The stochastic model

The state of the lattice site I (activated or unactivated) is represented by the lattice variable 7 with 

We define S = (7/) as the mean value of the activity of the catalyst and this is independent of I.  

We introduce a variable for the particles 0 0, H,N,A, B where 0 represents a vacant site, A represents a NH particle and B a NH2 particle. The state of a lattice point 7/ consists of the state of the catalyst (activated or unactivated) and its coverage with a particle. This leads to the following possible states  

The H and N particles are created on the surface by adsorption from of the gas phase with the rates 

At normal temperatures H atoms are very mobile on metal surfaces. We take this into account by the possibility of diffusion steps for the H atoms. A H atom jumps with rate Djz onto the nearest neighbour sites on the lattice. If this site is occupied by N, reaction occurs and an A particle (NH particle) is formed. The same holds if the site is occupied by A or B (NH2) where the products B or NH3 = 0 are formed, respectively. NH3 desorbs immediately from the surface and an empty site is formed. That is we deal with the diffusion-limited reaction system. It is important to note that all the reaction steps discussed above (with the exception of the N2-adsorption) are independent of 7/ and 7m. 

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The performance of the scheduler can be significantly improved by the use of a stochastic model. The stochastic model used here considers not only the probability distribution of the uncertain parameters but also the structure of decisions and observations that result from the moving horizon scheme. 

The mathematical model is often thought of as being composed of two parts (Mikhail, 1976) the functional model, and the stochastic model. 

Stochastic Model The stochastic model, on the other hand, designates and describes the nondeterministic or stochastic (probabilistic) properties of the variables involved, particularly those representing the measurements. 

Figure 11. Basic equations used in the stochastic model.

The greater number of folds in larger proteomes is intuitively obvious simply because the functioning of more complex organisms is expected to require a greater structural diversity of proteins. From a different perspective, the increase of diversity follows from a stochastic model, which describes a proteome as a finite sample from an infinite pool of proteins with a particular distribution of fold fractions ( a bag of proteins ). A previous random simulation analysis suggested that the stochastic model significantly (about twofold) underestimates the number of different folds in the proteomes (Wolf et al., 1999). In other words, the structural diversity of real proteomes does not seem to follow... 

The formulation outlined above allows for a simple stochastic implementation of the deterministic differential equation (35). Starting with an ensemble of trajectories on a given adiabatic PES W, at each time step At we (i) compute the transition probability pk k, (h) compare it to a random number ( e [0,1], and (iii) perform a hop if pt t > C- In Ih se of a pure A -level system (i.e., in the absence of nuclear dynamics), the assumption (37) holds in general, and the stochastic modeling of Eq. (35) is exact. Considering a vibronic problem with coordinate-dependent however, it can be shown that the electronic... 

As a consistency test of the stochastic model, one can check whether the percentage Nk t) of trajectories propagating on the adiabatic PES Wk is equal to the corresponding adiabatic population probability Pf t). In a SH calculation, the latter quantity may be evaluated by an ensemble average over the squared modulus of the adiabatic electronic coefficients [cf. Eq. (22)], that is. 

Combining these equation, the consistency condition of the stochastic model reads... 

The results from Fig. 14.1 show the developing turbulent flame zone. The nonsymmetries of the reaction rate field are due to inhomogeneity of the polydis-persed mixture, i.e., nonsymmetrical distribution of model particles and their velocities. The reaction front is under formation oxygen and partially the volatiles in the center are burnt out, but the reaction front is not sphere-shaped yet. The nonuniformity of the model particles distribution was induced initially due to the stochastic modeling of the particulate phase. 

To ensure that the original information structure associated with the decision process sequence is honored, for each of the products whose demand is uncertain, the number of new constraints to be added to the stochastic model counterpart, replacing the original deterministic constraint, corresponds to the number of scenarios. Herein lies a demonstration of the fact that the size of a recourse model increases exponentially since the total number of scenarios grows exponentially with the number of random parameters. In general, the new constraints take the form ... 

To interpret the solutions obtained from the stochastic models, we propose to investigate their corresponding coefficient of variation Cv. Cv for a set of values is defined as the ratio of the standard deviation to the expected value or mean and is usually expressed as a percentage. It is calculated as ... 

A 5% standard deviation from the mean value of market demand for the saleable products in the LP model is assumed to be reasonable based on statistical analyses of the available historical data. To be consistent, the three scenarios assumed for price uncertainty with their corresponding probabilities are similarly applied to describe uncertainty in the product demands, as shown in Table 6.2, alongside the corresponding penalty costs incurred due to the unit production shortfalls or surpluses for these products. To ensure that the original information structure associated with the decision process sequence is respected, three new constraints to model the scenarios generated are added to the stochastic model. Altogether, this adds up to 3 x 5 = 15 new constraints in place of the five constraints in the deterministic model. 

The stochastic model with recourse in the previous section takes a decision merely based on first-stage and expected second-stage costs leading to an assumption that the decision-maker is risk-neutral (Sahinidis, 2004). In order to capture the concept of risk in stochastic programming, Mulvey, Vanderbei and Zenios (1995) proposed the following amendment to the objective function ... 

Table 8.1 shows the stochastic model solution for the petrochemical system. The solution indicated the selection of 22 processes with a slightly different configuration and production capacities from the deterministic case, Table 4.2 in Chapter 4. For example, acetic acid was produced by direct oxidation of n-butylenes instead of the air oxidation of acetaldehyde. Furthermore, ethylene was produced by pyrolysis of ethane instead of steam cracking of ethane-propane (50-50 wt%). These changes, as well as the different production capacities obtained, illustrate the effect of the uncertainty in process yield, raw material and product prices, and lower product... 

This implies that if it were possible to know the future realization of the demand, prices and yield perfectly, the profit would have been 2 724 040 instead of 2 698 552, yielding savings of 25 488. However, since acquiring perfect information is not viable, we will merely consider the value of the stochastic solution as the best result. These results show that the stochastic model provided an excellent solution as the objective function value was not too far from the result obtained by the WS solution. 

However, as mentioned in the previous section, the stochastic model takes a decision based on first-stage and expected second-stage costs, and, hence, does not account for the decision-maker risk behavior (risk-averse or risk taker). For this reason, a more realistic approach would consider higher moments where the tradeoff between the mean value and the variations of different scenarios is appropriately reflected. 

The results of the model considered in this Chapter under uncertainty and with risk consideration, as one can intuitively anticipate, yielded different petrochemical network configurations and plant capacities when compared to the deterministic model results. The concepts of EVPI and VSS were introduced and numerically illustrated. The stochastic model provided good results as the objective function value was not too far from the results obtained using the wait-and-see approach. Furthermore, the results in this Chapter showed that the final petrochemical network was more sensitive to variations in product prices than to variation in market demand and process yields when the values of 0i and 02 were selected to maintain the final petrochemical structure. 

Table 9.3 Computational results with SAA for the stochastic model.

Before leaving the topic of modelling, it is pertinent to note that, at room temperature, the deterministic model [9] and the stochastic model [10b] both predict that ca. 50% ofg(H2) comes from the sum of reactions (R4) and (R5), and negligible amounts from reaction (R6). The values for the deterministic model are given in Table 1. The corresponding values for the stochastic model are 0.24 and 0.15 molecules (100 eV) respectively [10,b,], and 0.28 and 0.14 molecules (100 eV) respectively [20]. On the other hand, g(H202) is predicted to be almost entirely accounted for by reactions (R8) and (R9) in both types of model. 

The sources of g(H2) according to deterministic [9] and stochastic [10] modelling are processes (I), (III), and (V) and also the spur reactions (R4) (R5) (R6). As shown in Table 1, the deterministic model indicates that approximately 50% of g(H2) is produced in reactions (R4) and (R5), and similar results are predicted by the stochastic model [10b,20]. The models also predict that the extent of reactions (R4) and (R5) increases with LET. For an increase in LET from 0.2 to 60 eV nm g(H2) increases to 0.27 and 0.42 molecules (100 eV) due to reactions (R4) and (R5), respectively [21]. These data, together with numerous experimental results, are not consistent with the conclusion [17] that H2 is generated mainly through reaction of with H2O. There will, of course, be no produced under conditions where e is scavenged efficiently in sufficiently concentrated solution. However, it does not follow that reactions (R4) and (R5) do not contribute to g(H2) in dilute solution or pure water. 

In this section, we discuss briefly how the Langevin equation, which is a stochastic equation, can be derived from the molecular equations of motion. The stochastic model described by the Langevin equation has been of great use in interpreting a large number of experiments and physical systems. The stochastic model is extremely simple but, as always, its ultimate justification rests on the molecular dynamical laws. 

Recently, the stochastic models for the Mossbauer line shape problem have been discussed by several investigators.20 Such models can be treated in a systematic way as we have described in the above. For example, in a 57Fe nucleus, the spin in the excited state is / = and that in the ground state is / = i, so that the Hamiltonian is a 6 x 6 matrix. If a two-state-jump model is adopted, the dimension of the matrix equation, Eq. (63), is 6 x 2 = 12. If the stochastic operator is of the type (26), then the equation is a set of six differential equations. These equations can be solved, if necessary, by computers to yield the line shape functions for various values of parameters. 

Let the random variable Y(f) be the number of A molecules in the system at time f. The stochastic model is then completely defined by the following assumptions. 

Note that the mean value of the stochastic representation is the deterministic result, showing that the two representations are consistent in the mean. We shall see later that this is true only for unimolecular reactions. The stochastic model, however, also gives higher moments and so fluctuations can now be included in chemical kinetics. One sees that the stochastic approach is to chemical kinetics as statistical thermody-... 

Just as in the unimolecular cases, the basis for the stochastic approach is to consider the reaction 2A-> B as being a pure death process with a continuous time parameter and transition probabilities for the elementary events that make up the reaction process. Letting the random variable X(t) be the number of A molecules in the system at time t, the stochastic model is then completely defined by the following assumptions ... 

As for the multidimensional freely jointed chain, it is possible to relate a to the parameters which describe a Rouse chain by evaluating the translational diffusion constant D for the center of mass. In the stochastic model, we determine the square of the displacement per unit time of a single bead averaged over an equilibrium ensemble. For bead j,... 

We assume that all molecules spend total time tm in the mobile phase. This is the retention time of unretained solute. Important results of the stochastic model are ... 

The stochastic model applies to processes involving the stationary phase. To analyze the chromatogram, we need to subtract contributions to peak broadening from dispersion in the mobile phase and extra-column effects such as finite injection width and finite detector volume. These effects account for the width of the unretained peak. To subtract the unwanted effects, we write... 

Since these characteristics are time-dependent, let us assume particle birth-death and migration to be the Markov stochastic processes. Note that making use of the stochastic models, we discuss below in detail, does not contradict the deterministic equations employed for these processes. Say, the equations for nv t), Xu(r,t), Y(r,t) given in Section 2.3.1 are deterministic since both the concentrations and joint correlation functions are defined by equations (2.3.2), (2.3.4) just as ensemble average quantities. Note that the... 

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. 

In this Section we focus our attention on the development of the formalism for complex reactions with application to the formation of NH3. The results obtained (phase transition points and densities of particles on the surface) are in good agreement with the Monte Carlo and cellular automata simulations. The stochastic model can be easily extended to other reaction systems and is therefore an elegant alternative to the above-mentioned methods. 

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