@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.}
}
@article {1559,
	title = {On the use temperature parameterized rate coefficients in the estimation of non-equilibrium reaction rates},
	journal = {Physica a-Statistical Mechanics and Its Applications},
	volume = {365},
	number = {2},
	year = {2006},
	note = {ISI Document Delivery No.: 044QSTimes Cited: 11Cited Reference Count: 58},
	month = {Jun},
	pages = {317-332},
	type = {Article},
	abstract = {The present paper considers a detailed analysis of the nonequilibrium effects for a model reactive system with the Chapman-Eskog (CE) solution of the Boltzmann equation as well as an explicit time dependent solution. The elastic cross sections employed are a hard sphere cross section and the Maxwell molecule cross section. Reactive cross sections which model reactions with and without activation energy are used. A detailed comparison is carried out with these solutions of the Boltzmann equation and the approximation introduced by Cukrowski and coworkers [J. Chem. Phys. 97 (1992) 9086; Chem. Phys. 89 (1992) 159; Physica A 188 (1992) 344; Chem. Phys. Lett. A 297 (1998) 402; Physica A 275 (2000) 134; Chem. Phys. Lett. 341 (2001) 585; Acta Phys. Polonica B 334 (2003) 3607.] based on the temperature of the reactive particles. We show that the Cukrowski approximation has limited applicability for the large class of reactive systems studied in this paper. The explicit time dependent solutions of the Boltzmann equation demonstrate that the CE approach is valid only for very slow reactions for which the corrections to the equilibrium rate coefficient are very small. (c) 2005 Elsevier B.V. All rights reserved.},
	keywords = {BIMOLECULAR CHEMICAL-REACTION, Boltzmann equation, BOLTZMANN-EQUATION, Chapman-Enskog, CHAPMAN-ENSKOG METHOD, CHEMISTRY, DILUTE GAS, LORENTZ GAS, MULTICOMPONENT SYSTEMS, NITROGEN-ATOMS, nonequilibrium, PERTURBATION, REACTIONS, VELOCITY DISTRIBUTION},
	isbn = {0378-4371},
	url = {<Go to ISI>://000237687900006},
	author = {Shizgal, B. D. and Chikhaoui, A.}
}
@article {983,
	title = {Neutral particle collisional relaxation of suprathermal distributions in aeronomy},
	journal = {Planetary and Space Science},
	volume = {52},
	number = {10},
	year = {2004},
	note = {ISI Document Delivery No.: 858UKTimes Cited: 3Cited Reference Count: 46},
	month = {Aug},
	pages = {915-922},
	type = {Article},
	abstract = {There are numerous situations in laboratory experiments and in atmospheric science that are characterized by distributions of energetic neutral species with extended high-energy tails. This paper provides a detailed analysis of the relaxation of such isotropic nonequilibrium distributions of neutral species. We consider a minor constituent, referred to as the test-particle dilutely dispersed in a second component that acts as a heat bath at equilibrium. A hard sphere cross section is assumed for the collisions of the test particles and the heat bath particles, and collisions between test particles are not included in the analysis. We study the approach to equilibrium with a finite difference method of solution of the Boltzmann equation. The solution of the Boltzmann equation is also presented as an expansion in the eigenfunctions of the Boltzmann collision operator. The main objective of the paper is the calculation of the energy-dependent relaxation times for the distribution function. It is anticipated that these relaxation times will not exhibit a strong energy dependence owing to the energy-independent hard sphere cross section. This relaxation behaviour is important in the characterization of nonthermal populations of energetic atoms in aeronomy. The results are compared with a similar analysis for Coulomb collisions described in the preceeding paper. (C) 2004 Elsevier Ltd. All rights reserved.},
	keywords = {BINARY GASES, Boltzmann equation, CHARGE-EXCHANGE, distribution functions, ENERGY-TRANSFER, FAST NITROGEN-ATOMS, FOKKER-PLANCK EQUATION, HYDROGEN, KINETIC-THEORY, MARS, O(P-3) ATOMS, OXYGEN-ATOMS, relaxation times, suprathermal populations},
	isbn = {0032-0633},
	url = {<Go to ISI>://000224217300003},
	author = {Shizgal, B. D.}
}
@article {409,
	title = {Velocity distributions of energetic atoms in planetary exospheres from dissociative recombination},
	journal = {Journal of Geophysical Research-Planets},
	volume = {107},
	number = {E7},
	year = {2002},
	note = {ISI Document Delivery No.: 610UBTimes Cited: 3Cited Reference Count: 47},
	month = {Jul},
	pages = {9},
	type = {Article},
	abstract = {[1] A kinetic theory description of translational energetic atoms in the upper planetary atmosphere is presented. A new analytical result for the velocity distribution of the products of reactive collisions is described. Our calculation takes into account different temperatures of the reactants and arbitrary dependence of the cross section on the relative velocity of the colliding particles. The final result is applied to the production of hot oxygen and carbon by dissociative recombination of O-2(+) and CO+, respectively. The nascent distribution of hot atoms generated in this way is compared with the earlier Monte Carlo calculations. We use the Boltzmann equation to study the thermalization of the hot oxygen via collisions with the thermal oxygen population. The results of this calculation demonstrate quasi-steady state velocity distribution of high-energy oxygen atoms near the exobase of Venus for daytime conditions.},
	keywords = {Boltzmann equation, CROSS-SECTIONS, dissociative recombination, ESCAPE, gas kinetic, hot atoms, HOT OXYGEN-ATOMS, HYDROGEN, ION STORAGE-RING, MARS, MONTE-CARLO, NITROGEN-ATOMS, theory, UPPER-ATMOSPHERE, VENUS},
	isbn = {0148-0227},
	url = {<Go to ISI>://000178978400004},
	author = {Kabin, K. and Shizgal, B. D.}
}
@article {3526,
	title = {CONVERSION BETWEEN KINETIC-ENERGY AND POTENTIAL-ENERGY IN THE CLASSICAL NONLOCAL BOLTZMANN-EQUATION},
	journal = {Journal of Statistical Physics},
	volume = {80},
	number = {5-6},
	year = {1995},
	note = {ISI Document Delivery No.: RV585Times Cited: 6Cited Reference Count: 29},
	month = {Sep},
	pages = {1085-1117},
	type = {Article},
	abstract = {An important property of the classical Boltzmann equation is that kinetic energy is conserved. This is closely connected to the fact that the Boltzmann equation describes the nonequilibrium properties of an {\textquoteright}{\textquoteright}ideal{\textquoteright}{\textquoteright} gas. Generalizations of the Boltzmann equation to higher density involve, among other things, allowing the colliding particles to be at different positions. This spatial nonlocality is known to contribute to the density corrections of gas transport properties. For soft potentials such a spatial separation of the particles also leads to a conversion between kinetic and potential energy. In evaluating these effects the classical dynamics of the whole collision trajectory must be taken into account, involving also the time for the collision process. The resulting time nonlocality has usually been reinterpreted in terms of a spatial nonlocality. However, for a homogeneous system this is not possible and only the time nonlocality remains, this then being responsible for the conversion between kinetic and potential energy. This paper aims to clarify these properties of the nonlocal corrections to the classical mechanical Boltzmann collision term. Comments on the corresponding problem for the quantum Boltzmann equation are also made.},
	keywords = {Boltzmann equation, ENERGY CONSERVATION, GASES},
	isbn = {0022-4715},
	url = {<Go to ISI>://A1995RV58500006},
	author = {Snider, R. F.}
}
