Boillat & Ruggeri [6] 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
p
+
q
+
r
=
l
. Accordingly
may be written as
with
s
+
t
+
u
=
k
.
Table 1:
Pulse speed in extended thermodynamics of moments.
n
: Number of moments,
N
: Highest degree of moments,
: Pulse speed.
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
p
through
u
and we choose, rather arbitrarily
p
=
N,
s
=
N
-1 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
).
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Speeds of Propagation in Classical and Relativistic
Extended Thermodynamics
Ingo Müller http://www.livingreviews.org/lrr-1999-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |