Ginzburg criterion

Mean field theory gives sensible results as long as one is able to neglect fluctuations in the system under consideration.

If ${\displaystyle \phi }$ is the order parameter of the system, then mean field theory requires that the fluctuations in the order parameter are much smaller than the actual value of the order parameter near the critical point.

Quantitatively, this means that:

${\displaystyle \displaystyle {\mathcal {\langle }}(\delta \phi )^{2}\rangle \quad {\ll }\quad \langle \phi \rangle ^{2}}$

The Ginzburg criterion is a restatement of this inequality through measurable quantities, such as the magnetic susceptibility in the Ising model.

It also gives the idea of an upper critical dimension, a dimensionality of the system above which mean field theory gives proper results, and the critical exponents predicted by mean field theory match exactly with those obtained by numerical methods.