Физикалық мінез-құлқының жуықтауы

Approximation of physical behavior

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). Such models consider many individual components that interact with each other. 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. This reduces any many body problem into an effective one 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, including statistical inference, graphical models, neuroscience, artificial intelligence, epidemic models, queueing theory, computer network performance and game theory, as in the quantal response equilibrium.

The idea first appeared in physics (statistical mechanics) in the work of Pierre Curie and Pierre Weiss to describe phase transitions. MFT has been used in the Bragg–Williams approximation, models on Bethe lattice, Landau theory, Pierre–Weiss approximation, Flory–Huggins solution theory, and Scheutjens–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 Gaussian random field theories, the 1D Ising model). Often combinatorial problems arise that make things like computing the partition function of a system difficult. MFT is an approximation method that often makes the original solvable and open to calculation, and in some cases MFT may give very accurate approximations. In field 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 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. There is sometimes a critical 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). 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.

Mean field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions.

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 called dynamical mean field theory (DMFT), the mean field becomes a time dependent quantity. For instance, DMFT can be applied to the Hubbard model to study the metal–Mott insulator transition.