I'm reading an article that derives an expression related to the Curie-Weiss-Potts models. The question pertains to how Equation (7) in the article is derived. Below is my summary of the information they provide leading up to equation (7)
The model is defined using the Hamiltonian built on a fully connected (or complete) graph as:
$$ H = -\frac{J}{N} \sum_{i < j} \delta(\sigma_i, \sigma_j) $$
where
$$ \delta(\sigma, \sigma') $$
is the Kronecker delta function. Rewriting the Hamiltonian while neglecting certain terms for large N, we have:
$$ H = -\frac{J}{N} \sum_{\sigma} \left( \sum_i \delta(\sigma_i, \sigma) \right)^2 $$
This leads to the partition function:
$$ Z \propto \int \prod_{\sigma=1}^{q} dx_\sigma e^{-N \left[ \sum_{\sigma} \frac{\beta J x_\sigma^2}{2} - \log \left( \sum_{\sigma} e^{\beta J x_\sigma} \right) \right]} $$
They then use a saddle point argument to derive Equation (4), which gives the probability of an element being in particular state sigma:
$$ x_\sigma = \frac{e^{\beta J x_\sigma}}{\sum_{\sigma'} e^{\beta J x_{\sigma'}}} ,\sigma=1,\dots,q $$
Free-energy density:
$$ \beta f = -\log \left( \sum_{\sigma} e^{\beta J x_\sigma} \right) + \sum_{\sigma} \frac{\beta J x_\sigma^2}{2} $$
Next they argue that equation (4) is symmetric under the permutation of the components (x_1, \dots, x_q). We can find all possible solutions by setting (q-1) components equal to each other and solving one single equation. If (i_1, \dots, i_q) is any permutation of (1, \dots, q), we have:
$$ x_{i_1} = x, x_{i_j} = y, j = 2, \dots, q $$
where
$$ y = \frac{1}{q-1}(1-x) $$
Finally they arrive at equation (7):
$$ x = \frac{1}{1 + (q-1) \exp \left( \frac{\beta J (1-qx)}{q-1} \right)} $$
I'm unsure how they derive this given the previous equations. Any help would be much appreciated.