When such a system is cooled through the transition temperature, interconnected domains of the two equilibrium phases form and grow with time ("coarsen") so as to decrease the total interfacial area. The domain patterns show an interesting "scaling" dynamics, in which the patterns at later times are statistically similar to earlier times apart from an overall change of scale. Interest then naturally focuses on how the characteristic length scale L(t) changes with time. It turns out that L(t) typically grows as a power of time. This power is the "growth exponent" n.
The case of just two equilibrium phases has been qualitatively understood for some years: one finds n=1/3 or n=1/2 depending on whether or not the total "order parameter" (here a simple scalar quantity, the magnetization) is conserved by the dynamics. In recent years emphasis has turned to systems described by more general types of order parameter, e.g. an m-component vector ("XY" or "Heisenberg" ferromagnetic models, or superfluid helium), or a second-rank tensor (nematic liquid crystals). This general field is called "phase-ordering kinetics". Other related phenomena include grain growth or the coarsening of soap froths.
Our group has made substantial progress in determining n for a large class of systems in recent years. Our approach focuses on the domain walls, or "topological defects", which in many cases drive the dynamics. For systems with a vector or tensor order parameter, the relevant topological defects are "vortices", "monopoles" or "disclination lines". These objects are nucleated as the system is cooled through the transition, because the order parameter symmetry-breaking that accompanies the transition chooses different phases in different parts of the system. The topological defects are a consequence of the resulting "mismatch". These ideas extend across the whole range of phase transitions found in nature, including the symmetry breaking transitions believed to have occurred in the early universe, where the formation of topological defects (e.g cosmic strings) has been proposed as a possible mechanism for the formation of cosmic structure (i.e. to answer the question "why is the universe not spatially uniform?").
There are many open questions in this field. For example, although the growth laws are quite well understood, the associated scaling functions (which describe the spatial structure of the scaling state) are difficult to calculate in a systematic way. Indeed, the very existence of scaling in many of these systems has still to be conclusively demonstrated. Furthermore, fascinating new features of coarsening dynamics, such as the recently discovered "persistence phenomena" (see stochastic processes) are constantly being discovered.
For further information on this subject see my review article on phase-ordering kinetics.