Metropolis rates

The main difference between the force-bias and the smart Monte Carlo methods is that the latter does not impose any limit on the displacement that m atom may undergo. The displacement in the force-bias method is limited to a cube of the appropriate size centred on the atom. However, in practice the two methods are very similar and there is often little to choose between them. In suitable cases they can be much more efficient at covering phase space and are better able to avoid bottlenecks in phase space than the conventional Metropolis Monte Carlo algorithm. The methods significantly enhance the acceptance rate of trial moves, thereby enabling Icirger moves to be made as well as simultaneous moves of more than one particle. However, the need to calculate the forces makes the methods much more elaborate, and comparable in complexity to molecular dynamics. 

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... 

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). 

A more interesting problem is that the Metropolis Monte Carlo studies used a different (physically simplified) kinetic rate law for atomic motion than the KMC work. That is, the rules governing the rate at which atoms jump from one configuration to the next were fundamentally different. This can have serious implications for such dynamic phenomena as step fluctuations, adatom mobility, etc. In this paper, we describe the physical differences between the rate laws used in the previous work, and then present results using just one of the simulation techniques, namely KMC, but comparing both kinds of rate laws. 

The results shown in this study are limited to the KMC algorithm. In principle, to model a realistic system the set k, can be found using molecular dynamics simulations, or other similar techniques these can then be used as input into KMC. Here, our purposes are more general. A square lattice is examined, and there are two simplified rate laws of interest. The first is often used in KMC simulations of deposition, and is termed i-kinetics. A second set represents a kinetics that is analogous to the hopping probabilities used in the Metropolis simulations. We term the latter A/-kinetics. 

In the literature, some computer models describing the evolution of copolymer sequences have been proposed [26,28]. Most of them are based on a stochastic Monte Carlo optimization principle (Metropolis scheme) and aimed at the problems of protein physics. Such optimization algorithms start with arbitrary sequences and proceed by making random substitutions biased to minimize relative potential energy of the initial sequence and/or to maximize the folding rate of the target structure. 

The Metropolis method is derived by imposing the condition of microscopic reversibility at equilibrium the transition between two states occurs at the same rate. The rate of transition from a state m to a state n equals the product of the population (p ) and the appropriate element of the transition matrix (7t ). Thus, at equilibrium we can write ... 

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. 

Fig. 7.12. Metropolis algorithm, (a) If 2 is only a little higher than Ei, then the Metropolis criterion often leads to accepting the new conformation (of energy 2)- (b) On the other hand if the energy difference is large, then the new conformation is accepted only rarely. If the temperature increases, the acceptance rate increases too.

It is also possible to perform simultaneous random or biased moves on many degrees of freedom. This was shown not to be advantageous for the simulation of liquid phases of simple molecules [27,33]. However, polymeric systems are expected to behave differently. For example, in simple Lennard-Jones systems it was found that the maximum efficiency of a Metropolis MC algorithm (fastest sampling of configuration space) corresponded to an overall acceptance rate of roughly 50%. This was found not to be the case in MC simulations of a model protein, where maximum efficiency corresponded to an arx ptance rate of only 15% [41, 51]. 

All those in the know agreed that, had this been an attack with highly toxic bacterial or viral agents, within days infection rates among the people of the London metropolis would have been disproportionately high compared... 

In Bayesian analysis GTR+I+G model of nucleotide substitutions with four rate categories was used. Four Metropolis-coupled MCMC chains were run from randomly chosen starting trees for 3000000 generations, trees were saved once every 10 generations, 114000 first trees were ignored. The other options retained default values. Majority-rule consensus trees were constructed and Bayesian posterior probabilities as branch support values were calculated. This analysis will be referred to as MB144 where based on dataset 1, and as MB135 where based on dataset 2. 

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