  We resum the non-equilibrium gradient corrections to a single-particle
distribution function evolved by the Boltzmann equation in the relaxation time
approximation (RTA). We first study a system undergoing Bjorken expansion and
show that, for a constant relaxation time, the exact solution of the RTA
Boltzmann equation at late times (i.e. after the decay of non-hydrodynamic
modes) generates the Borel resummed Chapman-Enskog series. Extending this
correspondence to systems without Bjorken symmetry, we construct a
(3+1)-dimensional hydrodynamic generator for RTA kinetic theory, which is an
integral representation of the Chapman-Enskog series in the limit of vanishing
non-hydrodynamic modes. Relaxing this limit we find at earlier times a set of
non-hydrodynamic modes coupled to the Chapman-Enskog expansion. Including the
dynamics of these non-hydrodynamic modes is shown to control the emergence of
hydrodynamics as an effective field theory description of non-equilibrium
fluids, which works well even for far-off-equilibrium situations where the
Knudsen number is initially large.
