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.

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 Worldline master formulas for the dressed electron propagator. Part 2. On-shell amplitudes

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
N. Ahmadiniaz ◽  
V. M. Banda Guzmán ◽  
F. Bastianelli ◽  
O. Corradini ◽  
J. P. Edwards ◽  
...  
Keyword(s):  
Cross Sections ◽  
Drop Out ◽  
Matrix Elements ◽  
Fermion Propagator ◽  
Particle Path ◽  
Path Integral Representation ◽  
Fermion Line ◽  
Electron Propagator ◽  
The Matrix ◽  
Standard Linear

Abstract In the first part of this series, we employed the second-order formalism and the “symbol” map to construct a particle path-integral representation of the electron propagator in a background electromagnetic field, suitable for open fermion-line calculations. Its main advantages are the avoidance of long products of Dirac matrices, and its ability to unify whole sets of Feynman diagrams related by permutation of photon legs along the fermion lines. We obtained a Bern-Kosower type master formula for the fermion propagator, dressed with N photons, in terms of the “N-photon kernel,” where this kernel appears also in “subleading” terms involving only N − 1 of the N photons.In this sequel, we focus on the application of the formalism to the calculation of on-shell amplitudes and cross sections. Universal formulas are obtained for the fully polarised matrix elements of the fermion propagator dressed with an arbitrary number of photons, as well as for the corresponding spin-averaged cross sections. A major simplification of the on-shell case is that the subleading terms drop out, but we also pinpoint other, less obvious simplifications.We use integration by parts to achieve manifest transversality of these amplitudes at the integrand level and exploit this property using the spinor helicity technique. We give a simple proof of the vanishing of the matrix element for “all +” photon helicities in the massless case, and find a novel relation between the scalar and spinor spin-averaged cross sections in the massive case. Testing the formalism on the standard linear Compton scattering process, we find that it reproduces the known results with remarkable efficiency. Further applications and generalisations are pointed out.

Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Yufeng Xu ◽  
Om Agrawal
Keyword(s):  
Finite Difference ◽  
Diffusion Process ◽  
Weight Function ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Weight Functions ◽  
Linear Recurrence ◽  
Scale Function ◽  
Special Cases ◽  
Wide Range

AbstractIn this paper, numerical solutions of Burgers equation defined by using a new Generalized Time-Fractional Derivative (GTFD) are discussed. The numerical scheme uses a finite difference method. The new GTFD is defined using a scale function and a weight function. Many existing fractional derivatives are the special cases of it. A linear recurrence relationship for the numerical solutions of the resulting system of linear equations is found via finite difference approach. Burgers equations with different fractional orders and coefficients are computed which show that this numerical method is simple and effective, and is capable of solving the Burgers equation accurately for a wide range of viscosity values. Furthermore, we study the influence of the scale and the weight functions on the diffusion process of Burgers equation. Numerical simulations illustrate that a scale function can stretch or contract the diffusion on the time domain, while a weight function can change the decay velocity of the diffusion process.

Electromagnetic fields in the presence of cylindrical objects of arbitrary physical properties and cross sections

1970 ◽  
Vol 48 (15) ◽  
pp. 1789-1798 ◽  
Author(s):  
L. Shafai
Keyword(s):  
Physical Properties ◽  
Electromagnetic Fields ◽  
Cross Sections ◽  
Approximate Solutions ◽  
Scattering Matrix ◽  
Two Dimensional ◽  
Matrix Elements ◽  
Approximate Evaluation ◽  
Permittivity And Permeability ◽  
The Matrix

Approximate solutions for two-dimensional problems of electromagnetic fields in the presence of cylindrical objects have been found by approximate evaluation of a scattering matrix. The equations are derived for cylindrical objects of arbitrary physical properties and cross sections and a procedure for evaluation of the matrix elements is discussed. The elements of permittivity and permeability tensors are assumed to be analytic, but otherwise arbitrary functions of the transverse coordinates.

scholarly journals Transfer scattering matrix of non-uniform surface acoustic wave transducers

1987 ◽  
Vol 10 (3) ◽  
pp. 563-581
Author(s):  
N. C. Debnath ◽  
T. Roy
Keyword(s):  
Acoustic Wave ◽  
Equivalent Circuit ◽  
Surface Acoustic Wave ◽  
Scattering Matrix ◽  
Circuit Model ◽  
Complex Frequency ◽  
Matrix Elements ◽  
Special Cases ◽  
The Matrix ◽  
Scattering Matrix Elements

This paper is concerned with a general mathematical theory for finding the admittance matrix of a three-port non-uniform surface acoustic wave (SAW) network characterized bynunequal hybrid sections. The SAW interdigital transducer and its various circuit model representations are presented in some detail. The Transfer scattering matrix of a transducer consisting ofNnon-uniform sections modeled through the hybrid equivalent circuit is discussed. General expression of the scattering matrix elements for aN-section SAW network is included. Based upon hybrid equivalent circuit model of one electrode section, explicit formulas for the scattering and transfer scattering matrices of a SAW transducer are obtained. Expressions of the transfer scattering matrix elements for theN-section crossed-field and in-line model of SAW transducers are also derived as special cases. The matrix elements are computed in terms of complex frequency and thus allow for transient response determinations. It is shown that the general forms presented here for the matrix elements are suitable for the computer aided design of SAW transducers.

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.

Unified approach to the representations of groups SU(2) and SU(1, 1)

1970 ◽  
Vol 48 (10) ◽  
pp. 1272-1282
Author(s):  
Klang-Chuen Young
Keyword(s):  
Generalized Functions ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Unified Approach ◽  
Canonical Bases ◽  
Representations Of Groups ◽  
Special Cases ◽  
The Matrix ◽  
Finite Transformation ◽  
Explicit Expressions

A unified approach to the representations of groups SU(2) and SU(1, 1) is made. The method is based on the observation that SU(2) and SU(1, 1) can be considered as special cases of a group G(a). The representation of G(a) is realized in the space of homogeneous generalized functions. The canonical bases of the unitary irreducible representations are constructed explicitly. The matrix elements for the finite transformation are found. Explicit expressions for the Wigner coefficients are also obtained.

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