finite temperature the equilibrium state is disordered with a vanishing magnetization and a coherence length $\xi (T)$ that diverges in the $T\to 0$ limit.

We consider a system initially prepared in an high temperature uncorrelated state and then quenched, at time $t=0$, to a lower final temperature $T_f$. The dynamics is characterized by the ordering of the system over a characteristic length growing in time until, at time $\tau _p^{eq}(T_f)$ it becomes comparable to $\xi (T_f)$. At this point the final equilibrium state at $T_f$ is entered. Quenching to $T_f=0$, since $\xi (0)=\infty $, one has $\tau _p^{eq}(T_f)=\infty$; therefore an infinite system never reaches equilibrium and the phase-ordering kinetics continues indefinitely. If the system is quenched to a sufficiently low temperature, since $\xi (T_f)$ is very large, the same behavior, as for $T_f=0$, can be observed over the time window $t<\tau _p^{eq}(T_f)$.

The power law growth of the characteristic size of ordered regions depends on the specific mechanisms at work in the kinetic process. In the 1d Ising model with non conserved order parameter, i.e. single spin flip dynamics, ordering is determined by the Brownian motion of the interfaces between up and down domains, which annihilate upon meeting. This leads 