Reviews in Computational Chemistry

Cellular Automata 215

Movement (Transition) Rules

The movement of cells is based upon rules governing the events inherent in cellular automata dynamics. The rules describe the probabilities of two adjacent cells separating, two cells joining at a face, two cells displacing each other in a gravity simulation, or a cell with different designated edges rotating in the grid. These events are the essence of the cellular automata dynamics and produce configurations that may possibly mirror physical events. The following sections develop these topics, and the Appendix provides technical detail about the hardware and software used and defines the directional moving probability, MP(d), for each direction d.

General Conditions for Water Models

Several probabilistic rules are used to govern the trajectories of molecules moving in the grid. The first set of these rules, called transition functions, govern the movement of the occupied cells across the grid. The rules respond to the immediate neighborhood of the occupied cell, thus all events are local. A probabilistic or stochastic set of rules operating in a grid on the surface of a torus is most appropriate for the simulation of water and solution phenomena. This arrangement provides a near-infinitely large system with no boundary conditions.

The Free Movement Probability

The first rule is the movement probability Pm. This rule involves the probability that an occupant in an unbound cell, i, will move to one of four adjacent cells, j, if that space is unoccupied. An example is cell i in Figure 6 that may move (in its turn) to any unoccupied cell, j. If it moves to a cell whose neighbor is an occupied cell, k, then a bond will form between cells i and k. As

Figure 6 Possible movement of cell i occupant to unoccupied cell j.

216 Cellular Automata Models of Aqueous Solution Systems

a matter of course, this probability, Pm, is usually set at 1.0, which means that this event always happens (a rule).

Breaking Rules

Just as two cells can join together, so their tessellated state can be broken. The first of two trajectory or interaction rules is the breaking probability PB. To comprehend this, it is convenient to refer to the occupant of a cell as a molecule. The PB(AB) rule is the probability for a molecule A, bonded to molecule B, to break away from B, as shown in Figure 7(a). The value for PB lies between 0 and 1. If molecule A is bonded to two molecules, B and B, the simultaneous probability of a breaking away event from both B and B is PB(AB)*PB(AB), as shown in Figure 7(b). If molecule A is bound to three other molecules, B, B, and B, the simultaneous breaking probability of molecule A is PB(AB)*PB(AB)*(PB(AB), shown in Figure 7(c). If molecule A is surrounded by four molecules, it cannot move on this 2D grid.

Joining Parameter

A joining trajectory parameter, J(AB), describes the movement of a molecule at A to join with a molecule at B, when an intermediate cell is vacant (see Figure 8). This rule follows the rule to move or not to move described above. The parameter J is a nonnegative real number. When J ¼ 1, molecule A has the same probability of movement toward or away from B as for the case when the B cell is empty. When J > 1, molecule A has a greater probability of movement toward an occupied cell B than when cell B is empty. When 0 < J < 1, molecule A has a lower probability of such movement. When J ¼ 0, molecule A cannot make any movement.

Relative Gravity

The simulation of a ‘‘gravity’’ effect has been introduced into the cellular automata paradigm to model separating phenomena such as the demixing of immiscible liquids or the flow of solutions in a chromatographic separation. To accomplish this effect, a boundary condition is imposed at the upper and lower edges of the grid to simulate a vertical versus a horizontal relationship. The differential effect of gravity is simulated by introducing two new rules governing the preferences of two cells of different composition to exchange positions when they are in a vertically joined state. When molecule A is on top of a second molecule B, then two new rules are actuated. The first rule, GD(AB), is the probability that molecule A will exchange places with molecule B, assuming a position below B. The other gravity term is GD(BA), which expresses the probability that molecule B will occupy a position beneath molecule A. These rules are illustrated in Figure 9. In the absence of any strong evidence to support the model that the two gravity rules are complementary to each other in general, the treatment described above reflects the situation in which the gravity effects of A and B are two separate random events. Based

(c)

Figure 7 Illustration of the breaking parameter PB. (a) The breaking away of cell occupant A from cell B and movement to an unoccupied cell with probabilities: north 0.266, east 0.266, south 0.266, and west 0.000. (b) Occupant A breaking away from cell B and moving to an unoccupied cell with probabilities: north 0.000, east 0.320, south 0.320, and west 0.000. (c) Occupant A breaking away from cell B and moving to an unoccupied cell B with probabilities: north 0.000, east 0.512, south 0.000, and west 0.000. The parameters used are PBðABÞ ¼ 0:8; JðABÞ ¼ JðACÞ ¼ 1:0; absGðAÞ ¼ 0; and GDðABÞ ¼ 0:

on an assumption of complementarity, the equality GD(AB) ¼ 1 GD(BA) may be employed in the gravity simulation. Once again, the rules are probabilities of an event occurring. The choice of actuating this event made by each cell, in turn, is based on a random number selection within the boundaries used for that particular event. This process was mentioned in the earlier section on grid boundary cells.

218 Cellular Automata Models of Aqueous Solution Systems

B

B

(c)

Figure 8 Illustration of the joining parameter J. (a) Movement of cell occupant A in three directions with probabilities: north 0.153, east 0.421, south 0.266, and west 0.000. (b) Movement of A in two directions with probabilities: north 0.000, east 0.484, south 0.266, and west 0.000. (c) Movement of A in one direction with probabilities: north 0.000, east 0.677, south 0.000, and west 0.000. The parameters used are

PBðABÞ ¼ 0:8; JðABÞ ¼ 0:5; JðACÞ ¼ 2:0; absGðAÞ ¼ 0; and GDðABÞ ¼ 0.

Absolute Gravity

The absolute gravity measure of a molecule A, denoted by absG(A), is a nonnegative number. It is the adjustment needed in the computation to determine movement if A is to have a bias to move down (or up). This movement is shown in Figure 10.

Cell Rotation

In cases where an unsymmetrical cell is used (see Figure 4, for examples), it is necessary to insure there exists a uniform representation of all possible

Figure 9 Illustration of the relative gravity parameter. (a) Occupant A may move to an empty cell with probabilities: north 0.174, east 0.445, south 0.379, or west 0.000, or it may also exchange positions with cell B with probability 0.379. (b) The grid configuration after cell A exchanges with cell B. The parameters used are PBðABÞ ¼ 0:8; JðABÞ ¼ 0:5; JðABÞ ¼ 2:0; absGðAÞ ¼ 0; and GDðABÞ ¼ 1:5.

rotational states of that cell in the grid. To accomplish this, the unsymmetrical cells are randomly rotated 90 every iteration after beginning the run. This process is illustrated for four iterations in Figure 11.

Collection of Data

A cellular automata simulation of a dynamic system provides two classes of information. The first, a visual display, may be very informative of the char-

B

Figure 10 Illustration of the absolute gravity parameter. Occupant A may move to an empty cell with probabilities: north 0.153, east 0.421, south 0.421, or west 0.000. The parameters used are PBðABÞ ¼ 0:8; JðABÞ ¼ 0:5; JðACÞ ¼ 2:0; absGðAÞ ¼ 0; and GDðABÞ ¼ 0.

220 Cellular Automata Models of Aqueous Solution Systems

Figure 11 Illustration of rotation of variegated cells each iteration.

acter of a system as it develops from initial conditions. This visualization can be a dramatic portrayal of a process that opens the door to greater understanding of systems in the minds of students. The second source of information is the count of cells in different configurations such as those in isolation or those that are joined to another cell. This pattern of cells is called the configuration and is a rich source of information from which understanding of a process and the prediction of unforeseen events may be derived. These are illustrated in the following sections.

Number of Runs

It is customary to collect data from several runs, averaging the counts over those runs. The number of iterations performed depends on the system under study. The data collection may be over several iterations following the achievement of a stable or equilibrium condition. This stability is reckoned as a series of values that exhibit a relatively constant average value over a number of iterations. In other words, there is no trend observed toward a higher or lower average value.

Types of Data Usually Collected

From typical simulations used in the study of aqueous systems, several attributes are customarily recorded and used in comparative studies with properties. These attributes used singly or in sets are useful for analyses of different phenomena. Examples of the use and significance of these attributes will be offered in a later part of this chapter. The designations are

Figure 13 The tetrahedral pattern of a fragment of five water molecules in a cluster.

ensemble of discrete, orthogonal events occurring within that system. Other studies using this approximation have been reported.37,38

Significance of the Rules

The breaking and joining rules described above have a physical parallel in studies of water and solution phenomena. The breaking probability, PB(W), governs the self-affinity of a water molecule, W. This probability has a relationship to the boiling point, described by the equation:

The companion rule, J(W), reflects the affinity of two W molecules. For water, it is observed that the joining and breaking parameters are related as

These rules produce conditions in the cellular automata configuration that simulate physical properties.

The relationship of a water molecule, W, and a solute molecule, S, is governed by the parameters PB(WS) and J(WS). Higher values of PB(WS) and lower values of J(WS) reflect a weak interaction between water and the solute. The opposite pattern of parameters characterize a closer physical