MeanfieldTheorie

MeanFieldTheorie, or mean-field theory, is a set of analytical approximations in physics and statistical mechanics that replace the effect of all other particles on a given particle by an average, effective field. This reduces a complex many-body problem to a simpler single-body problem with a self-consistently determined field.

In lattice spin models such as the Ising model, the interaction of a spin with its neighbors is approximated by a local field h_eff = J z m, where J is the interaction strength, z the coordination number, and m the average magnetization. The self-consistency equation m = f(h_eff) (for example m = tanh(beta h_eff) in the classical Ising case) yields the possible ordered states and phase transitions. Similar decouplings appear in other systems, and analogous ideas underlie Landau theory of phase transitions.

The mean-field approach is exact in the limit of high dimensionality or infinite coordination, but generally neglects fluctuations. It often provides qualitative insight and simple quantitative estimates, yielding mean-field critical exponents (for example beta = 1/2, gamma = 1, nu = 1/2) that differ from actual values in low dimensions.

Applications and variants include the Weiss theory for ferromagnetism, the Bragg–Williams approximation, and extensions such as dynamical mean-field theory (DMFT) for correlated electrons. In quantum many-body physics, mean-field approximations appear as Hartree–Fock, BCS theory, and the Gross–Pitaevskii equation for Bose systems.

Originating with Weiss in the early 20th century, mean-field theory provides a tractable framework for understanding collective phenomena, while recognizing its limitations due to neglected fluctuations.