Mean field

In physics and probability theory, mean field theory (MFT also known as self-consistent field theory) studies the behavior of large and complex models by studying a simpler model. Such models consider a large number of small individual components which interact with each other. The effect of all the other individuals on any given individual is approximated by a single averaged effect, thus reducing a many-body problem to a one-body problem.

The ideas first appeared in physics in the work of Pierre Curie and Pierre Weiss to describe phase transitions. Approaches inspired by these ideas have seen applications in epidemic models,queueing theory, computer network performance and game theory, as in the Quantal response equilibrium.

A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases (random field theory, 1D Ising model). The n-body system is replaced by a 1-body problem with a chosen good external field. The external field replaces the interaction of all the other particles to an arbitrary particle. The great difficulty (e.g. when computing the partition function of the system) is the treatment of combinatorics generated by the interaction terms in the Hamiltonian when summing over all states. The goal of mean field theory is to resolve these combinatorial problems. MFT is known under a great many names and guises. Similar techniques include Bragg–Williams approximation, models on Bethe lattice, Landau theory, Pierre–Weiss approximation, Flory–Huggins solution theory, and Scheutjens–Fleer theory.

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