Algorithm, Metropolis

Several variations of this method go under different names. The Metropolis algorithm uses only symmetrical proposal distributions such that q(Qi Qi-i) = q(Qi-i Qi). The expression for 7t(6, 6, i) reduces to... 

The MC method can be implemented by a modification of the classic Metropolis scheme [25,67]. The Markov chain is generated by a three-step sequence. The first step is identical to the classic Metropolis algorithm a randomly selected molecule i is displaced within a small cube of side length 26r centered on its original position... 

A standard Metropolis algorithm is used, whereby an attempted move of a randomly selected particle in a random direction Ax e [-1/2,1/2], Ay e [-1/2,1/2], Az G [-1/2,1/2] is accepted according to the standard Metropolis procedure [57] described above. The above displacement widths ensure a reasonably high acceptance rate A of the moves, of the order of... 

Another widely used choice is the Metropolis Algorithm [metrop53j, given by... 

We now discuss in more detail the time unit, tmc(T) [Eq. (5.15)], for the dynamic MC simulation. Remember that the Metropolis algorithm [7,12] only involves a transition probability, and in converting that into a transition prob-... 

Monte Carlo simulations were performed in 6/V-dimensional phase space, where N = 120-500 atoms [5]. The Metropolis algorithm was used with umbrella sampling. The weight density was... 

Gaussian approximation, 61 Monte Carlo simulations, 67-81 dynamics, 75-81 Metropolis algorithms, 70-71 nonequilibrium molecular dynamics, 71-74 structure profiles, 74-75 system details, 67-70... 

Metropolis algorithm, Monte Carlo heat flow simulation, 70-71 molecular dynamics, 76-81... 

What we do instead is to adopt the famous Metropolis algorithm, in which the generation of configurations is biased towards those that make the most significant contribution to the integral. The method generates states with a probability proportional to exp(- (Z)/kBT) (equal to their Boltzmann probability) and then counts each of them equally. 

In the Metropolis algorithm the components of the displacement vector 8y are obtained by uniformly sampling from the domain D, centered in the coordinates of the molecule y in the i configuration, and defined by the maximum allowed displacement max an(j maximum allowed rotation 8(>MAX parameters (convergence celerity greatly depends on the values used for these two parameters). That is, all the positions inside domain D have the same probability to be chosen as new trial configurations. Thus ... 

From the expressions (20), equation (16) for the Metropolis algorithm can be written as ... 

The second problem can be overcome by using a stochastic search. Thus Corana s simulating annealing SA [89], a sophisticated global search method arising from the Metropolis algorithm [90] was employed. A synthetic seven species problem was constructed and the elements in T-jx were correctly determined using SA. The recovered spectra are essentially identical to the synthetic 7 pure component spectra. [91]... 

Metropolis are few, and suggest that correspondence between the two methods is weak at best. However, we have found that the results ofMurty, et ah, who used the Metropolis algorithm, can be essentially duplicated using KMC. Murty has also seen this correspondence. ... 

In this section, the mechanics ofthe kinetic Monte Carlo algorithm (KMC) are compared to the Metropolis algorithm. ... 

A i-kinetics is contrasted with i-kinetics in that the activation barrier contains information about both the initial and final state when the atom hops. One way to implement this is shown in Figure 1 (b). In this case, we can write the kinetic law in a form analogous to hopping probabilities in the Metropolis algorithm ... 

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