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 goes as follows. From the original hamiltonian,

${\frac {U}{k_{B}T}}=-K\sum _{i}S_{i}\sum _{j}S_{j},$

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 squate lattice), and ${\bar {s}}$ is the (unknown) magnetization:

${\bar {s}}={\frac {1}{N}}\sum _{i}S_{i}.$