= Mean-field approximation
A mean-field approximation replaces the fluctuating influence of many neighbors by a self-consistent average. For an <Ising model> with coordination number $z$, exchange $J$ and field $h$, a uniform treatment gives $m=\tanh[\beta(zJm+h)]$ when spins take values $\pm1$.
The assumed average magnetization appears on both sides: solving the <self-consistency equation> chooses a state compatible with its own effective field. This makes a many-body problem tractable and can estimate ordering temperatures and qualitative phases. It neglects correlations between fluctuations, so it can overestimate ordering and predict incorrect behavior in low dimensions. The <Ginzburg criterion> assesses when critical fluctuations are too large to justify the simple mean-field result.
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