Results of Computer Experiments

At low enough values of M the calculated values of the parameter Uq agree well with the results of computer experiments. For large M the relationships of (7.4.11) lead to a rapid growth in Uo(M), which qualitatively agrees with experiment. However, the no calculated by (7.4.12) increases with increasing M more rapidly than in the computer calculations, as explained by the mentioned approximation for determining rk-... 

This Section deals with the problem of MEIS comparison with the models of motion that was studied in the previous Section. However, whereas comparison was performed there on the basis of purely theoretical analysis, here it was made on the examples of specific objects. Compared are the attainable completeness and significance of the results of computing experiments, and the possibility of using these results, accuracy of the obtained estimates for the sought characteristics of the modeled system, laboriousness of calculations and preparation of initial information. 

Section II is devoted to reviewing the basic ideas of the RMT, having in mind a linear chain of particles coupled with each other via linear interactions. Section III is devoted to illustrating the results of computer experiments that we did to supplement those of the interesting paper of Bishop et al. with additional information as to whether or not the non-Gaussian features of the velocity variable are the same as those of the real three-dimensional fluids. We shall show that these are qualitatively similar, although in the one-dimensional case the non-Gaussian character is more intense and much more persistent. 

The results of computer experiments indicate that the virtual potential must be softer than the linear approximation. However, a simple analytical expression for this, which is also sound from a physical point of view, can be obtained only in the rotational case, where complete agreement between theory and experiment can be achieved. Comparably good agreement in the translational case requires further investigation. 

First, the results of computer experiments with a cubic piece of ice (Figures 2—4 and Table 1) will be examined. 

The results of computer experiments are often used to test approximate theories of liquids such as those based on the OZ equation discussed above. For this reason they are an important part of understanding the structure of liquids and how structure affects liquid properties. 

The values thus obtained are in better agreement with the results of computer experiments. 

As a result of computer experiments with model porous networks, the factors ( other than pore shape and size distribution ) determining the form of the adsorption-desorption hysteresis loop have been elucidated. 

Equations (7.11) and (7.14) predict adequately within H the results of computer experiments and, therefore, numerical integration of the dynamic model to compute the concentrations of ethane and ethylene can be replaced within ft by the empirical models (7.11) and (7.14). Developing similar equations for all experimental responses, parameter estimation for dynamic models is reduced to a more trivial problem. (Miller and Frenklach, 1983). 

A common approximation in many flow field computations at high fluid velocities is to consider that inertial forces dominate the flow and to neglect viscous forces (inviscid approximation). Since solvent viscosity is a variable in some of the experiments discussed here, the above approximation may be not be valid throughout and viscous forces are explicitly considered in the flow equations. Results of computations showed, nevertheless, that even with viscous solvents such as bis-(2-ethyl-hexyl)-phtalate with qi = 65 mPa s, viscous forces do not affect the flow field unless tbe fluid velocity drops below a few m s"1 at the orifice. This limit is generally more than one order of magnitude lower than the actual range used in the present investigations. 

It may turn out that the mathematical model is too rough, meaning numerical results of computations are not consistent with physical experiments, or the model is extremely cumbersome for everyday use and its solution can be obtained with a prescribed accuracy on the basis of simpler models. Then the same work should be started all over again and the remaining stages should be repeated once again. 

The variables 17, Ua, and are the corresponding uncertainty values for each parameter. They are computed to the 67% confidence interval by taking the standard error of each parameter in the regressions (i.e., ai, U2 and (73, and multiplying by their Student f-score ts (i.e., =ts SEo ), where ts is the Student t-score at the confidence level of interest and SEai is the corresponding standard error for the parameter ai. The period can be chosen based on the maximum value or another statistical parameter. The results of four experiments are given in Figure 9.8. 

A molecular dynamics trajectory is computed for methyl-terminated PIB at 400 K. Several time-dependent properties (mean-square end-to-end distance, averaged bond angles, and the number and locations of rotational isomeric states) deduced from the trajectory are in reasonable agreement with the results of earlier experiments and earlier theoretical investigations of the static properties of this polymer. 

A field experiment was conducted at the Canadian Air Forces Base Borden, Ontario, to study the behavior of organic pollutants in a sand aquifer under natural conditions (Mackay et al., 1986). Figure 25.9 shows the results of two experiments, the first one for tetrachloroethene, the second one for chloride. Both substances were added as short pulses to the aquifer. The curves marked as ideal were computed according to Eqs. 25-20 or 25-23. The measured data clearly deviate from the ideal curve. The nonideal curves were constructed by Brusseau (1994) with a mathematical model that includes various factors causing nonideal behavior. 

To test the hypotheses (7.4.17) and (7.4.18), the kinetics of accumulation was simulated on a computer by the method described in [110]. For each of the values vp = 10,16,24, and 50, the process of accumulation was performed independently 200 times until the stage of steady-state values of no was reached. The relationships n(N), N = pt, and a(n) were constructed from the mean values obtained in this series. It was shown that within the limits of error of computer experiment ( 5%), the slowly varying function a(n) can be well approximated by the linear dependence of (7.4.18), which confirms the suitability of this approach for describing the accumulation of point defects in the discrete model. Analogous results are obtained for vp = 16 and 50 for which the values were found respectively, of 1.092 and 1.625 for n0 and 0.463 and 0.478 for f3(oo) = a(oo)vono. 

Yet, apart from the purely quantitative criterion, one should also take into consideration general reasoning. When comparing the discussed methods, we note that the er 2 vs. E plot is frequency-independent indeed, this is because a is frequency-independent by definition [see Eq. (2)]. This plot is believed to be the most universal for comparing results of different experiments because the contribution, coming from the frequency-dependent factor proper, is separated in this case. Unfortunately, the dimensionality of a precludes a correct comparison of the results thus computed with the directly measured capacitance values published in the literature. 

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