Monte Carlo methods definition

They point out that at the heart of technical simulation there must be unreality otherwise, there would not be need for simulation. The essence of the subject linder study may be represented by a model of it that serves a certain purpose, e.g., the use of a wind tunnel to simulate conditions to which an aircraft may be subjected. One uses the Monte Carlo method to study an artificial stochastic model of a physical or mathematical process, e.g., evaluating a definite integral by probability methods (using random numbers) using the graph of the function as an aid. 

The most common applications of the Monte Carlo method in numerical computation are for evaluating integrals. Monte Cario methods can also be used in solving systems of equations. All instances of Monte Carlo simulation can be reduced to the evaluation of a definite integral like the following ... 

The Monte Carlo method permits simulation, in a mathematical model, of stochastic variation in a real system. Many industrial problems involve variables which are not fixed in value, but which tend to fluctuate according to a definite pattern. For example, the demand for a given product may be fairly stable over a long time period, but vary considerably about its mean value on a day-to-day basis. Sometimes this variation is an essential element of the problem and cannot be ignored. 

The multidimensional integrals in the definition of the potential of mean force can be evaluated directly using the Monte Carlo method (see Appendix I). 

Theoretical trends in the study of suspensions employ concepts and techniques originally developed in connection with theories of liquids, for example, equation hierarchies, closure problems, and Monte Carlo methods. In marked contrast with the definitive achievements reviewed in the previous section, the present section outlines a field currently under active development. 

In many cases, too, the semiclassical model provides a quantitative description of the quantum effects in molecular systems, although there will surely be situations for which it fails quantitatively or is at best awkward to apply. From the numerical examples which have been carried out thus far— and more are needed before a definitive conclusion can be reached—it appears that the most practically useful contribution of classical S-matrix theory is the ability to describe classically forbidden processes i.e. although completely classical (e.g. Monte Carlo) methods seem to be adequate for treating classically allowed processes, they are not meaningful for classically forbidden ones. (Purely classical treatments will not of course describe quantum interference effects which are present in classically allowed processes, but under most practical conditions these are quenched.) The semiclassical approach thus widens the class of phenomena to which classical trajectory methods can be applied. 

The first use of the Monte-Carlo method in a form similar to that used today was by Ulam, von Neumann and Metropolis. The goal is to calculate the expectation of some observable g(X) of a random variable X (assumed to be defined by a complicated process) having a given probability density f x). By definition... 

As you perhaps may have noticed, many Monte Carlo methods can be found in the literature. We have not described all of the Monte Carlo methods developed to date. Indeed, new methods are being developed every day, and methods better than those that now exist can be expected to appear in the future. Nonetheless we can make a few definite recommendations to the readers interested in writing their own Monte Carlo programs (especially for studying atomic and molecular clusters) based on existing methodology. 

A great deal of systematic information has been developed from model systemswhich can be qualitatively applicable to real systems. A series of definitive reviews of the Monte Carlo method and results on model systems have been prepared by Wood. The dynamics approach was Initially characterized in the series of papers by Alder, Wainwright and coworkers. A ccmprehenslve review of liquid state theory was recently published by Barker and Henderson. ... 

Thomas and Deemer, in a well-known paper,48 have set down useful definitions and thoughts on simulation, Monte Carlo, and operational gaming methods. 

As stated in Sec. 3.1, only ideal systems will be considered in this section. This definition implies that there is no intramolecular reaction, a condition which is satisfied in practice for very low concentrations of Af monomers (f >2), in the A2 + Af chainwise polymerization. To take into account intramolecular reactions it would be necessary to introduce more advanced methods to describe network formation, such as dynamic Monte Carlo simulations. 

Despite its bad reputation as an analytical tool, XRF is potentially a traceable method according to the CCQM definition and could be a primary method although it was not selected as such, and won t be for a long time. In fact, it is the only microanalytical method which can at present be considered as a candidate for accurate microscopic elemental analysis. Proof of this statement follows from Monte Carlo calculations in which experimental XRF spectra can be accurately modelled starting from first principles [23], This is not an easy approach but with computing power now available it is feasible, though not worth the effort for bulk chemical analysis where other alternatives are available. 

The usual approach to dynamic Monte Carlo simulations is not based on the master equation, but starts with the definition of some algorithm. This generally starts, not with the computation of a time, but with a selection of a site and a reaction that is to occur at that site. We will show here that this can be extended to a method that also leads to a solution of the master equation, which we call the random-selection method (RSM). [31]... 

Uncertainties inherent to the risk assessment process can be quantitatively described using, for example, statistical distributions, fuzzy numbers, or intervals. Corresponding methods are available for propagating these kinds of uncertainties through the process of risk estimation, including Monte Carlo simulation, fuzzy arithmetic, and interval analysis. Computationally intensive methods (e.g., the bootstrap) that work directly from the data to characterize and propagate uncertainties can also be applied in ERA. Implementation of these methods for incorporating uncertainty can lead to risk estimates that are consistent with a probabilistic definition of risk. 

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