Difference between revisions of "Mean field models"

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(I can't believe there was not an entry about this... work in progress)
 
(Mean field solution of the Ising model)
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==Mean field solution of the Ising model==
 
==Mean field solution of the Ising model==
  
A well-known mean field solution of the [[Ising model]] goes as follows. From the original hamiltonian,
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A well-known mean field solution of the [[Ising model]], known as the ''Bragg-Williams approximation'' goes as follows.
:<math> \frac{U}{k_B T} = - K \sum_i S_i \sum_j S_j , </math>
+
From the original Hamiltonian,
 +
:<math> U = - J \sum_i^N S_i \sum_{<j>} S_j , </math>
 
suppose we may approximate
 
suppose we may approximate
:<math> \sum_j S_j \approx N \bar{s}, </math>
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:<math> \sum_{<j>} S_j \approx n \bar{s}, </math>
where <math>N</math> is the number of neighbors of site <math>i</math> (e.g. 4 in a 2-D squate lattice), and <math>\bar{s}</math> is the (unknown) magnetization:
+
where <math>n</math> is the number of neighbors of site <math>i</math> (e.g. 4 in a 2-D square lattice), and <math>\bar{s}</math> is the (unknown) magnetization:
 
:<math> \bar{s}=\frac{1}{N} \sum_i S_i . </math>
 
:<math> \bar{s}=\frac{1}{N} \sum_i S_i . </math>
 +
 +
Therefore, the Hamiltonian turns to
 +
:<math> U = - J n \sum_i^N S_i \bar{s} , </math>
 +
as in the regular Langevin theory of magnetism: the spins are independent, but coupled to a constant field of strength
 +
:<math>H= J n \bar{s}.</math>
 +
The magnetization of the Langevin theory is
 +
:<math>  \bar{s} = \tanh( H/k_B T ). </math>
 +
Therefore:
 +
:<math>  \bar{s} = \tanh(J n\bar{s}/k_B T). </math>
 +
 +
This is a '''self-consistent''' expression for <math>\bar{s}</math>. There exists a critical temperature, defined by
 +
:<math>k_B T_c= J n .</math>
 +
At temperatures higher than this value the only solution is <math>\bar{s}=0</math>. Below it, however, this solution becomes unstable
 +
(it corresponds to a maximum in energy), whereas two others are stable. Slightly below <math>T_c</math>,
 +
:<math>\bar{s} =\pm\sqrt{3\left(1 - \frac{T}{T_c}\right)}. </math>

Revision as of 16:17, 3 May 2010

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 = - J n \sum_i^N S_i \bar{s} ,

as in the regular Langevin theory of magnetism: the spins are independent, but coupled to a constant field of strength

H= J n \bar{s}.

The magnetization of the Langevin theory is

  \bar{s} = \tanh( H/k_B T ).

Therefore:

  \bar{s} = \tanh(J n\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= J n .

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)}.