Polystochastic Process

The stochastic models can present discrete or continuous forms. The former discussion was centred on discrete models. The continuous models are developed according to the same base as the discrete ones. Example 4.3.1 has already shown this method, which leads to a continuous stochastic model. This case can be gen- 

We can notice that relation (4.68) describes the evolution of the particles having reached position z in time t -i- At which were originally positioned among the particles at the distance v At with respect to z. In the interval of time At, their velocity changes to Vj. In the majority of the displacement processes with v velocity, a complete system of events appears and, consequently, the matrix of passage from one velocity to another is of the stochastic type. This means that the addition of 

The development described above transforms system (4.69) into the following system of n equations (k= 1, n) with partial derivatives  

If the parameters a j have constant values, then the model described by system (4.71) corresponds to a Markov connection linking the process components. In this case, as in general, the process components represent the individual displacements which can be characterized globally through the convective mixing of their spectra of speeds ( v, k= 1,N). 

For k = 3, Vj = Vj,, V2 = -Vj, V3 = 0 (at V3 = 0 the particle keeps a stationary position) we have the model (4.74)-(4.76) which has been successfully used in the analysis of axial mixing for a fluid that flows in a packed bed column [4.28]  

---

Mathematical Models of Continuous and Discrete Polystochastic Processes... 

Mathematical Models of Contimious and Discrete Polystochastic Processes 231... 

The diffusion model can usually be used for the description of many stochastic distorted models. The equivalent transformation of a stochastic model to its associated diffusion model is fashioned by means of some limit theorems. The first class of limit theorems show the asymptotic transformation of stochastic models based on polystochastic chains the second class is oriented for the transformation of stochastic models based on a polystochastic process and the third class is carried out for models based on differential stochastic equations. 

When a stochastic model is described by a continuous polystochastic process, the numerical transposition can be derived by the classical procedure that change the derivates to their discrete numerical expressions related with a space discretisation of the variables. An indirect method can be used with the recursion equations, which give the links between the elementary states of the process. 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

Polystochastic models are used to characterize processes with numerous elementary states. The examples mentioned in the previous section have already shown that, in the establishment of a stochastic model, the strategy starts with identifying the random chains (Markov chains) or the systems with complete connections which provide the necessary basis for the process to evolve. The mathematical description can be made in different forms such as (i) a probability balance, (ii) by modelling the random evolution, (iii) by using models based on the stochastic differential equations, (iv) by deterministic models of the process where the parameters also come from a stochastic base because the random chains are present in the process evolution. 

A second continuous polystochastic model can be obtained from the transformation of the discrete model. As an example, we consider the case of the model described by Eqs. (4.62) and (4.63). If P (z,t) is the probability (or, more correctly, the probability density which shows that the particle is in the z position at time t with a k-type process) then, p j is the probability that measures the possibility for the process to swap, in the interval of time At, the elementary process k with a new elementary process (component) j. During the evolution with the k-type process state, the particle moves to the left with probability and to the right with probability (it is evident that we take into account the fact that Pk + Yk ) Por this evolution, the balance of probabilities gives relation (4.77), which is written in a more general form in Eq. (4.78) ... 

By computing the values of the generator function for 0 —> 0 (relations (4.123) and (4.124)), we can observe similarities (identities) between both relations. Indeed, we corroborate that these functions come from a process with identical behaviour and we have a correct asymptotic transformation of the original model. We can conclude that in the case when the transition matrix of probabilities has a regular state, the generator function of the polystochastic chain process when n —> goes from one generator function to a Markov chain related with the model that is, for the present discussion, characterized by relation (4.123)... 

All other discrete stochastic models, obtained from polystochastic chains, attached to an investigated process, present the capacity to be transformed into an asymptotic model. When the original and its asymptotic model are calculated numerically, we can rapidly observe if they converge by direct simulation. In this case, the comparison between the behaviour of the original model and the generator function of the asymptotic stochastic model is not necessary. 

For the derivation of one asymptotic variant of a given polystochastic model of a process, we can use the perturbation method. For this transformation, a new time variable is introduced into the stochastic model and then we analyze its behaviour. The new time variable is t = eT, which includes the time evolution t and an arbitrary parameter e, which allows the observation of the model behaviour when its values become very small (e—>0). Here, we study the changes in the operator 0(t, t) when e 0 whilst paying attention to having stable values for t/e or t/e. ... 

---