Models of Continuous and Discrete Polystochastic Processes

Mathematical Models of Continuous and Discrete Polystochastic Processes 

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. 

Some ideas and rather simple concepts, which are fundamental for the alphabet of stochastic modelling, will be described here for some particular cases. It is 

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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. 

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) ... 

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