Monte Carlo procedure sampling

This detailed balance condition makes sure that the path ensemble sg[z )] is stationary under the action of the Monte Carlo procedure and that therefore the correct path distribution is sampled [23, 25]. The specific form of the transition matrix tt[z(° 2 ) -> z(n, 9-) depends on how the Monte Carlo procedure is carried out. In general, each Monte Carlo step consists of two stages in the first stage a new path is generated from an old one with a certain generation probability... 

Figure 8. Li + H2 Ground-state population as a function of time for a representative initial basis function (solid line) and the average over 25 (different) initial basis functions sampled (using a quasi-classical Monte Carlo procedure) from the Lit2/j) + H2(v — 0, j — 0) initial state at an impact parameter of 2 bohr. Individual nonadiabatic events for each basis function are completed in less than a femtosecond (solid line) and due to the sloped nature of the conical intersection (see Fig. 7), there is considerable up-funneling (i.e., back-transfer) of population from the ground to the excited electronic state. (Figure adapted from Ref. 140.)...

Various deterministic and stochastic sampling techniques for path ensembles have been proposed [4-6]. Here we consider only Monte Carlo methods. It is important, however, to be aware that while the path ensemble is sampled with a Monte Carlo procedure each single pathway is a fully dynamical trajectory such as one generated by molecular dynamics. 

Monte Carlo procedures can be applied very generally to sample probability distributions [24]. In particular, Monte Carlo techniques can also be used to sample ensembles of pathways. In this case a random walk is carried out in the space of trajectories instead of configuration space. The basic step of this procedure consists of generating a new path, from an old one. 

We overcome this problem by using Metropolis Monte Carlo (MMC) sampling [42] within each macrostate. In contrast with Monte Carlo integration, which uses a fixed kernel to select each sample point Q(i), in MMC a trial sample conformation Q(iy is selected with a transition distribution centered about the previous accepted sample conformation 2, 1). We accept CI(,) with probability pMMC, which is determined as in standard MMC procedure except that the potential is modified to account for the restriction of sampling to the macrostate region defined by w ... 

Discrete multidimensional FT of randomly sampled signal may be considered as an estimation of continuous multidimensional integral (12) with Monte Carlo procedure. According to properties of Monte Carlo integration, associated with... 

Analysis) works by using a Monte Carlo procedure to randomly orient a given number of molecules in a trial unit cell. To accomplish this stochastic approach to sampling trial unit cells, the energy of the cell is minimized with a rigid body optimizer, Although this procedure imposes no space group symmetry, symmetry may be present in the minimized structures as permitted by the number of molecules in the cell. 

Bucher, C. G. Adaptive sampling - an iterative fast Monte Carlo procedure. Structural Safety 5(2) (1988), 119-126. 

When Monte Carlo work is mentioned, one thinks primarily of the conventional techniques described in Chapter 4, which are concerned with classical and macroscopic systems, and are restricted to thermodynamic and structural information of a mechanical nature. There are under development, however, Monte Carlo procedures seeking to remove some of these limitations, and this chapter samples some of this newer work as it relates to the theory of fluids. 

To simulate the structure of the as-quenched alloys one starts with an Initial monoatomic packing of 1500 spheres relaxed in a modified Johnson potential . Cu and Ti atoms are then chosen at random to give the desired composition CU50 Ti 50. After that one tries to reproduce the experimental results by a Monte-Carlo procedure. Cu-Ti pairs are taken at random and permuted with a given probability if this permutation favours chemical order (i.e. an increase of the number of Cu-Ti pairs). The sample is then relaxed in a set of modified Johnson potentials. The position r emd depth e of the minimum of each metal-metal potential are given by... 

BUCHER, C.6., Adaptive Sampling - An iterative Fast Monte Carlo Procedure, in Report 12-87, Institute of Engineering Mechanics, University of Innsbruck, to appear 1987. 

In the real time domain, the sampling of quantum paths involves oscillating positive and negative weights which are problematical for any currently known Monte Carlo procedure. Thus we must await new developments in time-dependent path integral quantum Monte Carlo. 

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