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

**Authors:** Alireza Behtash, C. N. Cruz-Camacho, M. Martinez  
**arXiv:** [1711.01745](https://arxiv.org/abs/1711.01745)  
**Date:** 2017-11-06  
**Summary page:** https://www.summarizepaper.com/en/paper/1711.01745v1/

## Key points

- The paper investigates the dynamical non-equilibrium attractors of systems undergoing Gubser flow within relativistic kinetic theory.
- Nonlinear dynamical systems methods are employed to find fixed points, investigate the structure of flow diagrams, and characterize the basin of attraction using a Lyapunov function near stable fixed points.
- The authors obtain the dynamical attractors of anisotropic hydrodynamics, Israel-Stewart (IS), and transient fluid (DNMR) theories for the Gubser flow.
- Attractors for all three hydrodynamical theories are non-planar and have a three-dimensional basin of attraction.
- Anisotropic hydrodynamics matches up to high numerical accuracy with the non-equilibrium attractor of the exact solution while second-order hydrodynamical theories fail to describe it accurately.
- IS and DNMR asymptotic series expansion diverge and resurgence techniques are used to perform resummation on these divergences.
- A possible link between manifold steepest descent paths in path integrals and basin of attraction for attractors via Lyapunov functions is commented on, opening a new horizon toward effective field theory description of hydrodynamics.
- Anisotropic hydrodynamics is an effective theory for far-from-equilibrium fluid dynamics which resums Knudsen and inverse Reynolds numbers to all orders.
- The study provides valuable insights into understanding complex fluid dynamics in extreme conditions with implications for various fields including astrophysics and cosmology.

## Summary

In their paper "Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow," Alireza Behtash, C. N. Cruz-Camacho, and M. Martinez investigate the dynamical non-equilibrium attractors of systems undergoing Gubser flow within relativistic kinetic theory. The authors employ well-established methods of nonlinear dynamical systems, which rely on finding fixed points, investigating the structure of the flow diagrams of the evolution equations, and characterizing the basin of attraction using a Lyapunov function near stable fixed points. The authors obtain the dynamical attractors of anisotropic hydrodynamics, Israel-Stewart (IS), and transient fluid (DNMR) theories for the Gubser flow. They show that attractors for all three hydrodynamical theories are non-planar and that the basin of attraction is essentially three-dimensional. The authors compare the attractors of each hydrodynamical model with one obtained from the exact Gubser solution of the Boltzmann equation within the relaxation time approximation. They observe that anisotropic hydrodynamics is able to match up to high numerical accuracy with the non-equilibrium attractor of the exact solution while second-order hydrodynamical theories fail to describe it accurately. They also demonstrate how IS and DNMR asymptotic series expansion diverge and use resurgence techniques to perform resummation on these divergences. In addition, they comment on a possible link between manifold steepest descent paths in path integrals and basin of attraction for attractors via Lyapunov functions that opens a new horizon toward effective field theory description of hydrodynamics. Overall, their findings indicate that anisotropic hydrodynamics is an effective theory for far-from-equilibrium fluid dynamics which resums Knudsen and inverse Reynolds numbers to all orders. This study provides valuable insights into understanding complex fluid dynamics in extreme conditions and may have implications for various fields, including astrophysics and cosmology.