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Apply the Jensen inequality to the convex exponential function under the trial equilibrium measure:The given identity therefore implies $e^{-F}\ge e^{-F_0-⟨H-H_0{⟩}_0}$. Taking minus the logarithm reverses the inequality and proves the Feynman-Bogoliubov inequalityThe units $\beta =1$ make $H$ and $F$ dimensionless here. Equality holds when $H-H_0$ is constant on the support of the trial measure. Optimizing the trial parameters tightens an upper bound, which is the basis of the Gaussian variational approximation.

In the usual three-dimensional isotropic setting, the transition is expected to be discontinuous, by the Brazovskii fluctuation-induced first-order transition mechanism.

If $\bar{\kappa }<0$, write $q_{\ast }^2=-\bar{\kappa }/(2\gamma )$ and $r=\bar{a}-{\bar{\kappa }}^2/(4\gamma )$. Then $J(q)=r+\gamma (q^2-q_{\ast }^2{)}^2$. A continuous finite-wavevector transition would require $r\to 0$ with $q_{\ast }>0$. But the soft-mode shell integral behaves asThe positive fluctuation corrections in the self-consistency equations diverge in that limit. In particular $\bar{\kappa }=\kappa +2B{I}_0$ cannot stay at a finite negative value while $I_0$ diverges. Thus the isotropic disordered state cannot lose stability through a zero-mass mode on a fixed nonzero shell in this approximation.

Ordering is instead expected through a free-energy crossing with a finite-amplitude smectic phase, giving a first-order phase transition. This is a reasoned fluctuation prediction for the finite-wavevector transition; the disordered Gaussian variational approximation alone does not locate the coexistence point or calculate the amplitude jump. A zero-wavevector instability or a lower-dimensional setting would require a separate analysis.

Choose the translationally invariant Gaussian variational approximationThe superscript $+$ means that one member of each pair $\{\mathbf{q},-\mathbf{q}\}$ is included; this is necessary because a real field obeys ${\varphi }_{-\mathbf{q}}={\varphi }_{\mathbf{q}}^{\ast }$. The constrained zero mode is omitted.

For each independent complex mode,Consequently,Substitution into the Gibbs--Bogoliubov--Feynman inequality giveswhich is the required upper bound.

In three dimensions, let $J(q)=r+\gamma (q^2-q_{\ast }^2{)}^2$, with $q_{\ast }>0$ and small $r>0$. Near the soft shell, $J(q)\simeq r+4\gamma q_{\ast }^2(q-q_{\ast }{)}^2$, soThe radial integral uses ${\int }_{-\infty }^{\infty }ds/(r+A{s}^2)=\pi /\sqrt{Ar}$. The $r^{-1/2}$ divergence obstructs the zero-mass limit of a self-consistent Gaussian variational approximation at a fixed nonzero ordering wavevector. This is the mechanism behind the Brazovskii fluctuation-induced first-order transition; Gross, Ignatiev and Chakraborty's study of fluctuation-driven ordering analyzes the corresponding standard Brazovskii model.