  We study the quench dynamics of the three-dimensional Kitaev (spin) model
under a linear drive using both exact numerical calculations and analytical
"independent crossing approximation". Unlike the two-dimensional Kitaev model,
the three-dimensional Kitaev model reduces to a multilevel Landau-Zener problem
for each momentum. We show that for the slow quench, the defect density is
proportional to the quench rate $1/\tau$. We find that the zeros of the
relevant coupling between the levels determine the non-adiabatic condition for
the production of defects. The contour on which the energy spectrum becomes
gapless does not play an active role. The asymptotic behavior of the defect
density crucially depends on the way the system reaches the non-adiabatic
regime during the quenching process. We analytically show that defect
correlation varies as $\tau^{-1} e^{-A/\tau}$, where $A$ is a constant
independent of $\tau$. For the slow quench, the qualitative dependence of the
entropy (produced during the quenching process) on the quench time is the same
as that of the defect correlation, indicating a close connection between the
defect correlation and the entropy content of the final state. Possible
experimental realization of such quench dynamics is also described briefly.
