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).
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, including statistical inference, graphical models, neuroscience,[2] artificial intelligence, epidemic models,[3] queueing theory,[4] computer-network performance and game theory,[5] as in the quantal response equilibrium[citation needed].
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 Gaussian random-field theories, the 1D Ising model).
Often combinatorial problems arise that make things like computing the partition function of a system difficult.
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 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 approximation.
In general, dimensionality plays an active role in determining whether a mean-field approach will work for any particular problem.
This is true in cases of high dimensionality, when the Hamiltonian includes long-range forces, or when the particles are extended (e.g. polymers).
The Ginzburg criterion is the formal expression of how fluctuations render MFT a poor approximation, often depending upon the number of spatial dimensions in the system of interest.
This inequality states that the free energy of a system with Hamiltonian has the following upper bound: where
In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as where
are the degrees 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.
is the set of pairs that interact, the minimising procedure can be carried out formally.
over the degrees of freedom of the single component (sum for discrete variables, integrals for continuous ones).
The end result is the set of self-consistency equations where the mean field is given by Mean field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions.
[8] The Bogoliubov inequality, shown above, can be used to find the dynamics of a mean field model of the two-dimensional Ising lattice.
A magnetisation function can be calculated from the resultant approximate free energy.
Using a non-interacting or effective field Hamiltonian, the variational free energy is By the Bogoliubov inequality, simplifying this quantity and calculating the magnetisation function that minimises the variational free energy yields the best approximation to the actual magnetisation.
The physical interpretation of the magnetisation function is then a field of mean values for individual spins.
Let us transform our spin variable by introducing the fluctuation from its mean value
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.
In addition, we expect that the mean value of each spin is site-independent, since the Ising chain is translationally invariant.
prefactor avoids double counting, since each bond participates in two spins.
Substituting this Hamiltonian into the partition function and solving the effective 1D problem, we obtain where
We may obtain the free energy of the system and calculate critical exponents.
Similarly, MFT can be applied to other types of Hamiltonian as in the following cases: Variationally minimisation like mean field theory can be also be used in statistical inference.
For instance, DMFT can be applied to the Hubbard model to study the metal–Mott-insulator transition.