Computational implementations

Several computational solutions may be developed to address a particular feature of the model. For example, there are many ways in which the energy available to the system can be implemented. One can perform a random draw to represent the level of energy available to all constituents in the system. Similarly a Maxwell [51] energy ``daemon'' may be employed which also distributes evenly, via a discretized transfer of the energy to all constituents. A variation in the approach may be used, and this is the one we have chosen for our model. We implement a Maxwellian demon lattice in our model, with one demon per spin site to allow for the concept of local temperature. This is referred to as a heat sink/source, since it gives up or takes in energy. The energy moves around via neighbour interaction. The energy is not randomly redistributed as in the case of a Maxwell demon or a random energy function.

Another useful computational implementation in a standard Ising is the wrap-around Cartesian grid as illustrated in figure 6.3.

  

Figure 6.3: The Cartesian grid is wrap-around to reduce boundary conditions. Because the grid is square it is difficult to illustrate that corners are also wraped so there is continuity between all edges.

A wrap-around grid is helpful in reducing the constraints that arise from having boundaries in a system. The problem with boundaries is that they introduce circumstances that set them apart from the rest of the system. So in the case of our model, we define the interaction as depending on the nearest-neighbours. There are four nearest neighbours on a Cartesian grid. However, at the boundary of a square grid, a constituent situated on one of the corners will only have two or three neighbours hence changing the nature of the interaction. In a wrap-around grid all constituents have four neighbours because it is akin to a sphere and there are no edges on a sphere.

Several such computational strategies are used to create a simulation. The ones highlighted here are the most interesting ones. These, however, should not distract us from our primary discussion about the model and how the behavior of language can be described in its terms. There is a distinction to be made between a theoretical model and the computational implementations that can be used to run a simulation. Again, in order to confirm that the computational implementations of our simulation are faithful to our model, we use a graphical representation to evaluate whether the simulation behaves appropriately. A grid of colored pixels can be used to display the states of the vector spins as binary rather than directionally. We can also display the Maxwell demon lattice as a grid to illustrate local temperature, which we have done in the case of our model. Additionally, the use of a graph is useful to illustrate the relationship between temperature of the system and the state of the system. In our model we include some modifications to the standard Ising model, modifications that we will describe in the next section.