Steady asymptotic equilibria in conformal relativistic fluids

When one considers a shock wave in the frame where the shock is at rest, on either side one has a steady flow which converges to equilibrium away from the shock. However, hydrodynamics is unable to describe this flow if the asymptotic velocity is higher than the characteristic speed of the theory. We obtain an exact solution for the decay rate to equilibrium for a conformal fluid in kinetic theory under the relaxation time approximation, and compare it to two hydrodynamic schemes, one accounting for the second moments of the distribution function and thus equivalent, in the small deviations from equilibrium limit, to an Israel-Stewart framework, and another accounting for both second and third moments. While still having a finite characteristic speed, the second model is a significant improvement on the first.

Figure 1

Exact decay rate derived from kinetic theory with an Anderson-Witting collision term and its asymptotic forms: (blue, full line) exact decay rate, defined parametrically by Eqs. (45) and (47); (red, dashes) asymptotic behavior for $v_L\to 1/\sqrt{3}$, Eq. (51); (green, dots and dashes) asymptotic behavior for $v_L\to 1$, Eq. (54). The divergence in the decay rate is stronger than predicted by holography [12].

Figure 2

The decay rates for the different theories discussed in this work: (blue, full line) exact decay rate, defined parametrically by Eqs. (45) and (47); (green, dashes) the decay rate from Landau-Lifshitz theory, Eq. (15); (black, dots and dashes) decay rate for an Israel-Stewart fluid with the constitutive relation derived from Grad’s ansatz, ${\eta }_0=4{\tau }_0/5$, Eq. (22); (red, dots) the decay rate derived from the DTT including the third momentum of the distribution function, Eqs. (72) and (73). The vertical grid lines show the characteristic speeds of the Israel-Stewart and DTT models, see Appendix pp5.

Figure 3

The decay rates for the different theories discussed in this work on the right-hand side of the shock: (blue, full line) exact decay rate, defined parametrically by Eq. (f9); (green, dashes) the decay rate from Landau-Lifshitz theory, Eq. (f1); (black, dots and dashes) decay rate for an Israel-Stewart fluid with the constitutive relation derived from Grad’s ansatz, ${\eta }_0=4{\tau }_0/5$, Eq. (f2); (red, dots) the decay rate derived from the DTT including the third momentum of the distribution function, Eqs. (f12) and (f16). The vertical grid lines show the asymptotic right side velocities corresponding to the characteristic speeds of the Israel-Stewart and DTT models, see Appendix pp5, which mark their applicability limit.