Automated Boltzmann collision integrals for moment equations

Starting from Boltzmann�like equations, moment equations for a general gas mixture are developed. The equations are closed, and the collision integrals evaluated, by using Grad's 13�moment approximations for the velocity distribution functions. The collision integrals are determined for all possible types of binary encounters, which include recombination and attachment, spontaneous fission and natural decay, charge exchange and elastic collisions, and excitation and fission� and fusion�like processes. The derivation of transport relationships is considered, while the extension of the results to include such direct radiative phenomena as absorption, photoionization, stimulated emission, and Compton scattering is also briefly discussed.

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Grad–Fokker–Planck plasma equations. Part 1. Coffision moments

Journal of Plasma Physics ◽

10.1017/s0022377800011004 ◽

1982 ◽

Vol 27 (3) ◽

pp. 437-452 ◽

Cited By ~ 2

Author(s):

K. A. Broughan

Keyword(s):

Boltzmann Equation ◽

Planck Equation ◽

Computer Language ◽

Collision Term ◽

Moment Equations ◽

Species Pair ◽

Collision Integrals ◽

Fokker Planck Equation ◽

Hot Plasma ◽

Fokker Planck

Thirteen moments are taken of the collision term in the Boltzmann–Fokker– Planck equation for a multi-species, multi-temperature, hot plasma, following the method first developed by Grad for neutral gases. The collision integrals are evaluated for each colliding species pair. These integrals give, in particular, the rate of exchange of momentum and energy produced by collisions. The set of integrals may be combined with moments of the remaining terms in the Boltzmann equation to give thirteen moment equations for each species of particle. To complete the calculation, extensive use was made of the symbolic computer language REDUCE.

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Fixed-End Moment Equations for Continuous Prestressed Concrete Beams

PCI Journal ◽

10.15554/pcij.02011966.75.94 ◽

1966 ◽

Vol 11 (1) ◽

pp. 75-94

Author(s):

Duryl M. Bailey ◽

Phil M. Ferguson

Keyword(s):

Prestressed Concrete ◽

Concrete Beams ◽

Moment Equations ◽

Fixed End

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Kinetic Moment Equations for a Gas of Polyatomic Molecules with Many Internal Degrees of Freedom

The Physics of Fluids ◽

10.1063/1.1693100 ◽

1970 ◽

Vol 13 (6) ◽

pp. 1446 ◽

Cited By ~ 7

Author(s):

Francis J. McCormack

Keyword(s):

Degrees Of Freedom ◽

Polyatomic Molecules ◽

Moment Equations ◽

Internal Degrees Of Freedom ◽

Kinetic Moment

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Hyperbolic moment equations using quadrature-based projection methods

10.1063/1.4902651 ◽

2014 ◽

Cited By ~ 9

Author(s):

J. Koellermeier ◽

M. Torrilhon

Keyword(s):

Projection Methods ◽

Moment Equations ◽

Hyperbolic Moment Equations

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Stochasticity and heterogeneity in host–vector models

Journal of The Royal Society Interface ◽

10.1098/rsif.2007.1064 ◽

2007 ◽

Vol 4 (16) ◽

pp. 851-863 ◽

Cited By ~ 57

Author(s):

Alun L Lloyd ◽

Ji Zhang ◽

A.Morgan Root

Keyword(s):

Branching Process ◽

Moment Equations ◽

Demographic Stochasticity ◽

The Face ◽

Process Methodology ◽

The Impact ◽

Malaria Model ◽

Heterogeneous Transmission ◽

Insight Into ◽

Stochastic Extinction

Demographic stochasticity and heterogeneity in transmission of infection can affect the dynamics of host–vector disease systems in important ways. We discuss the use of analytic techniques to assess the impact of demographic stochasticity in both well-mixed and heterogeneous settings. Disease invasion probabilities can be calculated using branching process methodology. We review the use of this theory for host–vector infections and examine its use in the face of heterogeneous transmission. Situations in which there is a marked asymmetry in transmission between host and vector are seen to be of particular interest. For endemic infections, stochasticity leads to variation in prevalence about the endemic level. If these fluctuations are large enough, disease extinction can occur via endemic fade-out. We develop moment equations that quantify the impact of stochasticity, providing insight into the likelihood of stochastic extinction. We frame our discussion in terms of the simple Ross malaria model, but discuss extensions to more realistic host–vector models.

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