Each
state of the system occurs with a specific probability. If we want to perform a simulation
of a system governed by a Hamiltonian in the canonical ensemble (constant temperature) we
need to devise a method to generate states that follow the Boltzmann distribution. This is
much like the generation of random numbers. For these we required that they follow the uniform
distributionTermsUniformDistribution@terms. Here the distribution is non-uniform and
has a non-trivial form.We are then required in a simulation to generate states or
configurations which follow a distribution other than the uniform. To pave the way to the
Monte Carlo method, let us see how we can generate a distribution using uniformly
distributed random numbers. Let
![](Image8.gif)
be the distribution that we want to generate. G(x) is some function in the open
interval (0,1). We obtain the distribution by devising an acceptance/rejection mechanism
based upon uniformly distributed random numbers. For this we draw a random number from a
uniform distribution. We rescale this number such that the number is in the interval over
which G is defined. Let t=G(x), where x is the number from the uniform distribution.
Earlier in this chapter we considered the occurrence for a run down or a run up in a
sequence of random numbers. We use a similar concept to accept or reject a drawn number x
to be a sample from the distribution f. Generate k numbers from a uniform distribution
where k is determined from the condition
![](Image9.gif)
If k is even reject x and start with a new x. If k is odd, we have a sample from the
desired distribution. Using this method it is easy to generate for example random numbers
that are distributed with a maxwell distribution or any other distribution.