scholarly journals The κ-cookbook: a novel generalizing approach to unify κ-like distributions for plasma particle modelling

2020 ◽  
Vol 497 (2) ◽  
pp. 1738-1756 ◽  
Author(s):  
K Scherer ◽  
E Husidic ◽  
M Lazar ◽  
H Fichtner
Keyword(s):  
Debye Length ◽  
Distribution Functions ◽  
Energy Distributions ◽  
Pickup Ions ◽  
Plasma Distribution ◽  
Special Cases ◽  
Power Law Exponent ◽  
Velocity Moments ◽  
Analytical Expressions ◽  
Generalized Distribution

ABSTRACT In the literature different so-called κ-distribution functions are discussed to fit and model the velocity (or energy) distributions of solar wind species, pickup ions, or magnetospheric particles. Here, we introduce a generalized (isotropic) κ-distribution as a ‘cookbook’, which admits as special cases, or ‘recipes’, all the other known versions of κ-models. A detailed analysis of the generalized distribution function is performed, providing general analytical expressions for the velocity moments, Debye length, and entropy, and pointing out a series of general requirements that plasma distribution functions should satisfy. From a contrasting analysis of the recipes found in the literature, we show that all of them lead to almost the same macroscopic parameters with a small standard deviation between them. However, one of these recipes called the regularized κ-distribution provides a functional alternative for macroscopic parametrization without any constraint for the power-law exponent κ.

Suprathermal plasma distribution functions with relativistic cut-offs

2019 ◽  
Vol 491 (3) ◽  
pp. 3967-3973
Author(s):  
H-J Fahr ◽  
M Heyl
Keyword(s):  
Distribution Function ◽  
Heat Transport ◽  
Distribution Functions ◽  
The Other ◽  
Pressure Reduction ◽  
Plasma Distribution ◽  
Order Of Magnitude ◽  
Low Mass ◽  
Velocity Moments ◽  
Free Plasma

ABSTRACT In typical plasma physics scenarios, when treated on kinetic levels, distribution functions with suprathermal wings are obtained. This raises the question of how the associated typical velocity moments, which are needed to arrive at magnetohydrodynamic plasma descriptions, may appear. It has become evident that the higher velocity moments in particular, for example the pressure or heat transport, which are constructed as integrations of the distribution function, contain unphysical contributions from particles with velocities greater than the velocity of light. In what follows, we discuss two possibilities to overcome this problem. One is to calculate a maximal, physically permitted, upper velocity, which can be realized in view of the underlying energization processes, and to stop the integration there. The other is to modify the distribution function relativistically so that no particles with superluminal (v ≥ c) velocities appear. On the basis of a typical collision-free plasma scenario, like the plasma in the heliosheath, we obtain the corresponding expressions for electron and proton pressures and can show that in both cases the pressures are reduced compared with their classical values; however, electrons experience a stronger reduction than protons. When calculating pressure ratios, it turns out that these are of the same order of magnitude regardless of which of the two methods is used. The electron, as the low-mass particle, undergoes the more pronounced pressure reduction. It may turn out that electrons and protons constitute about equal pressures in the heliosheath, implying that no pressure deficit need be claimed here.

scholarly journals Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

2010 ◽  
Vol 42 (1) ◽  
pp. 48-68 ◽  
Author(s):  
L. Muche
Keyword(s):  
Voronoi Tessellation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Dimensional Euclidean Space ◽  
Spherical Contact ◽  
Chord Length Distribution ◽  
Special Cases ◽  
Contact Distribution ◽  
Contact Distribution Function

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

scholarly journals Interaction of a Screw Dislocation with Interface and Wedge-Shaped Cracks in One-Dimensional Hexagonal Piezoelectric Quasicrystals Bimaterial

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Lian he Li ◽  
Yue Zhao
Keyword(s):  
Screw Dislocation ◽  
Electric Fields ◽  
Electric Displacement ◽  
Image Force ◽  
Coupling Constants ◽  
Geometrical Parameters ◽  
Intensity Factors ◽  
One Dimensional ◽  
Special Cases ◽  
Analytical Expressions

Interaction of a screw dislocation with wedge-shaped cracks in one-dimensional hexagonal piezoelectric quasicrystals bimaterial is considered. The general solutions of the elastic and electric fields are derived by complex variable method. Then the analytical expressions for the phonon stresses, phason stresses, and electric displacements are given. The stresses and electric displacement intensity factors of the cracks are also calculated, as well as the force on dislocation. The effects of the coupling constants, the geometrical parameters of cracks, and the dislocation location on stresses intensity factors and image force are shown graphically. The distribution characteristics with regard to the phonon stresses, phason stresses, and electric displacements are discussed in detail. The solutions of several special cases are obtained as the results of the present conclusion.

scholarly journals Quantile mechanics II: changes of variables in Monte Carlo methods and GPU-optimised normal quantiles

2014 ◽  
Vol 25 (2) ◽  
pp. 177-212 ◽  
Author(s):  
WILLIAM T. SHAW ◽  
THOMAS LUU ◽  
NICK BRICKMAN
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Processing Unit ◽  
Model Risk ◽  
Change Of Variables ◽  
Special Cases ◽  
Wide Range ◽  
Branch Divergence

With financial modelling requiring a better understanding of model risk, it is helpful to be able to vary assumptions about underlying probability distributions in an efficient manner, preferably without the noise induced by resampling distributions managed by Monte Carlo methods. This paper presents differential equations and solution methods for the functions of the form Q(x) = F−1(G(x)), where F and G are cumulative distribution functions. Such functions allow the direct recycling of Monte Carlo samples from one distribution into samples from another. The method may be developed analytically for certain special cases, and illuminate the idea that it is a more precise form of the traditional Cornish–Fisher expansion. In this manner the model risk of distributional risk may be assessed free of the Monte Carlo noise associated with resampling. The method may also be regarded as providing both analytical and numerical bases for doing more precise Cornish–Fisher transformations. Examples are given of equations for converting normal samples to Student t, and converting exponential to normal. In the case of the normal distribution, the change of variables employed allows the sampling to take place to good accuracy based on a single rational approximation over a very wide range of sample space. The avoidance of branching statements is of use in optimal graphics processing unit (GPU) computations as it avoids the effect of branch divergence. We give a branch-free normal quantile that offers performance improvements in a GPU environment while retaining the best precision characteristics of well-known methods. We also offer models with low probability branch divergence. Comparisons of new and existing forms are made on Nvidia GeForce GTX Titan and Tesla C2050 GPUs. We argue that in both single- and double-precisions, the change-of-variables approach offers the most GPU-optimal Gaussian quantile yet, working faster than the Cuda 5.5 built-in function.

An instrument for rapidly measuring plasma distribution functions with high resolution

1982 ◽  
Vol 2 (7) ◽  
pp. 67-70 ◽  
Author(s):  
C.W. Carlson ◽  
D.W. Curtis ◽  
G. Paschmann ◽  
W. Michel
Keyword(s):  
High Resolution ◽  
Distribution Functions ◽  
Plasma Distribution

Small amplitude transverse wave propagation in a weakly Langmuir turbulent plasma

1969 ◽  
Vol 3 (4) ◽  
pp. 593-601
Author(s):  
N. Bel ◽  
J. Heynaerts
Keyword(s):  
High Frequency ◽  
Small Amplitude ◽  
Cold Plasma ◽  
Plasma Frequency ◽  
Turbulent Plasma ◽  
Distribution Functions ◽  
Isotropic Plasma ◽  
Particle Correlation ◽  
Special Cases ◽  
Corrective Term

The high frequency conductivity tensor of an isotropic plasma is derived taking into account particle correlations at the lowest consistent order in the parameter ωp/ω these correlations describe a weakly Langmuir turbulent plasma. Two special cases are investigated in which the two-particle correlation function is related to the turbulent electrostatic field spectrum. Particular distribution functions and spectra are considered and approximate dispersion relations are derived in both cases in ‘the cold plasma limit’. The importance of the corrective term is discussed in terms of three dimensionless parameters measuring the strength of the turbulence, the shape of the spectrum, and the frequency. The effect could be important for frequencies not too far from the plasma frequency.

Relativistic formulation for non-linear waves in a non-uniform plasma

1968 ◽  
Vol 2 (2) ◽  
pp. 257-281 ◽  
Author(s):  
D. S. Butler ◽  
R. J. Gribben
Keyword(s):  
Mathematical Formulation ◽  
Collisionless Plasma ◽  
Debye Length ◽  
Type Equation ◽  
Distribution Functions ◽  
Small Scale ◽  
Relativistic Limit ◽  
Background Distribution ◽  
Non Linear ◽  
Special Case

The mathematical formulation for the problem of non-linear oscillations in a self-consistent, non-uniform, collisionless plasma is considered. The fully nonlinear treatment illuminates the effect of the wave on the background distribution of the plasma through which it is passing. It is assumed that, although the overall non-uniformity may be large, significant changes occur only over time or length scales which are large compared with the plasma period or Debye length respectively. Exclusion of secular terms from the solution leads to a Liouvile type equation, which must be satisfied by the background distribution, and to propagation laws for the waves.The theory is restricted to almost one-dimensional electrostatic waves and a general presentation is given from a relativistically-invariant point of view. Then the equations are derived in terms of physical variables for the special case in which: (i) the distribution functions and electrostatic potential depend on one space co-ordinate (that of propagation of the wave) and the former on the corresponding particle velocity component only, (ii) the wave is slowly-varying only with respect to this co-ordinate and time, and (iii) the magnetic field is zero. Finally, the non-relativistic limit of this case is considered in more detail. The boundary conditions satisfied by the distribution functions are discussed and this leads to the conclusion that in some circumstances thin sheets of probability fluid are formed in phase space and the background distribution cannot be strictly defined. This motivates a reformulation and subsequent re-solution of the problem (for this non-relativistic special case) in terms of weak functions, corresponding to the physical assumption of the presence of a small-scale mixing mechanism, which is excited by and smears the sheeted distribution but is otherwise dormant.The results of the investigation are given as a system of differentio-integral equations which must be solved if necessary conditions for the absence of nonsecular solutions (of the Vlasov and Maxwell equations) are to be satisfied. No solution of this system is attempted here.

On the scattering of water waves by a submerged slender barrier

1998 ◽  
Vol 40 (2) ◽  
pp. 171-189
Author(s):  
P. K. Kundu ◽  
N. K. Saha
Keyword(s):  
Finite Length ◽  
Water Waves ◽  
Vertical Plate ◽  
Perturbation Technique ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
First Order ◽  
Transmission Coefficients ◽  
Special Cases ◽  
Analytical Expressions

AbstractAn approximate analysis, based on the standard perturbation technique, is described in this paper to find the corrections, up to first order to the reflection and transmission coefficients for the scattering of water waves by a submerged slender barrier, of finite length, in deep water. Analytical expressions for these corrections for a submerged nearly vertical plate as well as for a submerged vertically symmetric slender barrier of finite length are also deduced, as special cases, and identified with the known results. It is verified, analytically, that there is no first order correction to the transmitted wave at any frequency for a submerged nearly vertical plate. Computations for the reflection and transmission coefficients up to O(ε), where ε is a small dimensionless quantity, are also performed and presented in the form of both graphs and tables.

scholarly journals Maximum-Entropy Based Estimates of Stress and Strain in Thermoelastic Random Heterogeneous Materials

2020 ◽  
Vol 141 (2) ◽  
pp. 321-348
Author(s):  
Maximilian Krause ◽  
Thomas Böhlke
Keyword(s):  
Maximum Entropy ◽  
Heterogeneous Materials ◽  
Entropy Method ◽  
Distribution Functions ◽  
Internal Stresses ◽  
Field Methods ◽  
Common Procedure ◽  
Full Field ◽  
Mean Values ◽  
Special Cases

Abstract Mean-field methods are a common procedure for characterizing random heterogeneous materials. However, they typically provide only mean stresses and strains, which do not always allow predictions of failure in the phases since exact localization of these stresses and strains requires exact microscopic knowledge of the microstructures involved, which is generally not available. In this work, the maximum entropy method pioneered by Kreher and Pompe (Internal Stresses in Heterogeneous Solids, Physical Research, vol. 9, 1989) is used for estimating one-point probability distributions of local stresses and strains for various classes of materials without requiring microstructural information beyond the volume fractions. This approach yields analytical formulae for mean values and variances of stresses or strains of general heterogeneous linear thermoelastic materials as well as various special cases of this material class. Of these, the formulae for discrete-phase materials and the formulae for polycrystals in terms of their orientation distribution functions are novel. To illustrate the theory, a parametric study based on Al-Al2O3 composites is performed. Polycrystalline copper is considered as an additional example. Through comparison with full-field simulations, the method is found to be particularly suited for polycrystals and materials with elastic contrasts of up to 5. We see that, for increasing contrast, the dependence of our estimates on the particular microstructures is increasing, as well.

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