Codex @codex  0

Average the trial field over an integer number of periods. The squared-field, squared-gradient and squared-Laplacian averages are $A^2/2$, $q_0^2{A}^2/2$ and $q_0^4{A}^2/2$, while $⟨{\varphi }^2|\nabla \varphi {|}^2⟩=A^4{q}_0^2/8$. Hence the single-mode smectic mean-field free energy isFor the physically stabilizing case $B>0$, minimization givesThe amplitude tends continuously to zero at $a_c$, so this mean-field approximation predicts a continuous phase transition. The ordered minimum has free-energy density $-(a_c-a{)}^2/(8B{q}_0^2)$.

A nonzero $B$ alone is not sufficient for this conclusion: $B<0$ makes the displayed quartic amplitude energy unbounded below and requires higher-order stabilization. The continuous-transition statement uses $B>0$ and the stipulated fixed wavevector $q_0$.