Simulation results approximate solution

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

Modem theoretical treatments of defects in semiconductors usually begin with an approximate solution of the Schrodinger equation appropriate to an approximate model of the defect and its environment (Pantelides, 1978 Bachelet, 1986). Both classes of approximation are described in the following subsection as they pertain to the computational studies addressed in this Chapter. If it were not necessary to make approximations, the computational simulation would faithfully reproduce the experimental result. This would be ideal, but unfortunately, it is not possible. As a consequence, contact with experiment is not always so conclusive or satisfying. A successful theory, however, may still extract from the computational results the important essential features that lead to simple and general models for the fundamental phenomena. 

Figure 2. MD simulation results for SD in response to electronic excitation of Cl 53 in room-temperature acetonitrile (left panel) and CO2 liquids. The solvent models and thermodynamic states are as in Ref. " and the solute model parameters are from Ref Nonequilibrium solvent response, S(f), and linear response approximations to it for the solute in the ground, Co(t), and excited, Q (f), electronic states are shown.

One way to bridge the gap between simple models used for insight, in the present case the Kramers model and its extensions, and realistic systems, is to use numerical simulations. Given a suitable force field for the molecule, the solvent, and their interaction we could run molecular dynamic simulations hoping to reproduce experimental results like those discussed in the previous section. Numerical simulations are also often used to test approximate solutions to model problems, for... 

In summary, the microscale description provides two important pieces of information needed for the development of mesoscale models. First, the mathematical formulation of the microscale model, which includes all of the relevant physics needed to completely describe a disperse multiphase flow, provides valuable insights into what mesoscale variables are needed and how these variables interact with each other at the mesoscale. These insights are used to formulate a mesoscale model. Second, the detailed numerical solutions from the microscale model are directly used for validation of a proposed mesoscale model. When significant deviations between the mesoscale model predictions and the microscale simulations are observed, these differences lead to a reformulation of the mesoscale model in order to improve the physical description. Note that it is important to remember that this validation step should be done by comparing exact solutions to the mesoscale model with the microscale results, not approximate solutions that result... 

The cell theory plus fluid phase equation of state has been extensively applied by Cottin and Monson [101,108] to all types of solid-fluid phase behavior in hard-sphere mixtures. This approach seems to give the best overall quantitative agreement with the computer simulation results. Cottin and Monson [225] have also used this approach to make an analysis of the relative importance of departures from ideal solution behavior in the solid and fluid phases of hard-sphere mixtures. They showed that for size ratios between 1.0 and 0.7 the solid phase nonideality is much more important and that using the ideal solution approximation in the fluid phase does not change the calculated phase diagrams significantly. 

A promising method, developed in recent years, is the use of first principles molecular dynamics as exemplified by the Car-Parrinello technique (8]. In these calculations the interatomic potentials are explicitly derived from the electronic ground-state within the density functional theory in local or non-local approximation. It combines quantum mechanical calculations with molecular dynamics simulations and, therefore, overcomes the limitations of both methods. Actual computers allow only simulations of aqueous solutions of about 60 water molecules for several ps (10 s). This limit is still at least one order of magnitude shorter than the fastest directly measured water exchange rate, k = 3.5 x 10 s for [Eu(H20)8], i.e. one exchange event every (8 x 3.5 x lO s ) = 36 ps [9]. Nevertheless, several publications appeared in the late 1990s on solvated Be [10], K+ [11] and Cu + [12] presenting mainly structural results. 

As computer power has increased it has become possible to incorporate explicitly some solvent molecules and thereby simulate a more realistic system. The simplest way to do this is to surroimd the molecule with a skin of solvent molecules. If the skin is sufficiently deep then the system is equivalent to a solute molecule inside a drop of solvent. The number of solvent molecules in such cases is usually significantly fewer than would be required in the analogous periodic boundary simulation, where the solute molecule is positioned at the centre of the ceU and the empty space is filled with solvent. Boundary effects should be transferred from the molecule-vacuum interface to the solvent-vacuum interface and so might be expected to result in a more realistic treatment of the solute. To illustrate these three situations, we can consider dihydrofolate reductase, which is a small enzyme that contains approximately 2500 atoms. If this enzyme is surrounded by water molecules in a cubic periodic system such that the surface of the protein is at least 10 A from any side of the box, then the number of atoms rises to almost 20000. If a shell 10 A thick is used then the number of atoms falls to 14 700, and with a 5 A shell the system contains 8900 atoms. 

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