Monte Carlo Statistics

The fact that the number of simulated events is limited leads to a systematic uncertainty on the measured -quark production cross-section. The extent of this effect can be estimated by considering the results of the validation of the fitting procedure (Fig. 4.15). An overview of the relative error of the fitted -fraction is presented in Appendix B. The error of the fitted -fraction takes into account the limited MC statistics as well as the limited data statistics. 

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A complete set of intermolecular potential functions has been developed for use in computer simulations of proteins in their native environment. Parameters have been reported for 25 peptide residues as well as the common neutral and charged terminal groups. The potential functions have the simple Coulomb plus Lennard-Jones form and are compatible with the widely used models for water, TIP4P, TIP3P and SPC. The parameters were obtained and tested primarily in conjunction with Monte Carlo statistical mechanics simulations of 36 pure organic liquids and numerous aqueous solutions of organic ions representative of subunits in the side chains and backbones of proteins... 

C. P. Robert and G. Casella, Monte Carlo Statistical Methods, Springer, New York, 2004. 

The quantum/classical procedures recover the nuclear fluctuation properties of the surrounding medium via the Monte Carlo statistical approach or by using molecular dynamics simulations. In the following section we examine the problem of energy exchange between solute and solvent from a quantum dynamical viewpoint. 

The theoretical model describes the break up of the coal macromolecular network under the influence of bond cleavage and crosslinking reactions using a Monte Carlo statistical approach (32-38). A similar statistical approach for coal decomposition using percolation theory has been presented by Grant et al. (39). 

The addition of Me3C+ to Me2C=CH2 has been investigated hr the gas phase and in solution with ab initio calculations and Monte Carlo statistical mechanics simulations. The reaction is exothermic by 20 kcal mol-1 and proceeds without activation energy in the gas phase. By contrast, solvation introduces a 3 1 kcal mol 1 barrier at a C C separation near- 5 A in CH2CI2, THF, and MeOH. An intermediate with a shallow energy well was found at near 3 A separation. Implications for sterol biosynthesis were also discussed in this paper 44... 

A different approach simulates the thermodynamic parameters of a finite spin system by using Monte Carlo statistics. Both classical spin and quantum spin systems of very large dimension can be simulated, and Monte Carlo many-body simulations are especially suited to fit a spin ensemble with defined interaction energies to match experimental data. In the case of classical spins, the simulations involve solving the equations of motion governing the orientations of the individual unit vectors, coupled to a heat reservoir, that take the form of coupled deterministic nonlinear differential equations.23 Quantum Monte Carlo involves the direct representation of many-body effects in a wavefunction. Note that quantum Monte Carlo simulations are inherently limited in that spin-frustrated systems can only be described at high temperatures.24... 

Robert CP, Casella G (2004) Monte Carlo statistical methods 2nd edn. Springer, New York... 

We suggest that a picture of this type can also include the dynamics of the alcohols if the features of the hydrogen bonding in these liquids are taken into account. A Monte Carlo statistical simulation of liquid methanol and ethanoP gives the following restilts ... 

Methods Using 3D Descriptors Advances in quantum-chemical calculations and the increasing power of personal computers have made a great impact on the development of methods to predict aqueous solubility directly from the 3D structures of molecules. Monte Carlo statistical mechanics simulations by Jorgensen and Duffy [45] were used to predict the solubility of 150 compounds (MAE = 0.56) using the equation... 

Robinson and Dalton use Monte Carlo statistical mechanics to explore concentration and shape dependencies of the chromophores. Monte Carlo methods provide valuable information about the distribution of a collection of chromophores but are not able to provide atomistic information about the systems. The Monte Carlo simulations performed by Robinson and Dalton employ an array of point dipoles on a periodic lattice with the given parameters for the shape of the chromophores and the chromophore spacing adjustable to achieve the desired chromophore concentration. The model system consisted of 1000 chromophores on a body-centered cubic... 

Robinson, B.H., Dalton, L.R. Monte carlo statistical mechanical simulations of the competition of intermolecular electrostatic and poling-field interactions in defining macroscopic electro-optic activity for organic chromophore/polymer materials, J. Phys. Chem. A 104(20), 4785 795 (2000)... 

Figure 10.14 Dependence of the scaling exponent a on temperature, (a) Variation of a with kgT/iiE for Regime I (squares) and Regime II (circles). Error bars represent Monte Carlo statistical errors. The solid line curve represents the fitting for Regime II given by Eqn. 10.31. (b) Overall behavior of a for different temperatures, represented through fe T/AE. Curves for the intermediate regime are obtained by application of Eqn. 10.31 to Eqn. 10.30, while circles represent Monte Carlo results for w/AE = 0.3. Adapted firom Ref. 32.

Computer simulation techniques have been applied to such solution systems. The Monte Carlo statistical mechanics have provided much useful information about the energetics, structure, and molecular interactions. The computations suggested that at the hexanol-water interface minimal water penetration into the hydrocarbon regions takes place. 

Data gaps can often be addressed by using a combination of exposure reconstruction techniques simultaneously this approach is common, and there are many examples of it in the peer-reviewed literature. A number of these studies are discussed and compared below. The use of a combination of methods may also be the most effective approach for exposure reconstruction for a particular scenario, given a unique dataset with certain robust elements and otho- relatively weak elements. Uncertainty analyses, such as a Monte Carlo statistical assessment, can also be used to address data gaps generated by uncertainty in existing data or information, as well as to increase the likelihood that true exposures are captured (Cohen Hubal et al. 

Chan, H.M., P.R. Berti, O. Receveur, and H.V. Kuhnlein. 1997. Evaluation of the population distribution of dietary contaminant exposure in an Arctic population using Monte Carlo statistics. Environ. Health Perspect. 105(3) 316-321. 

Interesting information can be obtained from maps of the Monte Carlo statistical distribution of the molecules in a cluster as illustrated by Figure 2. 

A wide variety of different models of the pure water/solid interface have been investigated by Molecular Dynamics or Monte Carlo statistical mechanical simulations. The most realistic models are constructed on the basis of semiempirical or ab initio quantum chemical calculations and use an atomic representation of the substrate lattice. Nevertheless, the understanding of the structure of the liquid/metal surface is only at its beginning as (i) the underlying potential energy surfaces are not known very well and (ii) detailed experimental information of the interfacial structure of the solvent is not available at the moment (with the notable exception of the controversial study of the water density oscillations near the silver surface by Toney et al. [140, 176]). 

Casella, G., and Robert, C. (2004). Monte Carlo Statistical Methods, 2nd ed. New York Springer-Verlag. Cho, H., et al. (2007). Induction of dendritic cell-like phenotype in macrophages during foam cell formation. Physiol. Genomics, 29 149-160. 

As an example, such a methodology has been applied by Essex and Jorgensen to the calculation of the dielectric constant of formamide and dimethylformamide using Monte Carlo statistical mechanics simulations (see Section 8.7.2 for details). The simulation result for dimethylformamide, 32 2, is reasonably in agreement with respect to the experimental value, 37. However, in the case of formamide, the obtained value, 56 2, underestimates the experimental value of 109.3. The poor performance here addresses the fact that force field models with fixed charges underestimate the dielectric constant for hydrogen-bonded liquids. 

V11.15. Computational methods that directly solve forms of the Boltzmann transport equation to obtain k j are preferred for use in the criticality safety analysis. The deterministic discrete ordinates technique and the Monte Carlo statistical technique are the typical solution formulations used by most criticality analysis codes. Monte Carlo analyses are prevalent because these codes can better model the geometry detail needed for most criticality safety analyses. Well documented and weU validated computational methods may require less description than a limited-use and/or unique... 

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