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Now, suppose we need to know what is the average distance between the molecules, or what is the most ‘preferred’ arrangement? We are not able to measure these quantities accurately because the molecular arrangement
is constantly evolving. Neither could we use mathematical means to solve the problem since: (a) there are a large number of ways (if not infinite) these molecules can be arranged and (b) we do not know the probability (or the chance) for a particular arrangement to occur.
This is where the MC technique comes into play. By using a random sampling technique, one can simulate statistical fluctuations which numerically generate probability distributions. In other words, the idea of MC method is to consider only a subset of arrangements which are used
as a statistical sample. To achieve this the following MC procedure can be carried out:
(a) Place all the molecules randomly in the square.
(b) Calculate the associated energy and the corresponding weighting factor. The latter relates to the probability of the arrangement which is defined
as: exp(-E) where E is the energy.
(c) The weighting factor is compared with a random number and the arrangement will be accepted if the weighting factor is greater then the random number. Otherwise, it will be rejected.
The procedure is repeated so that average results can be obtained from
the set of accepted arrangements. The results are accurate apart from statistical errors which can be made as small as possible by repeating the above procedures many more times.
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