Hydrodynamic Attractor in a Hubble Expansion.
2021
We analytically investigate hydrodynamic attractor solutions in both
M\"{u}ller-Israel-Stewart (MIS) and kinetic theories in a viscous fluid system
undergoing a Hubble expansion with a fixed expansion rate. We show that the
gradient expansion for the MIS theory and the Chapman-Enskog expansion for the
Boltzmann equation within the relaxation time approximation are factorially
divergent and obtain hydrodynamic attractor solutions by applying the Borel
resummation technique to those asymptotic divergent series. In both theories,
we find that the hydrodynamic attractor solutions are globally attractive and
only the first-order non-hydrodynamic mode exists. We also find that the
hydrodynamic attractor solutions in the two theories disagree with each other
when gradients become large, and that the speed of the attraction is different.
Similarities and differences from hydrodynamic attractors in the Bjorken and
Gubser flows are also discussed. Our results push the idea of
far-from-equilibrium hydrodynamics in systems undergoing a Hubble expansion.
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