The Need for Refined Monte Carlo Sampling

The statistical-mechanical applications of Monte Carlo nearly all involve special sampling methods known as importance sampling and ordinarily require a Markov chain of sample configurations rather than independent samples. In order to understand this it is helpful to begin by imagining a simpler Monte Carlo estimation of a quantity like U) of (2), and then to see why such an estimation would not be successful. 

To carry out this crude estimation one could use random numbers to choose for the particles of the system random positions uniformly distributed in the volume V of the system. That is easy. For each system configuration so generated one would calculate the energy I/(q ) [for the example (2)] and the probability density p(q ). The average of the product t/(q )p(q ) over many such configurations is evidently an unbiased estimate of t/ / according to (2). 

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