# 1. Mean-field theory

In physics and probability theory, mean field theory (MFT also known as self-consistent field theory) studies the behavior of large and complex stochastic 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. Approaches inspired by these ideas have seen applications in epidemic models,[3] queueing theory,[4] computer network performance and game theory.[5]

The main idea of MFT is to replace all interactions to any one body with an average or effective interaction, sometimes called a molecular field.[6] This reduces any multi-body problem into an effective one-body problem.

In general, dimensionality plays a strong role in determining whether a mean-field approach will work for any particular problem. In MFT, many interactions are replaced by one effective interaction. Then it naturally follows that if the field or particle exhibits many interactions in the original system, MFT will be more accurate for such a system. This is true in cases of high dimensionality, or when the Hamiltonian includes long-range forces. The Ginzburg criterion is the formal expression of how fluctuations render MFT a poor approximation, depending upon the number of spatial dimensions in the system of interest.

# 2. 模型描述

The mean-field approximation consists of neglecting this second order fluctuation term. These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.

# 3. 疑问

[1]博文：平均场方法及其它
[2]Wikipedia: Mean field theory