Mean-field theory

Mean-field theory approximates a many-component system by replacing interactions with an effective field representing the average influence of other components.

Self-consistency: A condition requiring a model’s assumed quantities to agree with values calculated from its equations. Mean fields are usually determined by requiring the assumed average to match the system’s predicted average.
Ising model: A lattice model of binary spins coupled through pairwise interactions. Its mean-field solution illustrates how collective magnetization becomes a self-consistent field.
Monte Carlo method: A computational method that estimates quantities using repeated random sampling. Sampling can capture fluctuations that a deterministic mean-field solution omits.
Pierre Curie: A French physicist whose research helped establish the study of magnetism and symmetry. His work on magnetic materials helped shape the problems later treated by molecular-field models.
Order parameter: A quantity that distinguishes phases by measuring the degree of an ordered state. Its average often supplies the effective field that defines a mean-field solution.
Curie–Weiss model: An Ising model in which every spin interacts equally with every other spin. This exactly specified all-to-all model makes the mean-field assumption explicit.
Renormalization group: A framework for tracking how physical descriptions change across length scales. It explains why mean-field predictions fail when long-range critical fluctuations dominate.
Pierre-Ernest Weiss: A French physicist who proposed the molecular-field theory of ferromagnetism. His effective internal field is a foundational historical form of mean-field reasoning.
Landau theory: A phenomenological framework that describes phase transitions through an expansion in an order parameter. Its free-energy expansions turn mean-field assumptions into predictions for phase behavior.
Hartree–Fock method: An approximation for many-fermion systems using a single antisymmetrized product of orbitals. It replaces interactions with an average potential while retaining exchange effects.

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