# Second-order (2+1)-dimensional anisotropic hydrodynamics

## TL;DR

This paper develops a comprehensive second-order (2+1)-dimensional anisotropic hydrodynamics framework that improves upon leading-order models by incorporating deviations from spheroidal distribution functions, validated against exact solutions in RTA.

## Contribution

It introduces a complete second-order formulation of anisotropic hydrodynamics with coupled PDEs for key variables, extending the leading-order spheroidal approximation.

## Key findings

- Accurately reproduces exact RTA solutions for various shear viscosities.
- Provides a set of coupled PDEs for anisotropy, temperature, velocity, and viscous tensors.
- Demonstrates excellent approximation in (0+1)-dimensional expansion.

## Abstract

We present a complete formulation of second-order (2+1)-dimensional anisotropic hydrodynamics. The resulting framework generalizes leading-order anisotropic hydrodynamics by allowing for deviations of the one-particle distribution function from the spheroidal form assumed at leading order. We derive complete second-order equations of motion for the additional terms in the macroscopic currents generated by these deviations from their kinetic definition using a Grad-Israel-Stewart 14-moment ansatz. The result is a set of coupled partial differential equations for the momentum-space anisotropy parameter, effective temperature, the transverse components of the fluid four-velocity, and the viscous tensor components generated by deviations of the distribution from spheroidal form. We then perform a quantitative test of our approach by applying it to the case of one-dimensional boost-invariant expansion in the relaxation time approximation (RTA) in which case it is possible to numerically solve the Boltzmann equation exactly. We demonstrate that the second-order anisotropic hydrodynamics approach provides an excellent approximation to the exact (0+1)-dimensional RTA solution for both small and large values of the shear viscosity.