Moments as variables

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

$p0$ 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 (27

) there is a relation between n and N, viz.

The kinetic theory of gases implies that the moments (26

) satisfy equations of balance of the type (1

) so that the foregoing analysis holds. In particular, we have (18

) which may now be written in the form

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

where, by (31

), $dF --- = f dχ$ holds. The field equations (23

) now read

Obviously the coefficient matrices are symmetric in $α$ , $β$ and $∫ d2F- pαp βdχ2 dp$ is negative definite, provided that

i.e. $F (χ)$ must be concave for the system (34

) to be symmetric hyperbolic.