Editing Mean field models
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Therefore, the Hamiltonian turns to | Therefore, the Hamiltonian turns to | ||
:<math> U = - J n \sum_i^N S_i \bar{s} , </math> | :<math> U = - J n \sum_i^N S_i \bar{s} , </math> | ||
as in the regular Langevin theory of magnetism | 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> | :<math>H= J n \bar{s}.</math> | ||
The magnetization of the Langevin theory is | The magnetization of the Langevin theory is | ||
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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. | *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. | ||
*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) | *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) | ||