Physics

Geology

Nonlinear system

Attractor

Boltzmann equation

Mathematics

Collision

Statistical physics

Astronomy

Oceanography

Mathematical analysis

Particle (ecology)

Computer science

Quantum mechanics

Computer security

SPHERES

Classical mechanics

Thermodynamics

Hard spheres

Physical Review Letters

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.

Exact Hydrodynamic Attractor of an Ultrarelativistic Gas of Hard Spheres