The distribution of nucleation times for the Ising model is determined using the Metropolis algorithm. I find that the nucleation rate is suppressed at early times even after global variables such as the magnetization and energy are apparently in metastable equilibrium. I relate the observed nonequilibrium behavior of the nucleation rate to the time required for clusters of spins in the stable phase to grow to a size comparable to the size of the nucleating droplet. I also find subtle structural differences between the nucleating droplets formed before and after metastable equilibrium has been established. My results suggest that using global variables as indicators of metastable equilibrium may not be appropriate in general, and distinguishing between equilibrium and transient nucleation is important in studying the structures of nucleating droplets.

The homogeneous and heterogeneous nucleation of a Lennard-Jones liquid is investigated using the umbrella sampling method. The free energy cost of forming a nucleating droplet is determined as a function of the quench depth, and the saddle point nature of the nucleating droplets is verified using an intervention technique. The structure and symmetry of the nucleating droplets is found for a range of temperatures. I find that at deep quenches the nucleating droplets become more diffuse and anisotropic with no well defined core. The environment of the nucleating droplets form randomly stacked hexagonal planes. This behavior is consistent with a spinodal nucleation interpretation. I also find that the free energy barrier for heterogeneous nucleation is a minimum when the lattice spacing of the impurity equals the lattice spacing of the equilibrium crystalline phase. If the lattice spacing of the impurity is different, the crystal grows into the bulk instead of wetting the impurity.

Some preliminary results regarding the non-ergodic behavior of the long-range FPU model are obtained. Although the system is non-ergodic for nearest-neighbor interaction, it becomes ergodic in the mean-field limit. It is possible to explain its ergodic behaviors by assuming that the system can be described by a free energy picture.