Metropolis acceptance criterion

Based on (1.12), we can implement any complement of MC moves and formulate appropriate acceptance criteria such that the progression of configurations satisfies this distribution. For simple moves in which the proposal probability equals that of its inverse - symmetric moves, such as single-particle displacements - the Metropolis acceptance criterion then reads [141] ... 

Another promising modification of the Metropolis acceptance criterion has been proposed by Hsu et al., who replaced the difference CF— Ci oid in Equation (17) by the term ... 

Again the limiting probabilities occur only as a ratio and the value of Qc is not required. In most computer simulations, the Metropolis acceptance criterion is chosen, because it appears to sample phase space more efficiently than the other schemes that have been proposed [9]. 

Accept or reject the new pathway according to a Metropolis acceptance criterion obeying detailed balance with respect to the transition path ensemble... 

Fig. 2.11. Schematic representation of a parallel tempering simulation, in which the N states of a stratified FEP calculation are run concurrently. After a predefined number of steps, iVSampie, the cells are swapped randomly across the different processors. A Metropolis-based acceptance criterion is used to determine which of the N / 2 exchanged A-states should be accepted. Pairs of boxes that fail the test are swapped back. Then additional sampling is performed until the next exchange of the replicas...

In the standard HMC method two ingredients are combined to sample states from a canonical distribution efficiently. One is molecular dynamics propagation with a large time step and the other is a Metropolis-like acceptance criterion [76] based on the change of the total energy. Typically, the best sampling of the configuration space of molecular systems is achieved with a time step of about 4 fs, which corresponds to an acceptance rate of about 70% (in comparison with 40-50% for Metropolis MC of pure molecular liquids). 

Molecular dynamics can be coupled to a heat bath (see below) so that the resulting ensemble asymptotically approaches that generated by the Metropolis Monte Carlo acceptance criterion (Eq. 10). Thus, molecular dynamics and Monte Carlo are in principle equivalent for the purpose of simulated annealing although in practice one implementation may be more efficient than the other. Recent comparative work (Adams, Rice, Brunger, in preparation) has shown the molecular dynamics implementation of crystallographic refinement by simulated annealing to be more efficient than the Monte Carlo one. 

The success and efficiency of simulated annealing depends on the choice of the annealing schedule [58], that is, the sequence of numerical values Ti > T2 > > T for the temperature. Note that multiplication of the temperature T by a factor s is formally equivalent to scaling the target E by 1/s. This applies to both the Monte Carlo as well as the molecular dynamics implementation of simulated annealing. This is immediately obvious upon inspection of the Metropolis Monte Carlo acceptance criterion (Eq. 10). For molecular dynamics this can be seen as follows. Let E be scaled by a factor 1/s while maintaining a constant temperature during the simulation. 

The geometric cluster algorithm described in the previous section is formulated for particles that interact via hard-core repulsions only. Clearly, in order to make this approach widely applicable, a generalization to other t3rpes of pair potentials must be found. Thus, Dress and Krauth [14] suggested to impose a Metropolis-type acceptance criterion, based upon the energy difference induced by the cluster move. Indeed, if a pair potential consists of a hardcore contribution supplemented by an attractive or repulsive tail, such as a... 

Each state of the expanded ensemble is equilibrated as usual (e.g., through molecular rearrangements). However, a new type of trial move is now introduced trial transitions between neighboring states. Such trial moves are subject to a Metropolis-like acceptance criterion ... 

Here, Boltzmann s constant is set equal to 1. Regardless of whether a move is accepted or rejected, one unit of time (one Monte Carlo step) is considered to have passed. This probabilistic acceptance criterion is known as the Metropolis Monte Carlo algorithm. Although no connection exists between physically relevant time scales and Monte Carlo time steps, Monte Carlo simulations can estimate the relative time scales of protein folding versus simulation time, as well as the time needed to reach equilibrium at a given temperature. Keep in mind, however, that any time scale extracted from a Monte Carlo simulation depends on the move set used. Even so, useful information can be extracted from such a simulation, such as relative transition times for two different sequences. 

However, the activation energy which was deduced by using the activation-relaxation technique, with a Metropolis accept-reject criterion and a fictitious temperature of 0.5eV, ranged from 0.22 to l.OeV. It also exhibited a steep increase at low temperature. The very large pre-exponential factor suggested that the interatomic forces which resulted from the Tersoff potential were very strong. These predictions were consistent, to some extent, with recent experimental results for liquid Si. 

Some new numerical results for fluid He, fluid He, and the hard-sphere fluid, under quantum diffraction effects, are given below to illustrate a number of the basic main points discussed in this chapter. The particle masses (amu) have been set to m( He) = 4.0026, m("He) = 3.01603, and m(hard sphere) = 28.0134. PIMC simulations in the canonical ensemble using the necklace normal-mode moves have been employed. The Metropolis algorithm has been apphed with the general acceptance criterion set to 50% of the attempted moves for each normal mode. In the helium simulations the propagator SCVJ (a = 1 / 3) has been utihzed. The quantum hard-sphere fluid results presented in this chapter have been obtained from a further processing of data reported in Ref. 96. Also, for fluid He Monte Carlo classical (CLAS) and effective potential QFH calculations have been performed by following the standard procedures. 

For each BH iteration, the energy difference AE between the initial and final isomers is compared using the Metropolis MC criterion to evaluate the newly found minimum structure. If AE<0, or for AE>0 if exp [—AE/(A b7 )] (A b is the Boltzmann constant and T is the simulation temperature) is larger than a random number between 0 and 1, the newly found minimum is accepted. Thus in BH algorithm, temperature as a crucial parameter must be carefully chosen and adjusted, which will affect the tradeoff between the acceptance ratio and the sampling efficiency. ... 

These Metropolis-accepted or -rejected moves are taken in the distant past or far future in imaginary time, and ensure that the Markov chain converges to its target distribution. The algorithm s failure to meet the assumed criterion of microscopic reversibility causes the time-step bias to accumulate at the middle of the reptile, where the desired pure distribution is being sampled. Therefore, RQMC variants were introduced to avoid these difficulties [20-24]. 

With this simple acceptance criterion, the Metropolis Monte Carlo method generates a Markov chain of states or conformations that asymptotically sample the XTT probability density function. It is a Markov chain because the acceptance of each new state depends only on the previous state. Importantly, with transition probabilities defined by Eqs. 15.23 and 15.24, the transition matrix has the limiting, equihbrium distribution as the eigenvector corresponding to the largest eigenvalue of 1. 

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