| This thesis considers two issues of statistical physics—the liquid-liquid phase transitions and the evidence for entropically stabilized quasicrystals.; In recent years, there were reports of liquid-liquid phase transitions for several materials at pressures much higher than their corresponding liquid-gas transitions. To study the fundamental properties of liquid-liquid phase transitions, we developed a simple model based on the Lennard-Jones model. To mimic complex local ordering, we used a spin-one-half variable in our model and assigned different interactions between parallel and anti-parallel spins. We did Monte Carlo simulations in both constant pressure and Canonical ensembles to show that our model exhibits first-order phase transitions between high- and low-density liquids.; An essential issue in quasicrystals is the degree of stability of quasicrystals at low temperatures. We performed extensive Monte Carlo simulations on quasicrystals and quasicrystal approximants based on a model proposed by Widom, Strandburg and Swendsen. We mapped out the energy-composition plot at zero temperature and zero pressure. Our results strongly support the view that the quasicrystal state for this model is entropically stabilized. |
