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Chapter 8: Q 8.25 (page 353)
In Problem 8.15 you manually computed the energy of a particular state of a 4 x 4 square lattice. Repeat that computation, but this time apply periodic boundary conditions.
Therefore, the energy is 4J.
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By changing variables as in the text, express the diagram in equation in terms of the same integral as in the equation. Do the same for the last two diagrams in the first line of the equation. Which diagrams cannot be written in terms of this basic integral?
In this problem you will use the mean field approximation to analyse the behaviour of the Ising model near the critical point.
(a) Prove that, when
(b) Use the result of part (a) to find an expression for the magnetisation of the Ising model, in the mean field approximation, when T is very close to the critical temperature. You should find (not to be confused with 1/kT) is a critical exponent, analogous to the f defined for a fluid in Problem 5.55. Onsager's exact solution shows that in two dimensions, while experiments and more sophisticated approximations show that in three dimensions. The mean field approximation, however, predicts a larger value.
(c) The magnetic susceptibility is defined as . The behaviour of this quantity near the critical point is conventionally written as , where y is another critical exponent. Find the value of in the mean field approximation, and show that it does not depend on whether T is slightly above or slightly below Te. (The exact value of y in two dimensions turns out to be 7/4, while in three dimensions .)
Problem 8.10. Use a computer to calculate and plot the second virial coefficient for a gas of molecules interacting via the Lennard-Jones potential, for values of ranging from to . On the same graph, plot the data for nitrogen given in Problem 1.17, choosing the parameters and so as to obtain a good fit.
Modify the ising program to compute the average energy of the system over all iterations. To do this, first add code to the initialise subroutine compute the initial energy of the lattice; then, whenever a dipole is flipped, change the energy variable by the appropriate amount. When computing the average energy, be sure to average over all iterations, not just those iterations in which a dipole is actually flipped (why?). Run the program for a 5 x 5 lattice for T values from 4 down to l in reasonably small intervals, then plot the average energy as a function of T. Also plot the heat capacity. Use at least 1000 iterations per dipole for each run, preferably more. If your computer is fast enough, repeat for a 10x 10 lattice and for a 20 x 20 lattice. Discuss the results. (Hint: Rather than starting over at each temperature with a random initial state, you can save time by starting with the final state generated at the previous, nearby temperature. For the larger lattices you may wish to save time by considering only a smaller temperature interval, perhaps from 3 down to 1.5.)
At T = 0, equation 8.50 says that . Work out the first temperature-dependent correction to this value, in the limit . Compare to the low-temperature behaviour of a real ferromagnet, treated in Problem 7.64.
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