Day 2: Simulation Methods ...

Sampling a Distribution, the Partition Function

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

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

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.

Exercises

Program the above outlined algorithm to generate a maxwell distribution.

Write a program which initially generates a random walk. From this starting point select randomly one of the two ends. Remove this end point and add the site to the other end-point by selecting one of the nearest neighbors of the former end point with equal probability. For the moment we are not concerned with double occupancy (no excluded volume). Watch the chain (walk) reptate on the screen.