4 Finite Speeds in Relativistic 3 Finite Speeds in Non-Relativistic 3.5 Pulse Speeds in a

3.6 A Lower Bound for the Pulse Speed of a Non-Degenerate Gas 

Since in (47Popup Equation) the integral tex2html_wrap_inline3899 is symmetric and tex2html_wrap_inline3901 is symmetric and positive definite, it follows from linear algebra Popup Footnote that

  equation688

Boillat & Ruggeri [6] have used this knowledge to derive an estimate for tex2html_wrap_inline3875 in terms of N, the highest tensorial degree of the moments. The estimate reads

  equation696

Therefore, indeed, as more and more moments are drawn into the scheme of extended thermodynamics, the pulse speed goes up and, if N tends to infinity, so does tex2html_wrap_inline3875 .

The proof of (49Popup Equation) rests on the realization that - because of symmetry - tex2html_wrap_inline3913 has only tex2html_wrap_inline3915 independent components and they are simply powers of tex2html_wrap_inline3917 and tex2html_wrap_inline3919, so that tex2html_wrap_inline3781 may be written as tex2html_wrap_inline3923 with p + q + r = l . Accordingly tex2html_wrap_inline3927 may be written as tex2html_wrap_inline3929 with s + t + u = k .

   table712
Table 1: Pulse speed in extended thermodynamics of moments. n : Number of moments, N : Highest degree of moments, tex2html_wrap_inline3949 : Pulse speed.

Therefore (48Popup Equation) assumes the form

  equation724

The elements of a semi-definite matrix tex2html_wrap_inline3953 satisfy the inequalities tex2html_wrap_inline3955 and therefore (50Popup Equation) implies

  equation738

Since tex2html_wrap_inline3879 is an even function of tex2html_wrap_inline3965 we obtain

  equation760

This estimate depends on the choice of the exponents p through u and we choose, rather arbitrarily p = N, s = N -1 and all others zero. Also we set tex2html_wrap_inline3981 . In that case (52Popup Equation) implies

  equation781

which proves (49Popup Equation).

An easy check will show that for each N the value tex2html_wrap_inline3985 lies below the corresponding values of Table  1, as they must. It may well be possible to tighten the estimate (49Popup Equation).



4 Finite Speeds in Relativistic 3 Finite Speeds in Non-Relativistic 3.5 Pulse Speeds in a

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