scholarly journals An analytic solution of the non-linear equation∇2λ(r)=f(λ)and its application to the ion-atmosphere theory of strong electrolytes

1981 ◽  
Vol 4 (2) ◽  
pp. 209-277 ◽  
Author(s):  
S. N. Bagchi
Keyword(s):  
Distribution Function ◽  
Analytic Solution ◽  
Extensive Study ◽  
Distribution Functions ◽  
A Posteriori ◽  
Consistent Theory ◽  
Generalized Poisson ◽  
Primary Object ◽  
Simplifying Assumptions ◽  
Strong Electrolytes

For a long time the formulation of a mathematically consistent statistical mechanical theory for a system of charged particles had remained a formidable unsolved problem. Recently, the problem had been satisfactorily solved, (see Bagchi [1] [2]) ,by utilizing the concept of ion-atmosphere and generalized Poisson-Boltzmann (PB) equation. Although the original Debye-Hueckel (DH) theory of strong electrolytes [3] cannot be accepted as a consistent theory, neither mathematically nor physically, modified DH theory, in which the exclusion volumes of the ions enter directly into the distribution functions, had been proved to be mathematically consistent. It also yielded reliable physical results for both thermodynamic and transport properties of electrolytic solutions. Further, it has already been proved by the author from theoretical considerations (cf. Bagchi [4])as well as from a posteriori verification (see refs. [1] [2]) that the concept of ion-atmosphere and the use of PB equation retain their validities generally. Now during the past 30 years, for convenice of calculations, various simplified versions of the original Dutta-Bagchi distribution function (Dutta & Bagchi [5])had been used successfully in modified DH theory of solutions of strong electrolytes. The primary object of this extensive study, (carried out by the author during 1968-73), was to decide a posteriori by using the exact analytic solution of the relevant PB equation about the most suitable, yet theoretically consistent, form of the distribution function. A critical analysis of these results eventually led to the formulation of a new approach to the statistical mechanics of classical systems, (see Bagchi [2]), In view of the uncertainties inherent in the nature of the system to be discussed below, it is believed that this voluminous work, (containing 35 tables and 120 graphs), in spite of its legitimate simplifying assumptions, would be of great assistance to those who are interested in studying the properties of ionic solutions from the standpoint of a physically and mathematically consistent theory.

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

scholarly journals Estimating a Distribution Function at the Boundary

2020 ◽  
Vol 49 (1) ◽  
pp. 1-23
Author(s):  
Shunpu Zhang ◽  
Zhong Li ◽  
Zhiying Zhang
Keyword(s):  
Distribution Function ◽  
Kernel Estimation ◽  
Kernel Estimator ◽  
Distribution Functions ◽  
Practical Application ◽  
Real World Applications ◽  
Distribution Kernel ◽  
Estimation Of Distribution ◽  
Simulation Results ◽  
Boundary Correction

Estimation of distribution functions has many real-world applications. We study kernel estimation of a distribution function when the density function has compact support. We show that, for densities taking value zero at the endpoints of the support, the kernel distribution estimator does not need boundary correction. Otherwise, boundary correction is necessary. In this paper, we propose a boundary distribution kernel estimator which is free of boundary problem and provides non-negative and non-decreasing distribution estimates between zero and one. Extensive simulation results show that boundary distribution kernel estimator provides better distribution estimates than the existing boundary correction methods. For practical application of the proposed methods, a data-dependent method for choosing the bandwidth is also proposed.

scholarly journals Transformation of circular random variables based on circular distribution functions

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5931-5947
Author(s):  
Hatami Mojtaba ◽  
Alamatsaz Hossein
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Real Data ◽  
Random Variable ◽  
Distribution Functions ◽  
Likelihood Method ◽  
Circular Distribution ◽  
Data Set ◽  
Trigonometric Moments ◽  
Von Mises

In this paper, we propose a new transformation of circular random variables based on circular distribution functions, which we shall call inverse distribution function (id f ) transformation. We show that M?bius transformation is a special case of our id f transformation. Very general results are provided for the properties of the proposed family of id f transformations, including their trigonometric moments, maximum entropy, random variate generation, finite mixture and modality properties. In particular, we shall focus our attention on a subfamily of the general family when id f transformation is based on the cardioid circular distribution function. Modality and shape properties are investigated for this subfamily. In addition, we obtain further statistical properties for the resulting distribution by applying the id f transformation to a random variable following a von Mises distribution. In fact, we shall introduce the Cardioid-von Mises (CvM) distribution and estimate its parameters by the maximum likelihood method. Finally, an application of CvM family and its inferential methods are illustrated using a real data set containing times of gun crimes in Pittsburgh, Pennsylvania.

scholarly journals How Many Monte Carlo Simulations Does One Need to Do? II. Lognormal, Binomial, Cauchy, and Exponential Distributions

2005 ◽  
Vol 23 (6) ◽  
pp. 429-461
Author(s):  
Ian Lerche ◽  
Brett S. Mudford
Keyword(s):  
Monte Carlo ◽  
Distribution Function ◽  
Probability Distribution ◽  
General Procedure ◽  
Mean Value ◽  
Distribution Functions ◽  
Considerable Uncertainty ◽  
Free Parameters ◽  
Two Parameters ◽  
Achievable Accuracy

This article derives an estimation procedure to evaluate how many Monte Carlo realisations need to be done in order to achieve prescribed accuracies in the estimated mean value and also in the cumulative probabilities of achieving values greater than, or less than, a particular value as the chosen particular value is allowed to vary. In addition, by inverting the argument and asking what the accuracies are that result for a prescribed number of Monte Carlo realisations, one can assess the computer time that would be involved should one choose to carry out the Monte Carlo realisations. The arguments and numerical illustrations are carried though in detail for the four distributions of lognormal, binomial, Cauchy, and exponential. The procedure is valid for any choice of distribution function. The general method given in Lerche and Mudford (2005) is not merely a coincidence owing to the nature of the Gaussian distribution but is of universal validity. This article provides (in the Appendices) the general procedure for obtaining equivalent results for any distribution and shows quantitatively how the procedure operates for the four specific distributions. The methodology is therefore available for any choice of probability distribution function. Some distributions have more than two parameters that are needed to define precisely the distribution. Estimates of mean value and standard error around the mean only allow determination of two parameters for each distribution. Thus any distribution with more than two parameters has degrees of freedom that either have to be constrained from other information or that are unknown and so can be freely specified. That fluidity in such distributions allows a similar fluidity in the estimates of the number of Monte Carlo realisations needed to achieve prescribed accuracies as well as providing fluidity in the estimates of achievable accuracy for a prescribed number of Monte Carlo realisations. Without some way to control the free parameters in such distributions one will, presumably, always have such dynamic uncertainties. Even when the free parameters are known precisely, there is still considerable uncertainty in determining the number of Monte Carlo realisations needed to achieve prescribed accuracies, and in the accuracies achievable with a prescribed number of Monte Carol realisations because of the different functional forms of probability distribution that can be invoked from which one chooses the Monte Carlo realisations. Without knowledge of the underlying distribution functions that are appropriate to use for a given problem, presumably the choices one makes for numerical implementation of the basic logic procedure will bias the estimates of achievable accuracy and estimated number of Monte Carlo realisations one should undertake. The cautionary note, which is the main point of this article, and which is exhibited sharply with numerical illustrations, is that one must clearly specify precisely what distributions one is using and precisely what free parameter values one has chosen (and why the choices were made) in assessing the accuracy achievable and the number of Monte Carlo realisations needed with such choices. Without such available information it is not a very useful exercise to undertake Monte Carlo realisations because other investigations, using other distributions and with other values of available free parameters, will arrive at very different conclusions.

The Method of Probabilistic Nodes Combination in Decision Making and Risk Analysis

2016 ◽  
pp. 32-60
Author(s):  
Dariusz Jacek Jakóbczak
Keyword(s):  
Decision Making ◽  
Distribution Function ◽  
Probability Distribution ◽  
Risk Analysis ◽  
Probability Distribution Function ◽  
Linear Interpolation ◽  
Distribution Functions ◽  
Probability Distribution Functions ◽  
Characteristic Points ◽  
Basic Probability

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve interpolation and extrapolation using the set of key points (knots or nodes). Nodes can be treated as characteristic points of data for modeling and analyzing. The model of data can be built by choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter ? as probability distribution function enables value anticipation in risk analysis and decision making. Two-dimensional curve is extrapolated and interpolated via nodes combination and different functions as discrete or continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function. Novelty of the paper consists of two generalizations: generalization of previous MHR method with various nodes combinations and generalization of linear interpolation with different (no basic) probability distribution functions and nodes combinations.

The influence of diffuse pollution on groundwater content patterns for the groundwater bodies of Germany

2007 ◽  
Vol 55 (3) ◽  
pp. 97-105 ◽  
Author(s):  
R. Kunkel ◽  
F. Wendland ◽  
S. Hannappel ◽  
H.J. Voigt ◽  
R. Wolter
Keyword(s):  
Distribution Function ◽  
Groundwater Monitoring ◽  
Distribution Functions ◽  
Monitoring Data ◽  
Hydrochemical Parameters ◽  
Federal States ◽  
Chemical Status ◽  
Water Problems ◽  
Natural Component ◽  
Monitoring Stations

Commissioned by Germany's Working Group of the Federal States on Water Problems (LAWA) the authors developed a procedure to define natural groundwater conditions from groundwater monitoring data. The distribution pattern of a specific groundwater parameter observed by a number of groundwater monitoring stations within a petrographically comparable groundwater typology is reproduced by two statistical distribution functions, representing the “natural” and “influenced” component. The range of natural groundwater concentrations is characterized by confidence intervals of the distribution function of the natural component. The applicability of the approach was established for 17 hydrochemical different groundwater typologies occurring throughout Germany. Based on groundwater monitoring data from ca. 26,000 groundwater-monitoring stations, 40 different hydrochemical parameters were evaluated for each groundwater typology. For all investigated parameters the range of natural groundwater concentrations has been identified. According to the requirements of the EC Water Framework Directive (article 17) (WFD) this study is a basis for the German position to propose criteria for assessing a reference state for a “good groundwater chemical status”.

scholarly journals Collisionless distribution functions for force-free current sheets: using a pressure transformation to lower the plasma beta

2018 ◽  
Vol 84 (3) ◽  
Author(s):  
F. Wilson ◽  
T. Neukirch ◽  
O. Allanson
Keyword(s):  
Distribution Function ◽  
Current Sheet ◽  
Plasma Phys ◽  
Distribution Functions ◽  
Hermite Functions ◽  
Slow Convergence ◽  
Nonlinear Force ◽  
Free Current ◽  
Transformed Distribution ◽  
General Method

So far, only one distribution function giving rise to a collisionless nonlinear force-free current sheet equilibrium allowing for a plasma beta less than one is known (Allansonet al.,Phys. Plasmas, vol. 22 (10), 2015, 102116; Allansonet al.,J. Plasma Phys., vol. 82 (3), 2016a, 905820306). This distribution function can only be expressed as an infinite series of Hermite functions with very slow convergence and this makes its practical use cumbersome. It is the purpose of this paper to present a general method that allows us to find distribution functions consisting of a finite number of terms (therefore easier to use in practice), but which still allow for current sheet equilibria that can, in principle, have an arbitrarily low plasma beta. The method involves using known solutions and transforming them into new solutions using transformations based on taking integer powers ($N$) of one component of the pressure tensor. The plasma beta of the current sheet corresponding to the transformed distribution functions can then, in principle, have values as low as$1/N$. We present the general form of the distribution functions for arbitrary$N$and then, as a specific example, discuss the case for$N=2$in detail.

scholarly journals Radial Velocity Profiles for Anisotropic Spherically Symmetric Clusters: An Example

1988 ◽  
Vol 126 ◽  
pp. 691-692
Author(s):  
Herwig Dejonghe
Keyword(s):  
Distribution Function ◽  
Radial Velocity ◽  
Velocity Dispersion ◽  
Mass Density ◽  
Distribution Functions ◽  
Spherically Symmetric ◽  
Line Profiles ◽  
Anisotropic Models ◽  
The Family

A 1-parameter family of anisotropic models is presented. They all satisfy the Plummer law in the mass density, but have different velocity dispersions. Moreover, the stars are not confined to a particular subset of the total accessible phase space. This family is mathematically simple enough to be explored analytically in detail. The family is rich enough though to allow for a 3-parameter generalization which illustrates that even when both the mass density and the velocity dispersion profiles are required to be the same, a degeneracy in the possible distribution functions persists. The observational consequences of the degeneracy can be studied by calculating the observable radial velocity line profiles obtained with different distribution functions. It turns out that line profiles are relatively sensitive to changes in the distribution function. They therefore can be considered to be more natural observables when a determination of the distribution function is desired.

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