= Scalar-field source Legendre transform
{title2=$\Gamma[m]=F[h]+\int hm,\quad m=-\delta F/\delta h$}
For a real statistical source convention $Z[h]=\int\mathcal D\phi\,e^{-H_0[\phi]+\int h\phi}$, define $F=-\log Z$ and the displayed <Legendre transform>. On a differentiable stable branch, $\delta\Gamma/\delta m=h$. In the <Landau approximation>, the branch expression is $\Gamma_{\rm L}[m]=H_0[m]$. The exact Legendre effective <free energy> is convex; a bare nonconvex <Landau free energy> is a local approximation and requires branch or coexistence interpretation.
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