# Difference between revisions of "Mean field models"

(I can't believe there was not an entry about this... work in progress) |
(→Mean field solution of the Ising model) |
||

Line 3: | Line 3: | ||

==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 | + | A well-known mean field solution of the [[Ising model]], known as the ''Bragg-Williams approximation'' goes as follows. |

− | :<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> \ | + | :<math> \sum_{<j>} S_j \approx n \bar{s}, </math> |

− | where <math> | + | 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,

suppose we may approximate

where is the number of neighbors of site (e.g. 4 in a 2-D square lattice), and is the (unknown) magnetization:

Therefore, the Hamiltonian turns to

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

The magnetization of the Langevin theory is

Therefore:

This is a **self-consistent** expression for . There exists a critical temperature, defined by

At temperatures higher than this value the only solution is . Below it, however, this solution becomes unstable (it corresponds to a maximum in energy), whereas two others are stable. Slightly below ,