@inbook {2381,
	title = {Pseudospectral Solution of the Boltzmann Equation: Quantum Cross Sections},
	booktitle = {Rarefied Gas Dynamics},
	series = {Aip Conference Proceedings},
	volume = {1084},
	year = {2009},
	note = {ISI Document Delivery No.: BJF97Times Cited: 0Cited Reference Count: 18Chang, Yongbin Shizgal, 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 = {421-426},
	publisher = {Amer Inst Physics},
	organization = {Amer Inst Physics},
	address = {Melville},
	abstract = {The relaxation of it nonequilibrium test particle population of mass m(1) in contact with a background gas of particles of mass m(2) at temperature T-b is studied with the spatially homogeneous Boltzmann equation. A pseudospectral method of solution is employed which is based on the discretization of the distribution function at a nonuniform grid that coincides with a quadrature. In order to implement this approach, the integral collision operator of the Boltzmann equation is expressed explicitly in terms of a kernel which is also discretized. This procedure fails if the classical differential collision cross section is used as it diverges for small scattering angles and the kernel is no longer well defined. In the present paper, we choose the interaction potential between the test particles and the background particles to be the Maxwell molecule interaction, that is, V (r) = V-o(d/r)(4) for which the eigenvalues and eigenfunctions are well known. We employ the quantum mechanical differential cross section for this potential, which is finite at zero scattering angle and we apply a pseudospectral approach based on speed polynomials. We compare the well known results for the eigenvalue spectrum of the Boltzmann collision operator for the classical differential cross section with the new results with the quantum differential cross section and the basis set defined by the speed polynomials. The relaxation to equilibrium of initial nonequilibrium distribution functions is studied with both methods.},
	keywords = {Boltzmann equation, DISTRIBUTIONS, DYNAMICS, GASES, Maxwell molecules, pseudospectral method, quantum cross section, relaxation, VELOCITY DISTRIBUTION},
	isbn = {0094-243X978-0-7354-0615-5},
	url = {<Go to ISI>://000265564800068},
	author = {Chang, Y. B. and Shizgal, B. D.},
	editor = {Abe, T.}
}
@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 = {<Go to ISI>://000265564800005},
	author = {Shizgal, B. D.},
	editor = {Abe, T.}
}
@article {1512,
	title = {A pseudospectral method of solution of Fisher{\textquoteright}s equation},
	journal = {Journal of Computational and Applied Mathematics},
	volume = {193},
	number = {1},
	year = {2006},
	note = {ISI Document Delivery No.: 046PTTimes Cited: 9Cited Reference Count: 59},
	month = {Aug},
	pages = {219-242},
	type = {Article},
	abstract = {In this paper, we develop an accurate and efficient pseudospectral solution of Fisher{\textquoteright}s equation, a prototypical reaction-diffusion equation. The solutions of Fisher{\textquoteright}s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient. There is an ongoing effort to better adapt pseudospectral methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations. The collocation method employed is based on Chebyshev-Gauss-Lobatto quadrature points. We compare results for a single domain as well as for a subdivision of the main domain into subintervals. Instabilities that occur in the. numerical solution for a single domain, analogous to those found by others, are attributed to round-off errors arising from numerical features of the discrete second derivative matrix operator. However, accurate stable solutions of Fisher{\textquoteright}s equation are obtained with a multidomain pseudospectral method. A detailed comparison of the present approach with the use of the sinc interpolation is also carried out. (c) 2005 Elsevier B.V. All rights reserved.},
	keywords = {ACCURACY, COLLOCATION METHOD, DISCRETE SINGULAR CONVOLUTION, error, Fisher equation, NUMERICAL-SOLUTION, PARABOLIC EQUATIONS, PARTIAL-DIFFERENTIAL-EQUATIONS, POINTS, pseudospectral method, REACTION-DIFFUSION-EQUATIONS, round-off, sinc interpolation, TRAVELING-WAVE SOLUTIONS},
	isbn = {0377-0427},
	url = {<Go to ISI>://000237824200015},
	author = {Olmos, D. and Shizgal, B. D.}
}
@article {6979,
	title = {SIMPLIFIED GREEN-FUNCTION APPROXIMATIONS - FURTHER ASSESSMENT OF A POLARIZATION MODEL FOR 2ND-ORDER CALCULATION OF OUTER-VALENCE IONIZATION-POTENTIALS IN MOLECULES},
	journal = {Physical Review A},
	volume = {44},
	number = {9},
	year = {1991},
	note = {ISI Document Delivery No.: GP388Times Cited: 5Cited Reference Count: 81},
	month = {Nov},
	pages = {5773-5783},
	type = {Article},
	abstract = {Ab initio methods for calculating the binding-energy spectra of large molecules have traditionally been restricted to primarily either Koopmans{\textquoteright}s theorem or the density-functional transition-orbital method. The limitations of the former are well known, and the density-functonal "band-gap problem" has led to a further realization of intrinsic difficulties with the latter. An increasingly popular alternative to these two methods is to seek a simple approximation for the Green-function self-energy. The Green-function self-energy is the optical potential seen by a scattering particle (hole). As such, the dominant many-body effects contributing to the self-energy result from polarization of the charge density at energies below the first excitation energy of the target molecule (quasiparticle regime), as well as excitations of the target at higher energies. The physical importance of polarization effects is apparent in Hedin{\textquoteright}s GW approximation, which treats the self-energy as a product of the Green function (G) and a screened interaction (W) that can be calculated (essentially) from the time-dependent linear response of the charge density. In the present paper, we examine the contribution of polarization to the usual second-order Green-function (GF2) approximation with respect to the calculation of outer-valence ionization potentials in small molecules. A simplified version (GW2) of the GW approximation is found to be an acceptable substitute for the GF2 approximation, provided a self-interaction correction is included to prevent an electron from polarizing itself. Polarization effects are further analyzed using the Coulomb-hole and screened-exchange (COHSEX) and modified-COHESEX (M-COHSEX) approximations. A second-order version (M-COHSEX2) of the M-COHSEX approximation is used to examine the origin of the incorrect ordering by Koopmans{\textquoteright}s theorem of the first three ionization potentials of the nitrogen molecule in terms of static polarization and retardation effects. Finite-basis-set errors are also explored. Although higher-order Green-function approximations must be examined before drawing final conclusions, we believe that the present work provides preliminary evidence that suitably modified versions of time-dependent density-functional, dielectric-function-based self-energy approximation can be useful for molecules.},
	keywords = {BAND-GAPS, COULOMB-HOLE, EXCHANGE-CORRELATION POTENTIALS, HARTREE-FOCK, KOOPMANS THEOREM, MANY-BODY THEORY, PI-ELECTRON, pseudospectral method, SCREENED-EXCHANGE, SELF-ENERGY OPERATORS, SYSTEMS},
	isbn = {1050-2947},
	url = {<Go to ISI>://A1991GP38800053},
	author = {Casida, M. E. and Chong, D. P.}
}
