Monte Carlo generation

Example 24 Consider as an example the set of 19 Monte Carlo generated, normally distributed values with a mean = 2.25 and a standard deviation = 1.43 used in Section 1.8.1 Table 1.24 is constmcted in six steps. The experimental (observed) frequencies are compared with the theoretical (expected) number. The critical x -value for p = 0.05 and/ = 4 is 9.49, thus no difference in distribution function is detected. Note that the first and last classes extend to infinity it might even be advisable to eliminate these poorly defined classes by merging them with the neighboring ones is found as... 

Simulation of the (n-Bu)3SnH reduction of PVC is carried out in a manner similar to that described for TCH. Instead of beginning with 100 TCH molecules we take a 1000 repeat unit PVC chain that has been Monte Carlo generated to reproduce the stereosequence composition of the experimental sample of PVC used in the reduction to E-V copolymers (2), ie. a Bernoullian PVC with P =0.45. At this point we have generated a PVC chain with a chain length and a stereochemical structure that matches our experimental starting sample of PVC. 

Figure 3.1. A typical Monte Carlo-generated tunneling path in the quartic double well. The smooth curve shows the quasiclassical solution.

The measured time spectra were then fitted using Monte Carlo generated time spectra for a grid of 65 kinetic energies between thermal energy and a few hundred eV. 

In this detailed balance condition 7r[x ° )(T) —> x T) is the probability to move from the old to the new path. According to the Monte Carlo generation and acceptance/rejection scheme, this probability is the product of a generation probability Pgen and an acceptance probability Pg.cc -... 

Using quantum chemical calculations for Monte-Carlo-generated conformers of... 

When dealing with realistic geometries, the accuracy of the computed view factors should be checked. For example, according to the enclosure rule (Chap. 7), the summation of view factors from an individual surface to the enclosure must equal unity in order to satisfy conservation of radiative energy. Sample predicted sums of the view factors from surface 1 (see Fig. 18.35) are shown in Table 18.6, along with the maximum error of A,F/ [171]. Monte Carlo-generated view factors are treated as the exact values, and only Monte Carlo-generated view factors whose estimated accuracy is better than 90 percent are used in the comparison exercise. 

Amorphous catalysts do not have the regular or structured morphology dealt with above. Yet, a more representative model of the structure is required for an accurate prediction of their performance, in particular when the structure is modified during its application, e.g., through pore blockage by poisons carried by the feed or by coke formed by the process itself. A pore medium can evidently be considered as a network of channels — preferably 3-dimensional — with a size distribution, but the disorder also has to be included. The problem is two-fold a structure has to be generated and the operation and performance of such a structure has to be described. Two frequently used methods are briefly described here the Monte-Carlo generation and simulation and the Effective Medium Approximation. 

Fig. 6. Pore size distributions of two different Lattice Monte Carlo-generated structures, (a) cylindrical SBA-15, (b) spherical MCF, (c) Pore size distribution of MCF material obtained by nitrogen...

Preliminary studies have been carried out to model the effects of physical adsorption of some of the chains onto the particle surfaces [70]. The goal was to determine the relative importance of the two major effects expected. These are the increase in the effective number of chains or cross links (which would certainly increase the elastic force, stress, and modulus), and the changes in the end-to-end distances of the chains that are adsorbed (which could conceivably either increase or decrease these elastomeric properties). Specifically, amorphous PE chains having 50 skeletal bonds were Monte Carlo generated in the presence of filler particles having 20 A diameters, with the first atom of each chain being attached to the particle surface. 

Figure 9.23. Seatterplot of model parameter eorrelations from Monte Carlo generated data for the data of Figure 9.21.

These values indicate that the Monte Carlo generated one dimensional probability bounds are almost identical to the 2-sigma standard error bounds. In addition, the joint bounds cover only slightly larger range for the C1-C2 and C2-C3 joint bounds. For this particular case, the standard error bounds provide a good estimate of the uncertainty with which the peak location can be determined from the experimental data and the model location. 

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