  While a large number of studies have focused on the nonequilibrium dynamics
of a system when it is quenched instantaneously from a disordered phase to an
ordered phase, such dynamics have been relatively less explored when the quench
occurs at a finite rate. Here we study the slow quench dynamics in two
paradigmatic models of classical statistical mechanics, {viz.}, one-dimensional
kinetic Ising model and mean-field zero-range process, when the system is
annealed slowly to the critical point. Starting from the time evolution
equations for the spin-spin correlation function in the Ising model and the
mass distribution in the zero-range process, we derive the Kibble-Zurek scaling
laws. We then test a recent proposal that critical coarsening which is ignored
in the Kibble-Zurek argument plays a role in the nonequilibrium dynamics close
to the critical point. We find that the defect density in the Ising model and a
scaled mass distribution in the zero-range process decay linearly to the
respective value at the critical point with the time remaining until the end of
the quench provided the final quench point is approached sufficiently fast, and
sublinearly otherwise. As the linear scaling for the approach to the critical
point also holds when a system following an instantaneous quench is allowed to
coarsen for a finite time interval, we conclude that critical coarsening
captures the scaling behavior in the vicinity of the critical point if the
annealing is not too slow.
