|
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. The ideas first appeared in physics in the work of Pierre Curie and Pierre Weiss to describe phase transitions. Approaches inspired by these ideas have seen applications in epidemic models, queueing theory, computer network performance and game theory. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases (random field theory, 1D Ising model). The n-body system is replaced by a 1-body problem with a chosen good external field. The external field replaces the interaction of all the other particles to an arbitrary particle. The great difficulty (e.g. when computing the partition function of the system) is the treatment of combinatorics generated by the interaction terms in the Hamiltonian when summing over all states. The goal of mean field theory is to resolve these combinatorial problems. MFT is known under a great many names and guises. Similar techniques include Bragg–Williams approximation, models on Bethe lattice, Landau theory, Pierre–Weiss approximation, Flory–Huggins solution theory, and Scheutjens–Fleer theory. 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 multi-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 relatively low cost. 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 an MFT system has no fluctuations, but this coincides with the idea that one is replacing all interactions with a "mean field". Quite often, in the formalism of fluctuations, MFT provides a convenient launch-point to studying first or second order fluctuations. 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. While MFT arose primarily in the field of statistical mechanics, it has more recently been applied elsewhere, for example in inference, graphical models theory, neuroscience, and artificial intelligence. ==Formal approach== The formal basis for mean field theory is the Bogoliubov inequality. This inequality states that the free energy of a system with Hamiltonian : has the following upper bound: : where is the entropy and where the average is taken over the equilibrium ensemble of the reference system with Hamiltonian . In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as : where is shorthand for 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 minimizing the right hand side of the inequality. The minimizing reference system is then the "best" approximation to the true system using non-correlated degrees of freedom, and is known as the mean field approximation. For the most common case that the target Hamiltonian contains only pairwise interactions, ''i.e.,'' : where is the set of pairs that interact, the minimizing procedure can be carried out formally. Define as the generalized sum of the observable over the degrees of freedom of the single component (sum for discrete variables, integrals for continuous ones). The approximating free energy is given by :_\mathcal(\xi_,\xi_,...,\xi_)P^_(\xi_,\xi_,...,\xi_) |- | | |} where is the probability to find the reference system in the state specified by the variables . This probability is given by the normalized Boltzmann factor : where is the partition function. Thus : In order to minimize we take the derivative with respect to the single degree-of-freedom probabilities using a Lagrange multiplier to ensure proper normalization. The end result is the set of self-consistency equations : where the mean field is given by : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「mean field theory」の詳細全文を読む スポンサード リンク
|