Thermodynamic derivatives of classical distribution functions

1985 ◽  
Vol 82 (9) ◽  
pp. 4385-4386 ◽  
Author(s):  
J. A. Hernando
Keyword(s):  
Distribution Functions ◽  
Classical Distribution ◽  
Derivatives Of

Probability and Random Processes

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Random Processes ◽  
Distribution Functions ◽  
Basic Logic ◽  
Heavy Tailed Distributions ◽  
Random Components ◽  
Classical Distribution ◽  
Heavy Tailed ◽  
Basic Facts ◽  
Stochastic Events ◽  
Stochastic Event

Phenomena, systems, and processes are rarely purely deterministic, but contain stochastic,probabilistic, or random components. For that reason, a probabilistic descriptionof most phenomena is necessary. Probability theory provides us with the tools for thistask. Here, we provide a crash course on the most important notions of probabilityand random processes, such as odds, probability, expectation, variance, and so on. Wedescribe the most elementary stochastic event—the trial—and develop the notion of urnmodels. We discuss basic facts about random variables and the elementary operationsthat can be performed on them. We learn how to compose simple stochastic processesfrom elementary stochastic events, and discuss random processes as temporal sequencesof trials, such as Bernoulli and Markov processes. We touch upon the basic logic ofBayesian reasoning. We discuss a number of classical distribution functions, includingpower laws and other fat- or heavy-tailed distributions.

scholarly journals Changes in the shape of cloud ice water content vertical structure due to aerosol variations

2016 ◽  
Vol 16 (10) ◽  
pp. 6091-6105 ◽  
Author(s):  
Steven T. Massie ◽  
Julien Delanoë ◽  
Charles G. Bardeen ◽  
Jonathan H. Jiang ◽  
Lei Huang
Keyword(s):  
Water Content ◽  
Vertical Structure ◽  
Distribution Functions ◽  
Ozone Monitoring Instrument ◽  
Modis Aod ◽  
Ice Water Content ◽  
Moderate Resolution Imaging Spectroradiometer ◽  
Cloud Ice ◽  
Ice Water ◽  
Derivatives Of

Abstract. Changes in the shape of cloud ice water content (IWC) vertical structure due to variations in Moderate Resolution Imaging Spectroradiometer (MODIS) aerosol optical depths (AODs), Ozone Monitoring Instrument (OMI) absorptive aerosol optical depths (AAODs), and Microwave Limb Sounder (MLS) CO (an absorptive aerosol proxy) at 215 hPa are calculated in the Tropics during 2007–2010 based upon an analysis of DARDAR IWC profiles for deep convective clouds. DARDAR profiles are a joint retrieval of CloudSat-CALIPSO data. Analysis is performed for 12 separate regions over land and ocean, and carried out applying MODIS AOD fields that attempt to correct for 3-D cloud adjacency effects. The 3-D cloud adjacency effects have a small impact upon our particular calculations of aerosol–cloud indirect effects. IWC profiles are averaged for three AOD bins individually for the 12 regions. The IWC average profiles are also normalized to unity at 5 km altitude in order to study changes in the shape of the average IWC profiles as AOD increases. Derivatives of the IWC average profiles, and derivatives of the IWC shape profiles, in percent change per 0.1 change in MODIS AOD units, are calculated separately for each region. Means of altitude-specific probability distribution functions, which include both ocean and land IWC shape regional derivatives, are modest, near 5 %, and positive to the 2σ level between 11 and 15 km altitude. Similar analyses are carried out for three AAOD and three CO bins. On average, the vertical profiles of the means of the derivatives based upon the profile shapes over land and ocean are smaller for the profiles binned according to AAOD and CO values, than for the MODIS AODs, which include both scattering and absorptive aerosol. This difference in character supports the assertion that absorptive aerosol can inhibit cloud development.

Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics

2007 ◽  
Vol 592 ◽  
pp. 335-366 ◽  
Author(s):  
T. ISHIHARA ◽  
Y. KANEDA ◽  
M. YOKOKAWA ◽  
K. ITAKURA ◽  
A. UNO
Keyword(s):  
Reynolds Number ◽  
High Resolution ◽  
Longitudinal Velocity ◽  
Second Order ◽  
Distribution Functions ◽  
Small Scale ◽  
Kolmogorov Length Scale ◽  
Velocity Gradients ◽  
In Series ◽  
Derivatives Of

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 40963 grid points. The DNS consist of two series of runs; one is with kmaxη ~ 1 (Series 1) and the other is with kmaxη ~ 2 (Series 2), where kmax is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number Rλ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on Rλ of the skewness and flatness factors (S and F). The DNS data suggest that the Rλ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ~ −0.32Rλ0.11 and F ~ 1.14Rλ0.34, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with Rλ similarly to each other at Rλ > 100, in contrast to Rλ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at Rλ ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with Rλ approximately proportional to RλαE and RλαL, respectively, where αE ≈ 0.5 and αL ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to RλβE and RλβL, respectively, where βE ≈ 1.5 and βL ≈ 3.0.

scholarly journals Changes in the shape of cloud ice water content vertical structure due to aerosol variations

2016 ◽  
Author(s):  
Steven T. Massie ◽  
Julien Delanoe ◽  
Charles G. Bardeen
Keyword(s):  
Water Content ◽  
Vertical Structure ◽  
Distribution Functions ◽  
Convective Clouds ◽  
Modis Aod ◽  
Ice Water Content ◽  
Moderate Resolution Imaging Spectroradiometer ◽  
Cloud Ice ◽  
Ice Water ◽  
Derivatives Of

Abstract. Changes in the shape of cloud ice water content vertical structure due to aerosol variations are calculated in the Tropics during 2007–2010 based upon an analysis of DARDAR ice water content (IWC) profiles for deep convective clouds. DARDAR profiles are a joint retrieval of CloudSat-CALIPSO data. Our analysis is performed for 12 separate regions over land and ocean, and carried out applying Moderate-Resolution Imaging Spectroradiometer (MODIS) aerosol optical depth (AOD) fields that attempt to correct for 3D cloud adjacency effects. The 3D cloud adjacency effects have a small impact upon our calculations of aerosol-cloud indirect effects. IWC profiles are averaged for three AOD bins individually for the 12 regions. The IWC average profiles are also normalized to unity at 5 km altitude in order to study changes in the shape of the average IWC profiles as AOD increases. Derivatives of the IWC average profiles, and derivatives of the IWC shape profiles, in percent change per 0.1 change in MODIS AOD units, are calculated separately for each region. Means of altitude-specific probability distribution functions, which include both ocean and land IWC shape regional derivatives, are modest, near 5 %, and positive to the 2σ level between 11 and 15 km altitude.

Sum rules for the pair-correlation functions of inhomogeneous fluids: results for the hard-sphere–hard-wall system

1987 ◽  
Vol 410 (1839) ◽  
pp. 409-420 ◽  
Keyword(s):  
Sum Rules ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Distribution Functions ◽  
Hard Wall ◽  
Chain Equation ◽  
Hypernetted Chain ◽  
Chain Approximation ◽  
Derivatives Of

Starting from well-known relations for the derivatives of the radial distribution functions of a mixture of fluids, and allowing the diameter of one particle to become exceedingly large, three sum rules for a fluid with density inhomogeneities are obtained. None of these sum rules are new. However, the relation between the Lovett–Mou–Buff–Wertheim and the Born–Green hierarchy of equations seems not well known. The accuracy of a recent parametrization of the pair correlation of hard spheres near a hard wall and of the solutions of the Percus–Yevick and hypernetted-chain equation for this same function are examined by determination of how well these functions satisfy these sum rules and the accuracy of their surface tension, calculated from the sum rule of Triezenberg and Zwanzig. Generally speaking, the Percus–Yevick theory gives the best results and the hypernetted-chain approximation gives the worst results with the parametrization being intermediate.

scholarly journals Determination of partition surface of grained material by means of non-classical approximation methods of distributions functions of particle size and density

2016 ◽  
Vol 32 (1) ◽  
pp. 137-154
Author(s):  
Tomasz Niedoba
Keyword(s):  
Particle Size ◽  
Distribution Function ◽  
Statistical Methods ◽  
Particle Density ◽  
Distribution Functions ◽  
Alternative Methods ◽  
Hard Coal ◽  
Classical Distribution ◽  
Partition Surface ◽  
Non Parametric

Abstract In this paper, the grained material analyzed was hard coal collected from one of the mines located in Upper Silesia. Material was collected from a dust jig where it was separated in industrial conditions by concentrate and waste. It was then screened in sieves and it was separated in dense media into density fractions. Both particle size distribution and particle density distribution for feed and concentrate were approximated by several classical distribution functions. The best results were obtained by means of the Weibull (RRB) distribution function. However, because of the unsatisfying quality of approximations it was decided to apply non-parametric statistical methods, which became more and more popular alternative methods in conducting statistical investigations. In the paper, the kernel methods were applied to this purpose and the Gauss kernel was accepted as the kernel function. Kernel method, which is relatively new, gave much better results than classical distribution functions by means of the least squared method. Both classical and non-parametric obtained distribution functions were evaluated by means of mean standard error, the values of which proved that they sufficiently well approximate the empirical data. Such function forms were then applied to determine the theoretical distribution function for vector (D, P), where D is the random variable describing particle size and P – its density. This approximation was sufficiently acceptable. That is why it served to determine the equation of partition surface dependent on particle size and particle density describing researched material. The obtained surface proves that it is possible to evaluate material separation which occurs during mineral processing operations, such as jigging, by means of more than one feature of researched material. Furthermore, its quality confirms that it is justified to apply non-parametric statistical methods instead of commonly used classical ones.

Evaluation of the distribution functions of spatial derivatives of an ionic electric microfield in plasmas by using only the central interactions approach

2013 ◽  
Vol 79 (5) ◽  
pp. 727-734
Author(s):  
S. CHIHI ◽  
S. GUERRICHA
Keyword(s):  
Distribution Functions ◽  
Spatial Derivatives ◽  
Derivatives Of ◽  
Theoretical Results

AbstractBy using some physical approximations, distribution functions of spatial derivatives of components of an electric ionic microfield in plasmas have been theoretically evaluated. The particles are considered quasistatic. Only interactions between charged emitter and perturber ions are taken into account. Our theoretical results have been compared with those found in the literature. The few differences found are discussed. Some properties of distribution functions of spatial derivatives have been deduced from curves.

Statistics of the fine structure of turbulent velocity and temperature fields measured at high Reynolds number

1970 ◽  
Vol 41 (1) ◽  
pp. 153-167 ◽  
Author(s):  
Carl H. Gibson ◽  
Gilbert R. Stegen ◽  
Robert Bruce Williams
Keyword(s):  
Distribution Functions ◽  
Temperature Fields ◽  
Inertial Subrange ◽  
Local Anisotropy ◽  
Probability Distribution Functions ◽  
Zero Values ◽  
Velocity And Temperature Fields ◽  
High Reynolds ◽  
Derivatives Of ◽  
Variable Probability

Derivatives of velocity and temperature in the wind over the ocean were found to be quite variable. Probability distribution functions of squared derivatives were consistent with lognormality predictions by Kolmogoroff, Obukhoff and Yaglom. Kurtosis values for velocity derivatives ranged from 13 to 26 and from 26 to 43 for temperature derivatives. Universal inertial subrange constants were evaluated from dissipation spectra and were found to be 40 to 300% larger than most values reported previously. Evidence for local anisotropy of the temperature field is provided by non-zero values of the measured derivative skewness.

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