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

## Abstract

The nonequilibrium 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 nonplanar 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 expansions 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 the basin of attraction for the attractors via Lyapunov functions that opens a new horizon towardmore »

- Authors:

- North Carolina State Univ., Raleigh, NC (United States). Dept. of Physics; Univ. of California, Santa Barbara, CA (United States). Kavli Inst. for Theoretical Physics
- Universidad Nacional de Colombia, Bogotá (Colombia). Dept. de Física
- North Carolina State Univ., Raleigh, NC (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- North Carolina State Univ., Raleigh, NC (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1422657

- Alternate Identifier(s):
- OSTI ID: 1498962

- Grant/Contract Number:
- FG02-03ER41260; SC0013036

- Resource Type:
- Journal Article: Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 97; Journal Issue: 4; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS

### Citation Formats

```
Behtash, Alireza, Cruz-Camacho, C. N., and Martinez, M..
```*Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow*. United States: N. p., 2018.
Web. doi:10.1103/physrevd.97.044041.

```
Behtash, Alireza, Cruz-Camacho, C. N., & Martinez, M..
```*Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow*. United States. doi:10.1103/physrevd.97.044041.

```
Behtash, Alireza, Cruz-Camacho, C. N., and Martinez, M.. Mon .
"Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow". United States. doi:10.1103/physrevd.97.044041.
```

```
@article{osti_1422657,
```

title = {Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow},

author = {Behtash, Alireza and Cruz-Camacho, C. N. and Martinez, M.},

abstractNote = {The nonequilibrium 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 nonplanar 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 expansions 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 the basin of attraction for the attractors via Lyapunov functions that opens a new horizon toward an effective field theory description of hydrodynamics. 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 thus, it can be understood as an effective theory for the far-from-equilibrium fluid dynamics},

doi = {10.1103/physrevd.97.044041},

journal = {Physical Review D},

issn = {2470-0010},

number = 4,

volume = 97,

place = {United States},

year = {2018},

month = {2}

}

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