Mean field theory In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of ...

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

\phi

is the

order parameter In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic s ...

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\mathcal \langle(\delta \phi)^2\rangle \quad\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.

Example: Ising model

One can prove that:

k_BT\chi \ll \langle M \rangle^2

Where

k_B

is the Boltzmann constant,

T

is the system temperature,

\chi

is the total magnetic susceptibility and

\langle M \rangle

is the total average magnetization of the system. Using this in the

Landau theory Landau theory (also known as Ginzburg–Landau theory, despite the confusing name) in physics is a theory that Lev Landau introduced in an attempt to formulate a general theory of continuous (i.e., second-order) phase transitions. It can also be ...

, which is identical to the mean field theory for the

Ising model The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...

, the value of the upper critical dimension comes out to be 4. If the dimension of the space is greater than 4, the mean-field results are good and self-consistent. But for dimensions less than 4, the predictions are less accurate. For instance, in one dimension, the mean field approximation predicts a phase transition at finite temperatures for the Ising model, whereas the exact analytic solution in one dimension has none (except for

T=0

and

T\rightarrow \infty

Example: Classical Heisenberg model

In the

classical Heisenberg model In statistical physics, the classical Heisenberg model, developed by Werner Heisenberg, is the n = 3 case of the ''n''-vector model, one of the models used to model ferromagnetism and other phenomena. Definition The classical Heisenberg model can ...

of magnetism, the order parameter has a higher symmetry, and it has violent directional fluctuations which are more important than the size fluctuations. They overtake to the Ginzburg temperature interval over which fluctuations modify the mean-field description thus replacing the criterion by another, more relevant one.

Footnotes

References

* * * * {{Cite journal , author=H. Kleinert , author-link=Hagen Kleinert , title=Criterion for Dominance of Directional over Size Fluctuations in Destroying Order , year=2000 , journal=Phys. Rev. Lett. , volume=84 , issue=2 , pages=286–289 , url=http://prl.aps.org/abstract/PRL/v84/i2/p286_1 , archive-url=https://archive.today/20130223113538/http://prl.aps.org/abstract/PRL/v84/i2/p286_1 , url-status=dead , archive-date=2013-02-23 , doi=10.1103/physrevlett.84.286 , arxiv=cond-mat/9908239 , bibcode=2000PhRvL..84..286K , pmid=11015892 , s2cid=24140115 Statistical mechanics Physical quantities