Stability analysis for a thermodynamically consistent model of relativistic fluid dynamics

In relativistic fluid mechanics, positive entropy production is known to be insufficient for guaranteeing stability. Much stronger criteria for thermodynamic admissibility have become available in nonequilibrium thermodynamics. We here perform a linear stability analysis for a model of relativistic hydrodynamics that is based on the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics. Assuming a quadratic entropy function near equilibrium, we find stability for the entire range of physically meaningful model parameters for relativistic fluid dynamics based on GENERIC. The search for thermodynamic admissibility moreover reveals a fundamental difference between liquids and gases in relativistic fluid dynamics.

Figure 1

(a) Real and (b) imaginary part of the roots $\mathrm{\Omega }$ as a function of $k$, for $Q=0.001$. Other parameters are $A=1/3, X=1, Y=1/2, Z=1, {\lambda }_0=1, {\lambda }_1=2/3$, and ${\lambda }_2=1/6$. The dashed lines represent the roots of the characteristic polynomial of the $3\times 3$ block $\mathit{R}$; the solid lines represent the roots of the characteristic polynomial of the $6\times 6$ block $\mathit{N}$. Both $\mathrm{\Omega }$ and $k$ are depicted in nondimensional units.

Figure 2

(a) Real and (b) imaginary part of the roots $\mathrm{\Omega }$ as a function of $k$, for $Q=0.1$. For other parameters and line style code see Fig. 1.

Figure 3

(a) Real and (b) imaginary part of the roots $\mathrm{\Omega }$ as a function of $k$, for $Q=1$. For other parameters and line style code see Fig. 1.