Carlo Importance Sampling

MODELING STRATEGIES Monte Carlo Importance Sampling  

For a system of N atoms or molecules, an instantaneous configuration represents a microstate in the phase space of the system (position and momentum), characterized by the set of state variables a (denoted as a for brevity). The probability for a system being in a given microstate is given by its Boltzmann weighting factor 

If AH 0, then pj, 1, and the move is accepted. That is, the new configuration is lower in energy than the old. The new configuration becomes 

The accepted configurations resulting from the random walk in phase space form a Markov chain.39 Although the successive configurations do not evolve according to a force law (as in MD), they can still be used to study the dynamics of the system by associating a time parameter such as Monte Carlo steps per particle. However, it should be borne in mind that this time is not the physical time, and the dynamics does not represent the true physical phenomenon. 

The setup of the initial configuration, the application of the interaction potential, and the periodic boundary conditions are identical in MD and MC methods. In the case of MD, a Boltzmann distribution of velocities appropriate to the temperature is also assigned to the atoms. The atoms move under the gradients (the negative of the gradients gives the force that is actually used in the MD simulation) of the potential for a time step (At) according to Newton s laws of motion (F = ma = -dVIdr) to obtain a new set of coordinates, and the process is repeated for N time steps to obtain a simulation time of NAt.42 

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Let us begin with the one-dimensional case in Eq. (1.2). Suppose we want to compute the definite integral by Monte Carlo importance sampling with sampling points... 

Hence, it would clearly be advisable to carry out a non-uniform Monte Carlo importance sampling of configuration space with a w approximately proportional to the Boltzmann factor. 

The Monte Carlo method includes both a random sampling scheme and an importance sampling scheme. Both sampling schemes have been used in Section 4.1 on classical trajectory calculations. 

It was recognized early that using information about a distribution function improves the efficiency of a Monte Carlo calculation [1], This is importance sampling, which carries over to QMC where the trial wave function serves as an importance function that produces improved sampling. Into the late 1980s, the most common form of QMC trial wave function was a product of an approximate HF function and a correlation function, i.e., a function explicitly dependent on... 

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. 

Importance sampling is central to Monte Carlo applications in statistical physics. In the subsections that follow, we describe the concepts of the schemes that are most frequently used in zeolite modeling. 

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 ... 

Projector Monte Carlo and Importance Sampling Matrix Elements... 

Diffusion Monte Carlo with Importance Sampling... 

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