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1 Introduction to Monte Carlo Methods     continued...

It should be kept in mind though that this general description of Monte Carlo methods may not directly apply to some applications. It is natural to think that Monte Carlo methods are used to simulate random, or stochastic, processes, since these can be described by pdf's. However, this coupling is actually too restrictive because many Monte Carlo applications have no apparent stochastic content, such as the evaluation of a definite integral or the inversion of a system of linear equations. However, in these cases and others, one can pose the desired solution in terms of pdf's, and while this transformation may seem artificial, this step allows the system to be treated as a stochastic process for the purpose of simulation and hence Monte Carlo methods can be applied to simulate the system. Therefore, we take a broad view of the definition of Monte Carlo methods and include in the Monte Carlo rubric all methods that involve statistical simulation of some underlying system, whether or not the system represents a real physical process.

To illustrate the diversity of Monte Carlo methods, Figure 2 lists applications that have been addressed with statistical simulation techniques. As can be seen, the range of applications is enormous, from the simulation of galactic formation to quantum chromodynamics to the solution of systems of linear equations.

Figure 2: Monte Carlo Applicationts.

This wide diversity of methods is the reason that ``Monte Carlo is not Monte Carlo is not Monte Carlo.''