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Procedure (c) ensures that the acceptance of an arrangement is proportional to its probability of occurrence. One should not confuse with the latter which refers to the real system, whereas the MC random sampling method can produce
all possible arrangements. To illustrate this point lets look at a particular arrangement, produced from a MC sampling as shown below.
This particular arrangement will most definitely be rejected since two disk overlap with each other. This results in high energy which carries a low weighting factor.
In this particular arrangement two molecules penetrate each other and this produces a high energy state (try to press your hand against the wall you will feel a force exerts against your hand. This is because atoms from your hand and the wall do not like to get too close to one another).
Obviously, the occurrence of such a state can be considered rare and therefore associated with a very low weighting factor. Consequently, such an arrangement is very likely to be rejected in the MC simulation.
In fact we have just illustrated a simple problem of a physic subject, namely, the statistical mechanics. It is the study of statistical behavior of component particles that determines the property of a system. For example,
the temperature that we measure is in fact an average effect of a velocity distribution of the particles (atoms or molecules) in a system. Indeed, the MC is an important technique in solving problems of statistical mechanics.
In statistical mechanics terms, the square within which the disks exist is called phase space. It contains all information
regarding a state of the system. For example, the coordinates that specify the location of each disk. The number of ways disks can be arranged is called the partition function and the weighting factor
that relates to the probability of a state is called the Boltzmann factor. On the other hand, the average results that we measure are called observables.
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