Our previous analysis was based on the assumption that the entropy current can be described as a linear combination of the various fluxes in the system, the four-velocity , the heat-flux and the diffusion . In a series of papers, Israel and Stewart [107, 57, 58] contrasted this “first-order” theory with relativistic kinetic theory. Following early work by Müller [81] and connecting with Grad’s 14-moment kinetic theory description [50], they concluded that a satisfactory model ought to be “second order” in the various fields. If we, for simplicity, work in the Eckart frame (cf. Lindblom and Hiscock [53]) this means that we would use the Ansatz

This expression is arrived at by asking what the most general form of a vector constructed from all the various fields in the problem may be. Of course, we now have a number of new (so far unknown) parameters. The three coefficients , , and have a thermodynamical origin, while the two coefficients and represent the coupling between viscosity and heat flow. From the above expression, we see that in the frame moving with the effective entropy density is given by Since we want the entropy to be maximized in equilibrium, when the extra fields vanish, we should have . We also see that the entropy flux is affected only by the parameters and .Having made the assumption (297) the rest of the calculation proceeds as in the previous case. Working out the divergence of the entropy current, and making use of the equations of motion, we arrive at

In this expression it should be noted that we have introduced (following Lindblom and Hiscock) two further parameters, and . They are needed because without additional assumptions it is not clear how the “mixed” quadratic term should be distributed. A natural way to fix these parameters is to appeal to the Onsager symmetry principle [58], which leads to the mixed terms being distributed “equally” and hence .Denoting the comoving derivative by a dot, i.e. using etc. we see that the second law of thermodynamics is satisfied if we choose

where the angular brackets denote symmetrization as before. In these expression we have added yet another two terms, representing the coupling to vorticity. These bring further “free” parameters and . It is easy to see that we are allowed to add these terms since they do not affect the entropy production. In fact, a large number of similar terms may, in principle, be considered (see note added in proof in [53]). The presence of coupling terms of the particular form that we have introduced is suggested by kinetic theory [58].What is clear from these complicated expressions is that we now have evolution equations for the dissipative fields. Introducing characteristic “relaxation” times

the above equations can be written A detailed stability analysis by Hiscock and Lindblom [53] shows that the Israel–Stewart theory is causal for stable fluids. Then the characteristic velocities are subluminal and the equations form a hyperbolic system. An interesting aspect of the analysis concerns the potentially stabilizing role of the extra parameters (). Relevant discussions of the implications for the nuclear equation of state and the maximum mass of neutron stars have been provided by Olson and Hiscock [86, 84]. A more detailed mathematical stability analysis can be found in the work of Kreiss et al. [64].Although the Israel–Stewart model resolves the problems of the first-order descriptions for near equilibrium situations, difficult issues remain to be understood for nonlinear problems. This is highlighted in work by Hiscock and Lindblom [56], and Olson and Hiscock [85]. They consider nonlinear heat conduction problems and show that the Israel–Stewart formulation becomes non-causal and unstable for sufficiently large deviations from equilibrium. The problem appears to be more severe in the Eckart frame [56] than in the frame advocated by Landau and Lifshitz [85]. The fact that the formulation breaks down in nonlinear problems is not too surprising. After all, the basic foundation is a “Taylor expansion” in the various fields. However, it raises important questions. There are many physical situations where a reliable nonlinear model would be crucial, e.g. heavy-ion collisions and supernova core collapse. This problem requires further thought.

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