Metropolis Monte Carlo simulation implementation

Implementations of the generation mechanism differ in the way they generate a transition or move from one set of parameters to another which is consistent with the Boltzmann distribution at given temperature. The two most widely used generation mechanisms are Metropolis Monte Carlo [47] and molecular dynamics [48] simulations. Metropolis Monte Carlo can be applied to both discrete and continuous optimization problems, but molecular dynamics is restricted to continuous problems. 

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 article has briefly considered the role of Monte Carlo and kinetic Monte Carlo simulations in understanding dissolution and selective dissolution processes that can occur spontaneously in the natural environment and under directed control in laboratories. Algorithms for both Metropolis Monte Carlo and KMC models were discussed, and some results from an implementation of the KMC algorithm were shown as examples. Last, the article surveyed several areas where KMC models have been used to study corrosion processes and where they can contribute in engineering applications. 

We should note that the Monte-Carlo simulation with tw = 0 effectively samples the EP Hamiltonian. This version of field-theoretic Monte Carlo is equivalent to the real Langevin method (EPD), and can be used as an alternative. Monte Carlo methods are more versatile than Langevin methods, because an almost unlimited number of moves can be invented and implemented. In our applications, the W and tw-moves simply consisted of random increments of the local field values, within ranges that were chosen such that the Metropolis acceptance rate was roughly 35%. In principle, much more sophisticated moves are conceivable, e.g., collective moves or combined EPD/Monte Carlo moves (hybrid moves [84]). On the other hand, EPD is clearly superior to Monte Carlo when dynamic properties are studied. This will be discussed in the next section. 

Thus, the question is whether the combination of Metropolis data obtained in simulations at different temperatures can yield an improved estimate g E). This is indeed possible by means of the multiple-histogram reweighting method [86], sometimes also called weighted histogram analysis method (WHAM) [87]. Even though the general idea is simple, the actual implementation is not trivial. The reason is that conventional Monte Carlo simulation techniques such as the Metropolis method cannot yield absolute estimates for the partition sum Z T)= g E) i. e., estimates for the density of states at differ-... 

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