@inbook {2604,
title = {Relaxation Behavior for the Lorentz Fokker Planck Equation: Model Cross Sections and Nonextensive Entropy},
booktitle = {Rarefied Gas Dynamics},
series = {Aip Conference Proceedings},
volume = {1084},
year = {2009},
note = {ISI Document Delivery No.: BJF97Times Cited: 0Cited Reference Count: 19Shizgal, Bernie D.Proceedings Paper26th International Symposium on Rarefied Gas Dynmaics (RGD26)JUN 20-JUL 25, 2008Kyoto, JAPANJapan Soc Promot Sci, Japan Aerpspace Explorat Agcy, Soc Promot Space Sci, Iwantani Naoji Fdn, Inoue Fdn Sci, Casio Sci Promot Fdn, Kaijma Fdn, IHI Corp, IHI Aerospac Engn Co Ltd, Osaka Vaccuun Ltd, Nissin Inc2 HUNTINGTON QUADRANGLE, STE 1NO1, MELVILLE, NY 11747-4501 USA},
pages = {39-44},
publisher = {Amer Inst Physics},
organization = {Amer Inst Physics},
address = {Melville},
abstract = {The relaxation behavior of a test particle in a background gas at equilibrium for which the test particle background particle mass ratio, m(t)/m(b) tends to zero, that is the Lorentz limit, is studied with the Lorentz-Fokker-Planck equation. This is the situation for electrons in inert gases. If there is also an applied electric field the stationary distribution is not the equilibrium Maxwellian. In the present paper, the evolution of the isotropic distribution function is studied with both a finite difference solution and one based on the expansion of the distribution function in the eigenfunctions of the Fokker-Planck operator. The eigenvalue spectrum depends strongly on the velocity dependence of the momentum transfer cross section in the Fokker-Planck equation. In particular, the spectrum is composed of a discrete spectrum and a continuum, and the continuum can dominate the spectrum. The role of the eigenvalue spectrum of this operator on the time evolution of the distribution function is studied. In particular, the time dependence of the entropy is determined and we show that the Kullback-Leibler entropy rationalizes the approach to a stationary distribution. The Tsallis non-extensive entropy is not required to explain the evolution of the distributions.},
keywords = {Boltzmann equation, DEGRADATION, electron, Maxwell molecules, METHOD QDM, NONCLASSICAL BASIS FUNCTIONS, pseudospectral method, QUADRATURE DISCRETIZATION METHOD, quantum cross section, RARE-GAS MODERATORS, SUPRATHERMAL PARTICLE DISTRIBUTIONS, THERMALIZATION},
isbn = {0094-243X978-0-7354-0615-5},
url = {://000265564800005},
author = {Shizgal, B. D.},
editor = {Abe, T.}
}
@article {4757,
title = {A direct spectral collocation Poisson solver in polar and cylindrical coordinates},
journal = {Journal of Computational Physics},
volume = {160},
number = {2},
year = {2000},
note = {ISI Document Delivery No.: 315BRTimes Cited: 22Cited Reference Count: 33},
month = {May},
pages = {453-469},
type = {Article},
abstract = {In this paper, we present a direct spectral collocation method for the solution of the Poisson equation in polar and cylindrical coordinates. The solver is applied to the Poisson equations for several different domains including a part of a disk, an annulus, a unit disk, and a cylinder. Unlike other Poisson solvers for geometries such as unit disks and cylinders, no pole condition is involved for the present solver. The method is easy to implement, fast, and gives spectral accuracy. We also use the weighted interpolation technique and nonclassical collocation points to improve the convergence. (C) 2000 Academic Press.},
keywords = {ARBITRARY ORDER ACCURACY, coordinates, cylindrical, EXPANSION, METHOD QDM, NONCLASSICAL BASIS FUNCTIONS, Poisson solver, polar coordinates, QUADRATURE DISCRETIZATION METHOD, SCHRODINGER-EQUATION, SINGULARITIES, spectral collocation, TSCHEBYSCHEFF POLYNOMIALS},
isbn = {0021-9991},
url = {://000087093200002},
author = {Chen, H. L. and Su, Y. H. and Shizgal, B. D.}
}
@article {4226,
title = {The Quadrature Discretization Method (QDM) in the solution of the Schrodinger equation},
journal = {Journal of Mathematical Chemistry},
volume = {24},
number = {4},
year = {1998},
note = {ISI Document Delivery No.: 165EGTimes Cited: 12Cited Reference Count: 63},
pages = {321-343},
type = {Article},
abstract = {The Quadrature Discretization Method (QDM) is employed in the solution of several one-dimensional Schrodinger equations that have received considerable attention in the literature. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of non-classical polynomials orthogonal with respect to a weight function. For a certain class of problems with potentials of the form that occur in supersymmetric quantum mechanics, the ground state wavefunction is known. In the present paper, the weight functions that are used are related to the ground state wavefunctions if known, or some approximate form. The eigenvalues and eigenfunctions of four different potential functions discussed extensively in the literature are calculated and the results are compared with published values.},
keywords = {ACCURATE EIGENVALUES, ANHARMONIC-OSCILLATORS, DISCRETE-ORDINATE, ENERGY-LEVELS, FOKKER-PLANCK EQUATION, HILL DETERMINANT APPROACH, INNER-PRODUCT TECHNIQUE, method, NONCLASSICAL BASIS FUNCTIONS, OPTICAL WAVE-GUIDES, QUANTUM-MECHANICAL MODELS},
isbn = {0259-9791},
url = {://000078506600002},
author = {Chen, H. L. and Shizgal, B. D.}
}