The key to the exploitation of the entropy inequality lies in
the fact that the inequality should hold for thermodynamic
processes, i.e. solutions of the field equations rather than for
all fields. By a theorem proved by Liu [30] this constraint may be removed by the use of Lagrange
multipliers
- themselves constitutive quantities, so that
holds. Indeed, the new inequality
is equivalent to (3).
Liu's proof proceeds from the observation that the field
equations and the entropy equation are linear functions of the
derivatives
. By the Cauchy-Kowalewski theorem these derivatives are local
representatives of an analytical thermodynamic process and
therefore the entropy principle requires that the field equations
and the entropy equation must hold for all
. It is then a simple problem of linear algebra to prove
that
Liu's proof is not restricted to quasilinear systems of first
order equations but here we need his result only in that
particularly simple case.
We may use the chain rule on
and
in (6
) and obtain
The left hand side is an explicit linear function of the
derivatives
and, since the inequality must hold for all fields
, it must hold in particular for arbitrary values of the
derivatives
. The entropy inequality could thus easily be violated by some
choice of
unless we have
and there remains the residual inequality
The differential forms (8) represent a generalization of the Gibbs equation of equilibrium
thermodynamics; the classical Gibbs equation for the entropy
density is here generalized into four equations for the entropy
four-flux. Relation (9
) is the residual entropy inequality which represents the
irreversible entropy production. Note that the entropy production
is entirely due to the production terms in the balance
equations.
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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 |