Problems

1. Show that (4.1) has the property that it generates all the integers from $1$ to $10$ in an apparently random order if $a=7$ and $b=11$, and that the sequence repeats itself thereafter.
2. A certain statistical quantity is distributed according to
   
   $$p(x) = 2x\quad\mbox{ for } 0 < x < 1$$
   
   
   Given a function which generates random numbers uniformly distributed between $0$ & $1$, show how to transform these into the distribution $p(x)$.
3. Suggest a Monte-Carlo integration procedure for the integral
   
   $$\int_{-\infty}^{+\infty}e^{- a^2 x^2 - b^4 x^4} \d x$$
   
   

4. Describe how you would use the Metropolis algorithm to generate a set of random numbers distributed according to the integrand of problem 3.

The following are essay type questions which are provided as examples of the sorts of questions which might arise in an examination of Monte-Carlo and related methods.

1. In an ionic conductor, such as $\mbox{Ag}\mbox{I}$, there are several places available on the lattice for each $\mbox{Ag}$ ion, and the ions can move relatively easily between these sites, subject to the Coulomb repulsion between the ions. Describe how you would use the Metropolis algorithm to simulate such a system.
2. Some quantum mechanics textbooks suggest that the ground state wave function for a Hydrogen atom is
   
   $$\psi(r) \propto e^{-\alpha r}\quad .$$
   
   
   Describe how you would use the variational quantum Monte-Carlo procedure to calculate the ground state energy for a range of values of $\alpha$ and hence how you would provide estimates of the true ground state energy and the value of $\alpha$.
3. Describe how you would simulate the melting (or sublimation) of Argon in vacuo using the molecular dynamics method. Pay particular attention to how you would calculate the specific and latent heats.