Metropolis

Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A FI and Teller E 1953 Equation of state calculations by fast computing machines J. Chem. Phys. 21 1087... 

The expense is justified, however, when tackling polymer chains, where reconstruction of an entire chain is expressed as a succession of atomic moves of this kind [121]. The first atom is placed at random the second selected nearby (one bond length away), the third placed near the second, and so on. Each placement of an atom is given a greater chance of success by selecting from multiple locations, as just described. Biasing factors are calculated for the whole multi-atom move, forward and reverse, and used as before in the Metropolis prescription. For fiirther details see [122, 123. 124. 125]. A nice example of this teclmique is the study [126. 127] of the distribution of linear and branched chain alkanes in zeolites. 

A similar algorithm has been used to sample the equilibrium distribution [p,(r )] in the conformational optimization of a tetrapeptide[5] and atomic clusters at low temperature.[6] It was found that when g > 1 the search of conformational space was greatly enhanced over standard Metropolis Monte Carlo methods. In this form, the velocity distribution can be thought to be Maxwellian. 

Thus, if an ensemble can be prepared that is at equilibrium, then one Metropolis Moni Carlo step should return an ensemble that is still at equilibrium. A consequence of this that the elements of the probability vector for the limiting distribution must satisfy ... 

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 particle is displaced, using the usual Metropolis method. 

Metropolis N, A W Rosenbluth, M N Rosenbluth, A H Teller and E Teller 1953. Equation of Sta Calculations by Fast Computing Machines. Journal of Chemical Physics 21 1087-1092. 

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