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Mean-field theory and approximations

时间:2010-03-04 12:23:58  来源:  作者:

Mean field theory (also known as self consistent field theory) and to illustrate the theory by applying it to the description of the Ising model.
The main idea of the mean field theory is to focus on one particle and assume that the most important contribution to the interactions of such particle with its neighboring particles is determined by the mean field due to the neighboring particles.

The goal of this section is to introduce the so-called mean field theory (also known as self consistent field theory) and to illustrate the theory by applying it to the description of the Ising model.

The main idea of the mean field theory is to focus on one particle and assume that the most important contribution to the interactions of such particle with its neighboring particles is determined by the mean field due to the neighboring particles. In the 1-dimensional Ising model, for instance, the average force $ overline{F_k}$ exerted on spin $ S_k$ is

$displaystyle overline{F_k} equiv -overline{frac{partial H}{partial S_k}} = bar{mu} B + J sum_{j} overline{S_j},$ (300)


where the index $ j$ includes all the nearest neighbors of spin $ S_k$. Therefore, the average magnetic field $ overline{B}$ acting on spin $ S_k$ is

$displaystyle overline{B} equiv frac{overline{F_k}}{bar{mu}} = B + Delta B,$ (301)


where

$displaystyle Delta B = J 2 overline{S_k}/ bar{mu},$ (302)


is the contribution to the mean field due to the nearest neighbors. Note that $ overline{S_k}=overline{S_j}$ 

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