 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 $\pm 1$.

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.