In my last post, I discussed phase transitions, and how
computing the free energy for a model would let you work out the
phase diagram. Today, I want to discuss in more detail some
methods for computing free energies. The most popular tool
physicists use for computing free energies is “mean-field
theory.” There seems to be at least one “mean-field theory” for
every model in physics. When I was a graduate student, I became
very unhappy with the derivations for mean-field theory, not
because there were not any, but because there were too many!
Every different book or paper had a different derivation, but I
didn’t particularly like any of them, because none of them told
you how to correct mean-field theory. That seemed strange
because mean-field theory is known to only give approximate
answers. It seemed to me that a proper derivation of mean-field
theory would let you systematically correct the errors. One
paper really made me think hard about the problem; the famous
1977 “TAP” spin glass paper by Thouless, And