2.2 Elements of the Constitutive 2 Scope and StructureCharacteristic 2 Scope and StructureCharacteristic

2.1 Thermodynamic Processes 

Thermodynamics, and in particular relativistic thermodynamics is a field theory with the primary objective to determine the thermodynamic fields. These are typically the 14 fields of the number density of particles, the particle flux vector and the fields of the stress-energy-momentum tensor. However, in extended thermodynamics we have generally more fields and therefore it is better - at least for the initial arguments - to leave the number of fields and their tensorial character unspecified. Therefore we consider n fields, combined in the n -vector tex2html_wrap_inline3329 . tex2html_wrap_inline3331 denotes the space-time components of an event. We have tex2html_wrap_inline3333 and tex2html_wrap_inline3335 Popup Footnote .

For the determination of the n fields u we need field equations - generally n of them - and these are based on the equations of balance of mechanics and thermodynamics. The generic form of these balance equations reads

  equation72

The comma denotes partial differentiation with respect to tex2html_wrap_inline3351, and tex2html_wrap_inline3353 is the n -vector of densities, while tex2html_wrap_inline3357 is the n -vector of flux components. Thus tex2html_wrap_inline3361 represents n four-fluxes, and tex2html_wrap_inline3365 is the n -vector of productions.

Obviously the balance equations (1Popup Equation) are not field equations for the fields tex2html_wrap_inline3369, at least not in this form. They must be supplemented by constitutive equations. These relate the four-fluxes tex2html_wrap_inline3361 and the productions tex2html_wrap_inline3365 to the fields u in a materially dependent manner. We write

  equation87

tex2html_wrap_inline3389 and tex2html_wrap_inline3391 denote the constitutive functions . Note that the constitutive quantities tex2html_wrap_inline3361 and tex2html_wrap_inline3395 at one event depend only on the values of u at that same event. In particular there is no dependence on gradients and time derivatives of tex2html_wrap_inline3369 .

If the constitutive functions tex2html_wrap_inline3389 and tex2html_wrap_inline3403 are explicitly known, we may eliminate tex2html_wrap_inline3361 and tex2html_wrap_inline3395 between the balance equations (1Popup Equation) and the constitutive relations (2Popup Equation) and obtain a set of explicit field equations for the fields tex2html_wrap_inline3369 . These are quasilinear partial differential equations of first order. Every solution of the field equations is called a thermodynamic process .



2.2 Elements of the Constitutive 2 Scope and StructureCharacteristic 2 Scope and StructureCharacteristic

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
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