## 5 Mathematical Models
### Models at the Meso Scale
#### 5.2.3 Cellular Automata Model
Movements and other actions on the grid are controlled by rules that are only dependent on the characteristics of the cells that are closest to the ingredient. The neighborhood of a cell is its immediate surroundings. The _\"Von Neumann neighborhood\"_ is the most often employed neighborhood in two-dimensional CA investigations (see Figure 7d, where the blue cell neighborhood is pictured by the four red cells). Another common neighborhood is the _\"Moore neighborhood\"_, pictured by the red cells surrounding the blue cell in Figure 7c. Another useful neighborhood is the _\"extended Von Neumann neighborhood\"_, shown in Figure 7d by the red and green cells surrounding the blue cell. Every grid cell's value will be impacted by how its neighborhoods are managed, including cells that are on the edge of the CA grid. Keeping the values in those cells constant is one approach that might be taken. Using distinct neighborhood definitions for these cells is an additional strategy. They might have fewer neighbors, but it is also possible to define more elaborate strategies for the cells that are closest to the edges. These cells are typically handled in a toroidal manner; for instance, in a bi-dimensional grid, one cell \"goes off the top\" and \"enters\" at the corresponding position on the bottom, and one cell \"enters on\" the right when it \"goes off\" the left (this is sometimes referred to as periodic boundary conditions in the field of partial differential equations). To illustrate this, consider taping the rectangle's left and right sides to create a tube, and then the top and bottom edges to create a torus (see Figure 7b). **The rules:**: The behaviors of the ingredients on the grid, and consequently the evolutions of the CA systems, are governed by a variety of rules. According to what is described in [69], a list of potential rule types is provided below. _Movement rules_: Movement rules define the condition that, during an iteration, the ingredients move to neighborhood cells. These rules take several forms [120]:  
* The _\"breaking\"_ probability, \(P_{B}(AB)\in[0,\,1]\), defines the condition that two adjacent ingredients \(A\) and \(B\) may remain linked to each other. * The _\"joining\"_ parameter, \(J(AB)\in\Re^{+}\), defines the condition that, if two ingredients \(A\) and \(B\) are separated by an empty cell, ingredient \(A\) moves toward or away from ingredient \(B\). * The _\"free-moving\"_ probability \(P_{m}(A)\in[0,\,1]\) of an ingredient \(A\) defines the ingredient's propensity to move on the grid more rapidly or slowly. _Transition rules_: Transition rules define the condition that, during an iteration, an ingredient will transform to some other species. These rules take several forms [120]:  
* The _\"simple first-order transition\"_ probability \(P_{T}(AB)\in[0,\,1]\) defines the condition that an ingredient of species \(A\) will change to species \(B\). Figure 7: (a) A grid of cellular automata in two dimensions. Two sets of occupied cells with various constituents, \(A\) and \(B\), are displayed. The empty cells are left blank. (b) Movement of cell constituents at the grid boundaries on a torus’s surface. It is possible for one of the ingredients in the blue cell to move off the grid to the right and then return to the grid’s left border. It is possible for another component of red cells to shift from the bottom to the top of the grid. (c,d) Representation (see the red cells) of the Moore (c) and von Neumann (d) neighborhood, respectively, for the blue cell. Green cells in Figure (d) represent the extended von Neumann neighborhood for the blue cell. The \"_reaction\"_ probability \(P_{R}(AB\to CD)\in[0,\ 1]\) defines the condition that ingredients \(A\) and \(B\) will transform into ingredients \(C\) and \(D\), respectively, in case they \"_encounter\"_ each other during their movements in the grid. Other types of rules may also be considered [69; 120].