The next step is to deduce the permissible form for the additional fields from the second law of thermodynamics. The requirement that the total entropy must not decrease leads to the entropy flux having to be such that

Assuming that the entropy flux is a combination of all the available vectors, we have where and are yet to be specified. It is easy to work out the divergence of this. Then using the component of Equation (272) along , i.e. and the thermodynamic relation which follows from assuming the equation of state , and we recall that , one can show that We want to ensure that the right-hand side of this equation is positive definite (or indefinite). An easy way to achieve this is to make the following identifications: and Here we note that , where is the Gibbs free energy density. We also identify where the “diffusion coefficient” , and the projection is needed in order for the constraint to be satisfied. Furthermore, we can use where is the coefficient of bulk viscosity, and with being the heat conductivity coefficient. To complete the description, we need to rewrite the final term in Equation (282). To do this it is useful to note that the gradient of the four-velocity can generally be written as where the acceleration is defined as the expansion is , and the shear is given by Finally, the “twist” follows fromThe model we have written down is quite general. In particular, it is worth noticing that we did not yet specify the four-velocity . By doing this we can obtain from the above equations both the formulation due to Eckart [39] and that of Landau and Lifshitz [66]. To arrive at the Eckart description, we associate with the flow of particles. Thus we take (or equivalently ). This prescription has the advantage of being easy to implement. The Landau and Lifshitz model follows if we choose the four-velocity to be a timelike eigenvector of the stress-energy tensor. From Equation (274) it is easy to see that, by setting , we get

This is equivalent to setting . Unfortunately, these models, which have been used in most applications of relativistic dissipation to date, are not at all satisfactory. While they pass the key test set by the second law of thermodynamics, they fail several other requirements of a relativistic description. A detailed analysis of perturbations away from an equilibrium state [54] demonstrates serious pathologies. The dynamics of small perturbations tends to be dominated by rapidly growing instabilities. This suggests that these formulations are likely to be practically useless. From the mathematical point of view they are also not acceptable since, being non-hyperbolic, they do not admit a well-posed initial-value problem.http://www.livingreviews.org/lrr-2007-1 |
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