The Monte Carlo and Molecular Dynamics Methods

Therefore, an infinite number of correct transition probability density sets, which differ from that of Eq. [34], can be defined. Notice that p-j depends on the ratio Pj/Pf hence it does not depend on the partition function Z, which is canceled. This simplification is the main strength of the MC method, making it very versatile. For example, if the distribution P is not Boltzmannian but proportional to w(x)exp[-E(x)/k T], where w(x) is a known function of the coordinates, one can generate a sample according to this PD with p,. satisfying 

Note that the normalization factor of P is canceled. As discussed later, this feature is useful in umbrella sampling (see also the discussion preceding and following Eqs. [32] and [33]). 

Whereas the MC method selects configurations correctly with the Boltzmann PD, it does not provide the value of this PD, and therefore the absolute entropy cannot be obtained in a direct manner with Eq. [29]. The difficulty stems from the fact that if the simulation starts from configuration i and after t MC steps reaches /, one knows the probability density of the specific avenue i j that was chosen in the simulation. However, to obtain the PD of / one has to sum up the probability densities of all the large number of possible avenues / in t MC steps. Yet as mentioned in the Introduction, P (x) [like E(x)j is also written on the configurations x and can be deciphered approximately using the local states method or the hypothetical scanning method (discussed later). A direct estimation of S (hence F) by Eq. [29] becomes possible with these methods. 

Molecular dynamics, like MC, is a dynamical procedure, but of a deterministic rather than stochastic nature. One starts from an arbitrary configuration and an initial set of particle velocities, and Newton s equations of motion of the system are integrated numerically as a function of time this time (unlike the MC time) corresponds to the real time. For fluids, MC and MD have comparable efficiency. For dense materials like proteins, MD is the more efficient because the random MC trial moves are rejected with high probability unless the moves are very small. The difficulty in obtaining the entropy with MC (discussed above) applies also to MD. 

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State, in less than 50 words, the essential principles behind the Monte Carlo and molecular dynamics methods of calculating the numerical values of pheuomeua in liquids. Why is it that such methods need prior experimental determinations in nearby systems ... 

The problem of HI as formulated in Section 8.3 can, in principle, be studied by various simulation techniques. Such an approach encounters some serious difficulties, however, in addition to those listed in Chapter 6, in connection with the study of pure liquid water. It is well known that the available techniques of simulation, such as the Monte Carlo and molecular dynamics methods, are not suitable for the computation of the free energy or the entropy of a system [see, for example. Wood (1968)]. Since the HI problem has been formulated in terms of a difference in the free energy of the system for two configurations of a pair of solutes, its direct computation by these methods is at present unfeasible. 

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

The Monte Carlo and molecular dynamics simulation methods can be used to explor the conformational space of molecules. During such a simulation the system is able t... 

Molecular simulation techniques, namely Monte Carlo and molecular dynamics methods, in which the liquid is regarded as an assembly of interacting particles, are the most popular... 

Describe the basic principles of the Monte Carlo and molecular dynamic simulation methods applied to electrolytes. How are the adjustable parameters determined ... 

As an alternative to the normal-mode method, Monte Carlo and molecular dynamics calculations have been performed on small clusters. Monte Carlo and molecular dynamics methods have the virtue of being exact, within calculable error bars, subject to the constraint of the approximate intermo-lecular interactions that are used. Prior to about six years ago both methods were restricted to systems projjerly described by classical mechanics. This restriction implied that systems for which tunneling or low-temjjerature vibrations were important at best could be treated approximately. 

Exact determination of entropy effects in enzymatic reactions is not an easy task even nowadays when sophisticated Monte Carlo and molecular dynamics methods are available for calculations (Warshel, 1991 Aqvist and Warshel, 1993). One way to examine the importance of entropy is to analyse the configuration space available to the system in its ground and transition states, both in the enzyme and in solution. The entropic contribution to the catalytic effect, relative to the uncatalysed solution can be expressed as... 

A number of studies have been conducted, however, involving the conformational searching of metal complexes. The geometries of flexible side chains have been investigated using molecular dynamics [247], and conformational searching using Monte Carlo and molecular dynamics methods has been carried out on transition metal complexes [72] and metalloproteins [248]. 

The first part of this chapter contains a short introduction to statistical mechanics of continuum models of fluids and macromolecules. The next section presents a discussion of basic sampling theory (importance sampling) and the Metropolis Monte Carlo and molecular dynamics methods. The remainder of the chapter is devoted to descriptions of methods for calculating F and S, including those that were mentioned above as well as others. 

This chapter will focus on a simpler version of such a spatially coarse-grained model applied to micellization in binary (surfactant-solvent) systems and to phase behavior in three-component solutions containing an oil phase. The use of simulations for studying solubilization and phase separation in surfactant-oil-water systems is relatively recent, and only limited results are available in the literature. We consider a few major studies from among those available. Although the bulk of this chapter focuses on lattice Monte Carlo (MC) simulations, we begin with some observations based on molecular dynamics (MD) simulations of micellization. In the case of MC simulations, studies of both micellization and microemulsion phase behavior are presented. (Readers unfamiliar with details of Monte Carlo and molecular dynamics methods may consult standard references such as Refs. 5-8 for background.)... 

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. 

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