Mean-field theory

Mean-field theory replaces the fluctuating influence of a system's many components on any one of them by a single averaged, effective field. The interacting many-body problem then collapses into a solvable one-body problem, usually closed by requiring self-consistency: the average that each component responds to must equal the average that all of them produce. Van der Waals's 1873 equation of state, the Weiss theory of ferromagnetism, and Landau's 1937 expansion of the free energy in an order parameter are all mean-field theories.

It is a superb approximation over most of a phase diagram and a systematically wrong one near a critical point. Its predicted exponents — β = 1/2, γ = 1, ν = 1/2, δ = 3 — are internally consistent, satisfying the Rushbrooke and Widom relations exactly, but they do not match measurement: the real value of β for three-dimensional systems is 0.326. The failure is diagnosable. Mean-field theory assumes fluctuations are small corrections about an average, whereas criticality consists precisely of fluctuations on every length scale at once; the approximation averages away the only thing that matters.

The theory even signals the dimension in which it is correct. Its exponents violate the hyperscaling relation 2 − α = dν except at d = 4, the upper critical dimension: above four dimensions each component has enough neighbours for averaging to work, and mean-field theory becomes exact. Below it — including the three dimensions of the world — fluctuations dominate and the renormalisation group is required. The framework nonetheless survived its own numbers, since Landau's free-energy expansion remains the standard language of spontaneous symmetry breaking.

Originates with van der Waals's 1873 doctoral thesis and Pierre Weiss's 1907 molecular-field theory of ferromagnetism; generalised into a universal framework by Lev Landau in 1937. Its breakdown near criticality was quantified by the Ginzburg criterion in 1960 and explained by Wilson's renormalisation group in the 1970s.