## 5 Mathematical Models
### Models at the Meso Scale
#### 5.2.1 Monte Carlo-Based Methods  
The collection of algorithms, based on Markov Chain Monte Carlo (MCMC) methods, are universally appreciated for their power and easy implementation beyond their versatility and flexibility. Such relevance inspired Dongarra et al. [110] to call the original Metropolis algorithm [111], which is the first implementation of an MCMC method, one of the ten most important algorithms of the twentieth century. For the story of the evolution of MCMC methods, we suggest reading the paper by Matthew Richey [112] that is also relevant for the accessible and synthetic introduction to MCMC methods. MCMC is a broad computing method for assessing sums and integrals or estimates of energy or other macro-parameters in a physical or biological system [113]. The original MCMC method, the so-called _\"Metropolis algorithm\"_, was inspired by nature and arose in physics. Its creators were aware that they had defined  
_...a general method, suitable for fast electronic computing machines, of calculating the properties of any substance which may be considered as composed of interacting individual molecules [111]._  
The original Metropolis algorithm proposed an approach to solve the basic problem of averaging in classical equilibrium statistical mechanics. Such an approach was intended to compute averages of any physical observable \(F\) using an arithmetic mean of the values of \(F\) obtained in different sampled states of a system composed of \(N\) particles. The sampling schema described in [111] was based on an approach that later was discovered to be a Marchov Chain stochastic process [112]. Let us briefly describe some of the fundamental ideas behind the Marchov Chain. **Definition 1**. : Markov Chains  
_Given a finite state space \(S=\{1,2,...,N\}\), a Markov Chain is a stochastic process defined by a sequence of random variables, \(X_{i}\in S,\ i\in\mathbb{N}\) such that_  
\[Prob(X_{k+1}=x_{k+1}|X_{1}=x_{1},\ldots,X_{k}=x_{k})=Prob(X_{k+1}=x_{k+1}|X_{k }=x_{k}). \tag{65}\]  
Equation (65) expresses the fact that the probability of being in a particular state at the \((k+1)\)-th step only depends on the state at the \(k\)-th step. When Markov Chains are considered for which this dependence is independent of \(k\) (that is, time-homogeneous Markov Chains), a \(N\times N\) transition matrix \(\mathbf{P}=\big{(}\mathbf{p}_{i,j}\big{)}\) can be defined by  
\[\mathbf{p}_{i,j}=Prob(X_{k+1}=j|X_{k}=i). \tag{66}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & _“Pros”_ & _“Cons”_ \\ \hline \multicolumn{3}{c}{An approach where the behaviors of the species under study are determined by rules which can be probabilistic or deterministic. CA could be 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.} \\ \multicolumn{3}{c}{The cellular automata method (Section 5.2.3)} \\ \multicolumn{3}{c}{An approach where the behaviors of the species under study are determined by rules which can be probabilistic or deterministic. CA could be 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.} \\ \multicolumn{3}{c}{The cellular particle dynamics (CPD) (Section 5.2.2)} \\ \multicolumn{3}{c}{The cellular particle dynamics (CPD) (Section 5.2.2)} \\ \multicolumn{3}{c}{The originant dynamics (CPD) (Section 5.2.2)} \\ \multicolumn{3}{c}{The originant dynamics (CPD) (Section 5. Note that  
\(1.\quad\sum_{i=1}^{N}\mathbf{p}_{i,j}=1\). \(2.\quad\) The \((i,j)\)-entry of the \(K\)-th power of \(\mathbf{P}\) gives the probability of transitioning from state \(i\) to state \(j\) in \(K\) steps. A Markov Chain is  
_Irreducible_ if, for all states \(i\) and \(j\), there exists \(K\) such that \(\big{(}\mathbf{P}^{K}\big{)}_{i,j}\
eq 0\).