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14.2 The Israel–Stewart approach

From the above discussion we know that the most obvious strategy for extending relativistic hydrodynamics to include dissipative processes leads to unsatisfactory results. At first sight, this may seem a little bit puzzling because the approach we took is fairly general. Yet, the formulation suffers from pathologies. Most importantly, we have not managed to enforce causality. Let us now explain how this problem can be solved.

Our previous analysis was based on the assumption that the entropy current μ s can be described as a linear combination of the various fluxes in the system, the four-velocity μ u, the heat-flux qμ and the diffusion νμ. In a series of papers, Israel and Stewart [107Jump To The Next Citation Point57Jump To The Next Citation Point58Jump To The Next Citation Point] 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 [53Jump To The Next Citation Point]) this means that we would use the Ansatz

1 1 ( ) α0τqμ α1τμνqν sμ = suμ + --qμ − --- β0τ2 + β1qαqα + β2ταβ ταβ u μ + ------+ --------. (297 ) T 2T T T
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 β 0, β 1, and β 2 have a thermodynamical origin, while the two coefficients α0 and α1 represent the coupling between viscosity and heat flow. From the above expression, we see that in the frame moving with μ u the effective entropy density is given by
1 ( ) − u μsμ = s −--- β0 τ2 + β1q αqα + β2ταβταβ . (298 ) 2T
Since we want the entropy to be maximized in equilibrium, when the extra fields vanish, we should have [β0,β1,β2] ≥ 0. We also see that the entropy flux
μ ν 1 μ μν ⊥ ν s = T-[(1 + α0 τ)q + α1τ qν] (299 )
is affected only by the parameters α0 and α1.

Having made the assumption (297View Equation) 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

[ ( )] μ 1 μ μ μ μ ( α0) τT β0uμ ∇ μs = − --τ ∇ μu + β0u ∇ μτ − α0∇ μq − γ0T q ∇ μ --- + ---∇ μ ----- T [ T 2 T -1 μ 1- ν ν ν − T q T ∇ μT + u ∇νu μ + β1u ∇ νqμ − α0∇ μτ − α1∇ ντ μ T (β uν) ( α ) (α )] + -qμ∇ ν --1-- − (1 − γ0)τT ∇ μ -0- − (1 − γ1)T τνμ∇ ν --1 [ 2 T T ( ) T ] 1 T β2uα (α1 ) − --τ μν ∇ μuν + β2uα ∇ ατμν − α1 ∇ μqν +-τμν∇ α ----- − γ1Tqμ∇ ν --- . (300 ) T 2 T T
In this expression it should be noted that we have introduced (following Lindblom and Hiscock) two further parameters, γ 0 and γ 1. 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 [58Jump To The Next Citation Point], which leads to the mixed terms being distributed “equally” and hence γ0 = γ1 = 1 ∕2.

Denoting the comoving derivative by a dot, i.e. using uμ∇ μτ = ˙τ etc. we see that the second law of thermodynamics is satisfied if we choose

[ ( ) ( μ )] μ μ μ α0- τT- β0u-- τ = − ζ ∇μu + β0˙τ − α0∇ μq − γ0T q ∇ μ T + 2 ∇ μ T , (301 ) [ ( α) qμ = − κT ⊥ μν 1-∇ νT + u˙ν + β1q˙ν − α0∇ ντ − α1 ∇ αταν + T-qν∇ α β1u-- T 2 T ( α0) ( α1 ) ] − (1 − γ0)τT ∇ ν --- − (1 − γ1)T ταν∇ α --- + γ2∇ [νuα ]qα , (302 ) [ ( α T) ⟨ T ( ) ⟩ ] T- β2u-- α1- α τμν = − 2η β2˙τμν + 2 τμν∇α T + ∇ μuν − α1 ∇ μqν − γ1T qμ∇ ν T + γ3∇ [μuα ]τν , (303 )
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 γ2 and γ3. 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 [53Jump To The Next Citation Point]). The presence of coupling terms of the particular form that we have introduced is suggested by kinetic theory [58Jump To The Next Citation Point].

What is clear from these complicated expressions is that we now have evolution equations for the dissipative fields. Introducing characteristic “relaxation” times

t0 = ζβ0, t1 = κβ1, t2 = 2η β2, (304 )
the above equations can be written
t0τ˙+ τ = − ζ [...], (305 ) t ⊥ μν ˙qν + qμ = − κT ⊥ μν [...], (306 ) 1 t2˙τμν + τμν = − 2 η[...]. (307 )
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 (β0,...,α0,...). 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 [8684]. 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 [56Jump To The Next Citation Point], and Olson and Hiscock [85Jump To The Next Citation Point]. 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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