3.4 Specific form of the phase density

For moments as variables the entropy four-flux A h follows from (19View Equation) and (33View Equation). We obtain
∫ hA = pA (χf (χ ) − F (χ )) dp. (36 )
On the other hand statistical mechanics defines the four-flux of entropy by (e.g. see Huang [20])
∫ ( f y ( f) ( f )) Fermions hA = − k pA ln --± -- 1 ± -- ln 1 ± -- f dp for . (37 ) y f y y Bosons
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,

f = ---y----, (38 ) eχ∕k ± 1
so that
( ) F = ∓ky ln 1 ± e−χ∕k . (39 )
f is the phase density appropriate to a degenerate gas in non-equilibrium. Differentiation of (39View Equation) with respect to χ proves the inequality (35View Equation).

Therefore symmetric hyperbolicity of the system (34View Equation) 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 (38View Equation) may be neglected. In that case we have

f = ye−χ∕k, (40 )
hence
d2F 1 F = − kf and ---2 = − --f, (41 ) dχ k
and therefore the field equations (23View Equation), (34View Equation) assume the form
[ 1-∫ A ] − k p pαpβf dp Λβ,A = πα. (42 )
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-----− 2pm2kT fE = √ -------3e . (43 ) 2πmkT
n and T denote the number density and the temperature of the gas in equilibrium. Comparison of (43View Equation) with (40View Equation) shows that only two Lagrange multipliers are non-zero in equilibrium, viz.
y√2-πmkT--3 1 ΛE = kln ------------ and ΛEii = -2-----. (44 ) n 3mkT

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