Therefore the concavity of the entropy density
in the variables
- the negative-definiteness of
- implies global invertibility between the field vector
and the Lagrange multipliers
.
The transformation
helps us to recognize the structure of the field equations and
to find generic restrictions on the constitutive functions.
Indeed, obviously, with
as field vector instead of
, or
, we may rephrase (8
) in the form
where
Thus we have
and
so that the constitutive quantities
and
result from
- defined by equation (19
) - through differentiation. Therefore the vector
is called the
thermodynamic vector potential
.
It follows from equation (20) that
which implies 4
n
(n
-1) restrictions on the constitutive functions
.
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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 |