Constrained order-parameter free energy ($F_c(M)=-k_{B}T{V}^{-1}\log Z_M$)

Fix the spatial mean of a scalar order parameter to $M$ and sum the Boltzmann weights of all configurations with that mean. The resulting constrained partition function $Z_M$ defines this free-energy density, with an additive normalization fixed consistently. For an applied conjugate field $h$, $Z(h)=\int dM{e}^{-{\beta }_{T}V[F_c(M)-hM]}$, up to the chosen measure normalization. The thermodynamic limit gives $f(h)={\inf }_M[F_c(M)-hM]$. A homogeneous Landau free energy is an approximation to this constrained potential; phase-separated configurations can convexify it in the full thermodynamic limit.

In a finite box, retain the constant mode $M$ while integrating all nonzero field modes using the Wilsonian coarse-grained statistical Hamiltonian. Below the smallest nonzero momentum, the remaining effective energy is $V{U}_{\Lambda }(M)$ and its Boltzmann weight is precisely the constrained order-parameter free energy weight. Thus the displayed equality holds with consistent field-independent normalization. It is not an identity between a bare statistical Hamiltonian and an unconstrained equilibrium free energy. An analytic polynomial Landau free energy further assumes a homogeneous branch and regular effective coefficients; long-distance fluctuations near and below an upper critical dimension can invalidate that analytic approximation.