# 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,

suppose we may approximate

where is the number of neighbors of site (e.g. 4 in a 2-D square lattice), and is the (unknown) magnetization:

Therefore, the Hamiltonian turns to

as in the regular Langevin theory of magnetism: the spins are independent, but coupled to a constant field of strength

The magnetization of the Langevin theory is

Therefore:

This is a **self-consistent** expression for . There exists a critical temperature, defined by

At temperatures higher than this value the only solution is . Below it, however, this solution becomes unstable (it corresponds to a maximum in energy), whereas two others are stable. Slightly below ,

## 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 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 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)