Editing Mean-field theory (section)

==Origins==
The idea first appeared in physics ([[statistical mechanics]]) in the work of [[Pierre Curie]]<ref>{{Cite journal | last1 = Kadanoff | first1 = L. P. | author-link1 = Leo Kadanoff| title = More is the Same; Phase Transitions and Mean Field Theories | doi = 10.1007/s10955-009-9814-1 | journal = Journal of Statistical Physics | volume = 137 | issue = 5–6 | pages = 777–797 | year = 2009 | arxiv = 0906.0653|bibcode = 2009JSP...137..777K | s2cid = 9074428 }}</ref> and [[Pierre Weiss]] to describe [[phase transitions]].<ref>{{cite journal | title = L'hypothèse du champ moléculaire et la propriété ferromagnétique | first = Pierre | last = Weiss | author-link = Pierre Weiss | journal = J. Phys. Theor. Appl. | volume = 6 | issue = 1 | year= 1907 | pages= 661–690 | doi = 10.1051/jphystap:019070060066100 | url = http://hal.archives-ouvertes.fr/jpa-00241247/en }}</ref> MFT has been used in the Bragg–Williams approximation, models on [[Bethe lattice]], [[Landau theory]], [[Curie-Weiss law]] for magnetic susceptibility, [[Flory–Huggins solution theory]], and [[Scheutjens–Fleer theory]].

[[Many-body system|Systems]] with many (sometimes infinite) degrees of freedom are generally hard to solve exactly or compute in closed, analytic form, except for some simple cases (e.g. certain Gaussian [[random-field]] theories, the 1D [[Ising model]]). Often combinatorial problems arise that make things like computing the [[Partition function (mathematics)|partition function]] of a system difficult. MFT is an approximation method that often makes the original problem to be solvable and open to calculation, and in some cases MFT may give very accurate approximations.

In [[classical field theory|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 that an MFT system has no fluctuations, but this coincides with the idea that one is replacing all interactions with a "mean-field”.

Quite often, MFT provides a convenient launch point for studying higher-order fluctuations. For example, when computing the [[partition function (statistical mechanics)|partition function]], studying the [[combinatorics]] of the interaction terms in the [[Hamiltonian mechanics|Hamiltonian]] can sometimes at best produce [[perturbation theory|perturbation]] results or [[Feynman diagram]]s that correct the mean-field approximation.