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.
Highlights
AND SUMMARYHydrodynamics is an effective theory which describes the long-wavelength and/or small-frequency phenomena of physical systems
Our findings indicate that the reorganization of the expansion series carried out by anisotropic hydrodynamics resums the Knudsen and inverse Reynolds numbers to all orders and it can be understood as an effective theory for the far-from-equilibrium fluid dynamics
Knowing that the system converges to a fixed point sometime at late times τ ≫ 0, it is certainly not physical to consider that the point to be reached has a negative temperature ð−38c/15Þ and we are led to a situation where we look for the stable steady state in the range π 1⁄4 πc < 2
Summary
Hydrodynamics is an effective theory which describes the long-wavelength and/or small-frequency phenomena of physical systems. The attractor is a set of points in the phase space of the dynamical variables to which a family of solutions of an evolution equation merge after transients have died out In relativistic hydrodynamics it has been found in recent years that for far-from-equilibrium initial conditions the trajectories in the phase space merge quickly towards a nonthermal attractor before the system reaches the full thermal equilibrium. This type of nonequilibrium attractor can be fully determined by very few terms of the gradient series of relatively large size which involves transient nonhydrodynamical degrees of freedom [19,20]. We choose the fluid velocity to be defined in the Landau frame, i.e., Tμνuν ≡ εuμ
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