Ising and Language

The Ising Model can be further refactored to model the dynamics of what we call lexical attenuation in natural language. Our model describes a possible history of the dynamics of a meta-vocable. This meta-vocable is a statistical representation of all vocables.

As in the standard model we use spin states to represent vocables on a two dimensional lattice. We also use a Maxwellian demon lattice to apply a local level of activity - the frequency of use - for each constituents. In addition, we introduce the concept of attenuation and we use it as a statistical bias. It is similar to degeneracy in that it can be introduced as a feature of a two-dimensional Ising model but with one difference: Usually a particular degeneracy value is assigned homogeneously to each spin. In our case we introduce a gradient of attenuation, that is, spins are assigned an increasing number of potential attenuated states, illustrated as a gradient. We will describe this process in detail a little later.

We use three Cartesian grids for our language model. The first grid represents the two dimensional lattice that hosts spins. These spins represent a set of instances or occurrences of one meta-vocable. A black (or dark) pixel for a constituent represents a functional state of the vocable while a blue (or light) pixel represents a lexical state. Figure 6.4 illustrates a typical representation of that grid. Notice the white line that travels the grid. This line highlights the ratio between functional (dark pixel) and lexical (light pixel) for all rows.

  

Figure 6.4: Vocable (spin) grid. Vocables can be in 1 of 2 states; light is lexical while dark is functional. The white line indicate the ratio of light to dark for every row.

A second grid represents the local potential for attenuation. A gradient from black to white describes the state of attenuation each meta-vocable can achieve. The changing levels of attenuation can be thought of as actual real time-line (versus simulation time) in the lifespan of a vocable. Early instances are found in the dark areas of the gradient while later instances are found in the light ones (see figure 6.5)

  

Figure 6.5: Attenuation grid. The different shades represent degrees of attenuation; in this case constituents can be attenuated up to a 7 to 1 ratio.

This gradient affects the critical point at which a first-order phase transition can occur within each area for each degeneracy. For our physicist audience, let us remember that, given a Hamiltonian $H_{i}€=-\sum_{j}€s_{i}€*s_{j}€-s_{i}€\Delta/2$, the field value is $\Delta$(flipping a spin changes the energy by $2*\Delta$ if spin values are +1/-1), and attenuation is $\delta$. Because $\delta$ changes along the length of the attenuation grid, we observe (figure 6.6) that the critical point (green or dash line) is pushed inwards as the level of attenuation augments.

  

Figure 6.6: This graph illustrates how the critical point (green or thick lines) at which a first-order phase transition occurs moves inwards for constituents ranging from 1 state of degeneracy to 3 states of degeneracy. The x's indicate the state of constituents. Those with 3 states of attenuation have become functionalized because the activity level is past their critical point. The other constituents remain lexical because their respective critical points have not been reached.

Consider a simulation where there are mixed attenuated states for constituents. The attenuation gradient is up to three states. No state of attenuation is represented as black while attenuated states from one to three is represented respectively as dark, medium and light grey (as shown in figure 6.6). The Meanfield solutions in figure 6.6 illustrates that some attenuated constituents remain lexical - x at the top of the curve - at a sub-critical level of activity. Once the level of activity in the system reaches the critical point of state change for particular states of attenuation, constituents become functional - x on the bottom curve. In this graph we illustrate the critical points at which constituents will become functionalized. Notice that only constituents that have three states of attenuation have become functionalized because their critical point has been reached.

As in the standard Ising model, we also use a bias that favours a particular state energetically. Lexical vocables tend to stay lexical for reasons energetically similar to the reasons why ice tends to remain in its solid state until environmental constraints are such that a liquid state becomes favoured. This feature, however, is not graphically represented.

A third grid, corresponding to the Maxwellian demon grid, illustrates an activity rate. The energy driving attenuation is the physical effort expended in the actual uses of the vocables in speech. The temperature variable used in the standard Ising model finds its equivalent in the activity variable of our model. As the activity level is increased, the likelyhood for vocables - spins - to become attenuated will also increase. Augmenting the activity value represents an increase in the use of vocables within a linguistic community. In the real world, linguistic activity increases with a growing number of linguistic participants in a population; however, since our model has a finite population - constituents, it is solely the frequency of use of a vocable that defines the activity variable. For example, given a system where there are only two linguistic participants, we can expect that some vocables can become very attenuated by the mere fact of overuse.

Consider the case of technical language used by small groups of people in the context of developing new technologies. As the use of such vocables becomes attenuated, we lose our capacity to say what it specifically means; we think that technical jargon is a product of similar dynamics. Figure 6.7 illustrates local variances in the activity levels of each constituent.

  

Figure 6.7: Activity grid. This is a "Maxwell" demon grid. A random level activity is assigned to every vocable constituent in the system. The overall average is the activity in the system. It is equivalent to the level of linguistic activity that can occur in a population of language users.

Each spin site has a local activity value assigned to it. In the application to language, a high activity value at a spin site can be thought of as representing extensive use of that vocable by a single user. The underlying theory suggests that higher usage eventually entails a higher rate of attenuation. So a low distribution of activity in the system can be taken to represent comparative lexical richness.