# Emergence of hydrodynamical behavior in expanding quark-gluon plasmas

## TL;DR

This paper demonstrates how simple angular moments of the Boltzmann equation can effectively describe the transition from free streaming to hydrodynamic behavior in an expanding quark-gluon plasma, highlighting the role of fixed points and moments.

## Contribution

It introduces a two-moment truncation approach to capture hydrodynamization in expanding plasmas, connecting kinetic theory with second and third order hydrodynamics.

## Key findings

- The first two moments suffice to describe the transition to hydrodynamics.
- Fixed points in the moment hierarchy underpin the emergence of hydrodynamic behavior.
- Late-time dynamics are dominated by monopole and quadrupole moments, with delayed isotropization.

## Abstract

We use a set of simple angular moments to solve the Boltzmann equation in the relaxation time approximation for a boost invariant longitudinally expanding gluonic plasma. The transition from the free streaming regime at early time to the hydrodynamic regime at late time is well captured by the first two-moments, corresponding to the monopole and quadrupole components of the momentum distribution, or equivalently to the energy density and the difference between the longitudinal and the transverse pressures. We relate this property to the existence of fixed points in the infinite hierarchy of equations satisfied by the moments. These fixed points are already present in the two-moment truncations and are only moderately affected by the coupling to higher moments. Collisions contribute to a damping of all the non trivial moments. At late time, when the hydrodynamic regime is entered, only the monopole and quadrupole moments are significant and remain strongly coupled, the decay of the quadrupole moment being delayed by the expansion, causing in turn a delay in the full isotropization of the system. The two-moment truncation contains second order viscous hydrodynamics, in its various variants, and third order hydrodynamics, together with explicit values of the relevant transport coefficients, can be easily obtained from the three-moment truncation.