# Exact solutions and attractors of higher-order viscous fluid dynamics for Bjorken flow

## TL;DR

This paper analyzes higher-order relativistic viscous hydrodynamics theories for Bjorken flow, demonstrating that third-order theories better approximate the exact kinetic theory attractor and characterizing the approach to equilibrium.

## Contribution

It provides analytical solutions and a detailed comparison of second- and third-order hydrodynamics theories with exact kinetic theory for Bjorken flow.

## Key findings

- Third-order theory outperforms MIS and DNMR in approximating the exact attractor.
- Exponential approach to the attractor at small Knudsen numbers.
- Power-law decay of deviations persists even at large Knudsen numbers.

## Abstract

We consider causal higher order theories of relativistic viscous hydrodynamics in the limit of one-dimensional boost-invariant expansion and study the associated dynamical attractor. We obtain evolution equations for the inverse Reynolds number as a function of Knudsen number. The solutions of these equations exhibit attractor behavior which we analyze in terms of Lyapunov exponents using several different techniques. We compare the attractors of the second-order M\"uller-Israel-Stewart (MIS), transient DNMR, and third-order theories with the exact solution of the Boltzmann equation in the relaxation-time approximation. It is shown that for Bjorken flow the third-order theory provides a better approximation to the exact kinetic theory attractor than both MIS and DNMR theories. For three different choices of the time dependence of the shear relaxation rate we find analytical solutions for the energy density and shear stress and use these to study the attractors analytically. By studying these analytical solutions at both small and large Knudsen numbers we characterize and uniquely determine the attractors and Lyapunov exponents. While for small Knudsen numbers the approach to the attractor is exponential, strong power-law decay of deviations from the attractor and rapid loss of initial state memory is found even for large Knudsen numbers. Implications for the applicability of hydrodynamics in far-off-equilibrium situations are discussed.