Metropolis Monte Carlo particle simulation

Molecular Dynamics simulation is one of many methods to study the macroscopic behavior of systems by following the evolution at the molecular scale. One way of categorizing these methods is by the degree of determinism used in generating molecular positions [134], On the scale from the completely stochastic method of Metropolis Monte Carlo to the pure deterministic method of Molecular Dynamics, we find a multitude and increasingly diverse number of methods to name just a few examples Force-Biased Monte Carlo, Brownian Dynamics, General Langevin Dynamics [135], Dissipative Particle Dynamics [136,137], Colli-sional Dynamics [138] and Reduced Variable Molecular Dynamics [139]. 

Another simulation approach often used that does not offer a simple deterministic time evolution of the system is the Metropolis Monte Carlo [MMC] method. Based on the Metropolis-Hasting algorithm [3, 4], MMC methods are weighted sampling techniques in which particles are randomly moved about to obtain a statistical ensemble of atoms with a particular probability distribution for some quantity. This is usually the energy but can also be other quantities such as experimental inputs that can be quickly calculated from an atomic... 

The output of simulations is a list of microscopic states in phase space. These are either a sample of points generated with Metropolis Monte Carlo, or a succession of points in time generated by molecular dynamics. Particle velocities, as well as higher time derivatives of particle positions, can also be saved during a molecular dynamics simulation. 

It is also straightforward to carry out Monte Carlo simulations with a constant number of particles, temperature and pressure (the NPTensemble). In such simulations, in addition to random moves of the atoms or molecules random changes in the volume of the simulation cell are also attempted, and in the Metropolis step O(Z) + pV replaces < >(Z). Monte Carlo calculations, both NVT and NPT, have thus been extremely useful in establishing equations of state. 

In the grand-canonical Monte Carlo method, the system volume, temperature, and chemical potential are kept fixed, while the number of particles is allowed to fluctuate.There exist three types of trial move (1) displacement of a particle, (2) insertion of a particle, and (3) removal of a particle. These trial moves are generated at random with equal probability. The acceptance probability of the Metropolis method can be used for the trial moves of type (1). For the two other types, the acceptance probabilities are different. Regarding zeolites, an adsorption isotherm can be calculated with the grand-canonical Monte Carlo method by running a series of simulations at varying chemical potentials. 

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). 

An alternative to molecular dynamics based simulated annealing is provided by Metropolis importance sampling Monte Carlo (Metropolis et al., 1953) which has been widely exploited in the evaluation of configurational integrals (Ciccotti et al., 1987) and in simulations of the physical properties of liquids and solids (Allen and Tildesley, 1987). Here, as outlined in Chapters 1 and 2, a particle or variable is selected at random and displaced both the direction and magnitude of the applied displacement within standard bounds are randomly selected. The energy of this new state, new, is evaluated and the state accepted if it satisfies either of the following criteria ... 

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. ... 

---