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Mean-field theory

In physics and probability theory, mean-field theory (MFT) or self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a statistic that are free to vary).

Version
v1 · 2026-09-28 · History
Domain-specific #
10628
Domain group
Natural Sciences
Origin domain
Physics
Subdomain
Statistical Mechanics → Physics

Core Idea

Mean-field theory is treated here as the recurring natural sciences, engineering, and health identity summarized by this source-grounded definition: In physics and probability theory, mean-field theory (MFT) or self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a statistic that are free to vary). In physics and probability theory, mean-field theory (MFT) or self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model.

Scope of Application

  • Origins. MFT has been used in the Bragg–Williams approximation, models on Bethe lattice, Landau theory, Curie-Weiss law for magnetic susceptibility, Flory–Huggins solution theory, and Scheutjens–Fleer theory.

  • Origins. Often combinatorial problems arise that make things like computing the partition function of a system difficult.

  • Origins. MFT is an approximation method that often makes the original problem to be solvable and open to calculation, and in some cases MFT may give very accurate approximations.

  • Origins. For example, when computing the partition function, studying the combinatorics of the interaction terms in the Hamiltonian can sometimes at best produce perturbation results or Feynman diagrams that correct the mean-field.

  • Ising modelFormal derivation. The Bogoliubov inequality, shown above, can be used to find the dynamics of a mean field model of the two-dimensional Ising lattice.

Clarity

A clear use of Mean-field theory names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In physics and probability theory, mean-field theory (MFT) or self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final.

Manages Complexity

Mean-field theory compresses multiple natural sciences, engineering, and health details into a stable diagnostic relation. The source shows both the central mechanism—where P^{(N)}0(\xi1, \xi2, \dots, \xiN) is the probability to find the reference system in the state specified by the variables (\xi1, \xi2, \dots, \xiN) .—and the practical consequence—t\text{c} is given by the following relation: T\text{c} = \frac{J z}{kB} .

Abstract Reasoning

  1. Type the carrier. Identify the natural sciences, engineering, and health entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In physics and probability theory, mean-field theory (MFT) or self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a statistic that are free to vary).
  3. Check operation and conditions.

Knowledge Transfer

Within the home domain. Knowledge about Mean-field theory transfers literally when a new case preserves the same carrier type, relation, and recognition test. MFT has been used in the Bragg–Williams approximation, models on Bethe lattice, Landau theory, Curie-Weiss law for magnetic susceptibility, Flory–Huggins solution theory, and Scheutjens–Fleer theory. Often combinatorial problems arise that make things like computing the partition function of a system difficult. Beyond the home domain. No canonical parent is asserted for Mean-field theory.

Relationships to Other Abstractions

Local relationship map for Mean-field theoryParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Mean-field theoryDOMAINPrime abstraction: Theory — is a kind ofTheoryPRIME

Current abstraction Mean-field theory Domain-specific

Parents (1) — more general patterns this builds on

  • Mean-field theory is a kind of Theory Prime

    Mean-field theory is a strict kind of Theory: its frozen identity entails the parent's defining structure while adding domain-specific restrictions.

Hierarchy paths (2) — routes to 2 parentless roots

Neighborhood in Abstraction Space

Mean-field theory sits in a crowded region of the domain-specific corpus (27th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Quantum States & Information Measures (25 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08