### 3.4 Specific form of the phase density

For moments as variables the entropy four-flux follows from (19) and (33). We obtain
On the other hand statistical mechanics defines the four-flux of entropy by (e.g. see Huang [20])
is the Boltzmann constant and is the smallest phase space element.
Comparison shows that we must have

and hence, by differentiation with respect to ,

so that
is the phase density appropriate to a degenerate gas in non-equilibrium. Differentiation of (39) with
respect to proves the inequality (35).
Therefore symmetric hyperbolicity of the system (34) and hence the concavity of the entropy density
with respect to the variables is implied by the moment character of the fields and the form of the
four-flux of entropy.

For a non-degenerate gas the term in the denominator of (38) may be neglected. In that case we
have

hence
and therefore the field equations (23), (34) assume the form
Note that the matrices of coefficients are composed of moments in this case of a non-degenerate
gas.
We know that a non-degenerate gas at rest in equilibrium exhibits the Maxwellian phase density

n and denote the number density and the temperature of the gas in equilibrium. Comparison
of (43) with (40) shows that only two Lagrange multipliers are non-zero in equilibrium, viz.