Mean field models

A mean field model, or a mean field solution of a model, is an approximation to the actual solution of a model in statistical physics. The model is made exactly solvable by treating the effect of all other particles on a given one as a mean field (hence its name). It appear in different forms and different contexts, but all mean field models have this feature in common.

Mean field solution of the Ising model

A well-known mean field solution of the Ising model, known as the Bragg-Williams approximation goes as follows. From the original Hamiltonian,

$U=-J\sum _i^N{S}_i{\sum }_{<j>}{S}_j,$

suppose we may approximate

${\sum }_{<j>}{S}_j\approx n\overline{s},$

where $n$ is the number of neighbors of site $i$ (e.g. 4 in a 2-D square lattice), and $\overline{s}$ is the (unknown) magnetization:

$\overline{s}=\frac{1}{N}{\sum }_i{S}_i.$

Therefore, the Hamiltonian turns to

$U=-Jn\sum _i^N{S}_i\overline{s},$

as in the regular Langevin theory of magnetism (see Curie's_law): the spins are independent, but coupled to a constant field of strength

$H=Jn\overline{s}.$

The magnetization of the Langevin theory is

$\overline{s}=\tanh (H/k_{B}T).$

Therefore:

$\overline{s}=\tanh (Jn\overline{s}/k_{B}T).$

This is a self-consistent expression for $\overline{s}$. There exists a critical temperature, defined by

$k_B{T}_c=Jn.$

At temperatures higher than this value the only solution is $\overline{s}=0$. Below it, however, this solution becomes unstable (it corresponds to a maximum in energy), whereas two others are stable. Slightly below $T_c$,

$\overline{s}=\pm \sqrt{3(1-\frac{T}{T_c})}.$

General discussion

The solution obtained shares a number of features with any other mean field approximation:

• It largely ignores geometry, which may be important in some cases. In particular, it reduces the lattice details to just the number of neighbours.
• As a consequence, it may predict phase transitions where none are found: the 1-D ising model $n=2$ is known to lack any phase transition (at finite temperature)
• In general, the theory underestimates fluctuations
• It also leads to classical critical exponents, like the ${(1-\frac{T}{T_c})}^{1/2}$ decay above. In 3-D, the magnetization follows a power law with a different exponent.
• Nevertheless, above a certain space dimension the critical exponents are correct. This dimension is 4 for the Ising model, as predicted by a self-consistency requirement due to E.M. Lipfshitz (similar ones are due to Peierls and L.D. Landau)