Monte Carlo simulations trial move

The molecular dynamics and Monte Carlo simulation methods differ in a variety of ways. The most obvious difference is that molecular dynamics provides information about the time dependence of the properties of the system whereas there is no temporal relationship between successive Monte Carlo configurations. In a Monte Carlo simulation the outcome of each trial move depends only upon its immediate predecessor, whereas in molecular dynamics it is possible to predict the configuration of the system at any time in the future - or indeed at any time in the past. Molecular dynamics has a kinetic energy contribution to the total energy whereas in a Monte Carlo simulation the total energy is determined directly from the potential energy function. The two simulation methods also sample from different ensembles. Molecular dynamics is traditionally performed under conditions of constant number of particles (N), volume (V) and energy (E) (the microcanonical or constant NVE ensemble) whereas a traditional Monte Carlo simulation samples from the canonical ensemble (constant N, V and temperature, T). Both the molecular dynamics and Monte Carlo techniques can be modified to sample from other ensembles for example, molecular dynamics can be adapted to simulate from the canonical ensemble. Two other ensembles are common ... 

The Monte Carlo simulation includes two different types of moves trial moves of W and co. The moves in co are straightforward. Implementing the moves in W is more involved. Every time that W is changed, the new saddle point i[7 [M j must be evaluated. In an incompressible blend, the set of self consistent equations... 

Keywords Charged coUoids Monte Carlo simulation - Mean force Ewald summation Cluster trial move... 

The simplest Monte Carlo simulations are carried out by sampling trial moves for the degrees of fieedom frran a uniform distribution. In the simplest case of a canonical ensranbte simulation (see Sect 2.3 below) a degree of freedom is... 

Figure 2. Schematic representation of a Metropolis Monte Carlo simulation. This scheme is suitable for most soft potentials, but constraints must be handled carefully (see footnote to text). N trial moves are attempted, with coordinate sampling carried out every N/N trial moves.

The Monte Carlo simulation comprises three distinct moves (i) Canonical Monte Carlo moves update the molecular conformations in the Mr repUca. In this specific application, we employ a Smart Monte Carlo algorithm [119] that utilizes strong bonded forces to propose a trial displacement [43, 87]. The amplitude of the trial displacement has been optimized in order to maximize the mean-square displacement of molecules [91], and the single-chain dynamics closely resembles the Rouse-dynamics of unentangled macromolecules [120]. (ii) Since each replica is an... 

Some of the earliest attempts to devise efficient algorithms for MC simulations of fluids in a continuum are due to Rossky et al. and Pangali et al., who proposed the so-called Smart Monte Carlo and Force-Bias Monte Carlo methods, respectively. In a Force-Bias Monte Carlo simulation, the interaction sites of a molecule are displaced preferentially in the direction of the forces acting on them. In a Smart Monte Carlo simulation, individual-site displacements are also proposed in the direction of the forces in this case, however, a small stochastic contribution is also added to the displacements dictated by the forces. In both algorithms, the acceptance criteria for trial moves are modified to take into account the fact that displacements are not proposed at random but in the direction of intersite forces. 

The Monte Carlo simulations were carried out in the canonical ensemble, at temperatures of T = 100 and 220 K with the 6-31+ G d) basis set and T = 220 K with the apVDZ basis set. Each simulation was carried out for l.l x 10 trial moves, with the first 1.0 x 10 moves being used for equilibration. 

Another important decision relates to the number of trials, iVtriais-Admittedly, there is no precise a-priori determination of the number of Monte Carlo moves necessary to sample the probability density. The answer is attained empirically, with test simulations. Properties must be monitored, such as the energy and the pair distribution function. When the average properties converge to stable values, the Monte Carlo simulation may be stopped. 

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. 

In the method discussed here, a running estimate of the weight factors can be computed and refined in a self-consistent and self-monitoring manner. At the beginning of a simulation, g i ) is assumed to be unity for all states. Trial Monte Carlo moves are accepted with probability... 

Fig. I. Schematic illustration of the implementation of the hyperparallel Monte Carlo method. Each box represents a distinct replica of the simulated system these replicas are simulated simultaneously in a single run. In addition to traditional Monte Carlo trial moves, these replicas can (1) change their state variables in the expanded dimension and (2) exchange configurations with each other, thereby visiting different values of T and fi.

Monte Carlo computer simulation methods require the energy of an assembly of molecules to determine whether a trial move is accepted or rejected, while in MD methods the forces on molecules along their trajectories are needed. Simple analytic forms, such as the Lennard-Jones potential, are commonly used to describe interaction potentials in which all atoms are treated explicitly with distinct parameters or a small collection of atoms is... 

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