$T_0$. Depending on the depth of the quench we observe two clearly distinct ordering–scenarios: thermally activated nucleation of the ordered phase for shallow quenches, $T_0>T_f>T_{sp}$, and spinodal ordering, when $T_f<T_{sp}$. Here, the spinodal temperature $T_{sp}$ was taken over from independent simulations at equilibrium.

In the case of thermally activated processes there is some characteristic incubation time, $\tau_{inc}$, after which a small fraction of the system is covered by overcritical nuclei. Detailed simulation results for $\tau_{inc}$, based on vacancy–atom exchange, were presented. The time period $t<\tau_{inc}$ manifests itself in plateau regions for energy relaxation and for the size of ordered domains, when plotted versus $t$. Clearly, $\tau_{inc}$ is expected to diverge as $T_f$ approaches $T_0$ from below. Within the simple model investigated here and the available maximum computing time, $\tau_{inc}$ grows more than 260 times in a narrow temperature interval above $T_{sp}$ – from about 90 MCS at $T_f=0.911\,T_0$ to roughly $24\cdot10^3$MCS at $T_f=0.976\,T_0$. In comparison with measurements [@Noda] of $\tau_{inc}$ this seems to constitute the correct order of magnitude when Monte Carlo times are converted to physical times with the help of experimental tracer diffusion coefficients. In the coarsening regime we observed growth of the characteristic domain–size that was clearly 