Bimodal Distribution Function Research Articles

Empirical Model of Equatorial ElectroJet (EEJ) Using Long‐Term Observations From the Indian Sector

AbstractThe Equatorial Electrojet (EEJ) is one of the important near‐earth space weather phenomena which exhibits significant diurnal, seasonal and solar activity variations. This paper investigates the EEJ variations at diurnal, seasonal and solar cycle time scales from the Indian sector and portrays a new empirical EEJ field model developed using the observations spanning over nearly two solar cycles. The Method of Naturally Orthogonal Components (MNOC), also known as Principal Component Analysis (PCA), was employed to extract the dominant patterns of principal diurnal, semi‐diurnal, and ter‐diurnal components contributing to the EEJ variation. The amplitudes of these diurnal, semi‐diurnal, and ter‐diurnal components in EEJ are found to vary significantly with the season and solar activity. The seasonal and solar activity dependencies of these principal components are modeled using suitable bimodal distribution functions. Finally, the empirical model for EEJ field was built by combining the principal components with their corresponding modeled amplitudes. This model accurately reproduces the diurnal, seasonal and solar activity variations of EEJ. The modeled monthly mean variations of EEJ field at ground exhibit excellent correlation of 0.96 with the observations with the root mean square error <5 nT. It also successfully captures the seasonal and solar activity variations of Counter Electrojet (CEJ). Finally, this model named “Indian Equatorial Electrojet (IEEJ) Model” is made publicly available for interested scientific users (https://iigm.res.in/system/files/IEEJ_model.html).

Incorporating flowpaths as an explicit measure of river-floodplain connectivity to improve estimates of floodplain sediment deposition

Variation in floodplain topography can lead to gradual flooding and increase river-floodplain connectivity. We show that incorporating flowpaths as an explicit measure of river-floodplain connectivity can improve estimates of floodplain sediment deposition. We focus on the floodplain of the South River, downstream of Waynesboro, Virginia, where measurements of mercury accumulation have been used to estimate decadal-scale sedimentation rates. We developed a two-dimensional Hydrologic Engineering Center's River Analysis System (2D HEC-RAS) hydrodynamic model and used simulated model results with sediment deposition data to create regression models describing sedimentation across the floodplain. All of our statistical models incorporated a flowpath length from the location on the floodplain downstream to the riverbank as an explicit measure of river-floodplain connectivity that improved our estimates of floodplain sediment deposition (r2 = 0.514). We applied our best regression model to our hydrodynamic model results to create a map of floodplain sedimentation rate and discuss differences of three separate sections of floodplain. We found that floodplains with variable topography had wider, bimodal probability distribution functions (PDFs) of sedimentation rate (aggregated spatially) than floodplains without this topographic relief (with narrower log-normal PDFs). Our work highlights how floodplain topography and river-floodplain connectivity affect sedimentation rates and can help inform the development of floodplain sediment budgets.

Two statistical regimes in the transition to filamentation.

We experimentally investigate fluctuations in the spectrum of ultrashort laser pulses propagating in air, close to the critical power for filamentation. Increasing the laser peak power broadens the spectrum while the beam approaches the filamentation regime. We identify two regimes for this transition: In the center of the spectrum, the output spectral intensity increases continuously. In contrast, on the edges of the spectrum the transition implies a bimodal probability distribution function for intermediate incident pulse energies, where a high-intensity mode appears and grows at the expense of the original low-intensity mode. We argue that this dual behavior prevents the definition of a univoquial threshold for filamentation, shedding a new light on the long-standing lack of explicit definition of the boundary of the filamentation regime.

Bimodal Distribution Function with Fuzzy Regression in Predicting Random Truckload Patterns

A bimodal distribution function with statistical parameters obtained using fuzzy regression is presented for predicting random truckload patterns that include all load ranges including overloads. Overload trucks often appear as a sizable portion of truck populations on highways. In applications when damage estimation of transportation facilities such as pavements and bridges is desired, theoretical models providing a reasonable representation of truckload populations including overloads will be useful. Load populations mostly exhibit inconsistent patterns, often with two or more distinct peaks. This is because of a combination of loaded and empty trucks as well as overloads in the population. As such, a mixed distribution model instead of a simple statistical distribution is used to portray a realistic representation of truckload populations. In this paper, using the bimodal model, theoretical distributions are developed to (1) represent the entire truckload population with weigh-in-motion (WIM) data, and (2) predict the population with limited data. If no truckload population is available, the bimodal distribution model can still be used, with certain assumptions, based on fuzzy regression using certain parameters pertaining to the traffic data such as the average daily truck traffic and an overall estimate of the percentage of overload data in the population. This paper represents new directions in modeling truckload patterns with capabilities to offer the continuity as well as the bimodal feature of the limited load data.

Capture of Dark Matter Particles by a Galaxy in the Case of a Bimodal Distribution of Their Velocities

We have analyzed the rate of capture of dark matter (DM) particles by the galaxy in the case of the existence of two different types of DM or a bimodal velocity distribution function for DM. It is shown that, in addition to the scenario considered in our previous work which is based on the assumption of an unimodal distribution, more complex scenarios are possible in which the transition to the state of intense capture and/or exit from it can occur in two stages. A detailed description is given of the change in the curve describing the rate of capture of dark matter particles as a function of the rate of increase in the baryon mass of the galaxy for various values of the rate of decrease of the DM density.

Verification of the master curve concept (ASTM E1921) and inhomogeneity analysis of a German RPV weld for various loading rates

Fracture-mechanical investigations were carried out on the weld metal of a German reactor pressure vessel steel 22NiMoCr3-7. The crack tip loading rate and test temperature were varied in the brittle-ductile transition region, while the results were evaluated using the Master Curve methodology (ASTME1921), including inhomogeneity analyses in Appendix X5. The highly inhomogeneous datasets could be uniformly described with a bimodal distribution function with good accuracy compared to the standard procedure. However, this agreement decreases with increasing crack tip loading rate. In contrast, a modified Master Curve with a shape factor of p = 0.03/°C progressively matches the data better with increasing crack tip loading rate. The weld metal has a significantly higher toughness than the base material. SEM analysis suggests that the inhomogeneity is due to the stochastic distribution of sharp microcracks, while the observed inhomogeneity is additionally superimposed by dynamic effects (adiabatic heating and local crack arrest) at increased crack tip loading rates. It is argued that the microstructural inhomogeneity is hereby »neutralized« and could instead be interpreted as a possible artifact of the standard Master Curve under dynamic conditions.

Phase-field model of grain boundary diffusion in nanocrystalline solids: Anisotropic fluctuations, anomalous diffusion, and precipitation

The anisotropic phase-filed model of grain boundary diffusion and precipitation of solute in nanocrystalline solids has been developed. In this model, the Cahn–Hilliard equation is generalized for the anisotropic phase-field diffusion of solute and anisotropic compositional fluctuations. It is found that dynamics of solute concentration profile demonstrates the anomalous diffusion behavior with scaling parameters depending on the mobility ratio and microstructure of a solid solution. It is noteworthy that the increase in source concentration can slow down the concentration front propagation due to uphill diffusion or formation of a new phase. Parameters of grain boundary diffusion control the precipitation dynamics. In particular, a decrease in transverse diffusion coefficient is responsible for longer incubation time, and lower rates of nucleation and nuclei growth in comparison with the case of isotropic solute transport near grain boundaries. Transport properties of boundary and bulk are responsible for the formation of the bimodal size distribution function of second phase particles and specific kinetics of average radius and number density.

Electric-field-dependent bimodal distribution functions for electrons in argon, xenon, and krypton owing to the Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections

This paper considers solutions of the linear Fokker-Planck equation for the steady velocity distribution functions of electrons dilutely dispersed in an excess of either argon, xenon, or krypton. Owing to the large Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections for these atoms, the steady electron distribution, often referred to as the Davydov distribution, exhibits a bimodal distribution which varies dramatically with the strength of the external electric field. This effect provides a unique verification of the location and shape of the Ramsauer-Townsend minima in the cross sections. It is anticipated that current experimental techniques that probe the details of nonequilibrium electron distributions will verify the results reported in this paper. In addition, these nonequilibrium, non-Maxwellian distributions cannot be rationalized in terms of either the Gibbs-Boltzmann entropy nor the Tsallis nonextensive entropy.

NEW DIRECTIONS IN PREDICTING AND MODELING TRUCK LOAD POPULATIONS INCLUDING OVERLOADS

Overload trucks often appear as a sizeable portion of truck populations on highways. Within most transportation jurisdictions in the United States, overloads refer to truck weights in excess of 356 kN (80 kips). The load populations exhibit an inconsistent pattern-often with two or more distinct peaks. This paper represents new directions in modeling truck load patterns with capabilities to offer the continuity as well as the bimodal feature of the load data. The method presented is data-driven and uses a bimodal distribution function. The extension of the model to predict the truck load population when data are incomplete or not available is also presented. The parameters required in the model are estimated via the fuzzy regression analysis in the proposed method. The fuzzy regression analysis allows the utilization of subjective and limited information. The Weigh-in-Motion (WIM) data obtained from several stations were used to quantify the parameters of this bimodal distribution function by employing the fuzzy regression analysis. The procedure on obtaining an appropriate distribution function for truck load population is explained along with illustrative examples.

Wind-tunnel experiments of Aeolian sand transport reveal a bimodal probability distribution function for the particle lift-off velocities

The probability distribution function (PDF) of the lift-off velocities of Aeolian (wind-blown) sand particles – i.e., the velocities with which the particles rebound or are ejected from the bed owing to a granular-bed collision – is an important parameter in models of sediment transport and dust emission. However, in order to reliably describe this distribution function, measurements of the individual particle trajectories near the sand bed during sediment transport are required. Such measurements are challenging due to the high particle concentration near the bed, which makes it difficult to track the dynamics of the particles involved in the lift-off process. Here we seek to overcome this difficulty by applying an experimental method for grain trajectory detection, which allows us to obtain a large number of near-bed lift-off particle trajectories during sand transport in a boundary layer wind tunnel.The PDFs of horizontal and vertical lift-off velocities all were found to display a bimodal distribution at a height smaller than 1 mm from the bed. The first and second maxima in the distribution curve of horizontal and vertical lift-off velocities occur at around 0.25 and 1.0 ms−1, and at 0 and 1.0 ms−1, respectively. Moreover, the first maximum is mainly associated with splashed particles, whereas the second peak is mainly caused by rebound particles. However, the lift-off velocities far from the bed follow a unimodal distribution, in agreement with previous experiments. Our results are providing, thus, unprecedented insights about near-bed transport processes over wind-blown soils, and have implication for modelling granular-bed interactions in Aeolian sand transport and geomorphodynamic models.

NEW DIRECTIONS IN PREDICTING AND MODELING TRUCK LOAD POPULATIONS INCLUDING OVERLOADS

Overload trucks often appear as a sizeable portion of truck populations on highways. Within most transportation jurisdictions in the United States, overloads refer to truck weights in excess of 356 kN (80 kips). The load populations exhibit an inconsistent pattern-often with two or more distinct peaks. This paper represents new directions in modeling truck load patterns with capabilities to offer the continuity as well as the bimodal feature of the load data. The method presented is data-driven and uses a bimodal distribution function. The extension of the model to predict the truck load population when data are incomplete or not available is also presented. The parameters required in the model are estimated via the fuzzy regression analysis in the proposed method. The fuzzy regression analysis allows the utilization of subjective and limited information. The Weigh-in-Motion (WIM) data obtained from several stations were used to quantify the parameters of this bimodal distribution function by employing the fuzzy regression analysis. The procedure on obtaining an appropriate distribution function for truck load population is explained along with illustrative examples.

Separate retrieval of microphysical characteristics in aerosol fractions from laser sensing data

A procedure is proposed for deriving the aerosol microphysical characteristics of the spherical particle, the complex refractivity index m = mreal + i*mimag and the bimodal size distribution function U(r), using laser sensing data at the wavelengths within 355-1064 nm. A possibility of a separate fraction-wise assessment of m + U(r) is studied for weakly absorbing particles for mimag < 0.015, when mfine ≠ mcoarse holds. The reconstruction of the mode-averaged mmean is used in the preliminary stage and provides information on the mmean + U(r, mmean) parameters and the mode difference of mreal in the current observation session. A subsequent transformation of the scattering efficiency factors (integral equation kernels) divides the refractive index with respect to the fractions and gives a correct estimation of mcoarse at different sought-for values of mrealfine and mrealcoarse. A change of the kernels and the region of allowed values of the sought-for U(r) ensures its admissible estimation for both fractions. The algorithms are tested for one mfine (1.50+i*0.01) and nine mcoarse (mreal = 1.40, 1.50, 1.60; mimag = 0.0001, 0.001, 0.01). In order to include the influence of the contribution from the modal particles into the total concentration, 462 empirical models of U(r) are used.

What Can We Learn from Exam Grade Distributions?

A course with good learning outcomes is one in which most of the enrolled students achieve the mastery specified in the predefined learning objectives. Since the enrolment is invariably a mix of students with heterogeneous capabilities, the class average grade is a poor indicator of how the class is divided into at least two groups, of high- and low-performers. Clearly, achieving the desired outcomes implies increasing the proportion of high-performing students and their mean grade by providing appropriately designed teaching protocols. In this paper, the actual class grade distribution is approximated by a bimodal probability distribution function, whose parameters enable the proportions and average performance of these two groups to be quantified. This paper describes the methodology to achieve this and demonstrates its usage to diagnose example exam grade distributions, as well as to provide quantification of the impact of pedagogic changes on the degree of achieving teaching objectives.

Evolution of spheroidal dust in electrically active sub-stellar atmospheres

Context. Understanding the source of sub-stellar polarimetric observations in the optical and near-infrared is key to characterizing sub-stellar objects and developing potential diagnostics for determining properties of their atmospheres. Differential scattering from a population of aligned, non-spherical dust grains is a potential source of polarization that could be used to determine geometric properties of the dust clouds. Aims. This paper addresses the problem of the spheroidal growth of dust grains in electrically activated sub-stellar atmospheres. It presents the novel application of a mechanism whereby non-spherical, elongated dust grains can be grown via plasma deposition as a consequence of the surface electric field effects of charged dust grains. Methods. We numerically solve the differential equations governing the spheroidal growth of charged dust grains via plasma deposition as a result of surface electric field effects in order to determine how the dust eccentricity and the dust particle eccentricity distribution function evolve with time. From these results, we determine the effect of spheroidal dust on the observed linear polarization. Results. Numerical solutions show that e ≈ 0.94 defines a watershed eccentricity, where the eccentricity of grains with an initial eccentricity less than (greater than) this value decreases (increases) and spherical (spheroidal) growth occurs. This produces a characteristic bimodal eccentricity distribution function yielding a fractional change in the observed linear polarization of up to ≈0.1 corresponding to dust grains of maximal eccentricity at wavelengths of ≈1 μm, consistent with the near infrared observational window. Order of magnitude calculations indicate that a population of aligned, spheroidal dust grains can produce degrees of polarization P ≈ 𝒪(10−2 − 1%) consistent with observed polarization signatures. Conclusions. The results presented here are relevant to the growth of non-spherical, irregularly shaped dust grains of general geometry where non-uniform surface electric field effects of charged dust grains are significant. The model described in this paper may also be applicable to polarization from galactic dust and dust growth in magnetically confined plasmas.

A Bimodal Lognormal Distribution Model for the Prediction of COVID-19 Deaths

The paper presents a phenomenological epidemiological model for the description and prediction of the time trends of COVID-19 deaths worldwide. A bimodal distribution function—defined as the mixture of two lognormal distributions—is assumed to model the time distribution of deaths in a country. The asymmetric lognormal distribution enables better data fitting with respect to symmetric distribution functions. Besides, the presence of a second mode allows the model to also describe second waves of the epidemic. For each country, the model has six parameters, which are determined by fitting the available data through a nonlinear least-squares procedure. The fitted curves can then be extrapolated to predict the future trends of the total and daily number of deaths. Results for the six continents and the World are obtained by summing those computed for the 210 countries in the Our World in Data (OWID) dataset. To assess the accuracy of predictions, a validation study is first conducted. Then, based on data available as of 30 September 2020, the future trends are extrapolated until the end of year 2020.

Static Disorder in Excitation Energies of the Fenna-Matthews-Olson Protein: Structure-Based Theory Meets Experiment.

Inhomogeneous broadening of optical lines of the Fenna–Matthews–Olson (FMO) light-harvesting protein is investigated by combining a Monte Carlo sampling of low-energy conformational substates of the protein with a quantum chemical/electrostatic calculation of local transition energies (site energies) of the pigments. The good agreement between the optical spectra calculated for the inhomogeneous ensemble and the experimental data demonstrates that electrostatics is the dominant contributor to static disorder in site energies. Rotamers of polar amino acid side chains are found to cause bimodal distribution functions of site energy shifts, which can be probed by hole burning and single-molecule spectroscopy. When summing over the large number of contributions, the resulting distribution functions of the site energies become Gaussians, and the correlations in site energy fluctuations at different sites practically average to zero. These results demonstrate that static disorder in the FMO protein is in the realm of the central limit theorem of statistics.

The inhibition of concentrated active baths.

Passive tracers in the active bath express fascinating behaviors. However, most studies are restricted to dilute active baths. Here, we use 2D simulation of suspensions consisting of active Brownian particles and a passive disk-shaped tracer to investigate tracers' diffusive behaviors in a wide range of volume fractions. Due to the competition between the thermal noise and collisions with active particles, tracers express a first transition from the normal diffusion to the superdiffusion at a short time scale and recur to normal diffusion at a long time scale. At a low volume fraction, infrequent active collisions retard the first transition of smaller tracers. At a high volume fraction, active particles with high activity aggregating around tracers induce a bimodal probability distribution function of tracer displacements during superdiffusion. Considering the enhancement of diffusion, the non-dimensional enhanced diffusivity increases asymptotically with the Peclet number. The asymptotic line gives an upper limit of non-dimensional enhanced diffusivity of tracers. Cases with lower enhanced diffusion have a high volume fraction and a low active velocity that indicates the inhibition of concentrated active baths. With the high negentropic work of these cases, the inhibition is explained as the change of the configuration of active baths for introducing tracers.

Dynamics of the one-dimensional isotropic Heisenberg model with Dzyaloshinskii–Moryia interaction in a random transverse field

The dynamics of the one-dimensional spin-1∕2 isotropic Heisenberg model with Dzyaloshinskii–Moriya (DM) interaction in the presence of a random magnetic field perpendicular to the DM axis (a transverse field) is investigated by using the method of recurrence relations. A bimodal probability distribution function for this transverse field is considered. The first four recurrants, which are used to obtain the short-time expansion for the time-dependent correlation functions, are exactly obtained as function of the probabilities and of the transverse field. A recent proposed extrapolation procedure for the higher-order recurrants is used in order to extend the results to longer times. In the limit of infinite temperature the relaxation and the spectral density functions are computed for several values of the external field and different probability distributions. Depending on the probabilities and on the external fields one has stronger and faster oscillations in the relaxation functions, and a suppression of the central peak in the spectral density.

Functionalization of porous siliceous materials, Part 2: Surface characterization by inverse gas chromatography

Surface modification of porous glass beads by ethanol-based 3-mercaptopropyltrimethoxysilane (MPTMS) grafting solutions is directly evidenced by nitrogen adsorption, elemental analysis, thermogravimetry, infrared and 29Si CP MAS NMR spectroscopy. Furthermore, the energetic characterization of the surface is essential to understand comprehensively the physico-chemical interactions between the pristine and MPTMS-modified surface and its gas/liquid-phase environment. In this study, inverse gas chromatography (IGC) is used to characterize the surface properties of porous glass (PG). By means of IGC at infinite dilution (IGC-ID), the dispersive component of the surface energy (γsd), the enthalpy and entropy of adsorption of C6-C10 hydrocarbon probes were determined at temperatures between 30 and 120 °C. The specific component of the surface energy (γssp) at the temperature of 120 °C has been obtained via the Van Oss theory and a least-squares procedure evaluating the IGC data of 8 polar probe molecules collectively. After surface silylation, the total surface energy (γst) decreased from 402 to 255 mJ/m² indicating both a reduced wettability and an increased hydrophobicity of the MPTMS-modified PG. Moreover, the acidity/basicity parameters according to the Van Oss and the Gutmann approach indicated that the acidity of the PG surface decreases by MPTMS grafting.Using n-octane and isopropanol probes, IGC at finite concentration (IGC-FC) was applied to obtain their adsorption isotherms and subsequently the BET specific surface areas. In addition, the surface heterogeneity of the studied PGs was also computed. The energy distribution functions of adsorption sites were monomodal (peak maximum at about 22 kJ/mol) for the n-octane probe, while isopropanol revealed a bimodal distribution function (maxima at about 18 and 25 kJ/mol) on both pristine and MPTMS-modified PG. Furthermore, the proportion of high energy sites (apparently assigned to SiOH groups) has been reduced by surface modification from 65% to only 35% despite a high surface coverage of ˜10 MPTMS species/nm2. These findings are in agreement with the results of 29Si CP MAS NMR measurements and are supported by DFT calculations on the adsorption of isopropanol and n-octane on the surface of a silica cluster model.

The bimodal [Mg/Fe] versus [Fe/H] bulge sequence as revealed by APOGEE DR14

Context. The Galactic bulge has a bimodal metallicity distribution function: different kinematic, spatial, and, potentially, age distributions characterize the metal-poor and metal-rich components. Despite this observed dichotomy, which argues for different formation channels for those stars, the distribution of bulge stars in the α-abundance versus metallicity plane has been found so far to be a rather smooth single sequence. Aims. We use data from the fourteenth data release of the APOGEE spectroscopic survey (DR14) to investigate the distribution in the Mg abundance (as tracer of the α-elements)-versus-metallicity plane of a sample of stars selected to be in the inner region of the bulge. Methods. A clean sample has been selected from the DR14 using a set of data- and pipeline-flags to ensure the quality of their fundamental parameters and elemental abundances. An additional selection made use of computed spectro-photometric distances to select a sample of likely bulge stars as those with RGC ≤ 3.5 kpc. We adopt magnesium abundance as an α-abundance proxy for our clean sample as it has been proven to be the most accurate α-element as determined by ASPCAP, the pipeline for data products from APOGEE spectra. Results. From the distribution of our bulge sample in the [Mg/Fe]-versus-[Fe/H] plane, we found that the sequence is bimodal. This bimodality is given by the presence of a low-Mg sequence of stars parallel to the main high-Mg sequence over a range of ∼0.5 dex around solar metallicity. The two sequences merge above [Fe/H] ∼ 0.15 dex into a single sequence whose dispersion in [Mg/Fe] is larger than either of the two sequences visible at lower metallicity. This result is confirmed when we consider stars in our sample that are inside the bulge region according to trustworthy Gaia DR2 distances.