### 3.3 Moments as variables

In a gas the most plausible choice for the four-fluxes are the moments of the phase density
of the atoms. Thus we have
is equal to , where is the atomic mass, while denotes the Cartesian coordinates of the
momentum of an atom. is a multi-index and stands for
so that the densities , 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 and note that by (30) the phase density depends on the single variable
only. Also (32) implies that the vector potential has the form

where, by (31), holds. The field equations (23) now read
Obviously the coefficient matrices are symmetric in , and is negative definite,
provided that
i.e. must be concave for the system (34) to be symmetric hyperbolic.