  We investigate the temporal evolution of a ferromagnetic system of Ising
spins evolving under Kawasaki dynamics from a random initial condition, in
spatial dimensions one and two. We examine in detail the asymptotic behaviour
of the two-time correlation and response functions. The linear response is
measured without applying a field, using a recently proposed algorithm. For the
chain at vanishingly small temperature, we introduce an accelerated dynamics
which has the virtue of projecting the system into the asymptotic scaling
regime. This allows us to revisit critically previous works on the behaviour at
large time of the two-time autocorrelation and response functions. We also
analyse the case of the two-dimensional system at criticality. A comparison
with Glauber dynamics is performed in both dimensionalities, in order to
underline the similarities and differences in the phenomenology of the two
dynamics.
