Asymptotic approximations to a linear transport equation

References

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• Throughout this paper, the symbol O (λn) in relations such as A = B + O (λn) signifies that the norm of A − B is of the order λn as λ → 0
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• Projection 𝒫 defined by (5.1) projects functions of variable s into the one-dimensional subspace of functions that are independent of s. The dimension of the subspace 𝒫X always equals the number of dependent variables we need to approximate the solution of the corresponding transport equation in the strong scattering limit; e.g., in the case of electromagnetic fields there are six of them, 1 in the case of the retarded Lorentz-gauge potentials or of a Dirac particle there are four of them,1 and in the case of a PN theory there are (N +I)2 of them.16,17,24
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• Since the approximations we derived are accurate only to the order of some power of the scattering scaling parameter λ, there is an infinite variety of similar approximations of the same order; a fact pointed out and used already by Pomraning in8, Sect II.
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• For examples of such computations see, e.5,7,8,14,16,17
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• This kind of approach has great flexibility and was used in effect by Larsen, McGhee and Morel to derive approximations to the transport equation beyond the usual P1 approximation through an expansion of the Peierls integral equation.3,4 Their key relations, (6) of 3 and (17) of 4, are special cases of (4.5) with n = 5; whereas the equations (18), (20) and (24) of 4, which they use to derive the simplified P1. P2 and P3 equations, are special cases of (4.25) with n = 1, 3, 5
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• In the case of one-speed linear transport of a vector1 or spinor field1 through an isotropic medium, we can apply an analogous procedure also to computing the corresponding relation (4.5), the key to approximations considered in Section IV
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• See, e.g.7,8 and the literature therein
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• Cf. Pomraning8, Eq. (63)
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• Pomraning's 8 Eq.(70) and diffusion equation (6.12) do not agree since they were derived using different scalings
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• Edmonds , A. R. 1974 . Angular momentum in quantum mechanics , 2nd edition Princeton, N. J. See, e.g. Eqs. (4.6.6), (2.5.4), (2.5.5), (2.5.14) and (4.6.5)
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• Williams , M. M. R. 1971 . Mathematical Methods in Particle Transport Theory , London : Butterworths . Sect. 11.2.1 See, e.g.
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