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| ==Mean field solution of the Ising model== | | ==Mean field solution of the Ising model== |
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| A well-known mean field solution of the [[Ising model]], known as the ''Bragg-Williams approximation'' goes as follows. | | A well-known mean field solution of the [[Ising model]] goes as follows. From the original hamiltonian, |
| From the original Hamiltonian, | | :<math> \frac{U}{k_B T} = - K \sum_i S_i \sum_j S_j , </math> |
| :<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> | | :<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 square 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 squate 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> |
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| Therefore, the Hamiltonian turns to
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| :<math> U = - J n \sum_i^N S_i \bar{s} , </math>
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| as in the regular Langevin theory of magnetism (see [[Curie's_law]]): the spins are independent, but coupled to a constant field of strength
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| :<math>H= J n \bar{s}.</math>
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| The magnetization of the Langevin theory is
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| :<math> \bar{s} = \tanh( H/k_B T ). </math>
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| Therefore:
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| :<math> \bar{s} = \tanh(J n\bar{s}/k_B T). </math>
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| This is a '''self-consistent''' expression for <math>\bar{s}</math>. There exists a critical temperature, defined by
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| :<math>k_B T_c= J n .</math>
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| At temperatures higher than this value the only solution is <math>\bar{s}=0</math>. Below it, however, this solution becomes unstable
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| (it corresponds to a maximum in energy), whereas two others are stable. Slightly below <math>T_c</math>,
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| :<math>\bar{s} =\pm\sqrt{3\left(1 - \frac{T}{T_c}\right)}. </math>
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| ==General discussion==
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| The solution obtained shares a number of features with any other mean field approximation:
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| *It largely ignores geometry, which may be important in some cases. In particular, it reduces the lattice details to just the number of neighbours.
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| *As a consequence, it may predict phase transitions where none are found: the [[1-dimensional_Ising_model|1-D ising model]] <math>n=2</math> is known to lack any phase transition (at finite temperature)
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| *In general, the theory ''underestimates fluctuations''
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| *It also leads to ''classical critical exponents'', like the <math>\left(1 - \frac{T}{T_c}\right)^{1/2}</math> decay above. In 3-D, the magnetization follows a power law with a different exponent.
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| *Nevertheless, above a certain space dimension the critical exponents are correct. This dimension is 4 for the Ising model, as predicted by a self-consistency requirement due to E.M. Lipfshitz (similar ones are due to Peierls and L.D. Landau)
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| ==References==
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| <references/>
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| [[Category: Statistical mechanics]]
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