 Solution

The mean-field approximation replaces the fluctuating environment of each microscopic variable by a uniform average field. For spins, write $S_i=S+\delta S_i$ and drop products $\delta S_i\delta S_j$ between different sites. The resulting single-site problem can be solved independently, and its average spin must reproduce the assumed order parameter $S$: this is the self-consistency equation.

It is reliable when correlated thermal fluctuations about the mean are small, for example with many interacting neighbours, long-range interactions or high spatial dimension. Near a thermodynamic critical point the quantitative test is the Ginzburg criterion. For short-range scalar ordering the upper critical dimension is four, with logarithmic qualifications at four; a two-dimensional nearest-neighbour lattice is below this and its asymptotic critical region of a phase transition is not described accurately by simple mean-field theory.^[ref]