  We analyze the effect of pairing on particle transport in time-dependent
theories based on the Hartree-Fock-Bogoliubov (HFB) or BCS approximations. The
equations of motion for the HFB density matrices are unique and the theory
respects the usual conservation laws defined by commutators of the conserved
quantity with the Hamiltonian. In contrast, the theories based on the BCS
approximation are more problematic. In the usual formulation of TDHF+BCS, the
equation of continuity is violated and one sees unphysical oscillations in
particle densities. This can be ameliorated by freezing the occupation numbers
during the evolution in TDHF+BCS, but there are other problems with the BCS
that make it doubtful for reaction dynamics. We also compare different
numerical implementations of the time-dependent HFB equations. The equations of
motion for the $U$ and $V$ Bogoliubov transformations are not unique, but it
appears that the usual formulation is also the most efficient. Finally, we
compare the time-dependent HFB solutions with numerically exact solutions of
the two-particle Schrodinger equation. Depending on the treatment of the
initial state, the HFB dynamics produces a particle emission rate at short
times similar to that of the Schrodinger equation. At long times, the total
particle emission can be quite different, due to inherent mean-field
approximation of the HFB theory.
