3.3 Moments as variables

In a gas the most plausible choice for the four-fluxes A F are the moments of the phase density f (x, p,t) of the atoms. Thus we have
∫ F A= pAp fdp, (α = 1,2, ...n),(A = 0,1, 2,3). (26 ) α α
p0 is equal to mc, where m is the atomic mass, while pa denotes the Cartesian coordinates of the momentum of an atom. α is a multi-index and pα stands for
( ||| 1 α = 1 { pi1 α = 2,3, 4 pα = || pi1pi2 α = 5,6, ...10 (27 ) |( pi1pi2 ...piN α = n − 1(N + 1)(N + 2),...,n 2
so that the densities F0α, (α = 1,2,...n ) form a hierarchy of moments of increasing tensorial degree up to degree N. Because of the evident symmetry of (27View Equation) there is a relation between n and N, viz.
1 n = -(N + 1)(N + 2)(N + 3). (28 ) 6
The kinetic theory of gases implies that the moments (26View Equation) satisfy equations of balance of the type (1View Equation) so that the foregoing analysis holds. In particular, we have (18View Equation) which may now be written in the form
′A A dh = Fα dΛα = (29 )
∫ A p d(Λ αpα)fdp = (30 )
∫ A p dF (Λαpα )dp = (31 )
∫ A d p F (Λ αpα)dp. (32 )

We introduce χ = Λαp α and note that by (30View Equation) the phase density depends on the single variable χ only. Also (32View Equation) implies that the vector potential has the form

∫ h ′A = pAF (χ)dp, (33 )
where, by (31View Equation), dF --- = f dχ holds. The field equations (23View Equation) now read
[∫ d2F ] pAp αpβ---2 dp Λ β,A = π α. (34 ) dχ
Obviously the coefficient matrices are symmetric in α, β and ∫ d2F- pαp βdχ2 dp is negative definite, provided that
d2F- dχ2 < 0, (35 )
i.e. F (χ) must be concave for the system (34View Equation) to be symmetric hyperbolic.
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