The Metropolis Sampling Scheme

With Monte Carlo methods, the adoption of the Metropolis sampling scheme intrinsically assumes equilibrium Boltzmann statistics, so special modifications are required to extend MC methods to non-equilibrium solvation as well. Fortunately, for a wide variety of processes, ignoring non-equilibrium solvation effects seems to introduce errors no larger than those already inherent from other approximations in the model, and thus both implicit and explicit models remain useful tools for studying chemical reactivity. 

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]. 

To emphasize how the Metropolis scheme works in practice, let us return once more to the hard-disk example. Here the matrix am represents the probability of moving a particle to a new trial positions, which is achieved by the generation of two random numbers for displacements in the x and directions. To make the underlying matrix symmetric, the displacements are calculated from dmlJ2 — 1), where [Pg.6]

In Monte Carlo simulations, ensembles of configurations of a system are generated using a Metropolis importance sampling scheme. Each configuration is sampled with the Boltzmann probability for the desired temperature of the system. From this set of configurations, or ensemble, average properties can... 

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