The physics of phase transition

At atmospheric pressure, water becomes ice at 0°C and becomes vapor at 100°C. Metal resistance becomes 0 at around 0 K. Some metals become magnet at low temperature... They are all called phase transition phenomena - one of the most mysterious and magical things in Physics. The phase transition is defined by the abrupt change of system properties. Why mysterious ? Until now, there are fairly a lot of phenomena which can not be explained quantitatively by an exact theory. For example, the para-ferromagnetic phase transition of Ising 3-dimensional model. In the other side, there exists also many theoretical predictions which haven’t been yet verified experimentally, like Bose-Einstein condensation. The most general theory until now for phase transition is the Landau theory. It can be considered like a qualitative approach to all phase transition phenomena. The main idea is to expand the (free) energy of the system into the power series of a so-called order parameter m which is defined to be vanish in one phase and nonzero in the others. The equilibrium states of system correspond to the minima of the free energy. The key point is, for some systems, the number of these minima can be different when temperature changes. This change can be considered as a phase transition and we can get the critical temperature from it. One word for the next part, take a stick and put it standing on the floor, then push vertically this stick. Obviously if we push strong enough the stick will be broken. Can you explain why ?