  Comparison of hydrodynamic calculations with experimental data inevitably
requires a model for converting the fluid to particles. In this work, nonlinear
$2\to 2$ kinetic theory is used to assess the overall accuracy of various shear
viscous fluid-to-particle conversion models, such as the quadratic Grad
corrections, the Strickland-Romatschke (SR) ansatz, self-consistent shear
corrections from linearized kinetic theory, and the correction from the
relaxation time approach. We test how well the conversion models can
reconstruct, using solely the hydrodynamic fields computed from the transport,
the phase space density for a massless one-component gas undergoing a 0+1D
longitudinal boost-invariant expansion with approximately constant specific
shear viscosity in the range $\sim 0.03 \le \eta/s \le \sim 0.2$.
  In general we find that at early times the SR form is the most accurate,
whereas at late times or for small $\eta/s\sim 0.05$ the self-consistent
corrections from kinetic theory perform the best. In addition, we show that the
reconstruction accuracy of additive shear viscous $f = f_{\rm eq} + \delta f$
models dramatically improves if one ensures, through "exponentiation", that $f$
is always positive. We also illustrate how even more accurate viscous $\delta
f$ models can be constructed if one includes information about the past
evolution of the system via the first time derivative of hydrodynamic fields.
Such time derivatives are readily available in hydrodynamic simulations, though
usually not included in the output.
