Models for interface dynamics

Interface dynamics describes the evolution of systems after phase segregation. In a quenching experiment a system is initially in thermodynamic equilibrium with a reservoir which is then cooled down below the critical temperature of the system. This process is usually so fast that the macroscopic state of the system does not change significantly. After the cooling it is no longer in equilibrium with the reservoir and one usually models the successive evolution by supposing its state still stationary but unstable. It is therefore very sensitive to external perturbations as those of stochastic nature coming from the reservoir. As soon as the state changes, because of its instability, the deterministic driving forces internal to the system take over, driving it toward the stable phases and the phase separation phenomena take place. Equilibrium is not reached yet at this stage. When the system is spatially extended it reaches locally a thermodynamic stable phase, but since there are several equally accessible phases, there is no reason for the equilibria of faraway regions to coincide. The typical picture at the end of this stage is then a collection of clusters of different phases with interfaces in between. The successive stage of the evolution is the interface dynamics which describes the competition between phases. A rigorous derivation of the whole picture from microscopic models has not yet been carried out in a systematic and rigorous theory as in equilibrium statistical mechanics. In the last years however some results have been obtained in the context of particular and simpler models. This thesis focuses on interface dynamics referring to two specific models with non conserved order parameter and with two symmetric stable phases. The evolution of the interfaces is ruled by the motion by mean curvature.