## 5 Mathematical Models
### Models at the Meso Scale
#### 5.2.3 Cellular Automata Model  
The previous sections discussed a number of standard categories of physical models. These models govern the evolution of physical systems formalized in terms of coupled sets of ordinary or partial differential equations. Most conventional models are deterministic, which means that the force fields that are employed and the initial conditions of the simulations fully determine the results of running simulations. This section offers an alternative approach where the behaviors of the species under study are determined by rules, which can be probabilistic or deterministic, as opposed to forces and energy. The foundation of this approach is the use of cellular automata (CA) [120] postulated for the first time by the mathematical physicist John von Neumann [121] and the mathematician Stanislaw Ulam [122], more than 70 years ago. Since then, different research teams have proven that CA is particularly significant in managing the behaviors of systems where the traditional, differential-equation-based methodologies are either useless or difficult to adopt due to the numerous complex interconnections between the system constituents. According to the mathematician Stephen Wolfram, the CA model can be defined as follows:  
_CA are simple mathematical idealizations of natural systems. They consist of a lattice of discrete identical sites, each site taking on a finite set of... values. The values of the sites evolve in discrete time steps according to... rules that specify the value of each site in terms of the values of neighboring sites. CA may thus be considered as discrete idealizations of the partial differential equations often used to describe natural systems_[123]. According to Kier et al. [120], a model made up of the following parts is referred to as a _cellular automaton_ (singular):  
* A grid of individual cells. * A group of ingredients. * A collection of local rules dictating how the constituents behave. * Identified starting conditions. A simulation can be run after the model's aforementioned elements have been defined. The system in the simulation changes through a sequence of discrete time steps, or iterations, when all of the system's components are subjected to the model's rules, and the system's configuration is updated as a result. Either synchronous or asynchronous rule application might lead to CA development. In the first scenario, every automaton cell changes on each iteration; in the second scenario, the algorithm merely processes a subset of the cells or even just one cell. One way to model the complex behavior of macroscopic systems, which arises from the interaction of all components on a small scale, is through the synchronous application of transition rules. However, there are serious practical implementation limitations associated with this approach, such as the potential for conflicts in certain situations (e.g., assigning two ingredients to move to the same empty cell). Due to this latter factor, the majority of automata currently in development can only be implemented asynchronously [124] (see algorithm described in Section 6.2.2 for the asynchronous version of the CA simulation). The above general CA framework can be used to construct a wide range of various models. The components of this CA model are then described in more detail. **The grid, the cells, and the ingredients:**: A cellular automaton consists of a regular grid of \(N_{Cells}\) cells. The grid can be in any finite number of dimensions \(N\). Every grid cell can typically exist in a limited number \(N_{States}\) of \"states\", which specify the cell's occupancy. The cell may be empty or hold a specified ingredient, which, if it is present, may be a type of particle, a specific molecule, or some other relevant thing for the topic under consideration. An illustration of a bi-dimensional CA grid with \(5\times 5\) cells each, whereby some cells are occupied by ingredient \(A\) (red cells) and other cells by ingredient \(B\) (blue cells), can be found in Figure 7a.