Bragg-Williams free energy of the Ising model ($A_{\mathrm{BW}}=⟨H⟩-TS$)

A product trial distribution with actual mean Ising spin $m$ givesIts entropy is the sum of independent-spin entropies, with $0\log 0=0$. Differentiation gives $k_{B}T\mathrm{arctanh}m=qJm+h$, the self-consistency equation. The variational mean parameter is distinct from the assumed field parameter of the Ising auxiliary mean-field free energy until the equilibrium condition is imposed.