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Exact hydrodynamic attractor of an ultrarelativistic gas of hard spheres

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.