The app above allows you to run simulations of a one dimensional Ising model using either the correct Hamiltonian or an
approximate mean field Hamiltonian. When you run your simulations information about the average behavior of the spins is
provided in the three graphs on the right. The top left graph shows how the estimate of the average magentisation of the spins
changes during the simulation. The top left graph gives the current estimate for the average spin-spin correlation function. The
x-axis of this figure shows the distance between the spins, while the y-axis gives the average degree to which spins this far apart
are correlated. The graph at the bottom gives a histogram and shows how often the magnetisation takes each of the possible values it
can take.
Use this tool to probe the behavior of the Ising model at different temperatures and different magentic field strengths. Write a short
report describing what you tried and why you tried those things. Then comment on the results that you obtained while making reference to
what you have learnt about the Ising model during this course. You should try to answer the following questions in your answer:
- When a magnetic field is present how do the spins behave at low and high temperature?
- When the magentic field is not present how do the spins behave at low and high temperature and how does their behavior differ from their behavior when the field is present? Hint: you are going to have to run multiple simulations in order to understand this difference
- How is the time it takes for the estimate of the average magentisation to converge to a near constant value affected by the temperature?
- Are the average fluctuations in the magnetisation larger at high temperature or at low temperature?
- How is the spin-spin correlation function affected by temperature in the absence of a field? Why is the correlation function affected by temperature in this way?
- If the mean field Hamiltonian is used instead of the exact Hamiltonian how is the spin-spin correlation function affected by temperature? Why does the spin-spin correlation function for the mean field Hamiltonian behave in a different way to the spin-spin correlation function for the exact Hamiltonian?
