Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow

Abstract

The non-equilibrium attractors of systems undergoing Gubser flow within relativistic kinetic theory are studied. In doing so we employ well-established methods of nonlinear dynamical systems which rely on finding the fixed points, investigating the structure of the flow diagrams of the evolution equations, and characterizing the basin of attraction using a Lyapunov function near the stable fixed points. We obtain the attractors of anisotropic hydrodynamics, Israel-Stewart (IS) and transient fluid (DNMR) theories and show that they are indeed non-planar and the basin of attraction is essentially three dimensional. The attractors of each hydrodynamical model are compared with the one obtained from the exact Gubser solution of the Boltzmann equation within the relaxation time approximation. We observe that the anisotropic hydrodynamics is able to match up to high numerical accuracy the attractor of the exact solution while the second order hydrodynamical theories fail to describe it. We show that the IS and DNMR asymptotic series expansion diverge and use resurgence techniques to perform the resummation of these divergences. We also comment on a possible link between the manifold of steepest descent paths in path integrals and basin of attraction for the attractors via Lyapunov functions that opens a new horizon toward effective field theory description of hydrodynamics. Our findings indicate that anisotropic hydrodynamics is an effective theory for far-from-equilibrium fluid dynamics which resums the Knudsen and inverse Reynolds numbers to all orders.