Ginzburg criterion and fluctuation dominance

Use a short-range scalar free energy containing a positive gradient coefficient c, quadratic coefficient r, and positive quartic coefficient u. Below the mean-field transition the uniform amplitude obeys $m_0^2=|r|/u$. Fluctuations averaged over a region of linear size comparable to the correlation length are the relevant comparison. Comparing the order parameter with the entire microscopic point variance would mix long-distance critical fluctuations with cutoff-dependent short-distance noise.

A quadratic expansion about one selected ordered minimum gives a fluctuation propagator with denominator $cq^2+2|r|$. Its correlation length is $\sqrt{c/(2|r|)}$. Integrating only momenta of order the inverse correlation length or smaller gives a coarse-region variance of order $k_BT/(c\xi^{d-2})$, up to dimension-dependent constants. The precise region shape and cutoff convention change those constants, so this estimate should be used as a scale criterion rather than an exact universal equality.

Mean-field consistency requires the dimensionless fluctuation ratio to be much smaller than one. Its growth as $\xi^{4-d}$ makes four the upper critical dimension of this short-range scalar quartic theory. Below four dimensions, approaching a generic continuous critical point eventually defeats a fixed Gaussian saddle description. Above four, the long-distance interaction becomes irrelevant for the leading critical fluctuations, although it still stabilizes the ordered state and may remain dangerously irrelevant for some observables.

At four dimensions the power estimate is marginal and cannot decide the flow by itself; logarithmic corrections require a more careful analysis. In three dimensions, writing the mean-field length as proportional to $|t|^{-1/2}$ gives a ratio proportional to $|t|^{-1/2}$, with material-dependent prefactor. A small prefactor creates a narrow fluctuation region. Thus observing apparent mean-field exponents over an accessible window does not prove that the asymptotic infinite-system exponents are mean-field values.

For unit Ising spins define total spin $M=\sum_i s_i$ and a physical conjugate energy h through the Hamiltonian term $-hM$. Differentiating the Boltzmann average gives the per-site susceptibility $\chi=\partial\langle M/N\rangle/\partial h=\operatorname{Var}(M)/(Nk_BT)$. The total response lacks the divisor N. Using a dimensionless field $H=h/(k_BT)$ instead removes the thermal-energy factor, so equations written in those two field conventions differ.

The variance contains connected pair correlations, including correlations between distinct sites. Independent zero-mean unit spins have total variance N and per-site susceptibility $1/(k_BT)$. Strong positive correlations can make the variance far larger. This fluctuation-response identity is an equilibrium statement; it does not predict relaxation time, guarantee independent measurements in a simulation, or authorize replacing correlated regions by independent individual spins. Statistical sampling error and physical order-parameter fluctuations are related questions with different normalizations.