Monte Carlo method Metropolis

Figure 7-1. Typical autocorrelation function of the energy. In this example it is calculated for the case of benzophenone in water simulated with Monte Carlo Metropolis method. The calculated auto-correlation function (circles) is fitted to the exponential decay (line) and the correlation time t is obtained using Eq. (7-7) applied in the fitted function shown in Eq. (7-8)...

The statistical ensemble framework of equilibrium statistical mechanics gives us the tools to analyze experimental data and to make theoretical predictions. The concept of entropy, its maximization, and the ensuing definition of intensive quantities such as pressure and temperature reduces the complexity of a statistical system of 10 particles to manageable proportions. In principle, the problem of predicting the collective behavior of equilibrium systems starting from microscopic interactions is solved. In practice, exact calculations are rarities. Computational tools such as the Monte Carlo Metropolis method can, however, fill in this void a priori knowledge of the probability distribution of microstates is at the heart of the Metropolis algorithm. 

A series of Monte Carlo computer simulation studies of the structure and properties of molecular liquids and solutions have recently been carried out in this Laboratory.The calculations employ the canonical ensemble Monte Carlo-Metropolis method based on analytical pairwise potential functions representative of ab initio quantum mechanical calculations of the intennolecular interactions. A number of thermodynamic properties including internal energies and radial distribution functions were determined and are reported herein. The results are analyzed for the structure of the statistical state of the systems by means of quasicomponent distribution functions for coordination number and binding energy. Significant molecular structures contributing to the statistical state of each system are identified and displayed in stereographic form. 

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

Metropolis Monte Carlo (MMC) method, 26 1035-1036 Met-spar, 4 577 Metsulfuron-methyl, 13 322 Mettler dropping point, 10 827 Mevacor, 5 142... 

The first molecular simulations were performed almost five decades ago by Metropolis et al. (1953) on a system of hard disks by the Monte Carlo (MC) method. Soon after, hard spheres (Rosenbluth and Rosenbluth, 1954) and Lennard-Jones (Wood and Parker, 1957) particles were also studied by both MC and molecular dynamics (MD). Over the years, the simulation techniques have evolved to deal with more complex systems by introducing different sampling or computational algorithms. Molecular simulation studies have been made of molecules ranging from simple systems (e.g., noble gases, small organic molecules) to complex molecules (e.g., polymers, biomolecules). 

RMC is a variation of the standard Metropolis Monte Carlo (MMC) method (Metropolis et al., 1953 see also Chapters l and 5). The principle is that we wish to generate an ensemble of atoms, i.e. a structural model, which corresponds to a total structure factor (set of experimental data) within its errors. These are assumed to be purely statistical and to have a normal distribution. Usually the level and distribution of statistical errors in the data is not a problem, but systematic errors can be. We shall initially consider materials that are macro-scopically isotropic and that have no long range order, i.e. glasses, liquids and gases. The basic algorithm, as applied to a monatomic system with a single set of experimental data, is as follows ... 

In Monte Carlo (MC) methods, a sequence of points in phase space is generated from an initial geometry by adding a random kick to the coordinates of a randomly chosen particle (atom or molecule). The new configuration is accepted if the energy decreases and with a probability of e if the energy increases. This Metropolis procedure ensures that the configurations in the ensemble obey a Boltzmann... 

The Monte Carlo (MC) method, used to simulate the properties of liquids, was developed by Metropolis et al. (1953). Without going into any detail, it should be pointed out that there are two important features of this MC method that make it extremely useful for the study of the liquid state. One is the use of periodic boundary conditions, a feature that helps to minimize the surface effects that are likely to be substantial in such a small sample of particles. The second involves the way the sample of configurations are selected. In the authors words Instead of choosing configurations randomly, then weighing them with exp[—/i ], we choose configurations with probability exp[—/6 ] and weight them evenly. ... 

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