= 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>.<Statistical Field Theory by David Tong>[§2.2.4, The Upper Critical Dimension: correlation-volume fluctuations and the Ginzburg criterion]{ref}
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