Energy of particles on a checkerboard (quals)

[ravx]

Hi, I'm new to the forum, and I hope I'm not posting this under the wrong thread. I'm studying for quals, and I've come across this past exam:

Homework Statement

http://www.physics.uiuc.edu/education/graduate/Qual/sm/SMSpring07B.pdf
This is a problem on the Landau theory of phase transitions, and I have a good idea on how to do all the parts except part (b), though I have a feeling the solution to that is embarrassingly simple.

Homework Equations

The Attempt at a Solution

Of course one way to attempt (b) is as a brute force counting problem; figure out the possible values of energy, and the number of configurations achieving those energies, and then sum to get the expectation value. But that's an insane way of doing it, since the number of configurations for achieving a certain energy is not so obvious (at least to me?).

Thankfully the rest of the problem can be done without solving (b), but still I would like to know if anyone has any ideas. Thanks a lot!

 

[rwturner]

This problem was fun. I really banged my head against part e until I remembered some very basic stuff about phase transitions...

[edit: I guess we're not supposed to post solutions, so here's what I did watered down]
I think it suffices to consider the average energy of a single A-B pair and scale up. There might be counting problems near the edges but you should be able to ignore these in the large N limit.

Good luck with your quals.