Metropolis importance sampling

In order to discuss schemes that allow us to include information about rejected states in our Monte Carlo sampling, it is useful to review the basic equations that underlie Metropolis importance sampling. Our aim is to sample a distribution p. The weight of state i is denoted by p i). The probability that the system that is initially in state m will undergo a transition to state n is denoted by TTmn- This probability is normalized as the system must end in some state (possibly the original state o). The normalization condition... 

An alternative to molecular dynamics based simulated annealing is provided by Metropolis importance sampling Monte Carlo (Metropolis et al., 1953) which has been widely exploited in the evaluation of configurational integrals (Ciccotti et al., 1987) and in simulations of the physical properties of liquids and solids (Allen and Tildesley, 1987). Here, as outlined in Chapters 1 and 2, a particle or variable is selected at random and displaced both the direction and magnitude of the applied displacement within standard bounds are randomly selected. The energy of this new state, new, is evaluated and the state accepted if it satisfies either of the following criteria ... 

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... 

Here, AE is the energy difference to transition from state A to B and jl the reciprocal thermal energy. Metropolis et al. [11] showed that such a scheme samples the Boltzmann distribution associated with the given Hamiltonian at the temperature specified by j>. For larger systems, such importance sampling is vastly superior to any systematic or random enumeration schemes, which scale extremely poorly with the number of degrees of freedom in the system [8]. 

The thermalization of the system follows the well-known Metropolis rules (importance sampling procedure) [14] obeying the hard core overlap, but no additional interactions. The degree of thermalization can be controlled by observing the melting factor distribution. The melting factor is a means of judging if the model system has already reached thermodynamic equilibrium. For our purposes we defined... 

Another procedure to overcome the inefficiency of Metropolis Monte Carlo is adaptive importance sampling.194-196 In this technique, the partition function (and quantities derived from it, such as the probability of a given conformation) is evaluated by continually upgrading the distribution function (ultimately to the Boltzmann distribution) to concentrate the sampling in the region (s) where the probabilities are highest. 

The Monte Carlo method is easily carried out in any convenient ensemble since it simply requires the construction of a suitable Markov chain for the importance sampling. The simulations in the original paper by Metropolis et al. [1] were carried out in the canonical ensemble corresponding to a fixed number of molecules, volume and temperature, N, V, T). By contrast, molecular dynamics is naturally carried out in the microcanonical ensemble, fixed (N, V, E), since the energy is conserved by Newton s equations of motion. This implies that the temperature of an MD simulation is not known a priori but is obtained as an output of the calculation. This feature makes it difficult to locate phase transitions and, perhaps, gave the first motivation to generalize MD to other ensembles. 

The basic importance-sampling MC algorithm is the Metropolis method... 

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