Thermodynamic processes

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 $u(xD )$ . $xD$ denotes the space-time components of an event. We have $x0 = ct$ and $xd = (x1, x2,x3)$ ² .

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

The comma denotes partial differentiation with respect to $A x$ , and $0 F$ is the n-vector of densities, while $a F$ is the n-vector of flux components. Thus $A F$ represents n four-fluxes, and $π$ is the n-vector of productions.

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

$^ A F$ and $^ π$ denote the constitutive functions. Note that the constitutive quantities $A F$ and $π$ 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 $u$ .

If the constitutive functions $F^A$ and $^π$ are explicitly known, we may eliminate $F A$ and $π$ between the balance equations (1 ) and the constitutive relations (2 ) and obtain a set of explicit field equations for the fields $u$ . These are quasilinear partial differential equations of first order. Every solution of the field equations is called a thermodynamic process.