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Direct Simulation Scheme Derived from the Boltzmann Equation. I. Monocomponent Gases
Abstract
The stochastic law that prescribes the velocities of simulated molecules after a small time increment was derived from the Boltzmann equation. The scheme to determine the velocity of a molecule after a small time increment is divided into three steps. The first step gives the collision probability of the molecule without specifying its collision partner. The second step gives a conditional probability distribution. If the molecule is accepted in the first step as a colliding molecule, its collision partner is sampled from this probability distribution. The last step gives a probability density from which the direction of the relative velocity after collision is sampled, and hence the step gives the post-collision velocity of the molecule. It is shown that the use of the present simulation scheme gives an exact solution of the Boltzmann equation.
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“Nevertheless, numerical tricks (e.g. enlarged vacuum permittivity and/or reduction of ion mass or thruster size scaling) are still used in many simulations to reduce the computational time (see Refs. 59-64 for E × B thrusters, and Refs. 65-67 for PATs).”
From Plasma propulsion modeling with particle-based algorithms (2023) · cites this paper as [65]
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Cites
The mathematical theory of non-uniform gases
York, N. (1953) · Journal of the Franklin Institute · doi:10.1016/0016-0032(53)90422-0 · Cited by 7,699
Introduction to Physical Gas Dynamics
Vincenti, W. G., Kruger, C. H., & Teichmann, T. (1966) · Physics Today · doi:10.1063/1.3047788 · Cited by 1,744
Bird, G. A. (1976) · Medical Entomology and Zoology · Cited by 1,267
Exact solutions of the Boltzmann equation
Krook, M., & Wu, T. T. (1977) · The Physics of Fluids · doi:10.1063/1.861780 · Cited by 214
Exact Solutions of the Boltzmann Equation
Cornille, H. (1990) · Springer eBooks · doi:10.1007/978-94-009-0591-7_2 · Cited by 176
D.P. (1976) · Journal of Molecular Structure · doi:10.1016/0022-2860(76)80184-6 · Cited by 62
Flagged first, then by citations
Research works printed but not linked: 3
- G. A.Bird: Annual Review of Fluid Mechanics, ed. M.van Dyke, J. V.Wehausen and J. L.Lumley (Annual Reviews, Palo Alto, California, 1978) Vol. 10, p. 11.
- K.Koura and J.Kondo: Rarefied Gas Dynamics, ed. L.Trilling and H. Y.Wachman (Academic, New York, 1969) Vol. 1, p. 181.
- Belotserkovskiy O. M., 1978, Fluid Mech. Sov. Res., 7, 42
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Degond, P., Mustieles, F. J., & Niclot, B. (1989) · Vieweg+Teubner eBooks · doi:10.1007/978-3-322-87869-4_9 · Cited by 1
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Metadata
- Authors
- Nanbu, K.
- Affiliations
- Tohoku University (JP)
- Corresponding author
- Nanbu, Kenichi
- Venue
- Journal of the Physical Society of Japan
- Publisher
- Physical Society of Japan
- Published
- 1980-11-15
- Volume
- 49
- Issue
- 5
- Pages
- 2042-2049
- ISSN
- 0031-9015 (ISSN-L)
- 1347-4073
- OpenAlex
- W1981163355
- Cited by
- 397
- References
- 6
- Citations by year
- 2026: 7 · 2025: 15 · 2024: 9 · 2023: 17 · 2022: 22 · 2021: 12 · 2020: 10 · 2019: 10 · 2018: 4 · 2017: 13 · 2016: 23 · 2015: 8 · 2014: 11 · 2013: 10 · 2012: 8
- FWCI
- 11.65
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of the Physical Society of Japan (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Gas Dynamics and Kinetic Theory
- Physical Sciences › Chemistry › Physical and Theoretical Chemistry › Advanced Physical and Chemical Molecular Interactions
- Physical Sciences › Engineering › Computational Mechanics › Lattice Boltzmann Simulation Studies
- Keywords
- direct simulation, Boltzmann equation, collision probability, conditional distributions, rarefied gas dynamics
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