Metropolis method

A particle is displaced, using the usual Metropolis method. 

Thermodynamically, the probability of finding a system in a state whose energy is AE above the ground state is proportional to exp(-AE/kT). HyperChem uses the Metropolis method,which chooses random configurations with this probability, to concen-... 

Essentially random atomic moves use the Metropolis method (98)... 

This method [24,26,30] is specifically suited for simulating solutions. A great deal of the more interesting properties of solutions are essentially determined by the solute-solvent and solvent-solvent interactions close the solute. This fact suggests that the convergence of many solution properties can be accelerated by mainly sampling in the vicinity of the solute in contrast with the Metropolis method that samples among all the solvent molecules with identical probability. 

In the cluster analysis, local molecular configurations of low energy in the equilibrium are presumed to be clusters. The cluster distribution is obtained using the equilibrium snapshots, when the following function F( n, ) is a minimum. We calculated F( n, ) and determined the cluster distribution using the Metropolis method. 

The MC method [56, 61] is used for the determination of equilibrium distributions and calculations of equilibrium properties. The method uses a Markovian sequence of stochastic steps in the phase space, and each step is characterized by some probability of occurrence. One of the possible methods for calculation of step probability is Metropolis method, in which the probability of transition from the state i to the state j is equal to ... 

A Metropolis method with umbrella sampling was employed [74,98-102]. For transition between states i and j, the acceptance ratio for moves is Fy = exp(—(Ej — Ei)/ksT), where ) is the energy of configuration i, kB is the Boltzmann constant, and T is the absolute temperature. The energy of conformation i is obtained by summing the Coulombic interactions over all charged species in a cell or its adjacent image cell [74, 101]. If h is the number of ion pairs that are deleted or inserted, then the acceptance ratio for insertions is... 

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)...

Dynamic Monte Carlo simulations were first used by Verdier and Stockmayer (5) for lattice polymers. An alternative dynamical Monte Carlo method has been developed by Ceperley, Kalos and Lebowitz (6) and applied to the study of single, three dimensional polymers. In addition to the dynamic Monte Carlo studies, molecular dynamics methods have been used. Ryckaert and Bellemans (7) and Weber (8) have studied liquid n-butane. Solvent effects have been probed by Bishop, Kalos and Frisch (9), Rapaport (10), and Rebertus, Berne and Chandler (11). Multichain systems have been simulated by Curro (12), De Vos and Bellemans (13), Wall et al (14), Okamoto (15), Kranbu ehl and Schardt (16), and Mandel (17). Curro s study was the only one without a lattice but no dynamic properties were calculated because the standard Metropolis method was employed. De Vos and Belleman, Okamoto, and Kranbuehl and Schardt studies included dynamics by using the technique of Verdier and Stockmayer. Wall et al and Mandel introduced a novel mechanism for speeding relaxation to equilibrium but no dynamical properties were studied. These investigations indicated that the chain contracted and the chain dynamic processes slowed down in the presence of other polymers. 

Steps 4 and 5 are often summarized by the expression Paccept = min[l, exp -AU/k T)]. Step 2 is usually carried out in such way that about 50% of the trial moves are accepted. The Metropolis method allows calculation of the structural and energetic properties of a system. 

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. 

Random displacement of each atom in the computational domain from its current position. The magnitude of this displacement is of the order 0.003A. Once each atom has been displaced the decision on acceptance of the new configuration is based on the standard Metropolis method ... 

As discussed above, Monte Carlo sampling can be performed on this sum using the Metropolis method. The simplest types of updates are local spin flips ... 

The extension of Gillespie s algorithm to spatially distributed systems is straightforward. A lattice is used to represent binding sites of adsorbates, which correspond to local minima of the potential energy surface. The discrete nature of KMC coupled with possible separation of time scales of various processes could render KMC inefficient. The work of Bortz et al. on the n-fold or continuous time MC CTMC) method can lead to computational speedup of the KMC method, which, however, has been underutilized most probably because of its difficult implementation. This method classifies all atoms in a finite number of classes according to their transition probability. Probabilities are computed a priori and each event is successful, in contrast to the Metropolis method (and other null event algorithms) whose fraction of unsuccessful (null) events increases drastically at low temperatures and for stiff problems. In conjunction with efficient search within a class and dynamic variation of atom coordi-nates, " the CPU time can be practically independent of lattice size. After each event, the time is incremented by a continuous amount. 

III. Single-Thread Monte Carlo A. Metropolis Method... 

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