2.4 Characteristic Speeds2 Scope and StructureCharacteristic 2.2 Elements of the Constitutive

2.3 Exploitation of the Entropy Inequality, Lagrange Multipliers 

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 tex2html_wrap_inline3481 - themselves constitutive quantities, so that tex2html_wrap_inline3483 holds. Indeed, the new inequality

  equation179

is equivalent to (3Popup Equation).

Liu's proof proceeds from the observation that the field equations and the entropy equation are linear functions of the derivatives tex2html_wrap_inline3493 . 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 tex2html_wrap_inline3493 . It is then a simple problem of linear algebra to prove that

displaymath195

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 tex2html_wrap_inline3503 and tex2html_wrap_inline3505 in (6Popup Equation) and obtain

  equation208

The left hand side is an explicit linear function of the derivatives tex2html_wrap_inline3493 and, since the inequality must hold for all fields tex2html_wrap_inline3369, it must hold in particular for arbitrary values of the derivatives tex2html_wrap_inline3493 . The entropy inequality could thus easily be violated by some choice of tex2html_wrap_inline3493 unless we have

  equation227

and there remains the residual inequality

  equation232

The differential forms (8Popup Equation) 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 (9Popup Equation) 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.



2.4 Characteristic Speeds2 Scope and StructureCharacteristic 2.2 Elements of the Constitutive

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
http://www.livingreviews.org/lrr-1999-1
© Max-Planck-Gesellschaft. ISSN 1433-8351
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