Hydrodynamic attractor and the fate of perturbative expansions in Gubser flow

Perturbative expansions, such as the well-known gradient series and the recently proposed slow-roll expansion, have been recently used to investigate the emergence of hydrodynamic behavior in systems undergoing Bjorken flow. In this paper we determine for the first time the large order behavior of these perturbative expansions in relativistic hydrodynamics in the case of Gubser flow. While both series diverge, the slow-roll series can provide a much better overall description of the system’s dynamics than the gradient expansion when both series are truncated at low orders. The truncated slow-roll series can also describe the attractor solution of Gubser flow as long as the system is sufficiently close to equilibrium near the origin (i.e., $\rho =0$) in $d{S}_3\otimes \mathbb{R}$. Differently than the case of Bjorken flow, here we show that the Gubser flow attractor solution is not solely a function of the effective Knudsen number ${\tau }_R\sqrt{{\sigma }_{\mu \nu }{\sigma }^{\mu \nu }}\sim {\tau }_R\tanh \rho$. Our results give further support to the idea that new resummed constitutive relations between dissipative currents and the gradients of conserved quantities can emerge in systems far from equilibrium that are beyond the regime of validity of the usual gradient expansion.

Figure 1

$|{\pi }_n{|}^{1/n}$ as a function of $n$ for $\rho =0.1$ (red), $\rho =1$ (bule), and $\rho =10$ (black).

Figure 2

$|{\pi }_n{|}^{1/n}$ as a function of $n$ for $\rho =0$ and $\rho \to \infty$.

Figure 3

Comparison between the exact result for $\pi$ defined in (13) obtained by solving Eqs. (14) and (15) and the gradient expansion truncated at different orders. In this plot $\eta /s=1/(4\pi )$.

Figure 4

Comparison between the exact result for $\pi$ defined in (13) obtained by solving Eqs. (14) and (15) and the gradient expansion truncated at different orders. In this plot $\eta /s=1$.

Figure 5

Comparison between the exact result for $\pi$ defined in (13) obtained by solving Eqs. (14) and (15) and the series in inverse powers of the relaxation truncated at different orders. In this plot $\eta /s=100$.

Figure 6

Large order behavior of the slow-roll series when $\rho \to 0$ and ${\tau }_R=0.1$, 1.

Figure 7

Large order behavior of the slow-roll series when $\rho =1$ and ${\tau }_R=1$, 10.

Figure 8

Comparison between the exact solution of Israel-Stewart (IS) equations and different truncations of the slow-roll series in Gubser flow for $\eta /s=1/(4\pi )$.

Figure 9

Detailed comparison when $\rho >0$ between the exact solution of the IS equations and the low order truncations of the slow-roll series in Gubser flow for $\eta /s=1/(4\pi )$.

Figure 10

Comparison between the exact solution of IS equations and different truncations of the slow-roll series in Gubser flow for $\eta /s=1$.

Figure 11

Detailed comparison when $\rho >0$ between the exact solution of the IS equations and the low order truncations of the slow-roll series in Gubser flow for $\eta /s=1$.

Figure 12

Comparison between the exact solution of IS equations undergoing Gubser flow for $\eta /s=1/(4\pi )$ and the $N=1$ and $N=2$ truncations of the slow-roll and gradient series, respectively.

Figure 13

Detailed comparison when $\rho >0$ between the exact solution of IS equations undergoing Gubser flow for $\eta /s=1/(4\pi )$ and the $N=1$ and $N=2$ truncations of the slow-roll and gradient series, respectively.

Figure 14

Comparison between the exact solution of IS equations undergoing Gubser flow for $\eta /s=1$ and the $N=1$ and $N=2$ truncations of the slow-roll and gradient series, respectively.

Figure 15

Attractor solution of IS equations undergoing Gubser flow (solid red) compared to other solutions of the equations (dashed black curves) for $\eta /s=1/(4\pi )$.

Figure 16

Attractor solution of IS equations undergoing Gubser flow (solid red) compared to other solutions of the equations (dashed black curves) for the very large value of $\eta /s=1000$.

Figure 17

Comparison between the attractor solutions of IS equations undergoing Gubser flow, for $\eta /s=1/(4\pi )$ and $\eta /s=1000$, and the corresponding approximations using the slow-roll series (dashed).

Figure 18

Comparison between the attractor solutions of IS equations undergoing Gubser flow, computed using different initial conditions for the temperature, and the corresponding approximations using the slow-roll series (dashed). In this plot $\eta /s=1/(4\pi )$.

Figure 19

Attractor solutions of IS equations undergoing Gubser flow, computed using different initial conditions for the temperature, vs the Knudsen numberlike quantity ${\tau }_R\tanh \rho$. The fact that these curves are distinct imply that in Gubser flow the attractor solution of IS theory is not just a simple function of ${\tau }_R\tanh \rho$. Rather, the attractor depends on both ${\tau }_R$ and $\tanh \rho$, separately. In this plot $\eta /s=1/(4\pi )$.