Mean field models

From SklogWiki
Revision as of 12:34, 29 April 2010 by Dduque (talk | contribs) (I can't believe there was not an entry about this... work in progress)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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 .