= Bragg-Williams free energy of the Ising model
{c}
{title2=$A_{\rm BW}=\langle H\rangle-TS$}
= Ising product-measure free energy
{c}
{synonym}
A product trial distribution with actual mean <Ising spin> $m$ gives
$$
\frac{A_{\rm BW}}N=-\frac{qJ}{2}m^2-hm+k_BT\left[\frac{1+m}{2}\log\frac{1+m}{2}+\frac{1-m}{2}\log\frac{1-m}{2}\right].
$$
Its <entropy> is the sum of independent-spin entropies, with $0\log0=0$. Differentiation gives $k_BT\operatorname{arctanh}m=qJm+h$, 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.
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