Hydrodynamic attractor and the fate of perturbative expansions in Gubser flow

Abstract

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 $dS_3 \otimes \mathbb{R}$. 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 \emph{resummed} constitutiv 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.