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## Mean field theoryA many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases (Gaussian field theory, 1D Ising model.) The great difficulty (e.g. when computing the partition function of the system) is the treatment of combinatorics generated by the interaction terms in the Hamiltonian when summing over all states. The goal of ## Additional recommended knowledgeThe main idea of MFT is to replace all interactions to any one body with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a relatively low cost. In field theory, the Hamiltonian may be expanded in terms of the magnitude of fluctuations around the mean of the field. In this context, MFT can be viewed as the "zeroth-order" expansion of the Hamiltonian in fluctuations. Physically, this means a MFT system has no fluctuations, but this coincides with the idea that one is replacing all interactions with a "mean field". Quite often, in the formalism of fluctuations, MFT provides a convenient launch-point to studying first or second order fluctuations. In general, dimensionality plays a strong role in determining whether a mean-field approach will work for any particular problem. In MFT, many interactions are replaced by one effective interaction. Then it naturally follows that if the field or particle exhibits many interactions in the original system, MFT will be more accurate for such a system. This is true in cases of high dimensionality, or when the Hamiltonian includes long-range forces. The Ginzburg criterion is the formal expression of how fluctuations render MFT a poor approximation, depending upon the number of spatial dimensions in the system of interest. While MFT arose primarily in the field of Statistical Mechanics, it has more recently been applied elsewhere, for example for doing Inference in Graphical Models theory in artificial intelligence. ## Formal approachThe formal basis for mean field theory is the Bogoliubov inequality. This inequality states that the free energy of a system with Hamiltonian has the following upper bound: where the average is taken over the equilibrium ensemble of the reference system with Hamiltonian . In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as
For the most common case that the target Hamiltonian contains only pairwise interactions, where is the set of pairs that interact. The minimizing procedure can be carried out formally. Define Tr where is the probability to find the reference system in the state specified by the variables (ξ - .
Thus In order to minimize we take the derivative with respect to the single degree-of-freedom probabilities using a Lagrange multiplier to ensure proper normalisation. The end result is the set of self-consistency equations where the mean field is given by ## ExampleConsider the Ising model on an where the Σ Let's transform our spin variable by introducing the fluctuation from its mean value . We may rewrite the Hamiltonian: where we define ; this is the If fluctuations are small, we may neglect this last term. As per the above arguments, when the fluctuations are small, then Again, the summand can be reexpanded to The only term that matters from the partition function's point of view is the first product. By symmetry arguments, the mean value of each spin is site-independent. We can replace with . We are still stuck with a double summation over neighboring spins, yet the summand involves only one site of each neighbor. Roughly speaking, we count 2 At this point, the Hamiltonian has been reduced to that of a single-body problem. The drawback is that now the effective coupling constant contains the mean value of the summand. Substituting this Hamiltonian into the partition function, and solving the effective 1D problem, we obtain where MFT is known under a great many names and guises. Similar techniques include Bragg-Williams approximation, models on Bethe lattice, Landau theory, Flory-Huggins solution theory, and Scheutjens-Fleer theory. |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Mean_field_theory". A list of authors is available in Wikipedia. |