and the minimum fracture diameter. It is shown that, when the trace lengths are uniform, exponential, gamma, lognormal or power law, none of fracture diameter distributions is lognormal, exponential or gamma, as assumed in the literature. Power law trace length distribution yields a power law fracture size. The minimum fracture diameter cannot be equal to zero and plays an important role in determining the fracture diameter distribution. When the trace length distribution is defined on an interval, the diameter lower bound must be contained in the trace length interval, and the upper bound must be equal to the trace length upper bound.

Classification and powerlaws: The logarithmic transformation

AbstractLogarithmic transformation of the data has been recommended by the literature in the case of highly skewed distributions such as those commonly found in information science. The purpose of the transformation is to make the data conform to the lognormal law of error for inferential purposes. How does this transformation affect the analysis? We factor analyze and visualize the citation environment of the Journal of the American Chemical Society (JACS) before and after a logarithmic transformation. The transformation strongly reduces the variance necessary for classificatory purposes and therefore is counterproductive to the purposes of the descriptive statistics. We recommend against the logarithmic transformation when sets cannot be defined unambiguously. The intellectual organization of the sciences is reflected in the curvilinear parts of the citation distributions while negative powerlaws fit excellently to the tails of the distributions.

Aerosol size spectra and CCN activity spectra: Reconciling the lognormal, algebraic, and power laws

Empirical power law expressions of the form NCCN = Csk have been used in cloud physics for over 4 decades to relate the number of cloud condensation nuclei (CCN) and droplets formed on them to cloud supersaturation, s. The deficiencies of this parameterization are that the parameters C and k are usually constants taken from the empirical data and not directly related to the CCN microphysical properties and this parameterization predicts unbounded droplet concentrations at high s. The activation power law was derived in several works from the power law Junge‐type aerosol size spectra and parameters C and k were related to the indices of the power law, but this still does not allow to describe observed decrease of the k‐indices with s and limited NCCN. Recently, new parameterizations for cloud drop activation have been developed based upon the lognormal aerosol size spectra that yield finite NCCN at large s, but does not explain the activation power law, which is still traditionally used in the interpretation of CCN observations, in many cloud models and some large‐scale models. Thus the relation between the lognormal and power law parameterizations is unclear, and it is desirable to establish a bridge between them. In this paper, algebraic and power law equivalents are found for the lognormal size spectra of partially soluble dry and wet interstitial aerosol, and for the differential and integral CCN activity spectra. This allows derivation of the power law expression for cloud drop activation from basic thermodynamic principles (Köhler theory). In the new power law formulation, the index k and coefficient C are obtained as continuous analytical functions of s and expressed directly via parameters of aerosol lognormal size spectra (modal radii, dispersions) and physicochemical properties. This approach allows reconciliation of this modified power law and the lognormal parameterizations and their equivalence is shown by the quantitative comparison of these models as applied to several examples. The advantages of this new power law relationship include bounded Nd at high s, quantitative explanation of the experimental data on the k‐index and possibility to express k(s) and C(s) directly via the aerosol microphysics. The modified power law provides a framework for using the wealth of data on the C, k parameters accumulated over the past decades, both in the framework of the power law and the lognormal parameterizations.

Structure of Stieltjes classes of moment-equivalent probability laws

A Stieltjes class is a one-parameter family of moment-equivalent distribution functions constructed by modulation of a given indeterminate distribution function F, called the center of the class. Members of a Stieltjes class are mutually absolutely continuous, and conversely, any pair of moment-equivalent and mutually absolutely continuous distribution functions can be joined by a Stieltjes class. The center of a Stieltjes class is an equally weighted mixture of its extreme members, and this places restrictions on which distributions can belong to a Stieltjes class with a given center. The lognormal law provides interesting illustrations of the general ideas. In particular, it is possible for two moment equivalent infinitely divisible distributions to be joined by a Stieltjes class, and random scaling can be used to construct new Stieltjes classes from a given Stieltjes class.

Distributions of Wave Steepness and Surf Parameter

Predictions of wave action on slopes, and various wave- and depth-induced processes in the surf zone rely on two principal parameters: Wave steepness and surf parameter. For irregular waves, the theoretical distributions of these parameters are not known. Some studies suggest that wave steepness is described well by the Weibull and lognormal probability distributions. These distributions and thus their predictive utility depend on two key statistics, namely the mean and root-mean-square values of wave steepness. This study develops approximate theoretical forms of these statistics for narrow-band long-crested waves. For waves with steepness less than a Miche-Stokes type upper limit, comparisons with measurements from two severe storms show that the theoretical forms so derived and the resulting lognormal law in particular describes empirical data well. These results then permit the distribution of surf parameter to be approximated in a lognormal form also. Consequently, the nature and implications of such an approximation, resulting statistics and their plausible applications to breaker heights and types on slopes are explored.

Use of a lognormal distribution model for estimating soil water retention curves from particle-size distribution data

Particle-size distribution (PSD) of soils has been widely used to estimate water retention curves (WRCs). Lognormal distribution model, which estimates the WRC directly from the PSD data by applying a lognormal distribution law to both PSD and void-size distribution (VSD), has been recently developed and successfully applied to sandy soils. The objective of this study was to investigate in detail the applicability of the lognormal distribution model for a broader range of soil textures. Performance of the model was evaluated on five aggregated USDA texture classes (coarse-, moderately coarse-, medium-, moderately fine-, and fine-texture soils) using 229 ‘undisturbed’ soils (159 soils for model calibration and 70 soils for validation). The model performed worst for moderately fine-textured soils, since the PSDs and VSDs of soils included in this texture did not strongly follow the lognormal distribution. In general, performance of the model decreased with increase in fine particle fraction, possibly because of the surface sorption behavior of clay particles.

Fracture trace length and number distributions from fracture mapping

Statistical distributions of fracture trace length and density determined from two‐dimensional fracture mapping generally provide biased estimates of the underlying distributions due in part to edge effects of the finite window of observation (censoring) and the protocol adopted for recording short traces (lower truncation limit). Although methods for estimating the parameters of an assumed underlying distribution exist, validation of the inferred models by forward prediction of the observed length distribution is rarely undertaken since mathematical formulations of the required distributions are not reported in the literature. This paper presents formulae that can be used simply to obtain these distributions, with results for rectangular windows given as a specific example. Formulae relating the observed and underlying fracture density are also given. The formulae are presented for underlying trace lengths that can be characterized by exponential, lognormal, or finite range power law distributions. Results for the semi‐infinite range power law can also be obtained simply. The formulae are based upon the assumption that fracture locations can be adequately described by a uniform Poisson process, although the methodology is described more generally. The distributions of uncensored, singly censored and doubly censored trace lengths are also derived. The effects of the window of observation and the lower truncation limit for the frequently assumed semi‐infinite range power law are outlined.

Interaction of a femtosecond laser-produced plasma plume with a time delayed pulse

The use of a sequence of two ultra short light pulses enabled us to produce ions, neutrals and nanoparticles from a Ti metallic target and successively irradiate them. Efficient formation of nanoparticles was observed as well as their subsequent fragmentation into neutral atoms. The size distribution of the nanoparticles obeys a log–normal law, characterized by a statistical median diameter size of 70 nm. Measurements of their spectral emission provides information regarding their temperature.

Truncated fractal frequency distribution of element abundance data: A dynamic model for the metasomatic enrichment of base and precious metals

The frequency distributions of elements in rocks and ore deposits do not conform to a universal scaling law, but vary in dependence of the mechanism of element enrichment. In the present contribution a dynamic model is proposed that describes element enrichments taking place by metasomatic processes causing the formation of hydrothermal ore deposits. It is shown that metasomatic element enrichment can result in a fractal distribution of element abundance data in the low concentration region with a smooth truncation at higher concentration values due to the existence of physical upper concentration limits imposed by the mineralogical nature of the ore. The developed model is tested on the basis of base and precious metal concentration data determined on diamond drill core samples collected from the Waterloo massive sulfide deposit, Australia. It is shown that the frequency distributions of Zn, Pb, Cu, and Ag conform relatively well to truncated power–law relationships as predicted by the model. In contrast, the frequency distribution of the Au data was found to be better described by the log-normal law, presumably due to syn-genetic processes of element redistribution that have not been modeled by the proposed dynamic model.

A Brief History of Generative Models for Power Law and Lognormal Distributions

Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a lognormal distribution. In trying to learn enough about these distributions to settle the question, I found a rich and long history, spanning many fields. Indeed, several recently proposed models from the computer science community have antecedents in work from decades ago. Here, I briefly survey some of this history, focusing on underlying generative models that lead to these distributions. One finding is that lognormal and power law distributions connect quite naturally, and hence, it is not surprising that lognormal distributions have arisen as a possible alternative to power law distributions across many fields.

A GENERATIVE MODEL OF INCOME DISTRIBUTION 2: INEQUALITY OF THE ITERATED INVESTMENT GAME

This paper attempts to develop the model formalized by Hamada (1999b), which generates a distribution of incomes obeying the lognormal law by starting with an individual's rational choice at the micro level. As in the previous work, we employ an iterated investment game as a baseline model in which each player has a binary choice between investing and not investing. Hamada (1999b) showed only that if the game time n is sufficiently large and γ > 1/(R + 1), then the distribution of the profit of an iterated investment game is subject to approximately the normal distribution without cumulative effect on one hand and the lognormal distribution with cumulative effect on the other hand. The present paper develops the previous work considerably and hence specifies the mathematical structure of the model to release special conditions. We succeed in showing that the proposition holds for every γ in (0, 1) by the change of variables theorem with respect to the probability density function of the multidimensional normal distribution. This generalization enables us to exhibit the relation between the Gini coefficient of the lognormal distribution derived from the model and the parameters determining a structure of the game, such as prize density γ and the rate of return R. As a result of analysis, we show some remarkable implications.

On the log-normal distribution of stock market data

We present our recent studies on the development of a statistical model of stock market data. For some stock market data, the statistical distribution of closing prices normalized by the corresponding traded volumes, fits well a log-normal law. For other stocks, the log-normal law is obtained after application of a detrending procedure. Different schemes for the trend determination are considered.

Special Features of the Statistical Processing of Seal Monitoring Data on Fuel-Cell Cladding in the ÉGP-6 Reactors at the Bilibino Nuclear Power Plant

The results of a statistical analysis of the indications of the seal-monitoring system for the outer cladding of the fuel elements in the fuel assemblies of the EGP-6 reactors at the Bilibino nuclear power plant are presented. It is shown that the distribution of the indications deviates substantially from the normal law, approaching the log-normal law. It is suggested that this phenomenon is due to seepage of reactor gas into the gas channels of a fuel assembly through the gaps between the graphite sleeves of the assemblies.

Evolution of Distribution of Dislocations in Ni3Ge Single Crystals During Deformation

The evolution of the distribution of dislocations in Ni3Ge single crystals subjected to deformation in uniaxial compression is studied. The dislocation ensemble in the material under review is found to be of a chaotic homogeneous type. Contact interactions between dislocations prevail, and a linear relation of the spacing between dislocations to the length of dislocation segments is observed for stoppers of an arbitrary type. An equation is derived for the probability density function of the fraction of mobile dislocation segments. The solution to the equation is the normal distribution law. This solution can be extended to parameters that are functions of the dislocation density or spacing between dislocations. The experimental histograms of the spacing between dislocations and of that between arbitrary stoppers with a high significance level obey the lognormal law for all degrees of reduction studied.

In situ investigation of the first stages of growth of cluster-assembled carbon films by scanning tunnelling microscopy

The first stages of aggregation of low energy deposited carbon clusters on Ag(1 1 0) surface have been observed in situ by scanning tunnelling microscopy (STM). Sub-monolayer coverages of the substrate at room temperature have been obtained by using a supersonic neutral cluster beam with a mass distribution centered at an average of 900 atoms/cluster. We observe that clusters are able to diffuse on the surface and along steps, and organise themselves in compact islands principally located at step edges. Size of supported islands ranges from a few nm up to 40 nm. A statistical analysis based on several STM images shows that the island size follows a lognormal distribution law. Possible mechanisms involved in the diffusion and coalescence of carbon clusters on metallic surfaces are discussed.

Uncertainties of the transfer of radionuclides in terrestrial ecosystems

The Nord-Cotentin radioecological study was conducted in different stages. First, an assessment tool was developed and implemented to estimate the number of leukaemia radio-induced by the operation of nuclear facilities of the La Hague site (Nord-Cotentin region, France). Then, an uncertainty analysis was initiated to verify if the confidence intervals around the best estimate previously calculated allows to confirm the low impact of the nuclear facilities. The modelling of the terrestrial ecosystems was especially studied because of the paucity of results of measurements. That is also why the methodology built to reconstruct the statistical laws for each parameter of the model mixes the bibliographic knowledge and exploitation of a national database of environmental measurements. The application of this methodology for up to forty parameters shows that the possibility of fitting an accurate distribution such as a normal or a lognormal law is strongly linked with the availability and the number of measured data.

Changes of surface and volume properties of calcite during a batch wet grinding process

An experimental study on the ultra-fine wet grinding of calcite in a batch stirred bead mill is reported. The evolution of the size distribution and of the specific surface area of the particles versus grinding time was measured revealing that an agglomeration phenomenon takes place during the comminution process. The X-ray diffraction profiles versus milling time did not show any polymorphic phase transformation but an intensity reduction and a broadening of diffraction peaks. The size distributions can be properly fitted by a linear combination of statistical log-normal laws, taking the agglomeration of particles into account. The analysis of rheological properties of the ground suspensions revealed a plastic thinning behaviour which can be described by the Hershel–Bulkley equation.