3.4 Specific Form of the 3 Finite Speeds in Non-Relativistic 3.2 Symmetric Hyperbolicity

3.3 Moments as Variables 

In a gas the most plausible choice for the four-fluxes tex2html_wrap_inline3361 are the moments of the phase density tex2html_wrap_inline3767 of the atoms. Thus we have

  equation483

tex2html_wrap_inline3771 is equal to mc, where m is the atomic mass, while tex2html_wrap_inline3777 denotes the Cartesian coordinates of the momentum of an atom. tex2html_wrap_inline3779 is a multi-index and tex2html_wrap_inline3781 stands for

  equation487

so that the densities tex2html_wrap_inline3801, tex2html_wrap_inline3803 form a hierarchy of moments of increasing tensorial degree up to degree N . Because of the evident symmetry of (27Popup Equation) there is a relation between n and N, viz.

  equation505

The kinetic theory of gases implies that the moments (26Popup Equation) satisfy equations of balance of the type (1Popup Equation) so that the foregoing analysis holds. In particular, we have (18Popup Equation) which may now be written in the form

  equation513

  equation517

  equation521

  equation525

We introduce tex2html_wrap_inline3817 and note that by (30Popup Equation) the phase density depends on the single variable tex2html_wrap_inline3819 only. Also (32Popup Equation) implies that the vector potential has the form

  equation531

where, by (31Popup Equation), tex2html_wrap_inline3823 holds. The field equations (23Popup Equation) now read

  equation540

Obviously the coefficient matrices are symmetric in tex2html_wrap_inline3779, tex2html_wrap_inline3829 and tex2html_wrap_inline3831 is negative definite, provided that

  equation550

i.e. tex2html_wrap_inline3833 must be concave for the system (34Popup Equation) to be symmetric hyperbolic.



3.4 Specific Form of the 3 Finite Speeds in Non-Relativistic 3.2 Symmetric Hyperbolicity

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
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