For the determination of the n fields 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 readsn-vector of densities, while is the n-vector of flux components. Thus represents n four-fluxes, and is the n-vector of productions.
Obviously the balance equations (1) are not field equations for the fields , at least not in this form. They must be supplemented by constitutive equations. These relate the four-fluxes and the productions to the fields in a materially dependent manner. We writeconstitutive functions. Note that the constitutive quantities and at one event depend only on the values of at that same event. In particular there is no dependence on gradients and time derivatives of .
If the constitutive functions and are explicitly known, we may eliminate and between the balance equations (1) and the constitutive relations (2) and obtain a set of explicit field equations for the fields . These are quasilinear partial differential equations of first order. Every solution of the field equations is called a thermodynamic process.
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