Now we establish the rules by which the interactions of the model are defined. We are interested in the change from lexical vocabulary to functional vocabulary. We propose that this change can be simulated as a first-order phase transition and that enough of what we know about language will find descriptions in these terms.
Local interactions lie in nearest-neighbour interactions in which energy is transferred from constituent to constituent. Standard nearest-neighbour interaction is described in the following; each individual spin is in contact with the spins on the four sites surrounding it and is influenced by them. When most immediate neighbours are in a given state, it is energetically favorable for a given spin to be in a similar state.
A phase transition will occur only when a critical point in the activity of the system is reached. This critical point of activity allow for attenuated states to be statistically favoured.
Consider the functional use of have in its auxiliary role. At a stage of the history of the language in which interlocutors use have as a relationship of possession, its auxiliary role would not generate the neurophysiological effects appropriate to that functional use. In every-day cases, less dramatic innovations - even if they cannot be fully understood - nevertheless increase the extension - attenuate - for the neighbours' continued lexical use, even if they are not yet in a position to use have in its functional role. Eventually, a critical level of attenuation will be reached. As this dynamic prevails in all neighbourhoods of the system, we observe a sudden change in the state of the system. Functional uses will have come about. The sudden change occurs because, initially one of the states, is energetically favoured by the system. In low activity, the influence of nearest neighbours lock the state of constituents by forcing each other to remain coherent. As the activity increases in the system, the rate of change in the local states of constituents augments. Because of our attenuation bias, attenuated states will be statistically favoured and eventually the energy bias will lose its hold.
Obviously there is no solid state for language but there is a state in which language cannot propagate any further. Like a crystal, the structure of world associations is semantically circumscribed and, as such, easy to understand but limited in its use. Attenuated language is much like an amorphic state, in which vocables are in a structurally loose state: Easy to use but difficult to understand. The structure of liquid water is correspondingly difficult to describe at a molecular level. Attenuated language is syntactically circumscribed and has lost most of the perceptual connections present early on. This phenomenon is a direct consequence of its propagation. The mere fact of use produces a degree of attenuation, howsoever minute. And the loss of perceptual circumscription that would be required for stability in a language, itself promotes new uses, and therefore greater instability.
Changes in the rate of attenuation can be occasioned in a number of ways; certainly through an increase in the biological population of its users; certainly by changes in the rate of environmental changes such as average age of the users and their longevity, by immigration, by technological developments and so on. The model is not dependent upon the specifics of propagation; it gives only a mathematical model of behavior at and near equilibrium.