= Solution
The <Ginzburg criterion> tests whether fluctuations on the scale of one <correlation volume> are small compared with the squared equilibrium <order parameter>. In the scalar quartic <Landau-Ginzburg theory> of part (a), the ordered mean-field value is $M^2=|r|/u$, and its longitudinal <correlation length> is proportional to $(c/|r|)^{1/2}$.
Average the fluctuation field over a region of size $\xi$. Its smoothing factor selects momenta of order $\xi^{-1}$ or smaller. The Gaussian <correlation function> then gives, up to a positive dimension-dependent constant,
$$
\langle(\delta\phi_\xi)^2\rangle
\asymp\int_{|q|\lesssim\xi^{-1}}\frac{d^dq}{(2\pi)^d}\frac1{cq^2+2|r|}
\asymp\frac1c\xi^{2-d}
\asymp c^{-d/2}|r|^{(d-2)/2}.
$$
The last step follows directly by setting $q=\sqrt{|r|/c}\,k$. Dividing by $M^2$ proves the <correlation-volume derivation of the scalar Ginzburg ratio>:
$$
\boxed{\mathcal G=\frac{\langle(\delta\phi_\xi)^2\rangle}{M^2}
\asymp u c^{-d/2}|r|^{(d-4)/2}\ll1.}
$$
For $d<4$, the ratio diverges as the transition is approached, so there is a fluctuation-dominated <critical region of a phase transition> in which <mean-field critical exponents> cannot be trusted. With $r=r_0t$, an estimate of its width is
$$
|t|\lesssim t_G,\qquad t_G\asymp\frac1{r_0}\left(\frac{u^2}{c^d}\right)^{1/(4-d)},
$$
in fixed microscopic units. Its numerical coefficient depends on the averaging convention and microscopic normalization.
For $d>4$, the ratio tends to zero and the long-distance <mean-field approximation> becomes self-consistent. Four is therefore the ordinary quartic <upper critical dimension>. At $d=4$ this test is marginal: its thermal power is zero, and the running quartic interaction produces logarithmic corrections. The <marginal Ginzburg criterion> alone cannot decide those logarithms. Short-distance contributions to an unaveraged variance renormalize the coefficients; they should not be mistaken for the long-wavelength fluctuations used in this criterion. Nor does the criterion by itself prove that a proposed ordered phase exists below its <lower critical dimension>.
Back to article page
