This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Mean-field theory" – news ·newspapers ·books ·scholar ·JSTOR(September 2022) (Learn how and when to remove this message)

Inphysics andprobability theory,Mean-field theory (MFT) orSelf-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging overdegrees of freedom (the number of values in the final calculation of astatistic that are free to vary). Such models consider many individual components that interact with each other.

The main idea of MFT is to replace allinteractions to any one body with an average or effective interaction, sometimes called amolecular field.^[1] This reduces anymany-body problem into an effectiveone-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a lower computational cost.

MFT has since been applied to a wide range of fields outside of physics, includingstatistical inference,graphical models,neuroscience,^[2]artificial intelligence,epidemic models,^[3]queueing theory,^[4]computer-network performance andgame theory,^[5] as in thequantal response equilibrium^{[citation needed]}.

The idea first appeared in physics (statistical mechanics) in the work ofPierre Curie^[6] andPierre Weiss to describephase transitions.^[7] MFT has been used in the Bragg–Williams approximation, models onBethe lattice,Landau theory,Curie-Weiss law formagnetic susceptibility,Flory–Huggins solution theory, andScheutjens–Fleer theory.

Systems with many (sometimes infinite) degrees of freedom are generally hard to solve exactly or compute in closed, analytic form, except for some simple cases (e.g. certain Gaussianrandom-field theories, the 1DIsing model). Often combinatorial problems arise that make things like computing thepartition function of a system difficult. 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.

Infield theory, the Hamiltonian may be expanded in terms of the magnitude of fluctuations around the mean of the field. In this context, MFT can be viewed as the "zeroth-order" expansion of the Hamiltonian in fluctuations. Physically, this means that an MFT system has no fluctuations, but this coincides with the idea that one is replacing all interactions with a "mean-field”.

Quite often, MFT provides a convenient launch point for studying higher-order fluctuations. For example, when computing thepartition function, studying thecombinatorics of the interaction terms in theHamiltonian can sometimes at best produceperturbation results orFeynman diagrams that correct the mean-field approximation.

In general, dimensionality plays an active role in determining whether a mean-field approach will work for any particular problem. There is sometimes acritical dimension above which MFT is valid and below which it is not.

Heuristically, many interactions are replaced in MFT by one effective interaction. So if the field or particle exhibits many random interactions in the original system, they tend to cancel each other out, so the mean effective interaction and MFT will be more accurate. This is true in cases of high dimensionality, when the Hamiltonian includes long-range forces, or when the particles are extended (e.g.polymers). TheGinzburg criterion is the formal expression of howfluctuations render MFT a poor approximation, often depending upon the number of spatial dimensions in the system of interest.

The formal basis for mean-field theory is theBogoliubov inequality. This inequality states that thefree energy of a system with Hamiltonian

${\displaystyle \mathcal{H}={\mathcal{H}}_0+\mathrm{\Delta }\mathcal{H}}$

has the following upper bound:

${\displaystyle F\le F_0\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}⟨\mathcal{H}{⟩}_0-T{S}_0,}$

where${\displaystyle S_0}$ is theentropy, and${\displaystyle F}$ and${\displaystyle F_0}$ areHelmholtz free energies. The average is taken over the equilibriumensemble of the reference system with Hamiltonian${\displaystyle {\mathcal{H}}_0}$. In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as

${\displaystyle {\mathcal{H}}_0=\sum _{i=1}^N{h}_i({\xi }_i),}$

where${\displaystyle {\xi }_i}$ are thedegrees of freedom of the individual components of our statistical system (atoms, spins and so forth), one can consider sharpening the upper bound by minimising the right side of the inequality. The minimising reference system is then the "best" approximation to the true system using non-correlated degrees of freedom and is known as themean field approximation.

For the most common case that the target Hamiltonian contains only pairwise interactions, i.e.,

${\displaystyle \mathcal{H}=\sum _{(i,j)\in \mathcal{P}}{V}_{i,j}({\xi }_i,{\xi }_j),}$

where${\displaystyle \mathcal{P}}$ is the set of pairs that interact, the minimising procedure can be carried out formally. Define${\displaystyle {\mathrm{Tr}}_{i}f({\xi }_i)}$ as the generalized sum of the observable${\displaystyle f}$ over the degrees of freedom of the single component (sum for discrete variables, integrals for continuous ones). The approximating free energy is given by

where${\displaystyle P_0^{(N)}({\xi }_1,{\xi }_2,\dots ,{\xi }_N)}$ is the probability to find the reference system in the state specified by the variables${\displaystyle ({\xi }_1,{\xi }_2,\dots ,{\xi }_N)}$. This probability is given by the normalizedBoltzmann factor

where${\displaystyle Z_0}$ is thepartition function. Thus

In order to minimise, we take the derivative with respect to the single-degree-of-freedom probabilities${\displaystyle P_0^{(i)}}$ using aLagrange multiplier to ensure proper normalization. The end result is the set of self-consistency equations

${\displaystyle P_0^{(i)}({\xi }_i)=\frac{1}{Z_0}{e}^{-\beta h_i^{MF}({\xi }_i)},i=1,2,\dots ,N,}$

where the mean field is given by

${\displaystyle h_i^{\text{MF}}({\xi }_i)=\sum _{\{j\mid (i,j)\in \mathcal{P}\}}{\mathrm{Tr}}_j{V}_{i,j}({\xi }_i,{\xi }_j)P_0^{(j)}({\xi }_j).}$

Mean field theory can be applied to a number of physical systems so as to study phenomena such asphase transitions.^[8]

The Bogoliubov inequality, shown above, can be used to find the dynamics of a mean field model of the two-dimensionalIsing lattice. A magnetisation function can be calculated from the resultant approximatefree energy.^[9] The first step is choosing a more tractable approximation of the true Hamiltonian. Using a non-interacting or effective field Hamiltonian,

${\displaystyle -m\sum _i{s}_i}$,

the variational free energy is

${\displaystyle F_V=F_0+{⟨\left(-J\sum s_i{s}_j-h\sum s_i\right)-\left(-m\sum s_i\right)⟩}_0.}$

By the Bogoliubov inequality, simplifying this quantity and calculating the magnetisation function thatminimises the variational free energy yields the best approximation to the actual magnetisation. The minimiser is

${\displaystyle m=J\sum ⟨s_j{⟩}_0+h,}$

which is theensemble average of spin. This simplifies to

${\displaystyle m=\text{tanh}(zJ\beta m)+h.}$

Equating the effective field felt by all spins to a mean spin value relates the variational approach to the suppression of fluctuations. The physical interpretation of the magnetisation function is then a field of mean values for individual spins.

Consider theIsing model on a${\displaystyle d}$-dimensional lattice. The Hamiltonian is given by

${\displaystyle H=-J\sum _{⟨i,j⟩}{s}_i{s}_j-h\sum _i{s}_i,}$

where the${\displaystyle \sum _{⟨i,j⟩}}$ indicates summation over the pair of nearest neighbors${\displaystyle ⟨i,j⟩}$, and${\displaystyle s_i,s_j=\pm 1}$ are neighboring Ising spins.

Let us transform our spin variable by introducing the fluctuation from its mean value${\displaystyle m_i\equiv ⟨s_i⟩}$. We may rewrite the Hamiltonian as

${\displaystyle H=-J\sum _{⟨i,j⟩}(m_i+\delta s_i)(m_j+\delta s_j)-h\sum _i{s}_i,}$

where we define${\displaystyle \delta s_i\equiv s_i-m_i}$; this is thefluctuation of the spin.

If we expand the right side, we obtain one term that is entirely dependent on the mean values of the spins and independent of the spin configurations. This is the trivial term, which does not affect the statistical properties of the system. The next term is the one involving the product of the mean value of the spin and the fluctuation value. Finally, the last term involves a product of two fluctuation values.

The mean field approximation consists of neglecting this second-order fluctuation term:

${\displaystyle H\approx H^{\text{MF}}\equiv -J\sum _{⟨i,j⟩}(m_i{m}_j+m_i\delta s_j+m_j\delta s_i)-h\sum _i{s}_i.}$

These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.

Again, the summand can be re-expanded. In addition, we expect that the mean value of each spin is site-independent, since the Ising chain is translationally invariant. This yields

${\displaystyle H^{\text{MF}}=-J\sum _{⟨i,j⟩}(m^2+2m(s_i-m))-h\sum _i{s}_i.}$

The summation over neighboring spins can be rewritten as${\displaystyle \sum _{⟨i,j⟩}=\frac{1}{2}\sum _i\sum _{j\in nn(i)}}$, where${\displaystyle nn(i)}$ means "nearest neighbor of${\displaystyle i}$", and the${\displaystyle 1/2}$ prefactor avoids double counting, since each bond participates in two spins. Simplifying leads to the final expression

${\displaystyle H^{\text{MF}}=\frac{J{m}^{2}Nz}{2}-\underset{h^{\text{eff.}}}{\underbrace{(h+mJz)}}\sum _i{s}_i,}$

where${\displaystyle z}$ is thecoordination number. At this point, the Ising Hamiltonian has beendecoupled into a sum of one-body Hamiltonians with aneffective mean field${\displaystyle h^{\text{eff.}}=h+Jzm}$, which is the sum of the external field${\displaystyle h}$ and of themean field induced by the neighboring spins. It is worth noting that this mean field directly depends on the number of nearest neighbors and thus on the dimension of the system (for instance, for a hypercubic lattice of dimension${\displaystyle d}$,${\displaystyle z=2d}$).

Substituting this Hamiltonian into the partition function and solving the effective 1D problem, we obtain

${\displaystyle Z=e^{-\frac{\beta J{m}^{2}Nz}{2}}{\left[2\cosh \left(\frac{h+mJz}{k_{\text{B}}T}\right)\right]}^N,}$

where${\displaystyle N}$ is the number of lattice sites. This is a closed and exact expression for the partition function of the system. We may obtain the free energy of the system and calculatecritical exponents. In particular, we can obtain the magnetization${\displaystyle m}$ as a function of${\displaystyle h^{\text{eff.}}}$.

We thus have two equations between${\displaystyle m}$ and${\displaystyle h^{\text{eff.}}}$, allowing us to determine${\displaystyle m}$ as a function of temperature. This leads to the following observation:

• For temperatures greater than a certain value${\displaystyle T_{\text{c}}}$, the only solution is${\displaystyle m=0}$. The system isparamagnetic.
• For, there are two non-zero solutions:${\displaystyle m=\pm m_0}$. The system isferromagnetic.

${\displaystyle T_{\text{c}}}$ is given by the following relation:${\displaystyle T_{\text{c}}=\frac{Jz}{k_B}}$.

This shows that MFT can account for the ferromagnetic phase transition.

Similarly, MFT can be applied to other types of Hamiltonian as in the following cases:

• To study the metal–superconductor transition. In this case, the analog of the magnetization is the superconducting gap${\displaystyle \mathrm{\Delta }}$.
• The molecular field of aliquid crystal that emerges when theLaplacian of the director field is non-zero.
• To determine the optimalamino acidside chain packing given a fixedprotein backbone inprotein structure prediction (seeSelf-consistent mean field (biology)).
• To determine theelastic properties of a composite material.

Variationally minimisation like mean field theory can be also be used instatistical inference.

In mean field theory, the mean field appearing in the single-site problem is a time-independent scalar or vector quantity. However, this isn't always the case: in a variant of mean field theory calleddynamical mean field theory (DMFT), the mean field becomes a time-dependent quantity. For instance, DMFT can be applied to theHubbard model to study the metal–Mott-insulator transition.

• Dynamical mean field theory
• Mean field game theory

• Statistical mechanics
• Concepts in physics
• Electronic structure methods

• Articles with short description
• Short description is different from Wikidata
• Articles needing additional references from September 2022
• All articles needing additional references
• All articles with unsourced statements
• Articles with unsourced statements from June 2022