  We study the warming process of a semi-infinite cylindrical Ising lattice
initially ordered and coupled at the boundary to a heat reservoir. The adoption
of a proper microcanonical dynamics allows a detailed study of the time
evolution of the system. As expected, thermal propagation displays a diffusive
character and the spatial correlations decay exponentially in the direction
orthogonal to the heat flow. However, we show that the approach to equilibrium
presents an unexpected slow behavior. In particular, when the thermostat is at
infinite temperature, correlations decay to their asymptotic values by a power
law. This can be rephrased in terms of a correlation length vanishing
logarithmically with time. At finite temperature, the approach to equilibrium
is also a power law, but the exponents depend on the temperature in a
non-trivial way. This complex behavior could be explained in terms of two
dynamical regimes characterizing finite and infinite temperatures,
respectively. When finite sizes are considered, we evidence the emergence of a
much more rapid equilibration, and this confirms that the microcanonical
dynamics can be successfully applied on finite structures. Indeed, the slowness
exhibited by correlations in approaching the asymptotic values are expected to
be related to the presence of an unsteady heat flow in an infinite system.
