Bragg-Williams approximation for the Potts model

We now apply the same approximation that we have just seen to a slightly more complex situation: the Potts model. This is defined exactly as the Ising model, but with an essential difference: the degrees of freedom of the system, which we now call , instead of just two values can assume integer values: . We can therefore write the Hamiltonian of such a system as:

(where we have supposed that the external magnetic field tends to favour the situation where the degrees of freedom assume the value 1; of course we could have done otherwise). As can be expected the Potts model with is equivalent to an Ising model, as can be seen from the following equivalence:
where and are the degrees of freedom of the Ising model (). However, a Potts model with is not equivalent to an Ising model where , as one could have expected. In fact it is not possible in this case to write a of the three-state variable with "simple" terms involving -s (namely, only quadratic terms); in particular it turns out that:

We therefore want to apply the Bragg-Williams approximation to a -state Potts model. First of all, we call our order parameter, and write the probability distribution of a single degree of freedom as[1]:

Therefore, from:
we get:
From now on one can proceed like we have previously seen.

  1. As we have stated in the footnote on page , we write in general as the sum of the probability that the degree of freedom assume a particular value and of that of all the remaining values; in this case we have chosen as this particular value, but of course we could have done otherwise.
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