Monte Carlo method constant number

An analysis of (5.98), in the case where the number of inequalities considerably exceeds the number of constants Tk/T, shows [42] that it is possible to make use of the Monte-Carlo method, i.e. one must generate by... 

Fig.12. Computation by Monte Carlo methods of the first four order parameters of an ensemble of 1000 chromophores (of dipole moment 13 Debye) existing in a medium of uniform dielectric constant. At the beginning of the calculation, the chromophores are randomly ordered thus, ==O. During the first 400 Monte Carlo steps, an electric poling field (600 V/micron) is on but the chromophore number density (=10 7 molecules/cc) is so small that intermolecular electrostatic interactions are unimportant. The order parameters quickly evolve to well-known equilibrium values obtained analytically from statistical mechanics (black dots in figure also see text). During steps 400-800 the chromophore number density is increased to 5xl020 and intermolecular electrostatic interactions act to decrease order parameters consistent with the results of equilibrium statistical mechanical calculations discussed in the text. Although Monte Carlo and equilibrium statistical mechanical approaches described in the text are based on different approximations and mathematical methods, they lead to the same result (i.e., are in quantitative agreement)...

Many computational studies of the permeation of small gas molecules through polymers have appeared, which were designed to analyze, on an atomic scale, diffusion mechanisms or to calculate the diffusion coefficient and the solubility parameters. Most of these studies have dealt with flexible polymer chains of relatively simple structure such as polyethylene, polypropylene, and poly-(isobutylene) [49,50,51,52,53], There are, however, a few reports on polymers consisting of stiff chains. For example, Mooney and MacElroy [54] studied the diffusion of small molecules in semicrystalline aromatic polymers and Cuthbert et al. [55] have calculated the Henry s law constant for a number of small molecules in polystyrene and studied the effect of box size on the calculated Henry s law constants. Most of these reports are limited to the calculation of solubility coefficients at a single temperature and in the zero-pressure limit. However, there are few reports on the calculation of solubilities at higher pressures, for example the reports by de Pablo et al. [56] on the calculation of solubilities of alkanes in polyethylene, by Abu-Shargh [53] on the calculation of solubility of propene in polypropylene, and by Lim et al. [47] on the sorption of methane and carbon dioxide in amorphous polyetherimide. In the former two cases, the authors have used Gibbs ensemble Monte Carlo method [41,57] to do the calculations, and in the latter case, the authors have used an equation-of-state method to describe the gas phase. 

Effects of the gas - solid potential corrugation on the behaviour of monolayers formed on the (100) face of an fee crystal at finite temperatures have been recently studied by Patrykiejew et al. [163] with the help of Monte Carlo method. They have considered three-dimensional systems of constant volume and containing fixed number of particles interacting via the Lennard-Jones potential (1). The gas - solid interaction potential has been assumed to be represented by the two-fold Fourier series [88]... 

It is worthy of note that An has similarities with the Madelung constant or crystal structures. It is an easy electrostatic problem to verify that, in the limit of a spherically symmetric surface charge distribution, Aqo has the value 0.5. In the results of reference [8], An has been minimized for fixed Z and i by a Monte-Carlo method. Values of An for Boron cages are tabulated in reference [8] for numbers of atoms n from 30 to 54. All we need to note here is that these numerical results can be fitted by... 

The simulation procedure for the constant number Monte Carlo method applied to coalescence processes consists of the following key steps ... 

Standard molecular dynamics calculations, i.e., those that solve Hamilton s equation, are performed on NVE ensembles, i.e., samples with a constant number of atoms N), fixed volume (V), and constant energy ( ). In standard Monte Carlo simulations the more widely applicable NVT ensembles are used, i.e., constant temperature (T) rather than energy, although both schemes can be modified to work in different ensembles. In particular, free energies can be directly evaluated using Monte Carlo methods in the Grand Canonical ensemble, although technical difficulties involved... 

From a theoretical point of view, the Lion et al. model has the merit to approach the DSS effect by applying constitutive laws formulated on the basis of fractional calculus, in other terms by formulating the behavior of materials with respect to fractional time derivatives of stress and strain an approach that in principle requires only a small number of material constants to express the material properties in the time or the frequency domain. However, deriving model parameters from experimental data is not straightforward and, for instance Lion et al. had to use a stochastic Monte Carlo method to estimate the model parameters for a comparison with experimental data on 60 phr CB filled rubber compound. Moreover, mathematical handling of the above equations (see Appendix 5.5) shows that, like the Kraus model, this one exhibits also horizontal symmetry for the G curve and vertical symmetry for the G" curve, and is therefore not expected to perfectly meet experimental data, at least in its present state of development. 

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