 Solution

In mean-field theory all sites have the same normalized single-site distribution $e^{x\cos \theta }/Z_1(x)$. Consequentlywhere $\mathcal{L}$ is the Langevin function. Requiring the average used to construct the field to equal the average in that field givesSetSince $S=\tau x$, the self-consistency equation is exactly $\tau x=\coth x-1/x$. It also follows by stationarity of $F_{\mathrm{MF}}/N=z_{c}J{S}^2/2-k_{B}T\log Z_1(\beta z_{c}JS)$. This independent route checks the constant term and all neighbor-counting factors in the partition function.