Mean field models - Revision history

Carl McBride: Added a category

2010-05-14T13:17:51Z

Added a category

← Older revision		Revision as of 15:17, 14 May 2010
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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)
			==References==
			<references/>
			[[Category: Statistical mechanics]]

Dduque: Added link to Curie's_law

2010-05-03T14:28:10Z

Added link to Curie's_law

← Older revision		Revision as of 16:28, 3 May 2010
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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: the spins are independent, but coupled to a constant field of strength		as in the regular Langevin theory of magnetism (see [[Curie's_law]]): 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

Dduque: Completed solution, section added.

2010-05-03T14:26:32Z

Completed solution, section added.

← Older revision		Revision as of 16:26, 3 May 2010
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	(it corresponds to a maximum in energy), whereas two others are stable. Slightly below <math>T_c</math>,		(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>		:<math>\bar{s} =\pm\sqrt{3\left(1 - \frac{T}{T_c}\right)}. </math>

			==General discussion==
			The solution obtained shares a number of features with any other mean field approximation:
			*It largely ignores geometry, which may be important in some cases. In particular, it reduces the lattice details to just the number of neighbours.
			*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)
			*In general, the theory ''underestimates fluctuations''
			*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)

Dduque: /* Mean field solution of the Ising model */

2010-05-03T14:17:02Z

Mean field solution of the Ising model

← Older revision		Revision as of 16:17, 3 May 2010
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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~~,		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>		:<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>

Dduque: I can't believe there was not an entry about this... work in progress

2010-04-29T10:34:36Z

I can't believe there was not an entry about this... work in progress

New page

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]] goes as follows. From the original hamiltonian,
:<math> \frac{U}{k_B T} = - K \sum_i S_i \sum_j S_j , </math>
suppose we may approximate
:<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:
:<math> \bar{s}=\frac{1}{N} \sum_i S_i . </math>