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

$\sum _{<j>}S_{j}\approx n{\bar {s}},$

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

$U=-Jn\sum _{i}^{N}S_{i}{\bar {s}},$

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

$H=Jn{\bar {s}}.$

The magnetization of the Langevin theory is

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

Therefore:

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

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

$k_{B}T_{c}=Jn.$

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)