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
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
Denoting the comoving derivative by a dot, i.e. using etc. we see that the second law of
thermodynamics is satisfied if we choose
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 ( 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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