Specific form of the phase density

3.4 Specific form of the phase density

For moments as variables the entropy four-flux $A h$ 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 ])

$k$ is the Boltzmann constant and $1∕y$ is the smallest phase space element.

Comparison shows that we must have

( ( ) ( ) ) f y f f χf (χ) − F (χ) = − k ln --± -- 1 ± -- ln 1 ± -- f, y f y y

and hence, by differentiation with respect to $χ$ ,

so that

$f$ is the phase density appropriate to a degenerate gas in non-equilibrium. Differentiation of (39

) with respect to $χ$ proves the inequality (35

Therefore symmetric hyperbolicity of the system (34 ) and hence the concavity of the entropy density with respect to the variables $0 Fα$ 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 $±1$ 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 $T$ 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.