Computational methods Monte Carlo

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

Such Bayesian models could be couched in terms of parametric distributions, but the mathematics for real problems becomes intractable, so discrete distributions, estimated with the aid of computers, are used instead. The calculation of probability of outcomes from assumptions (inference) can be performed through exhaustive multiplication of conditional probabilities, or with large problems estimates can be obtained through stochastic methods (Monte Carlo techniques) that sample over possible futures. 

Mezei, M., Excess free energy of different water models computed by Monte Carlo methods, Mol. Phys. 1982, 47, 1307-1315... 

This section has illustrated a relatively simple application of the Monte Carlo technique for simulating atmospheric diffusion. With the availability of large-scale computing capacities, Monte Carlo methods can be envi-... 

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)...

The same computer revolution that started in the middle of the last century also plays an important, in fact crucial, role in the development of methods and algorithms to study solvation problems. Dealing, for instance, with a liquid system means the inclusion of explicit molecules, in different thermodynamic conditions. The number of possible arrangements of atoms or molecules is enormous, demanding the use of statistical mechanics. Here is where computer simulation, Monte Carlo (MC) or molecular dynamics (MD), makes its entry to treat liquid systems. Computer simulation is now an important, if not central, tool to study solvation phenomena. The last two decades have seen a remarkable development of methods, techniques and algorithms to study solvation problems. Most of the recent developments have focused on combining quantum mechanics and statistical mechanics using MC or... 

Followed by the rapid development of computer power, Monte Carlo (MC) and molecular dynamics (MD) simulation methods have been applied to many fields so as to connect the microscopic interaction model with the macroscopic properties, such as pVT relation, phase equilibria and so on [6]. They have also been used to analyze the adsorption characteristics of supercritical fluid [7-9] however, the simulation studies for adsorption phenomena in supercritical fluid mixtures are still limited. 

For this reason, computer simulation methods (Monte Carlo and molecular and Brownian dynamic methods) have been developed not only for solving the problems of polymer statistics, but also for the investigation of the dynamic properties and the intramolecular mobility of polymers. 

Computer simulations, Monte Carlo or molecular dynamics, in fact appear to be the actual most effective way of introducing statistical averages (if one decides not to pass to continuous distributions), in spite of their computational cost. Some concepts, such as the quasi-structure model introduced by Yomosa (1978), have not evolved into algorithms of practical use. The numerous versions of methods based on virial expansion, on integral equation description of correlation functions, on the application of perturbation theory to simple reference systems (the basic aspects of these... 

The technique has been described in Section 10.15. In summary, a model of solvation is decided upon. The computer uses Monte Carlo or molecular dynamics methods and a simulation of the solvation pattern emerges. The beauty of the method lies in the capacity to vary the model at will by varying the type and number of interactions considered for each model. This will give a simulation for each set of conditions which can then be compared with each other and with results from all the experimental methods described earlier. In effect the computer is used to help find a model which fits experiment. 

Several methods have been developed for generating models of the structure of amorphous materials at the atomic level. These include random network models (which may be made by hand or, as discussed later, generated on a computer), energy minimization (or relaxation) methods, Monte Carlo (MC), Molecular Dynamics (MD), Reverse Monte Carlo (RMC), and Quantum Mechanical (QM) methods. The aim of this chapter is to discuss the generation of models for amorphous solids mainly using the most widely used and successful MD and RMC techniques, giving examples from the literature. 

The basic idea behind an atomistic-level simulation is quite simple. Given an accurate description of the energetic interactions between a collection of atoms and a set of initial atomic coordinates (and in some cases, velocities), the positions (velocities) of these atoms are advanced subject to a set of thermodynamic constraints. If the positions are advanced stochastically, we call the simulation method Monte Carlo or MG [10]. No velocities are required for this technique. If the positions and velocities are advanced deterministically, we call the method molecular dynamics or MD [10]. Other methods exist which are part stochastic and part deterministic, but we need not concern ourselves with these details here. The important point is that statistical mechanics teUs us that the collection of atomic positions that are obtained from such a simulation, subject to certain conditions, is enough to enable aU of the thermophysical properties of the system to be determined. If the velocities are also available (as in an MD simulation), then time-dependent properties may also be computed. If done properly, the numerical method that generates the trajectories... 

M, Mezei, Mol. Phys., 47, 1307 (1982). Excess Free Energy of Different Water Models Computed by Monte Carlo Methods. 

The problem of predicting the thermodynamic properties of such a classical system becomes the problem of evaluation of the configuration integral, the integral over exp(— O). This is still a difficult task. In general, it can be done only through computer simulations (Monte Carlo and molecular dynamics methods). However, there are a few simple approximations which are helpful. 

Membrane filtration is an important technology for ensuring the purity, safety and/or efiticiency of the treatment of water or effluents. In this study, various types of membranes are reviewed, first. After that, the states of the computational methods are applied to membranes processes. Many studies have foeused on the best ways of using a particular membrane process. But, the design of new membrane systems requires a considerable amoimt of proeess development as well as robust methods. Monte Carlo and molecular dynamics methods ean specially provide a lot of interesting information for the development of polymer/carbon nanotube membrane processes. 

Kremer K 1996 Computer simulation methods for polymer physics Monte Carlo and Molecular Dynamics of Condensed Matter Systems vol 49, ed K Binder and G Ciccotti (Bologna Italian Physical Society) pp 669-723... 

Swendsen R H 1993 Modern methods of analyzing Monte Carlo computer simulations Physica A 194 53-62... 

A comprehensive introduction to the field, covering statistical mechanics, basic Monte Carlo, and molecular dynamics methods, plus some advanced techniques, including computer code. 

Agranovich V M, Efremov N A and Kirsanov V V 1980 Computer simulation of kinetics of excitation bimolecular quenching by Monte-Carlo method Fiz. Tverd. Tela 22 2118-27... 

What has been developed within the last 20 years is the computation of thermodynamic properties including free energy and entropy [12, 13, 14]. But the ground work for free energy perturbation was done by Valleau and Torrie in 1977 [15], for particle insertion by Widom in 1963 and 1982 [16, 17] and for umbrella sampling by Torrie and Valleau in 1974 and 1977 [18, 19]. These methods were primarily developed for use with Monte Carlo simulations continuous thermodynamic integration in MD was first described in 1986 [20]. 

The basic scheme of this algorithm is similar to cell-to-cell mapping techniques [14] but differs substantially In one important aspect If applied to larger problems, a direct cell-to-cell approach quickly leads to tremendous computational effort. Only a proper exploitation of the multi-level structure of the subdivision algorithm (also for the eigenvalue problem) may allow for application to molecules of real chemical interest. But even this more sophisticated approach suffers from combinatorial explosion already for moderate size molecules. In a next stage of development [19] this restriction will be circumvented using certain hybrid Monte-Carlo methods. 

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