  We present a detailed phenomenological study of the multiple soft gluon
radiation for the $t$-channel single top and anti-top quark production at the
Large Hadron Collider (LHC). By applying the transverse momentum dependent
factorization formalism, large logarithms introduced by small total transverse
momentum $q_\perp$ of the single-top (anti-top) plus one-jet final state system
are resummed to all orders in the expansion of the strong interaction coupling
at the accuracy of Next-to-Leading Logarithm. We discuss various kinematical
distributions which are sensitive to this effect and find that soft gluon
radiation become more important when the final state jet is required to be in
the forward region. We show that the main difference from PYTHIA prediction
lies on the inclusion of the exact color coherence effect between the initial
and final states in our resummation calculation. We further propose to apply
the experimental observable $\phi^*$ to test the effect of multiple gluon
radiation in the single-top and anti-top events. The bottom quark mass effect
and jet rapidity distribution are also discussed.
