# 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, $U = - J \sum_i^N S_i \sum_{} S_j ,$

suppose we may approximate $\sum_{} S_j \approx n \bar{s},$

where $n$ is the number of neighbors of site $i$ (e.g. 4 in a 2-D square lattice), and $\bar{s}$ is the (unknown) magnetization: $\bar{s}=\frac{1}{N} \sum_i S_i .$

Therefore, the Hamiltonian turns to $U = - J n \sum_i^N S_i \bar{s} ,$

as in the regular Langevin theory of magnetism (see Curie's_law): the spins are independent, but coupled to a constant field of strength $H= J n \bar{s}.$

The magnetization of the Langevin theory is $\bar{s} = \tanh( H/k_B T ).$

Therefore: $\bar{s} = \tanh(J n\bar{s}/k_B T).$

This is a self-consistent expression for $\bar{s}$. There exists a critical temperature, defined by $k_B T_c= J n .$

At temperatures higher than this value the only solution is $\bar{s}=0$. Below it, however, this solution becomes unstable (it corresponds to a maximum in energy), whereas two others are stable. Slightly below $T_c$, $\bar{s} =\pm\sqrt{3\left(1 - \frac{T}{T_c}\right)}.$

## 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 $\left(1 - \frac{T}{T_c}\right)^{1/2}$ 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)