Monte Carlo method procedure

This section is used to introduce the momentum-enhanced hybrid Monte Carlo (MEHMC) method that in principle converges to the canonical distribution. This ad hoc method uses averaged momenta to bias the initial choice of momenta at each step in a hybrid Monte Carlo (HMC) procedure. Because these average momenta are associated with essential degrees of freedom, conformation space is sampled effectively. The relationship of the method to other enhanced sampling algorithms is discussed. 

The difficulty arises from the fact that the one-step transition probabilities of the Markov chain involve only ratios of probability densities, in which Z(N,V,T) cancels out. This way, the Metropolis Markov chain procedure intentionally avoids the calculation of the configurational integral, the Monte Carlo method not being able to directly apply equation (31). 

The conformation of atactic polymers, with any value of the m/r ratio, must be treated as that of a copolymer, wherein the monomer unit statistics are compounded with those of the rotational states. In this case we may either refer to Monte-Carlo type procedures, as done by Floiy, Mark, and Abe (194), or to the pseudostereochemical equilibrium method used by Allegra (195) and Briick-ner (196). In the latter case the atactic polymer is formally considered as a homopolymer that may assume the conformations of either the m or r dyads, with suitable adjusted statistical weights. 

The experimental method and apparatus, and a procedure of the Monte Carlo method that simulates the geminate ion recombination are described, and the time-dependent distribution is elucidated. 

Monte Carlo method a procedure for solving problems by constructing an artificial model representing a process and performing sampling experiments on it. 

The synthesis of occurrences or events in the Monte Carlo method makes use of random numbers and a cumulative-distribution function. In effect the random numbers are transformed, by means of the distribution function, into a simulated sequence of events. Figure 2 shows the general procedure followed. A random number is selected and transformed... 

The Monte Carlo method is especially suited for use on a digital computer, particularly one of the stored-program type. The mathematical model and the distribution function, even if quite complicated, can be expressed on the computer and the necessary calculations are highly repetitive. Also, random numbers (or rather pseudorandom numbers) can be synthesized so that the computer procedure becomes fully automatic and self-contained (M9, S5). 

The fit of the model packing fraction to the macroscopic density is the essential point of our model, as already mentioned. That is why we chose a Monte-Carlo method to obtain two-dimensional liquid-like distance statistics of hard discs. The procedure we used is exactly the same as used by Metropolis et al. [14] with the addition of averaging a large number of system configurations. 

However, it is normally assumed that the conformers that bind to target sites will be those with a minimum potential energy. Since molecules may have large numbers of such metastable conformers a number of techniques, such as the Metropolis Monte Carlo method and comparative molecular field analysis (CoMFA), have been developed to determine the effect of conformational changes on the effectiveness of docking procedures. 

Over the past ten years the numerical simulation of the behavior of complex reaction systems has become a fairly routine procedure, and has been widely used in many areas of chemistry, [l] The most intensive application has been in environmental, atmospheric, and combustion science, where mechanisms often consisting of several hundred reactions are involved. Both deterministic (numerical solution of mass-action differential equations) and stochastic (Monte-Carlo) methods have been used. The former approach is by far the most popular, having been made possible by the development of efficient algorithms for the solution of the "stiff" ODE problem. Edelson has briefly reviewed these developments in a symposium volume which includes several papers on the mathematical techniques and their application. [2]... 

For some variables, for example, the relative collision velocity, the cumulative distribution function does not have closed form, and then a third Monte Carlo method must be adopted. Here, another random number R is used to provide a value of v, but a decision on whether to accept this value is made on the outcome of a game of chance against a second random number. The probability that a value is accepted is proportional to the probability density in the statistical distribution at that value. The procedure is repeated until the game of chance is won, and the successful value of v is then incorporated into the set of starting parameters. 

The amount of Monte Carlo selection that has been employed in different studies has varied. For example, Blais and Bunker [48, 305] used a complete Monte Carlo procedure in their studies of the K + CH3I reaction, although the distribution of one or more parameters could be suppressed, allowing them to observe how particular results depended on different features of the collisions. On the other hand, Karplus et al. [20] adopted a rather different approach in their investigation of the H + Ha system. A batch of trajectories was calculated with particular values of v, b and vibrational and rotational energies of H2. The remainder of the variables were chosen by Monte Carlo methods. The vibrational and rotational energies corresponded to individual rotational states in the zero-point vibrational level. By averaging the results... 

The remaining problem is to evaluate the terms (2.12) for all collision processes to be considered. Due to the special notation chosen here, however, these terms are already exactly in the format to which Monte Carlo kinetic particle transport codes can be applied directly. The probabilistic formulation is particularly suitable for these procedures. We refer to standard literature on Monte Carlo methods for linear transport, such as [19]. Here it is only important to note that one may write equation (2.12) as linear functional of the neutral particle distribution function / ... 

The shape of a zeolite sorption uptake isotherm, a quantitation of the amount of a given sorbate taken up as a function of its partial pressure in the gas phase in equilibiitun with the zeolite sorbent, depends both on the zeolite sorbate interaction and on the sorbate - sorbate interactions. Simulation of such isotherms for one or more sorbates is accomplished by the Grand Canonical Monte Carlo method. Additional to the molecular reorientation and movement attempts is a particle creation or annihilation, the probability of which scales with the partial pressure [100,101]. This procedure thus simulates the eqmlibrium between the sorbed phase in the zeolite and an infinite gas / vapor bath. Reasonable reproduction of uptake isotherms for simple gases has been achieved for a small number of systems (e.g. [100,101]), and the molecular simulations have, for example, explained at a molecular level the discontinuity observed in the Ar - VPI-5 isotherm. 

Many quality assurance procedures in diagnostic radiology require the use of a phantom to simulate the X-ray attenuation of the patient. The phantom should transmit the same quantity and quality (i.e. spectrum) of radiation as that transmitted by the patient. The knowledge of photon spectra at the position of measurement on the surface, inside and behind standard dosimetric and imaging phantoms is helpful for performing dosimetric or calibration measurements and hence helpful in the course of quality control. The experimental approach is limited however, simulation with Monte Carlo methods has been proved to be a very powerful technique (Petoussi et al., 1992). 

Various deterministic and stochastic sampling techniques for path ensembles have been proposed [4-6]. Here we consider only Monte Carlo methods. It is important, however, to be aware that while the path ensemble is sampled with a Monte Carlo procedure each single pathway is a fully dynamical trajectory such as one generated by molecular dynamics. 

In the more realistic calculations of radiative exchange in furnaces and combustion chambers, a non-isothermal gas space has to be considered. H.C. Hottel and A.F. Sarohm [5.48] developed the so-called zone method for this case, cf. also [5.37], p. 647-652. Other procedures for the consideration of the temperature fields in the gas space have been extensively dealt with by R. Siegel and J.R. Howell [5.37], Chapter 15. The application of the Monte-Carlo method is suggested in particular, cf. [5.37] and [5.66], which despite being mathematically complex, produces results without making highly simplified assumptions. 

---