scholarly journals Techniques for computing the numerical values of the matrix elements of the collision operator in the collisional plasma model

1976 ◽  
Author(s):  
H. K. Meier ◽  
W. I. van Rij ◽  
Jr., C. O. Beasley
Keyword(s):  
Collision Operator ◽  
Collisional Plasma ◽  
Matrix Elements ◽  
Plasma Model ◽  
The Matrix

Matrix Elements of the Linearized Collision Operator for Multi-temperature Gas-mixtures

1978 ◽  
Vol 33 (4) ◽  
pp. 480-492
Author(s):  
Ulrich Weinert
Keyword(s):  
Collision Operator ◽  
Basis Functions ◽  
Inelastic Collisions ◽  
Matrix Elements ◽  
Flux Vector ◽  
Balance Equations ◽  
Heat Flux Vector ◽  
The Matrix ◽  
Boltzmann Collision Operator ◽  
Linearized Collision Operator

For a multi-component and multi-temperature gas-mixture the matrix elements of the linearized Boltzmann collision operator are investigated for isotropic interaction potentials. The representation by means of Burnett basis functions simplifies the algebraic structure and enables closed expressions for the general results, which can also be used for an investigation of inelastic collisions. For the elastic case those collision terms are given explicitely which appear in the balance equations for mass, momentum, energy and heat flux-vector.

scholarly journals Matrix Elements of the Boltzmann Collision Operator in a Basis Determined by an Anisotropic Maxwellian Weight Function including Drift

1980 ◽  
Vol 33 (2) ◽  
pp. 449 ◽  
Author(s):  
Kailash Kumar
Keyword(s):  
Weight Function ◽  
Cross Sections ◽  
Collision Operator ◽  
Weight Functions ◽  
Anisotropic Pressure ◽  
Matrix Elements ◽  
Methods Of Calculation ◽  
Special Cases ◽  
The Matrix ◽  
Boltzmann Collision Operator

The matrix elements of the linear Boltzmann collision operator are calculated in a Burnett-function basis determined by a weight function which itself describes a velocity distribution with a net drift and an anisotropic pressure (or temperature) tensor. Three different methods of calculation are described, leading to three different types of formulae. Two of these involve infinite summations, while the third involves only finite sums, but at the cost of greater complications in the summands and the integrals over cross sections. Both elastic and inelastic collisions are treated. Special cases arising from particular choices of the parameters in the weight functions are pointed out. The structure of the formulae is illustrated by means of diagrams. The work is a contribution towards establishing efficient methods of calculation based upon a better understanding of the matrix elements in such bases.

Kinetic Theory of Molecular Gases I: Models of the Linear Waldmann–Snider Collision Operator

1975 ◽  
Vol 53 (13) ◽  
pp. 1266-1278 ◽  
Author(s):  
G. Tenti ◽  
Rashmi C. Desai
Keyword(s):  
Kinetic Theory ◽  
Boundary Value Problems ◽  
Transport Properties ◽  
Boundary Value ◽  
Kinetic Equations ◽  
Collision Operator ◽  
Matrix Elements ◽  
Molecular Gases ◽  
The Matrix ◽  
Modeling Theory

Using a method closely akin to the Gross–Jackson–Sirovich procedure, we present a modeling theory of the linear Waldmann–Snider collision operator. The resulting model kinetic equations are applicable to all regions of wavelength and frequency consistent with the original equation itself. The theory is made parameter free by relating the matrix elements of the collision operator to measured transport properties. It is sophisticated enough to afford a study of both scalar and tensorial phenomena and can be applied to the analysis of a variety of initial and boundary value problems.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

Notizen: The Dipole Moment Function of HF Molecule Using Morse Potential

1977 ◽  
Vol 32 (8) ◽  
pp. 897-898 ◽  
Author(s):  
Y. K. Chan ◽  
B. S. Rao
Keyword(s):  
Wave Equation ◽  
Dipole Moment ◽  
Potential Function ◽  
Electric Dipole ◽  
Morse Potential ◽  
Moment Function ◽  
Matrix Elements ◽  
The Matrix ◽  
Vibration Rotation ◽  
Experimental Values

Abstract The radial Schrödinger wave equation with Morse potential function is solved for HF molecule. The resulting vibration-rotation eigenfunctions are then used to compute the matrix elements of (r - re)n. These are combined with the experimental values of the electric dipole matrix elements to calculate the dipole moment coefficients, M 1 and M 2.

Leach Behavior of SRL TDS-131 Defense Waste Glass in Water at High&Low Flow Rates

1983 ◽  
Vol 26 ◽  
Author(s):  
Aaron Barkatt ◽  
William Sousanpour ◽  
Alisa Barkatt ◽  
Morad A. Boroomand ◽  
Pedro B. Macedo
Keyword(s):  
Contact Time ◽  
Protective Layer ◽  
High Flow ◽  
Low Flow ◽  
Matrix Elements ◽  
Flow Rates ◽  
Low Concentrations ◽  
Controlling Mechanism ◽  
The Matrix ◽  
Leach Behavior

ABSTRACTLeach tests carried out on SRL TDS-131 Defense Waste Class indicate that at high flow rates the controlling mechanism is simple corrosion. The matrix elements (Si, Al) are leached out at rates similar to those of the leaching of the alkalis and of boron, and the leaching process is nearly linear with time. At slow flow rates (below 1 m/yr) leaching becomes controlled by the build-up of a protective layer. Al and most of the Si remain in the leached surface layer. The leach rates decrease in the course of the test before leveling off at constant values which are almost inversely proportional to the contact time, indicating that leachate concentrations have become solubility-limited. The low concentrations observed at this stage indicate the formation of alteration products.

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