@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 {4316,
title = {The coupling of electron thermalization and electron attachment in CCl4/Ar and CCl4/Ne mixtures},
journal = {Journal of Chemical Physics},
volume = {108},
number = {4},
year = {1998},
note = {ISI Document Delivery No.: YR488Times Cited: 11Cited Reference Count: 61},
month = {Jan},
pages = {1587-1600},
type = {Article},
abstract = {The relaxation of a nonequilibrium distribution of electrons in a mixture of CCl4 with either Ar or Ne is studied. In this paper, electron-CCl4 and electron-inert gas elastic collisions, vibrationally inelastic collisions between electrons and CCl4, as well as the electron attachment reaction with CCl4, are included in the analysis. The time dependent electron energy distribution function is determined from the Boltzmann equation and the energy relaxation times are determined. The coupling of the thermalization process and the attachment process are discussed in detail. The results from the calculations are analyzed analogous to experimental studies, and the methodology of the experimental reduction of the data is studied. (C) 1998 American Institute of Physics.},
keywords = {ARGON, CONDUCTIVITY, CROSS-SECTIONS, DEGRADATION, development, FIELD DEPENDENCE, MONTE-CARLO SIMULATION, RARE-GAS MODERATORS, SF6, SWARM, TRANSPORT},
isbn = {0021-9606},
url = {://000071500300035},
author = {Kowari, K. and Leung, K. and Shizgal, B. D.}
}
@article {4328,
title = {The Quadrature Discretization Method (QDM) in comparison with other numerical methods of solution of the Fokker-Planck equation for electron thermalization},
journal = {Journal of Mathematical Chemistry},
volume = {24},
number = {4},
year = {1998},
note = {ISI Document Delivery No.: 165EGTimes Cited: 7Cited Reference Count: 67},
pages = {291-319},
type = {Article},
abstract = {The determination of the relaxation of electrons in atomic gases continues to be an important physical problem. The main interest is the determination of the time scale for the thermalization of electrons in different moderators and the nature of the time-dependent electron energy distribution. The theoretical basis for the study of electron thermalization is the determination of the electron distribution function from a solution of the Lorentz-Fokker- Planck equation. The present paper considers a detailed comparison of different numerical methods of solution of the Lorentz-Fokker- Planck equation for the electron distribution function. The methods include a pseudospectral method referred to as the Quadrature Discretization Method (QDM) which is based on non-standard polynomial basis sets, a finite-difference method, and a Lagrange interpolation method. The Fokker-Planck equation can be transformed to a Schrodinger equation, and methods developed for the solution of either equation apply to the other.},
keywords = {ACCELERATION, BOLTZMANN-EQUATION, DISCRETE-ORDINATE, ENERGY-DISTRIBUTION, FIELD DEPENDENCE, method, MULTITERM CALCULATIONS, QUANTUM-MECHANICS, RARE-GAS MODERATORS, STOCHASTIC, TRANSPORT-COEFFICIENTS, VELOCITY DISTRIBUTION FUNCTION},
isbn = {0259-9791},
url = {://000078506600001},
author = {Leung, K. and Shizgal, B. D. and Chen, H. L.}
}
@article {7109,
title = {COMPARISON OF WKB (WENTZEL-KRAMERS-BRILLOUIN) AND SWKB SOLUTIONS OF FOKKER-PLANCK EQUATIONS WITH EXACT RESULTS - APPLICATION TO ELECTRON THERMALIZATION},
journal = {Canadian Journal of Physics},
volume = {69},
number = {6},
year = {1991},
note = {ISI Document Delivery No.: FZ294Times Cited: 5Cited Reference Count: 41},
month = {Jun},
pages = {712-719},
type = {Article},
abstract = {A comparison of WKB (Wentzel-Kramers-Brillouin) and SWKB eigenfunctions of the Schrodinger equation for potentials in the class encountered in supersymmetric quantum mechanics is presented. The potentials that are studied are those that result from the transformation of a Fokker-Planck eigenvalue problem to a Schrodinger equation. Linear Fokker-Planck equations of the type considered in this paper give the probability distribution function for a large number of physical situations. The time-dependent solutions can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker-Planck operator. The specific Fokker-Planck operator considered is the one that describes the thermalization of electrons in the inert gases. The WKB and SWKB semiclassical approximations are compared with exact numerical results. Although the eigenvalues can be very close to the exact values, we find significant departures for the eigenfunctions.},
keywords = {ACTIVATED, APPROXIMATION, DISCRETE-ORDINATE METHOD, EIGENVALUES, INVARIANCE, ISOMERIZATION, NUCLEATION, RARE-GAS MODERATORS, RATE-PROCESSES, SHAPE, SOLVABLE POTENTIALS, SUPERSYMMETRIC QUANTUM-MECHANICS},
isbn = {0008-4204},
url = {://A1991FZ29400011},
author = {Shizgal, B. and Demeio, L.}
}
@article {7059,
title = {THE PARTIAL CONTINUOUS SLOWING DOWN APPROXIMATION (PCSDA) IN THE SOLUTION OF THE SPENCER-FANO EQUATION},
journal = {Chemical Physics Letters},
volume = {176},
number = {6},
year = {1991},
note = {ISI Document Delivery No.: EX468Times Cited: 1Cited Reference Count: 24},
month = {Feb},
pages = {575-580},
type = {Article},
abstract = {The Spencer-Fano equation (SFE) is often used to study the energy distribution of energetic electrons and their moderation in gases. Although accurate results can be obtained from a numerical solution of the SFE, the calculations are time-consuming and instead the continuous slowing down approximation (CSDA), applied to all the collision terms in the SFE, is often used as an alternate more efficient methodology. The work in the present paper considers a modification of the usual CSDA which involves the application of the CSDA to specific collision terms in the SFE rather than to all the terms. This new approximation scheme referred to as the partial continuous slowing down approximation (PCSDA) is applied to the degradation of electrons in N2. The new method is found to give better results than the conventional CSDA and provides a more efficient determination of the degradation spectrum than does the solution of the SFE.},
keywords = {100 KEV ELECTRONS, ELECTRON DEGRADATION SPECTRA, EXCITATION, G-VALUES, INITIAL PRODUCTS, IONIZATION, MOLECULAR-HYDROGEN, RARE-GAS MODERATORS, SUBEXCITATION ELECTRONS, YIELDS},
isbn = {0009-2614},
url = {://A1991EX46800015},
author = {Kowari, K. and Shizgal, B.}
}
@article {7058,
title = {TIME-DEPENDENT ELECTRON-DISTRIBUTION FUNCTIONS AND DEGRADATION SPECTRA - A COMPARISON OF THE SPENCER-FANO EQUATION AND THE BOLTZMANN-EQUATION},
journal = {Applied Radiation and Isotopes},
volume = {42},
number = {10},
year = {1991},
note = {ISI Document Delivery No.: GL858Times Cited: 8Cited Reference Count: 43SYMP ON THE PHYSICS OF ELECTRON TRANSPORTAPR 02-03, 1990GAITHERSBURG, MDUS DOE, OFF HLTH \& ENVIRONM RES},
pages = {985-994},
type = {Proceedings Paper},
abstract = {The thermalization of electrons in gaseous moderators for initial nonequilibrium distributions not far removed from thermal energies has been studied previously with a Fokker-Planck equation for electron distribution. For electron energies in the keV to eV energy range an approach based on the Spencer-Fano equation has also been used. The present paper considers a detailed comparison of the two approaches to electron thermalization. The Spencer-Fano equation is analogous to the Fokker-Planck equation but with the moderator temperature set to zero, that is, with the cold gas approximation. The effect of the cold gas approximation on the time evolution of the nonequilibrium distributions and on the calculation of thermalization times is discussed in detail. The cold gas approximation is shown to be the basis for most of the concepts used in radiation chemistry and physics to study electron degradation. Applications to the thermalization of electrons in the inert gases are considered. The analysis is restricted to elastic electron-moderator collisions.},
keywords = {ARGON, HELIUM, HOT-ATOM REACTIONS, INITIAL PRODUCTS, NITROGEN, RARE-GAS MODERATORS, SUBEXCITATION ELECTRONS, THERMALIZATION, TRANSPORT, YIELDS},
isbn = {0969-8043},
url = {://A1991GL85800010},
author = {Kowari, K. and Shizgal, B.}
}