Mean field models

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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:

$\bar{s}=\frac{1}{N} \sum_i S_i .$