Shape Of Probability Distribution Function Research Articles

A Two-Component Probability Distribution Function Describes the Mid-IR Emission from the Disks of Star-forming Galaxies

High-resolution JWST-MIRI images of nearby spiral galaxies reveal emission with complex substructures that trace dust heated both by massive young stars and the diffuse interstellar radiation field. We present high angular (0.″85) and physical resolution (20–80 pc) measurements of the probability distribution function (PDF) of mid-infrared (mid-IR) emission (7.7–21 μm) from 19 nearby star-forming galaxies from the PHANGS-JWST Cycle 1 Treasury. The PDFs of mid-IR emission from the disks of all 19 galaxies consistently show two distinct components: an approximately lognormal distribution at lower intensities and a high-intensity power law component. These two components only emerge once individual star-forming regions are resolved. Comparing with locations of H ii regions identified from Very Large Telescope/MUSE Hα mapping, we infer that the power-law component arises from star-forming regions and thus primarily traces dust heated by young stars. In the continuum-dominated 21 μm band, the power law is more prominent and contains roughly half of the total flux. At 7.7–11.3 μm, the power law is suppressed by the destruction of small grains (including PAHs) close to H ii regions, while the lognormal component tracing the dust column in diffuse regions appears more prominent. The width and shape of the lognormal diffuse emission PDFs in galactic disks remain consistent across our sample, implying a lognormal gas column density N(H) ≈ 1021 cm−2 shaped by supersonic turbulence with typical (isothermal) turbulent Mach numbers ≈5−15. Finally, we describe how the PDFs of galactic disks are assembled from dusty H ii regions and diffuse gas and discuss how the measured PDF parameters correlate with global properties such as star formation rate and gas surface density.

Метод расчета надежности шарнирно- стержневых систем при интервальной оценке случайных величин

The article describes a method for the reliability analysis of hinge-rod systems (trusses) with an interval estimation of the variability limits of random variables. In practical tasks of structural reliability analysis, it is often not possible to obtain complete statistical data that is necessary for a probabilistic and statistical approach in accordance with GOST 27751-2014. The use of alternative theories of data analysis makes it possible to obtain a define assessment of reliability with incomplete statistical data without introducing unconfirmed hypotheses about the form and shape of probability distribution functions of random variables. In this paper, it is proposed to use data on the limits of variability of random variables (an interval analysis) to assess the reliability of hinge-rod systems on the truss example. A comparative analysis of the truss reliability is given by two approaches (the classical probabilistic-statistical and the proposed interval approach). By the interval approach, the reliability of the truss is lower due to the fact that the interval estimation takes into account other possible distributions within a given interval that differ from the normal probability distribution. In the case when it is impossible to determine the type of probability distribution, but the limits of variability for random variables are known, the use of the presented method is a more cautious for the safety of truss operation.

The density structure of supersonic self-gravitating turbulence

ABSTRACT We conduct numerical experiments to determine the density probability distribution function (PDF) produced in supersonic, isothermal, self-gravitating turbulence of the sort that is ubiquitous in star-forming molecular clouds. Our experiments cover a wide range of turbulent Mach number and virial parameter, allowing us for the first time to determine how the PDF responds as these parameters vary, and we introduce a new diagnostic, the dimensionless star formation efficiency versus density [ϵff(s)] curve, which provides a sensitive diagnostic of the PDF shape and dynamics. We show that the PDF follows a universal functional form consisting of a lognormal at low density with two distinct power-law tails at higher density: the first of these represents the onset of self-gravitation, and the second reflects the onset of rotational support. Once the star formation efficiency reaches a few per cent, the PDF becomes statistically steady, with no evidence for secular time evolution at star formation efficiencies from about 5 to 20 per cent. We show that both the Mach number and the virial parameter influence the characteristic densities at which the lognormal gives way to the first power law, and the first to the second, and we extend (for the former) and develop (for the latter) simple theoretical models for the relationship between these density thresholds and the global properties of the turbulent medium.

Towards a non-Gaussian model of redshift space distortions

ABSTRACT To understand the nature of the accelerated expansion of the Universe, we need to combine constraints on the expansion rate and growth of structure. The growth rate is usually extracted from 3D galaxy maps by exploiting the effects of peculiar motions on galaxy clustering. However, theoretical models of the probability distribution function (PDF) of galaxy pairwise peculiar velocities are not accurate enough on small scales to reduce the error on theoretical predictions to the level required to match the precision expected for measurements from future surveys. Here, we improve the modelling of the pairwise velocity distribution by using the Skew-T PDF, which has non-zero skewness and kurtosis. Our model accurately reproduces the redshift space multipoles (monopole, quadrupole, and hexadecapole) predicted by N-body simulations, above scales of about $10\, h^{-1}{\rm Mpc}$. We illustrate how a Taylor expansion of the streaming model can reveal the contributions of the different moments to the clustering multipoles, which are independent of the shape of the velocity PDF. The Taylor expansion explains why the Gaussian streaming model works well in predicting the first two redshift space multipoles, although the velocity PDF is non-Gaussian even on large scales. Indeed, any PDF with the correct first two moments would produce precise results for the monopole down to scales of about $10\, h^{-1}{\rm Mpc}$, and for the quadrupole down to about $30\, h^{-1}{\rm Mpc}$. An accurate model for the hexadecapole needs to include higher order moments.

Scaling behavior in measured keystroke time series from patients with Parkinson’s disease

Parkinson has remained as one of the most difficult diseases to diagnose, as there are no biomarkers to be measured, and this requires one patient to do neurological and physical examinations. As Parkinson is a progressive disease, accurate detection of its symptoms is a crucial factor for therapeutic reasons. In this study, we perform Multifractal Detrended Fluctuation Analysis (MFDFA) on measured keystroke time series for three different categories of subjects: healthy, early-PD, and De-Novo patients. We have observed different scaling behavior in terms of multifractality of the measured time series, which can be used as a practical tool for diagnosis purposes. Additionally, the source of the multifractality has been studied which shows that in healthy and early-PD subjects, multifractality due to the long-range correlations is stronger than the influence of its probability distribution function (PDF) fatness, while in De-Novo patients, both shape of PDF and long-range correlations are contributing to observed multifractality.

Detailed Balance and Exact Results for Density Fluctuations in Supersonic Turbulence

Abstract The probabilistic approach to turbulence is applied to investigate density fluctuations in supersonic turbulence. We derive kinetic equations for the probability distribution function (PDF) of the logarithm of the density field, s, in compressible turbulence in two forms: a first-order partial differential equation involving the average divergence conditioned on the flow density, , and a Fokker–Planck equation with the drift and diffusion coefficients equal to and , respectively. Assuming statistical homogeneity only, the detailed balance at steady state leads to two exact results, , and . The former indicates a balance of the flow divergence over all expanding and contracting regions at each given density. The exact results provide an objective criterion to judge the accuracy of numerical codes with respect to the density statistics in supersonic turbulence. We also present a method to estimate the effective numerical diffusion as a function of the flow density and discuss its effects on the shape of the density PDF.

Molecular Gas Properties in M83 from CO PDFs

Abstract We have obtained 12CO(1–0) data of the nearby barred spiral galaxy M83 from Atacama Large Millimeter/submillimeter Array and Nobeyama 45 m observations. By combining these two data sets, the total CO flux has been recovered, and a high angular resolution ( corresponding to ∼40 pc at the distance of M83) has been achieved. The field of view is corresponding to ∼3.4 kpc and covers the galactic center, bar, and spiral arm regions. In order to investigate how these galactic structures affect gas properties, we have created a probability distribution function (PDF) of the CO integrated intensity ( ), peak temperature, and velocity dispersion for a region with each structure. We find that the PDF for the bar shows a bright-end tail while that for the arm does not. As the star formation efficiency is lower in the bar, this difference in PDF shape is contrary to the trend in Milky Way studies where the bright-end tail is found for star-forming molecular clouds. While the peak temperature PDFs are similar for the bar and arm regions, velocity dispersion in the bar is systematically larger than in the arm. This large velocity dispersion is likely a major cause of the bright-end tail and of suppressed star formation. We also investigate an effect of stellar feedback to PDF profiles and find that the different PDFs between bar and arm regions cannot be explained by the feedback effect, at least at the current spatial scale.

THE LOGNORMAL PROBABILITY DISTRIBUTION FUNCTION OF THE PERSEUS MOLECULAR CLOUD: A COMPARISON OF HI AND DUST

The shape of the probability distribution function (PDF) of molecular clouds is an important ingredient for modern theories of star formation and turbulence. Recently, several studies have pointed out observational difficulties with constraining the low column density (i.e. Av <1) PDF using dust tracers. In order to constrain the shape and properties of the low column density probability distribution function, we investigate the PDF of multiphase atomic gas in the Perseus molecular cloud using opacity-corrected GALFA-HI data and compare the PDF shape and properties to the total gas PDF and the N(H2) PDF. We find that the shape of the PDF in the atomic medium of Perseus is well described by a lognormal distribution, and not by a power-law or bimodal distribution. The peak of the atomic gas PDF in and around Perseus lies at the HI-H2 transition column density for this cloud, past which the N(H2) PDF takes on a powerlaw form. We find that the PDF of the atomic gas is narrow and at column densities larger than the HI-H2 transition the HI rapidly depletes, suggesting that the HI PDF may be used to find the HI-H2 transition column density. We also calculate the sonic Mach number of the atomic gas by using HI absorption line data, which yields a median value of Ms=4.0 for the CNM, while the HI emission PDF, which traces both the WNM and CNM, has a width more consistent with transonic turbulence.

Constraining the halo bispectrum in real and redshift space from perturbation theory and non-linear stochastic bias

We present a method to produce mock galaxy catalogues with efficient perturbation theory schemes, which match the number density, power spectra and bispectra in real and in redshift space from N-body simulations. The essential contribution of this work is the way in which we constrain the bias parameters in the PATCHY-code. In addition of aiming at reproducing the two-point statistics, we seek the set of bias parameters, which constrain the univariate halo probability distribution function (PDF) encoding higher-order correlation functions. We demonstrate that halo catalogues based on the same underlying dark matter field with a fix halo number density, and accurately matching the power spectrum (within 2%), can lead to very different bispectra depending on the adopted halo bias model. A model ignoring the shape of the halo PDF can lead to deviations up to factors of 2. The catalogues obtained additionally constraining the shape of the halo PDF can significantly lower the discrepancy in the three-point statistics, yielding closely unbiased bispectra both in real and in redshift space; which are in general compatible with those corresponding to an N-body simulation within 10% (deviating at most up to 20%). Our calculations show that the constant linear bias of ~2 for Luminous Red Galaxy (LRG) like galaxies seen in the power spectrum, mainly comes from sampling halos in high density peaks, choosing a high density threshold rather than from a factor multiplying the dark matter density field. Our method contributes towards an efficient modelling of the halo/galaxy distribution required to estimate uncertainties in the clustering measurements from galaxy redshift surveys. We have also demonstrated that it represents a powerful tool to test various bias models.

TURBULENCE-INDUCED RELATIVE VELOCITY OF DUST PARTICLES. III. THE PROBABILITY DISTRIBUTION

Motivated by its important role in the collisional growth of dust particles in protoplanetary disks, we investigate the probability distribution function (PDF) of the relative velocity of inertial particles suspended in turbulent flows. Using the simulation from our previous work, we compute the relative velocity PDF as a function of the friction timescales, tau_p1 and tau_p2, of two particles of arbitrary sizes. The friction time of particles included in the simulation ranges from 0.1 tau_eta to 54T_L, with tau_eta and T_L the Kolmogorov time and the Lagrangian correlation time of the flow, respectively. The relative velocity PDF is generically non-Gaussian, exhibiting fat tails. For a fixed value of tau_p1, the PDF is the fattest for equal-size particles (tau_p2~tau_p1), and becomes thinner at both tau_p2<tau_ p1 and tau_p2>tau_p1. Defining f as the friction time ratio of the smaller particle to the larger one, we find that, at a given f in 1/2<f<1, the PDF fatness first increases with the friction time, tau_ph, of the larger particle, peaks at tau_ph~tau_eta, and then decreases as tau_ph increases further. For 0<f<1/4, the PDF shape becomes continuously thinner with increasing tau_ph. The PDF is nearly Gaussian only if tau_ph is sufficiently large (>>T_L). These features are successfully explained by the Pan & Padoan model. Using our simulation data and some simplifying assumptions, we estimated the fractions of collisions resulting in sticking, bouncing, and fragmentation as a function of the dust size in protoplanerary disks, and argued that accounting for non-Gaussianity of the collision velocity may help further alleviate the bouncing barrier problem.

A probabilistic approach to heterogeneity in space plasmas: the case of magnetic field intensity in solar wind

Abstract. Since the early 90s it was shown that Probability Distribution Functions (PDFs) of small scale differences (fluctuations) of several quantities in space plasmas display significant departures from Gaussianity. The non-Gaussian shape of PDFs was ascribed to intermittency and discussed in the framework of intermittent MHD turbulence. Here, we put the attention to the PDF of magnetic field intensity instead of its differences showing how the PDF of such quantity in a quiet solar wind can be related with the occurrence of heterogeneity. In detail, we derive the shape of the PDFs by simple statistical considerations based on the concept of subordination in probability theory. An interpretation and a discussion in terms of existing coherent magnetic structures in a mechanical near-equilibrium state are also presented.

The non-linear probability distribution function in models with local primordial non-Gaussianity

We use the spherical evolution approximation to investigate non-linear evolution from the non-Gaussian initial conditions characteristic of the local fnl model. We provide an analytic formula for the non-linearly evolved probability distribution function (PDF) of the dark matter which shows that the underdense tail of the non-linear PDF in the fnl model should differ significantly from that for Gaussian initial conditions. Measurements of the underdense tail in numerical simulations may be affected by discreteness effects, and we use a Poisson counting model to describe this effect. Once this has been accounted, our model is in good quantitative agreement with the simulations. In principle, our calculation is an important first step in programs which seek to reconstruct the shape of the initial PDF from observations of large-scale structures in the Lyα forest and the galaxy distribution at later times.

Measure, Simulation and Spectral Efficiency Analysis for MIMO Channels in Indoor Environments over the 2GHz Band

The space-time wireless communication systems will use the advantages of the MIMO (Multiple Input Multiple Output) channels. Precedent studies have focused less attention on the capacity analysis considering correlated indoor channels under different frequency bands and polarizations. In practice, there is certain correlation degree between the different propagation sub-channels, which makes interesting to evaluate the impact that this correlation has on the system performance under different frequency bands and polarizations. The paper shows a detail study around of real MIMO channel and its spectral efficiency behavior when change both the number of antenna elements and SNR (Signal to Noise Ratio). The results showed in the paper demonstrate that to increase the diversity order could not be a good technique to improve the performance of the MIMO systems in some highly correlated cases when picocells or indoor environments are considered. Besides, increasing the number of array antenna elements, both in the transmitter and receiver side, does not offer linear capacity increases under correlated cases. The results derived from a frequency measurement campaigns performed between 2 and 2.6 GHz show polarization dependence and less frequency band dependence in the MIMO channel correlation matrix for indoor environments. In other hand, under the WSS (Wide Sense Stationary) channel condition and guaranteeing a minimum SNR level, remarkable changes on the capacity probability distribution function shape, and on the cumulative distribution function of the MIMO system eigenvalues, were observed in different correlated cases.

Increased Occurrence of Stratospheric Sudden Warmings during El Niño as Simulated by WACCM

Abstract Experiments with Whole Atmosphere Community Climate Model (WACCM) under perpetual January conditions indicate that stratospheric sudden warmings (SSWs) are twice as likely to occur in El Niño winters than in La Niña winters, in basic agreement with the limited observational dataset. Tropical SST anomalies that mimic El Niño and La Niña lead to changes in the shape of probability distribution functions (PDFs) of stratospheric day-to-day variability, resulting in a warmer pole and weaker vortex on average for El Niño conditions. The tropical SST forcing induces a response similar to the observed response in the enhancement of the planetary wave of zonal wavenumber 1 (wave 1) and the weakening of wave 2 in the upper troposphere and stratosphere of high latitudes. The enhanced wave 1 contributes to a shift of the PDFs of poleward eddy heat flux in the lower stratosphere, or wave forcing entering the stratosphere. The shift of the PDFs includes an increase of strong wave events that induce more frequent SSWs.

Front propagation in population dynamics with dispersive variability and delayed growth

On the basis of the Cook model, we propose a delayed-growth reaction–diffusion model with an age-dependent disperser–nondisperser transition. We compare the speed of migration fronts between our model and the hyperbolic generalization of the Cook model. In particular, we study for both models the dependence of the migratory fronts speed on the shape of probability distribution function of adult ages for the Neolithic transition in Europe. The migratory fronts speed in both models behaves in a very similar qualitative way, exhibiting both a good agreement with observational data.

Statistical Model of Multicomponent-Fuel Drop Evaporation for Many-Drop Flow Simulations

A statistical formulation is developed describing the composition in an evaporating multicomponent-fuel liquid drop and in the gas phase surrounding it. When a complementary discrete-component model is used, it is shown that, when drops are immersed in a carrier gas containing fuel vapor, condensation of species onto the drop results in the development of a minor peak in the liquid composition probability distribution function (PDF). This peak leads to a PDF shape that can be viewed as a combination of two gamma PDFs, which is determined by five parameters. A model is developed for calculating the parameters of the two combined gamma PDFs. Extensive tests of the model for both diesel and gasoline show that the PDF results replicate accurately the discrete model predictions. Most important, the mean and variance of the composition at the drop surface are in excellent agreement with the discrete model. Results from the model show that although the second peak is minor for the liquid PDF, its corresponding peak for the vapor distribution at the drop surface has a comparable magnitude to and sometimes exceeds that corresponding to the first peak. Four-parameter models are also exercised, and it is shown that they are unable to capture the physics of the problem.

The Probability Distribution Function of Column Density in Molecular Clouds

We discuss the probability distribution function (PDF) of column density resulting from density fields with lognormal PDFs, applicable to isothermal gas (e.g., probably molecular clouds). For magnetic and nonmagnetic numerical simulations of compressible, isothermal turbulence forced at intermediate scales ( of the box size), we find that the autocorrelation function (ACF) of the density field decays over relatively short distances compared to the simulation size. We suggest that a "decorrelation length" can be defined as the distance over which the density ACF has decayed to, for example, 10% of its zero-lag value, so that the density "events" along a line of sight can be assumed to be independent over distances larger than this, and the central limit theorem should be applicable. However, using random realizations of lognormal fields, we show that the convergence to a Gaussian is extremely slow in the high-density tail. As a consequence, the column density PDF is not expected to exhibit a unique functional shape, but to transit instead from a lognormal to a Gaussian form as the ratio η of the column length to the decorrelation length (i.e., the number of independent events in the cloud) increases. Simultaneously, the variance of the PDF decreases. For intermediate values of η, the column density PDF assumes a nearly exponential decay. For cases with a density contrast of 104, as found in intermediate-resolution simulations, and expected from giant molecular clouds (GMCs) to dense molecular cores, the required value of η for convergence to a Gaussian is at least a few hundred, or, for 106, several thousand. We then discuss the density power spectrum and the expected value of η in actual molecular clouds, concluding that they are uncertain since they may depend on several physical parameters. Observationally, our results suggest that η may be inferred from the shape and width of the column density PDF in optically thin line or extinction studies. Our results should also hold for gas with finite-extent power-law underlying density PDFs, which should be characteristic of the diffuse, nonisothermal neutral medium (with temperatures ranging from a few hundred to a few thousand degrees). Finally, we note that for η ≳ 100, the dynamic range in column density is small (less than a factor of 10), but this is only an averaging effect, with no implication on the dynamic range of the underlying density distribution.

Power spectrum tests of random numbers

Three nonrandomness tests having very low correlation between them and utilizing some statistical properties of the spectral estimates of the random number sequences having any shape of probability distribution function, are given. The spectral density is estimated as 1/ N times the periodogram (magnitude square of the discrete Fourier transform) of an N-term sequence of the random numbers. From M such spectral estimates, M peak values, M spacings between the ordered (sorted) frequencies of the peak values, and the peak value of the ensemble average of the estimates, are found and tested to decide if they closely fit their cumulative probability distributions as derived for random numbers. The sensibility thresholds of the tests are measured on random binary sequences artificially contaminated by substitution of periodic binary signals or substitution of random binary signals having periodic mean or shortening long runs.

Structure of a swirl-stabilized spray flame relevant to gas turbine and furnaces

To investigate the structure of a hollow-cone spray flame similar to those occurring in the primary zone of gas-turbine combustors or within furnaces, a swirl-stabilized combustor was used. A kerosene spray was produced by a simplex atomizer with a nominal included-cone-angle of 30 deg. Swirling air had a calculated swirl number of 0.36 and was produced by a swirl plate having an exit air velocity vector of 30 deg with respect to the chamber axis. A phase Doppler particle analyzer was used to measure drop size; mean and root mean square (rms) values of axial, radial, and tangential components; and drop velocity and size probability distribution functions (PDFs) at five different axial positions from the nozzle. A mean spray flame structure and terminology are proposed. It is found that a large portion of the drop flowfield within this spray flame is dominated by radial and axial components of the velocity vector. It appears that sufficiently away from the nozzle the direction of the mean droplet velocity vector is primarily established by the design of the swirl plate. Based on the bimodal behavior of drop axial velocity PDF, pulsation of the central recirculation zone is proposed and the extent of its effect on the droplet field is investigated. The rms of droplet flowfield is not isotropic in this spray flame. At each axial position, axial rms has the highest value among the three components and occurs near the maximum temperature point. Moving away from the nozzle near and along the locus of maximum mean temperature, strong preferential evaporation and/or burning of small drops drastically affect the shape of the size PDFs by progressively shifting its profile into larger size ranges.

Properties of the cosmological density distribution function

The properties of the probability distribution function (PDF) of the cosmological continuous density field are studied. We focus our analysis on the quasi-linear regime where various calculations, based on dynamically motivated methods, have been presented: either by using the Zel'dovich approximation (ZA) or by using the perturbation theory to evaluate the behavior of the moments of the distribution function. We show how these two approaches are related to each other and that they can be used in a complementary way. For that respect the one-dimensional dynamics, where the ZA is exact solution, has first been used as a testing ground. In particular we show that, when the density PDF obtained with the ZA is regularized, its various moments exhibit the behavior expected by the perturbation theory applied to the ZA. We show that ZA approach can be used for arbitrary initial conditions (not only Gaussian) and that the nonlinear evolution of the moments can be obtained. The perturbation theory can be used for the exact dynamics. We take into account the final filtering of the density field both for ZA and perturbation theory. Applying these techniques we got the generating function of the moments for the one-dimensional dynamics, the three- dimensional ZA, with and without smoothing effects. We also suggest methods to build PDFs. One is based on the Laplace inverse transform of the moment generating function. The other, the Edgeworth expansion, is obtained when the previous generating function is truncated at a given order and allows to evaluate the PDF out of limited number of moments. It provides insight on the relationship between the moments and the shape of the density PDF. In particular it provides an alternative method to evaluate the skewness and kurtosis by measuring the PDF around its maximum. Eventually, results obtained from a numerical simulation with CDM initial conditions have been used to validate the accuracy of the considered approximations. We explain the successful log- normal fit of the PDF from that simulation at moderate σ as mere fortune, but not as a universal form of density PDF in general.