The sudden emergence of features in physical phenomena is spectacular but not a rare occurrence in Nature. The spontaneous magnetization in ferrous materials and the spontaneous structural change of a liquid to a solid state are well documented phenomena and mathematically described facts. An everyday example might be H2O and its clearly distinguishable states as ice, water, and vapor. In the general parlance of phase, H2O is understood to exhibit characteristics associated with different phases, and moreover, to have identifiable transitions between them. An important feature, that is well-defined mathematically, is the point at which one state of a system changes to another - this is referred to as the critical point. For example, as the temperature of a system of liquid water is decreased, the state will spontaneously change from a liquid with no structure into a crystalline configuration - ice - at the critical temperature, 0 Celsius. These apparently spontaneous changes are transformations that are well illustrated by the physical concept of phase transition.
A phase transition describes how a system moves from one phase to another, relative to a state variable that is unique to the type of system under observation. In the case of the water/ice transition, the state variable is a coherence length - a measure of the strength of a defined set of relationships between the objects under observation [51] - that reflects the presence of inter-molecular structure. There are several types of phase transitions, chief amongst them are the first- and second-order transitions. A first-order phase transition demonstrates a discontinuity in its state variable at the critical point - there is an abrupt change from one state to another. A second-order phase transition describes a smooth variation in the state variable. Water/ice follows a first-order phase transition.
First-order phase transitions is a well developed area in physics. Statistical mechanics has developed approaches that give a detailed microscopic level description of interactions amongst a large number of individual constituents, say H2O molecules, and their repercussion at a macroscopic level such as the structural change in a system, say, from a liquid configuration (water) to a solid configuration (ice). Such a system can display a phase transition if some intrinsic macroscopic quantity variable is varied. This variable - temperature in the case of H2O example - is defined at a macro level. Elevating or decreasing the temperature corresponds to a comparable variation in the mean energy in the (equilibrium) distribution of energies. For example, a phase change from liquid to solid for H2O involves a decrease in temperature. This is equivalent to a decrease in the average energy available to each water molecule.
The modeling of phase transitions involves the definition of the ensemble of energy states of the molecules and the macroscopic consequences for the entire system. Any related dynamical behaviours are assumed to take place on very long time scales so that the state of the system can always be viewed in terms of equilibrium or at least quasi-equilibrium. Any dynamics are then accounted for as slow, independent processes. We will describe more specifically how the model is implemented in the section The Ising Model.
We suggest that the dynamics of linguistic transcategorial changes leading to functionalization can be successfully compared to the processes that leads to a structural change in, say, freezing water. We suggest that these changes occur spontaneously. In this suggestion we are borrowing from physical models, used in physics, to formalize our language-related claims. In our model, transcategorial changes constitute the macro feature of the language system and vocables are the micro constituents.
What we model are the interactions that lead to attenuation and critical structural changes in vocables in a population of language users. The macro variable is the activity that exists in the system. In other words, it expresses how linguistically interactive the overall population is, both in propensity and in number of individuals. The concept of activity is a variable and is similar, in role, to the temperature in our H2O example. The micro feature that is monitored in our model is the capacity to generate effects in a population of language users. The capacity to generate effects is akin, in role, to the energy available to the physical system of H2O. However, though these effects are observables of a population of users we presume that, ultimately, they are of a neural kind and can be measured as physical energy.