Crystal structure stability and melting at high temperature-high pressure are fundamental issues in material physics and geophysics. Special attention has been devoted to transition metals, and to iron in particular because of its abundance in the Earth's core.
Several statistical mechanics methods have been proposed to find lines of coexistence between phases in pressure-temperature space. In this lecture I will describe some of these methods, and I will show how a hierarchy of computational methods of progressively increasing accuracy/computer-demand (from classical potentials, to density functional theory, to quantum Monte Carlo) can be used to effectively predict high temperature phase stability. In particular, I will discuss i) the free energy method, in which free energies of competing phases are explicitly computed; ii) the coexistence method, mainly applicable to solid-liquid transitions,
in which solid and liquid are directly simulated in coexistence with each other; and iii) the recently proposed Z method for solid-liquid transitions, where I will show that special care and statistical analysis is required in order to obtain reliable melting temperatures, and that in general the method only provides upper bounds to the melting temperature. I will discuss advantages and disadvantages of each of these methods, and present specific examples in which they have been applied, including high-pressure
high-temperature phase transitions in iron, molybdenum and other materials.