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 (see Curie's_law): the spins are independent, but coupled to a constant field of strength
The magnetization of the Langevin theory is
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 ,
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)