Mean Field Approximation (MFA) in the theory of magnetism.

MFA was constructed by P. Weiss and P. Curie for explaining the behavior of ferromagnetic materials. Such systems behave like paramagnets at high temperatures, but below certain critical temperature Tc they exhibit “spontaneous magnetization” — i.e., a sizable macroscopic magnetic moment even in the absence of an external magnetic field.

 In its original version MFA was a phenomenological theory. Microscopic justification for the model postulates was added later, when the nature of interactions between individual magnetic atoms (Heissenberg exchange) became understood on the grounds of quantum mechanics.

 MFA is often regarded as a quite “primitive” model. In fact, some of its predictions — especially, concerning the system behavior at T → 0 and in the region close to Tc (the so-called “critical region”) — appear to be in “less than perfect” agreement with the experiment. It would be too much to say that the MFA results are completely wrong in these two regions; however, they definitely cannot be used for fine-tuned quantitative data interpretation. The main reason for the poor model performance for T → 0 is that simple averaging is not a good tool for describing properties associated with propagating excitations, especially in the case of long wavelengths — and it is the long-λ propagating spin wave modes that play the principal role in the ferromagnet behavior at the lowest temperature region1. The same applies to fluctuations: averaging, for obvious reasons, “smoothens out” any fluctuations. Consequently, a method based on averaging cannot describe absollutely correctly any phase transistion phenomena — because, as we know, fluctuations play a crucial role in them.

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http://physics.oregonstate.edu/~giebultt/COURSES/ph674/mfa4.pdf