### 3.6 A lower bound for the pulse speed of a non-degenerate gas

Since in (47) the integral is symmetric and is symmetric and positive definite, it follows from linear algebra that
Boillat & Ruggeri [5] have used this knowledge to derive an estimate for in terms of N, the highest tensorial degree of the moments. The estimate reads
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 .

The proof of (49) rests on the realization that – because of symmetry – has only independent components and they are simply powers of and , so that may be written as with . Accordingly may be written as with .

Table 1: Pulse speed in extended thermodynamics of moments. n: Number of moments, N: Highest degree of moments, : Pulse speed.
 n N n N 4 1 0.77459667 2600 23 6.59011627 10 2 1.34164079 2925 24 6.75262213 20 3 1.80822948 3276 25 6.91176615 35 4 2.21299946 3654 26 7.06774631 56 5 2.57495874 4060 27 7.22074198 84 6 2.90507811 4495 28 7.37091629 120 7 3.21035245 4960 29 7.51841807 165 8 3.49555791 5456 30 7.66338362 220 9 3.76412372 5984 31 7.80593804 286 10 4.01860847 6545 32 7.94619654 364 11 4.26098014 7140 33 8.08426549 455 12 4.26098014 7770 34 8.22024331 560 13 4.71528716 8436 35 8.35422129 680 14 4.92949284 9139 36 8.48628432 816 15 5.13625617 9880 37 8.61651144 969 16 5.33629130 10660 38 8.74497644 1140 17 5.53020569 11480 39 8.87174833 1330 18 5.71852112 12341 40 8.99689171 1540 19 5.90168962 13244 41 9.12046722 1771 20 6.08010585 14190 42 9.24253184 2024 21 6.25411673 15180 43 9.36313918 2300 22 6.42402919

Therefore (48) assumes the form

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

Since is an even function of we obtain

This estimate depends on the choice of the exponents through and we choose, rather arbitrarily , and all others zero. Also we set . In that case (52) implies
which proves (49).

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