Stochastic models description

The above-mentioned stochastic model description considers an amorphous system of N matrix units, containing a small concentration of dye molecules. Each matrix unit will shift the electronic absorption line of the dye molecule by some amoimt v Ri where Ri is the position of the matrix unit with respect to the dye molecule. The contributions of all matrix units are assumed to be additive. The total inhomogeneous distribution of absorption lines can then be written as [2-4] ... 

Standard procedures that are used for testing of construction materials are based on square pulse actions or their various combinations. For example, small cyclic loads are used for forecast of durability and failure of materials. It is possible to apply analytical description of various types of loads as IN actions in time and frequency domains and use them as analytical deterministic models. Noise N(t) action as a rule is represented by stochastic model. 

So far, the effects of the chain ends were neglected in our stochastic model for the restricted chain. Therefore, n must be much larger than the number of steps needed to form the largest excluded polygon. The partition function, which incorporates the chain-end effects and which could be also employed for exact statistical description of short non-self-intersecting chains can be obtained as follows Assume, as before, that we eliminate only lowest-order polygons of t steps. Therefore, the first t — 1 steps in the chain are described as a sequence of independent events. Eq (9), then, will be replaced by... 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

The results obtained for the stochastic model show that surface reactions are well-suited for a description in terms of the master equations. Since this infinite set of equations cannot be solved analytically, numerical methods must be used for solving it. In previous Sections we have studied the catalytic oxidation of CO over a metal surface with the help of a similar stochastic model. The results are in good agreement with MC and CA simulations. In this Section we have introduced a much more complex system which takes into account the state of catalyst sites and the diffusion of H atoms. Due to this complicated model, MC and in some respect CA simulations cannot be used to study this system in detail because of the tremendous amount of required computer time. However, the stochastic ansatz permits to study very complex systems including the distribution of special surface sites and correlated initial conditions for the surface and the coverages of particles. This model can be easily extended to more realistic models by introducing more aspects of the reaction mechanism. Moreover, other systems can be represented by this ansatz. Therefore, this stochastic model represents an elegant alternative to the simulation of surface reaction systems via MC or CA simulations. 

Their theory, based on the classical Bloch equations, (31) describes the exchange of non-coupled spin systems in terms of their magnetizations. An equivalent description of the phenomena of dynamic NMR has been given by Anderson and by Kubo in terms of a stochastic model of exchange. (32, 33) In the latter approach, the spectrum of a spin system is identified with the Fourier transform of the so-called relaxation function. 

Stochastic modeling is used when a measurable output is available but the inputs or causes are unknown or cannot be described in a simple fashion. The black-box approach is used. The model is determined from past input and output data. An example is the description of incomplete mixing in a stirred tank reactor, which is done in terms of contributions of dead zones and short circuiting. In these cases, a sequence of output called a time series is known, but the inputs or causes are numerous and not known in addition, they may be unobservable. Though the causes for the response of the system are unknown, the development of a model is important to gain understanding of the process, which may be used for future planning. 

Lombardo and Bell (1991) reviewed stochastic models of the description of rate processes on the catalyst surface, such as adsorption, diffusion, desorption, and surface reaction, which make it possible to account for surface structure of crystallites, spatial inhomogeneities, and local fluctuations of concentrations. Comparison of dynamic MC and mean-field (effective) description of the problem of diffusion and reaction in zeolites has been made by Coppens et al. (1999). Gracia and Wolf (2004) present results of recent MC simulations of CO oxidation on Pt-supported catalysts. 

As analyzed in the preceding chapters concerning the description of a process evolution, stochastic modelling follows the identification of principles or laws related to the process evolution as well as the establishment of the best mathematical equations to characterize it. 

The stochastic modelling of the phenomena studied here can be described by one standard physical model (descriptive model) which can be defined by the following statement ... 

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. 

If we consider the example described at the beginning of this chapter, the element of study in stochastic modelling is the particle which moves in a trajectory where the local state of displacement is randomly chosen. The description for this discrete displacement and its associated general model, takes into consideration the... 

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. 

J. Mai, V. N. Kuzovkov, and W. von Niessen, A General Stochastic Model for the Description of Surface Reaction Systems, Physica A, 203... 

Z. Alexandrowicz,/. Chem. Phys., 55,2765 (1971). Stochastic Models for the Statistical Description of Lattice Systems. 

For example, the output rate of a simple SISO reactor depends on various conditions. To model the transformation of input to output, knowledge about the chemical reaction and the chemical reactor can be used. E.g., a linear model might be used to describe this relationship properly on an aggregated level (e.g. the hourly production rate). Neglecting minor influences leads to a simplification of the process model. Additionally, measurement errors may hinder a perfect description of the process and lead to uncertainty in the observed process measures. This uncertainty is expressed e.g. by a (normal) error process. The resulting linear regression model can be verified using historical records of the process. Often historical records allow analysts to deduce a proper stochastic model of such a process. For more complex production processes more sophisticated stochastic models (as described in section 2.3) can be necessary. 

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