We present a basic background on phase transitions.
Physics describes many emergent features as the result of phase transitions, that is, matter transforming, according to a certain order parameter, say, from one structural spatial state to an other as in the case of solid/liquid/vapor. Phase transition occurs in large composite systems. These phenomena are emergent features, emergent because they are observable at the macroscopic level and not at the constituent microscopic level. Phase transitions have been described for centuries but the consistent microscopic description has only been developed in the past three decades [3]. The microscopic description of phase transitions, though deterministic in its model, accentuates the emergent character by articulating clearly the conditions under which macro features emerge. Experimentalists have also observed that disparate microsystems have nearly identical macrolevel behavior for certain aspects of phase transition [3]. It is within this perspective that the language of critical phenomena can apply to the dynamics of language.
But first let us describe phase transitions in the context of physical phenomena. There are two basic kinds of phase transition, first and second order. We have already established the nature of first order as a discontinuous phase transition, while a second order is a continuous phase transition. In the condensation of gases the phase transition can be discontinuous under low pressure but continuous at a critical point. The critical point is the specification of macro variables at which the transition occurs. In the first case, the system changes its state from one phase to the other in a spontaneous way, that is one phase is replaced by another with an obvious interface between phases at a critical point. The second case describes a gradual change from one phase to the other such that there is no distinct phase, at a critical point, between liquid and gas. Phase transitions describe systems going from a disordered state to an ordered state or vice versa. Order in a system is a macro feature that has been characterized by an order parameter as introduced by Landau [51]. Order is a macroscopic variable that describes the variations between macro states, that is, a finite value in the ordered phase and zero in the disordered phase. The ordered phase usually occurs at low temperature and is destroyed at high temperature such that the order parameter vanishes. The order parameter is sometimes difficult to establish; for example, order in the case of the liquid-gas transition is based in the difference of density.
Here are various physical examples. Phase transitions occur in many material and describe phases other than the condensation of gases or the freezing of liquids. Iron is a good example: the ferromagnetic ordered phase is below 771c€ and is characterized by a net spontaneous magnetization that produces a magnet bar. Above 771c€, iron is in the paramagnetic phase which does not display macroscopic magnetization. Another example is the order-disorder phases in binary alloy such as brass. The copper and zinc atoms occupy alternate sites in a regular lattice in the ordered phase of brass and arbitrary ones in the disordered phase [3]. We will elaborate on the modeling of phase transition in the Ising Model chapter.