 Single-mode smectic mean-field free energy

For $\varphi =A\cos (q_{0}z)$, the gradient-quartic Brazovskii model has volume-averaged free-energy densityThe averages are $⟨{\cos }^2⟩=⟨{\sin }^2⟩=1/2$ and $⟨{\cos }^2{\sin }^2⟩=1/8$. For $B>0$, the stable amplitude is zero for $a\ge a_c$ and satisfies $A^2=(a_c-a)/(B{q}_0^2)$ below it. The mean-field approximation therefore predicts a continuous phase transition.