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  and connecting with Grad’s 14-moment kinetic theory description , 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 ) this means that we would use the Ansatz
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, 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). The presence of coupling terms of the particular form that we have introduced is suggested by kinetic theory .
What is clear from these complicated expressions is that we now have evolution equations for the dissipative fields. Introducing characteristic “relaxation” times 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. .
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 , and Olson and Hiscock . 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  than in the frame advocated by Landau and Lifshitz . 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.
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License.