Monte Carlo results

Figure A2.3.12 The osmotic coefficient of a 1-1 RPM electrolyte compared with the Monte Carlo results of...

Figure A2.3.13 The excess energy of 1-1, 2-1, 3-1 and 2-2 RPM electrolytes in water at 25°C. The frill and dashed curves are from the HNC and MS approximations, respectively. The Monte Carlo results of Card and Valleau [63] for the 1-3 and 2-2 charge types are also shown.

FIG. 21 Dependence of the average density on the configurational chemical potential. The solid line denotes the grand canonical Monte Carlo data, the long dashed fine corresponds to the osmotic Monte Carlo results for ZL = 40, and the dotted line for ZL = 80. (From Ref. 172.)... 

FIG. 19 Scaling plot for the relaxation of the mean chain length L t) after a T-jump from Tq = 0.35 to a series of final temperatures, given as a parameter along with the respective L o s. The same Monte Carlo results [64] as in Fig. 5 are used. Full line denotes the scaling function f x = = (0.215 + 8x) . In the inset the... 

Equilibrium Systems. Magda et al (12.) have carried out an equilibrium molecular dynamics (MD) simulation on a 6-12 Lennard-Jones fluid In a silt pore described by Equation 41 with 6 = 1 with fluid particle Interactions given by Equation 42. They used the Monte Carlo results of Snook and van Me gen to set the mean pore density so that the chemical potential was the same In all the simulations. The parameters and conditions set In this work were = 27T , = a, r = 3.5a, kT/e = 1.2, and... 

The results of the simple DHH theory outlined here are shown compared with DH results and corresponding Monte Carlo results in Figs. 10-12. Clearly, the major error of the DH theory has been accounted for. The OCP model is greatly idealized but the same hole correction method can be applied to more realistic electrolyte models. In a series of articles the DHH theory has been applied to a one-component plasma composed of charged hard spheres [23], to local correlation correction of the screening of macroions by counterions [24], and to the generation of correlated free energy density functionals for electrolyte solutions [25,26]. The extensive results obtained bear out the hopeful view of the DHH approximation provided by the OCP results shown here. It is noteworthy that in... 

The first analysis is one with AS-level precision, the second with TIMS-level precision. The first order 2a error for the resulting 331 ka age is 96 ka, but examination of the distribution of a Monte Carlo simulation (Fig. 2) shows that the actual age distribution is strongly asymmetric, with 95% confidence limits of 158/-79 ka. For either younger ages or more-precise analyses, however, the first-order age errors are more than adequate, as shown by the Monte Carlo results for the same data, but with TIMS-level precision (Fig. 2B). 

Figure 3 shows the profiles induced in a bulk system by an applied temperature gradient. These Monte Carlo results [ 1 ] were obtained using the static probability distribution, Eq. (246). Clearly, the induced temperature is equal to the applied temperature. Also, the slopes of the induced density and energy profiles can be obtained from the susceptibility, as one might expect since in the linear regime there is a direct correspondence between the slopes and the moments [1]. 

Figure 3. Induced temperature (top), number density (middle), and energy density (bottom) profiles for Pj = 0.0031 and To = 2, p = 0.5. The symbols are Monte Carlo results using the static probability distribution, Eq. (246), and the lines are either the applied temperature or the profiles predicted from the simulated susceptibility. (From Ref. 1.)...

Figure 4. Susceptibility of the energy moment at To — 2. The symbols are static Monte Carlo results [1] and the curve is obtained from a local thermodynamic approximation [1] using the bulk susceptibilities from a Lennard-Jones equation of state [90], (From Ref. 1.)...

Figure 7. Nonequilibrium Monte Carlo results for the thermal conductivity (To = 2). The circles and squares are the present steady-state results for bulk and inhomogeneous systems, respectively (horizontally offset by 0.015 for clarity), and the triangles are NEMD results [89, 91]. (From Ref. 5.)...

It is clear that the different procedures of handling reactive products are based on different approximations therefore, somewhat different results are expected. On the whole, since the IRT methodology is based on the conceived independence of pairwise bimolecular reactions, it needs validation by comparison with well-known examples for which Monte Carlo results are available. Such validations have in fact been made (see Figures 4 and 7 in Clifford et ah, 1986, and Figures 12, 27, and 28 in Pimblott and Green, 1995). 

Besides we have shown the possibility to apply the results of model calculations to some experimental data and consider a simple approximate approach to the calculation of dimensions of partially cross-linked coils and cross-linking kinetics. The accuracy of this approximation is evaluated bv comparison with Monte Carlo results. 

Some readers may have seen pharmaceutical portfolio analysis designed and implemented in a spreadsheet or other computational applications, including the ability to compute Monte Carlo results. While these approaches can indeed be of value, our clients have seen a number of additional insights into pipeline d5mamics, which only a dynamic modeling simulation can provide. 

Monte Carlo sampling is discussed extensively in Hammersley and Morton (1956), Hammersley and Handscomb (1964), Kloek and Van Dijk (1978), and Wilson (1984). For Monte Carlo results to be believable, the convergence properties of the Monte Carlo estimators must be met. Several statistical and practical limitations exist in this regard. The most important practical limitations of Monte Carlo are the following ... 

For example, the distribution from which the samples are drawn is assumed to be the true distribution of the parameter of interest. To the degree that the sample distribution differs from the actual distribution (which is generally assumed unknown by the classical statistician), the confidence in the Monte Carlo results is decreased. Just how close these distributions must be is a complicated statistical issue that is frequently unclear. In a practical sense, if misspecification of a sampling distribution occurs for a very sensitive parameter in a multiparameter model, then the confidence in the Monte Carlo results is greatly diminished because the model prediction is greatly influenced by that parameter. 

The analyst is better equipped to combine the Monte Carlo results with other analyses in a complex risk framework (e.g., combining exposure and effects distributions into a risk distribution). 

Determine whether or not actual measurements fall within the model prediction error. This may be the best way to justify the Monte Carlo result. 

The output is similar to the previous simulation except that the deviations from the nominal value are smaller. Only one run did not pass the specification. The run had a lower deviation of 0.0115, which corresponds to a gain of 0.5 - 0.0115 = 0.4885. Smaller deviations should be expected since, in the Gaussian distribution, the bulk of the resistors were within plus or minus one standard deviation. From the results of the two previous simulations, we conclude that the tolerance distribution has a large impact on the Monte Carlo results. 

The Monte-Carlo method is utilized to investigate the conformational distribution in the central section of a PIB decamer at various temperatures. It is checked that a six-state RIS model based on the two matrices P and Pj constitutes a description of the conformational distribution in PIB. The Monte-Carlo results are in excellent agreement with the experimental data on the average dimensions of PIB chains, as well as with the molecular scattering functions of this polymer in solution and in bulk. 

Monte Carlo results for the average frequency spacing between maxima . J. Acoust. Soc. Am., 34(1) 76—80. 

The SM2/AM1 model was used to examine anomeric and reverse anomeric effects and allowed to state that aqueous solvation tends to reduce anomeric stabilization [58]. Moreover, SM2/AM1 and SM3/PM3 models were accounted for in calculations of the aqueous solvation effects on the anomeric and conformational equilibria of D-glucopy-ranose. The solvation models put the relative ordering of the hydroxymethyl conformers in line with the experimentally determined ordering of populations. The calculations indicated that the anomeric equilibrium is controlled primarily by effects that the gauche/trans 0-C6-C5-0 hydroxymethyl conformational equilibrium is dominated by favorable solute-solvent hydrogen bonding interactions, and that the rotameric equilibria were controlled mainly by dielectric polarization of the solvent [59]. On the other hand, Monte Carlo results for the effects of solvation on the anomeric equilibrium for 2-methoxy-tetrahydropyran indicated that the AM1/SM2 method tends to underestimate the hydration effects for this compound [60]. 

Ebert U, Baumgartner A, Schafer L (1996) Universal short-time motion of a polymer in a random environment Analytical calculations, a blob picture, and Monte Carlo results. 

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