= Correlation-volume derivation of the scalar Ginzburg ratio
{title2=$\mathcal G\asymp u c^{-d/2}|r|^{(d-4)/2}$}
For the dimensionless scalar <Landau-Ginzburg theory> with stiffness $c$, quadratic coefficient $r<0$ and quartic term $u\phi^4/4$, the <mean-field approximation> gives $\phi_0^2=|r|/u$ and <correlation length> $\xi\asymp(c/|r|)^{1/2}$. Restricting the Gaussian <Fourier transform> covariance $(cq^2+2|r|)^{-1}$ to $|q|\lesssim\xi^{-1}$ estimates the variance of the field averaged over a <correlation volume> as $c^{-d/2}|r|^{(d-2)/2}$, up to a dimension-dependent constant. Dividing by $\phi_0^2$ proves the displayed <Ginzburg criterion> ratio. It diverges below four dimensions, vanishes above four and is marginal at four. Short-distance fluctuations renormalize microscopic coefficients and should not be confused with this long-distance critical test.
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