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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,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: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 $r=r_{0}t$, an estimate of its width isin 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.

Let $t=(T-T_c)/T_c$, and suppose $r=r_{t}t$ with $r_t>0$ while $u>0$ and $\kappa >0$ remain finite at the transition. Critical exponents describe the leading singular powers: $M\sim (-t{)}^{\beta }$ below the transition, $\chi \sim |t{|}^{-\gamma }$, $\xi \sim |t{|}^{-\nu }$, $C_{V,s}\sim |t{|}^{-\alpha }$, $M(t=0)\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$, and $G(R,t=0)\sim R^{-(D-2+\eta )}$.

The quartic equation of state $h=rM+u{M}^3$ immediately gives $M=\sqrt{-r/u}$ at zero field below the transition, so $\beta =1/2$. Differentiating with respect to the conjugate field gives ${\chi }^{-1}=r+3u{M}^2$, which is $r$ above and $2|r|$ below; therefore $\gamma =1$. At $r=0$, $M=(h/u{)}^{1/3}$ with its real sign, giving $\delta =3$. The quadratic correlation function has $\xi \propto |r{|}^{-1/2}$ and critical Ornstein--Zernike correlation function proportional to $q^{-2}$, giving $\nu =1/2$ and $\eta =0$.

Finally the minimized potential below the transition is $V_s=-r^2/(4u)$, while it is zero above. Its second temperature derivative is a finite step, so the mean-field heat-capacity critical exponent is $\alpha =0$. In particular,These are derived mean-field critical exponents. In the fluctuation-dominated critical region of a phase transition, one instead derives exponents from the scaling eigenvalues of the renormalization-group fixed point, as in the next question. A zero heat-capacity power exponent can also accompany logarithms in other theories, so specifying the power alone does not completely describe that singularity.