3.5 Pulse Speeds in a 3 Finite Speeds in Non-Relativistic 3.3 Moments as Variables

3.4 Specific Form of the Phase Density 

For moments as variables the entropy four-flux tex2html_wrap_inline3425 follows from (19Popup Equation) and (33Popup Equation). We obtain

  equation560

On the other hand statistical mechanics defines the four-flux of entropy by (e.g. see Huang [20])

  equation565

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

Comparison shows that we must have

displaymath583

and hence, by differentiation with respect to tex2html_wrap_inline3819,

  equation593

so that

  equation599

f is the phase density appropriate to a degenerate gas in non-equilibrium. Differentiation of (39Popup Equation) with respect to tex2html_wrap_inline3819 proves the inequality (35Popup Equation).

Therefore symmetric hyperbolicity of the system (34Popup Equation) and hence the concavity of the entropy density with respect to the variables tex2html_wrap_inline3801 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 tex2html_wrap_inline3853 in the denominator of (38Popup Equation) may be neglected. In that case we have

  equation608

hence

  equation613

and therefore the field equations (23Popup Equation), (34Popup Equation) assume the form

  equation623

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

  equation629

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

  equation638



3.5 Pulse Speeds in a 3 Finite Speeds in Non-Relativistic 3.3 Moments as Variables

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
http://www.livingreviews.org/lrr-1999-1
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