  We present a detailed phenomenological study of the next-to-next-to-leading
order (NNLO) QCD corrections for $t$-channel single top (anti-)quark production
and its semi-leptonic decay at the CERN Large Hadron Collider (LHC). We find
the NNLO corrections for the total inclusive rates at the LHC with different
center of mass energies are generally smaller than the NLO corrections,
indicative of improved convergence. However, they can be large for differential
distributions, reaching a level of $10\%$ or more in certain regions of the
transverse momentum distributions of the top (anti-)quark and the
pseudo-rapidity distributions of the leading jet in the event. In all cases the
perturbative hard scale uncertainties are greatly reduced after the NNLO
corrections are included. We also show a comparison of the normalized
parton-level distributions to recent data from the 8 TeV measurement of the
ATLAS Collaboration. The NNLO corrections tend to shift the theoretical
predictions closer to the measured transverse momentum distribution of the top
(anti)-quark. Importantly, for the LHC at 13 TeV, we present NNLO cross
sections in a fiducial volume with decays of the top quark included.
