The overall aim is to generalize the variational formulation described in Section 8 in such a way that viscous “stresses” are accounted for. Because the variational foundations are the same, the number currents play a central role ( is a constituent index as before). In addition we can introduce a number of viscosity tensors , which we assume to be symmetric (even though it is clear that such an assumption is not generally correct ). The index is “analogous” to the constituent index, and represents different viscosity contributions. It is introduced in recognition of the fact that it may be advantageous to consider different kinds of viscosity, e.g. bulk and shear viscosity, separately. As in the case of the constituent index, a repeated index does not imply summation.
The key quantity in the variational framework is the master function . As it is a function of all the available fields, we will now have . Then a general variation leads to
As in the non-dissipative case, the variational framework suggests that the equations of motion can be written as a force-balance equation,
For reasons that will become clear shortly, it is useful to introduce a set of “convection vectors”. In the case of the currents, these are taken as proportional to the fluxes. This means that we introduce according to
In order to facilitate a similar decomposition for the viscous stresses, it is natural to choose the conduction vector as a unit null eigenvector associated with . That is, we take
Finally, let us suppose that we choose to work in a given frame, moving with four-velocity (and that the associated projection operator is ). Then we can use the following decompositions for the conduction vectors:
So far the construction is quite formal, but we are now set to make contact with the physics. First we note that the above results allow us to demonstrate that
At this point, the general formalism must be completed by some suitably simple model for the various terms. A reasonable starting point would be to assume that each term is linear. For the chemical reactions this would mean that we expand each according to
A detailed comparison between Carter’s formalism and the Israel–Stewart framework has been carried out by Priou . He concludes that the two models, which are both members of a large family of dissipative models, have essentially the same degree of generality and that they are equivalent in the limit of linear perturbations away from a thermal equilibrium state. Providing explicit relations between the main parameters in the two descriptions, he also emphasizes the key point that analogous parameters may not have the same physical interpretation.
In developing his theoretical framework, Carter argued in favor of an “off the peg” model for heat conducting models . This model is similar to that introduced in Section 10, and was intended as a simple, easier to use alternative to the Israel–Stewart construction. In the particular example discussed by Carter, he chooses to set the entrainment between particles and entropy to zero. This was done in order to simplify the discussion. But, as a discussion by Olson and Hiscock  shows, it has disastrous consequences. The resulting model violates causality in two simple model systems. As discussed by Priou  and Carter and Khalatnikov , this breakdown emphasizes the importance of the entrainment effect in these problems.
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