Lognormal Law Research Articles

Statistical analysis of asperities on a rough surface

Main asperities on isotropic rough surfaces are analysed using their two-dimensional characteristics: the altitude of each asperity’s summit and its radius. Each of these geometrical characteristics is described using statistical analysis with probability density functions such as Gaussian or lognormal laws. These distributions are redimensioned with characteristic values deduced using the recent standard procedure for roughness and waviness parameter determination in accord with the so-called “motif” procedure. After a short review of the standardised procedure for determining roughness and waviness parameters, we present a detailed description of the characteristic values — mean and root mean square — of the dimensional parameters and their distributions. Finally, we apply the model herein developed using measurements from three different types of microgeometry ranging from quasi-periodic to quasi-random.

Preparation and characterization of nanocrystalline powders of Cu–Zn alloy by wire electrical explosion method

Nanocrystalline particles of Cu–Zn alloy in the size range from 10 to 140 nm with a mean particle size of 85 nm have been prepared from α-Cu–Zn alloy wire containing 39.8 at.% Zn by a wire electrical explosion method by adopting appropriate process parameters. The particles are identified as a mixture of α, β, γ, and ε phases of Cu–Zn alloy from X-ray powder diffraction (XRD). Most of the powders are hexagonal in shape, with only a small part being spherical and cubic. The particle size of the product obeys the log-normal distribution law, which indicates that the formation of particles with different sizes corresponds with probability dynamics. A possible mechanism for the formation of the alloy nanoparticles is proposed.

One-step synthesis of the nanometer particles of γ-Fe 2O 3 by wire electrical explosion method

Nanoparticles of γ-Fe 2O 3 with a mean particle size of 9 nm (an average nanograin size of 4 nm) have been prepared by an electrical explosion method through one-step. The particles were nearly equiaxed. The particle size distribution fits with log-normal distribution law as identified by SAXS. The coercivity and the saturation magnetization of the sample were determined to be as about 30% of those of the bulk materials by VSM. A possible mechanism for the formation of the γ-Fe 2O 3 was proposed.

The influence of gas (Ar) pressure on the formation of a bubble structure in silicate melt film

The formation of a bubble structure in a film of silicate melt obtained by the melting of glass powder layer has been studied using a computer-aided complex consisting of a high-temperature microscope, a digital video camera, an “input-output” card, and an IBM-486 personal computer. It is shown that the specific surface area of bubbles and their volume decrease with time at all studied pressures (0.02, 0.1, and 0.2 MPa). A decrease in the number of bubbles is reliably recorded only at pressures of 0.1 and 0.02 MPa. At a pressure of 0.2 MPa, the number of bubbles decreases only at the final stage. The size distribution of bubbles at pressures of 0.02 and 0.1 MPa is described by a lognormal law, whereas similar distribution at a pressure of 0.2 MPa considerably differs from this law.

Size reduction of polystyrene in a shaker bead mill—kinetic aspects

This paper reports an experimental study of polystyrene grinding in a shaker bead mill. The objective is to characterize the influence of the operating conditions on the evolution of the particle size. The effects of temperature, vibration frequency, and bead and polymer loads have been examined. Grinding kinetics have been predicted by combinations of four log-normal laws. The number of sub-populations used for the prediction has been defined after SEM observations. The kinetic parameters (mean size, standard deviation and proportionality coefficient) of the different laws have been determined and their evolution during grinding has been analysed.

Diagenetic evolution of crystallite thickness distribution of illitic material in Carpathian shales, studied by the Bertaut-Warren-Averbach XRD method (MudMaster computer program)

AbstractThe MudMaster computer program, based on the modified Bertaut-Warren- Averbach (BWA) XRD method, was used in the study of diagenetic evolution of illitic material in shales. The illitic material was characterized by XRD as a mixture of illite-smectite (I-S) and discrete illite. The experimental conditions for complete dehydration of swelling clays, necessary in this method, were established for a climate of high relative humidity (RH >40%). It was found that the distribution of crystallite thickness of dehydrated illitic material of shales is described by the lognormal law, as was established earlier by the BWA method for pure I-S from pyroclastic rocks. The parameters characterizing this distribution evolve with depth: the mean crystallite thickness (Te), the distribution broadening (DW) and the parameter of a lognormal distribution (β2) increase, whereas the percentage of the most frequent crystallite thickness decreases. The observed scatter of values is not random but indicates fluctuations in the characteristics of the illitic material of shales. The results imply that the modified Scherrer technique of measuring mean crystal thickness from the broadening of XRD reflections can be extended at least to some shales.

Scaling the saturated hydraulic conductivity of a vertic ustorthens soil under conventional and minimum tillage

With the aim of setting up a simplified method to measure hydraulic conductivity of structured soils at saturation, a method reported by Ahuja et al. (Soil Sci. Soc. Amer. J. 48 (1984) 699–702; Soil Sci. 148 (1989) 404–411), which also refers to the generalised equation of Kozeny–Carman and the scaling theory, is tested in this paper. Data were elaborated from hydraulic conductivity measurements K s on undisturbed soil cores taken from three plots of a sloping vertic soil which underwent various tillage practices for many years. In particular, following the proposed methodology, the spatial distribution effective soil porosity φ was first obtained. Once the distributions of hydraulic conductivity at saturation were obtained, it was possible to correlate the spatial variability of hydraulic conductivity with the variability of the unique factor α scaled by effective soil porosity. Statistical elaboration indicated that the log-normal law is the one which best approximates the frequency distribution of scale factors α k and α φ . No significant differentiations were noted between the distributions of parameters relative to different soil tillage systems.

Option pricing impact of alternative continuous-time dynamics for discretely-observed stock prices

In the present paper we construct stock-price processes with the same marginal lognormal law as that of a geometric Brownian motion and also with the same transition density (and returns' distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes', in that option prices can assume any value in-between the no-arbitrage lower and upper bounds. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants that are relevant for trading. The findings of the paper are inspired by a theoretical result, linking density-evolution of diffusion processes to exponential families. Such result is briefly reviewed in an appendix.

Fine grinding of polymers in a vibrated bead mill

Three polymers, having various glass temperatures, have been ground in a shaker bead mill. One is a friable polymer (poly(vinyl acetate), PVA) and two are thermoplastic polymers (polystyrene (PS) and polyethylene (PE)). These materials have been ground for various operating conditions (bead and polymer loads, vibration frequency, temperature). It was found that, in the case of the friable polymer, the size reduction occurs mainly according to a cleavage mechanism. Nevertheless, in specific operating conditions, attrition may happen. In the case of the thermoplastic polymers, different steps appear during the size reduction: abrasion, creation of cracks and fragmentation. Then, the fragmentation mechanism depends on the polymer nature. It is shown that it is necessary to work with a temperature below the glass temperature of polymer. Generally, the rate of size reduction increases with a decrease of the bead load and of the powder load. In the case of PVA, the evolution of the size distributions has been predicted by two statistical laws (the normal law at the beginning of the experiments and a combination of the normal and the log-normal laws at the end), whereas for PS, only log-normal laws have permitted to predict the evolution of the size distributions.

A differentiation method for separating a mixture of suspended particle size distributions

Abstract. A simple method is proposed to partition a mixture of two populations in suspended particle size data. The method, termed here "the differentiation method" is based on the function of the lognormal distribution. Suspended material in marine or estuarine situations often consists of difficult-to-interpret complex populations. The treatment of particle size data by the method described enables the confirmation of the lognormal law and also the demonstration of the occurrence of a combination of a number of populations which may not be distinguished by the classical Gaussian transformation or automatic methods. A simple combination of graphical and numerical techniques permits the decomposition and the easy determination of the various statistical parameters (median diameter, mean diameter, etc...). The method is applied to interpret observed size distributions of suspended particulate matter in the Seine estuary. The method enables the determination of the relative sizes of the constituent sub-populations that comprise the total suspended matter. In the example used to illustrate the method, particles are shown to be resuspended as a function of different hydrodynamic parameter.

Establishment of the background levels of some trace elements in soils of NE Spain with probability plots

The total content of Ba, Cr, Cu, Ni, Pb, V and Zn, determined by means of X-ray fluorescence was studied in soils of an agro-industrial area near Barcelona (NE Spain) in order to establish their background levels. A total of 363 samples were analysed, of which 268 belonged to 67 slightly disturbed and representative soil profiles. The remainder were taken at the soil surface (0–5 cm) in a polluted area located near a composting plant. The background ranges were established with probability plots and the upper limits of the resulting ranges were compared with those reported in the literature. In order to study the statistical distribution of the trace elements in the soils of the area, the Kolmogorov-Smirnov test of distribution fitting was applied. The upper limits of the established background ranges were: Ba, 1762 mg/kg; Cr, 275 mg/kg; Cu, 145 mg/kg; Ni, 83 mg/kg; Pb, 91 mg/kg; V, 183 mg/kg; and Zn, 326 mg/kg. It was concluded that probability plots seemed to be a reliable statistical method for establishing the background contents of these elements in the soils of the area studied. The trace elements investigated were distributed in these soils following the log-normal law.

Grain growth kinetics of Li 3PO 4-doped calcium carbonate

Grain growth mechanisms of calcium carbonate doped with lithium phosphate were determined. With small lithium phosphate addition, an abnormal needle-like grain growth was observed. Addition of the dopant as solid solution in calcite follows the equation: Li 3PO 4 CaCO 3 →3(Li i +) + + V Ca 2+ 2− + (PO 4 3−) CO 3 2+ −. In slightly higher quantities (2 wt.%) above the solubility limit of Li 3PO 4, a second phase segregated at the grain boundaries. Whatever the thermal treatment (temperature, CO 2 partial pressure), the grain distribution was self-similar ( G( t)/ G ( t))=time independent) and followed a single log normal law. The grain growth kinetics can be described by the equation: d G /d t= k( G −1- G c −1). The grain growth was controlled by the mobility of the second phase.

Nickel nanoparticles in silica gel: Preparation and magnetic properties

The reduction of nickel in silica gels was measured as a function of nickel content within the temperature range 600–900°C. This leads to the formation of metallic nickel nanoparticles homogeneously dispersed in this amorphous matrix. Furthermore, a Transmission Electron Microscopy (TEM) study has revealed that the size of these particles is distributed following a log-normal law. Moreover, this study has shown that the particles size is nearly independent of the nickel content while its distribution becomes wider when it increases. Magnetic measurements have shown, whatever nickel concentration and treatment temperature are, a superparamagnetic behavior. Finally, magnetization simulations, as a function of applied magnetic field, have been performed considering Langevin function in order to calculate a theoretical particles-size distribution and to compare it with the experimental one.

Tensile failure of filament-wound pipes under long-term creep loading: A probabilistic analysis

Creep tests exhibit wide scatter in time-to-failure results. One mean value is generally insufficient and, in order accurately to define the entire distribution, several tests are necessary. This study presents an experimental device designed to collect data on filament-wound pipes loaded in axial tension. It has been shown that experimental values can be fitted to a two-parameter log normal law or to a Weibull law. Two different methods are proposed for the evaluation of the law parameters. Both of them give similar estimates. A method is proposed for reducing the number of tests without losing the accuracy of estimators and is applied to the two-parameter ( m, σ 0) log normal law. Two equations suffice to determine the time-to-failure distributions in their entirety. In the medium stress range (between 74% and 83% of the static strength), σ 0 remains constant and m depends linearly on the applied stress. For lower stresses, our latest tests (72.5% of the static strength) prove the existence of a reliability area.

Effect of Pore Radius Distribution of Forest Soils on Vertical Water Movement in Soil Profile.

In order to clarify the effect of soil pore radius distribution g(r) on the water movement in soil profile, one-dimensional vertical flow equation was solved by using the model for soil hydraulic properties which was derived by applying lognormal distribution laws to g(r). Results showed that the recession hydrograph from a soil profile becomes gradual as the median of g (r) increases. For soil with a relatively large median of g (r), the recession hydrograph becomes steep as the width of g (r) increases. When the median of g(r) is relatively small, the hydrograph has more gradual decreases in recession flow as the width of g(r) increases. This critical median value increases as the length of soil profile (L) becomes small. By comparing pore radius distributions of various soils, it was shown that the undisturbed forest soils yield greater amount of recession flow than the disturbed loamy soils. 70 percent of the disturbed sandy soils yield greater amount of recession flow than the forest soils with L of 100cm. When L is 50cm, hydrographs of half of the sandy soils have greater decreases in recession flow than those of the forest soils. Hydrographs from the crumb-structure forest soils have more gradual decreases in recession flow than those from the granular-structure forest soils. Recession hydrographs from the massive forest soils are steeper than the those from the crumb and granular-structure soils. As a result, it was indicated that well-developed forest soil is effective as the surface soil to increase baseflow from hillslope.

Statistical Approach to Discriminate Background and Anthropogenic Input of Trace Elements in Soils of Catalonia, Spain

A study was conducted in Catalonia (NE Spain) to determine the baseline levels and the polluted contents of Ba, Cd, Cr, Cu, Ni, Pb, V and Zn (aqua regia-extractable) in soils of an agroindustrial area near Barcelona. The samples analyzed proceeded from two soil surveys. 268 samples belonged to 67 representative, and slightly disturbed, soil profiles sampled during a previous soil survey, and 95 superficial samples were taken near a composting plant (heavily disturbed and presumably polluted). Probability plots were used to establish the reference (or background) levels and a literature comparison was made to check the validity of the background ranges established. The raw data sets and the uncontaminated resulting populations were studied by means of frequency histograms and of statistical tests in order to verify the general assumption, which states that trace element distributions in soils follow the log-normal law.

Multifractal modelling of the sizes and grades of giant and supergiant deposits

The high-value tails of the sizes and size-grade distributions for sets of mineral deposits can often be modelled as Pareto distributions plotting as straight lines on log-log paper. The multifractal model for size-grade distributions proposed here reflects self-similarity (approximate scale independence) of the underlying spatial distributions of the deposits. It predicts that the majority of deposits of the same type can be described by a lognormal distribution but the largest deposits in the tail are controlled by a hyperbolic instead of the lognormal law. This topic is important because most ore comes from the relatively few giants and supergiants in a population of mineral deposits of the same type.

Once again, what does it mean? (On the analytical representation of the observed logN–logSrelation for radio sources)

Abstract The observed log N–log S relation for radio sources is well represented by a log-normal law.

Indoor 222Rn concentrations in Central Asturias.

A survey of the 222Rn concentrations in 106 homes in the four main towns of the Central Asturias region was carried out over three years. A total of 1,014 measurements was obtained using passive radon charcoal canisters. The 222Rn concentrations fit a log-normal distribution law, with a geometric mean of 23 Bq m(-3). There is a marked difference between the 222Rn concentration for the ground inhabited floors and first floors. For the other floors, the 222Rn concentrations remain practically constant. The 222Rn concentrations are lower in summer than in winter time. The annual equivalent dose for the general public due to the inhalation of 222Rn is equal to 0.81 mSv (81 mrem).

Logarithmic operators and hidden continuous symmetry in critical disordered models

We study the model of (2 + 1)-dimensional relativistic fermions in a random non-Abelian gauge potential at criticality. The exact solution shows that the operator expansion contains a conserved current — a generator of a continuous symmetry. The presence of this operator changes the operator product expansion and gives rise to logarithmic contributions to the correlation functions at the critical point. We calculate the distribution function of the local density of states in this model and find that it follows the famous log-normal law.