Metropolis Monte Carlo importance sampling

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 first part of this chapter contains a short introduction to statistical mechanics of continuum models of fluids and macromolecules. The next section presents a discussion of basic sampling theory (importance sampling) and the Metropolis Monte Carlo and molecular dynamics methods. The remainder of the chapter is devoted to descriptions of methods for calculating F and S, including those that were mentioned above as well as others. 

Importance Sampling with Metropolis-Monte Carlo Algorithm.312... 

With this simple acceptance criterion, the Metropolis Monte Carlo method generates a Markov chain of states or conformations that asymptotically sample the XTT probability density function. It is a Markov chain because the acceptance of each new state depends only on the previous state. Importantly, with transition probabilities defined by Eqs. 15.23 and 15.24, the transition matrix has the limiting, equihbrium distribution as the eigenvector corresponding to the largest eigenvalue of 1. 

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... 

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. 

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... 

The Monte Carlo (MC) method, used to simulate the properties of liquids, was developed by Metropolis et al. (1953). Without going into any detail, it should be pointed out that there are two important features of this MC method that make it extremely useful for the study of the liquid state. One is the use of periodic boundary conditions, a feature that helps to minimize the surface effects that are likely to be substantial in such a small sample of particles. The second involves the way the sample of configurations are selected. In the authors words Instead of choosing configurations randomly, then weighing them with exp[—/i ], we choose configurations with probability exp[—/6 ] and weight them evenly. ... 

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