@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 {2203,
	title = {Sound dispersion in single-component systems},
	journal = {Physica a-Statistical Mechanics and Its Applications},
	volume = {387},
	number = {16-17},
	year = {2008},
	note = {ISI Document Delivery No.: 312TETimes Cited: 0Cited Reference Count: 46Napier, Duncan G. Shizgal, Bernie D.},
	month = {Jul},
	pages = {4099-4118},
	type = {Article},
	abstract = {The present paper considers the theoretical description of the propagation of sound waves in a one component monatomic gas. The interatomic potential is assumed to vary as the inverse fourth power of the interatomic separation, that is for so-called Maxwell molecules. The eigenvalues and eigenfunctions of the linearized Boltzmann collision operator are known for this model. We emphasize the behaviour of this system in the rarefied, large Knudsen number regime for which the convergence of solutions of the Boltzmann equation can be very slow. We carry out a detailed comparison of the previous formalisms by Wang Chang and Uhlenbeck [C.S. Wang Chang, G.E. Uhlenbeck, The kinetic theory of gases, in: G.E. Uhlenbeck, De Boer, (Eds.), Studies in Statistical Mechanics, vol. 5, Elsevier, New York, 1970, pp. 43-75], Alexeev [B.V. Alexeev, Philos. Trans. R. Soc. A 349 (1994) 357] and Sirovich and Thurber [L. Sirovich, J. K. Thurber, J. Math. Phys. 10 (1969) 239]. The latter exploit a general method of solution of the Boltzmann equation developed by Gross and Jackson. We demonstrate that the Generalized Boltzmann Equation proposed by Alexeev is not appropriate and we show the reasoning for the success of the Sirovich Thurber approach over the Wang Chang and Uhlenbeck calculations. Comparisons are made with experimental data. (c) 2008 Elsevier B.V. All rights reserved.},
	keywords = {BINARY GAS-MIXTURES, Boltzmann equation, BOLTZMANN-EQUATION, KINETIC THEORY, LIGHT-SCATTERING, Maxwell molecules, MODEL, MONATOMIC GASES, SLOW SOUND, sound dispersion, WAVE-PROPAGATION},
	isbn = {0378-4371},
	url = {<Go to ISI>://000256692900007},
	author = {Napier, D. G. and Shizgal, B. D.}
}
