Exact hydrodynamic attractor of an ultrarelativistic gas of hard spheres

TL;DR

This paper derives an exact analytical solution for the hydrodynamic attractor of an ultrarelativistic hard-sphere gas undergoing Bjorken expansion, demonstrating convergence of the gradient expansion and the applicability of hydrodynamics at large gradients.

Contribution

It provides the first exact analytical solution for the hydrodynamic attractor in a system modeled by the Boltzmann equation with a nonlinear collision kernel.

Findings

Gradient expansion converges in this system.

Hydrodynamics accurately describes the late-time attractor.

Exact solution matches numerical simulations.

Abstract

We derive the general analytical solution of the viscous hydrodynamic equations for an ultrarelativistic gas of hard spheres undergoing Bjorken expansion, taking into account effects from particle number conservation, and use it to analytically determine its attractor at late times. Differently than all the cases considered before involving rapidly expanding fluids, in this example the gradient expansion converges. We exactly determine the hydrodynamic attractor of this system when its microscopic dynamics is modeled by the Boltzmann equation with a fully nonlinear collision kernel. The exact late time attractor of this system can be reasonably described by hydrodynamics even when the gradients are large.