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- Mean_field_theory abstract "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.".
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- Mean_field_theory wikiPageRevisionID "686655021".
- Mean_field_theory wikiPageWikiLink Amino_acid.
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- Mean_field_theory wikiPageWikiLink Bethe_lattice.
- Mean_field_theory wikiPageWikiLink Biomolecular_structure.
- Mean_field_theory wikiPageWikiLink Boltzmann_distribution.
- Mean_field_theory wikiPageWikiLink Bragg–Williams_approximation.
- Mean_field_theory wikiPageWikiLink Category:Concepts_in_physics.
- Mean_field_theory wikiPageWikiLink Category:Statistical_mechanics.
- Mean_field_theory wikiPageWikiLink Classical_field_theory.
- Mean_field_theory wikiPageWikiLink Combinatorics.
- Mean_field_theory wikiPageWikiLink Compartmental_models_in_epidemiology.
- Mean_field_theory wikiPageWikiLink Coordination_number.
- Mean_field_theory wikiPageWikiLink Critical_exponent.
- Mean_field_theory wikiPageWikiLink Degrees_of_freedom_(physics_and_chemistry).
- Mean_field_theory wikiPageWikiLink Dynamical_mean_field_theory.
- Mean_field_theory wikiPageWikiLink Entropy.
- Mean_field_theory wikiPageWikiLink Epidemic_model.
- Mean_field_theory wikiPageWikiLink Flory–Huggins_solution_theory.
- Mean_field_theory wikiPageWikiLink Ginzburg_criterion.
- Mean_field_theory wikiPageWikiLink Graphical_model.
- Mean_field_theory wikiPageWikiLink Hamiltonian_mechanics.
- Mean_field_theory wikiPageWikiLink Helmholtz_free_energy.
- Mean_field_theory wikiPageWikiLink Hubbard_model.
- Mean_field_theory wikiPageWikiLink Inference.
- Mean_field_theory wikiPageWikiLink Ising_model.
- Mean_field_theory wikiPageWikiLink Lagrange_multiplier.
- Mean_field_theory wikiPageWikiLink Landau_theory.
- Mean_field_theory wikiPageWikiLink Laplace_operator.
- Mean_field_theory wikiPageWikiLink Liquid_crystal.
- Mean_field_theory wikiPageWikiLink Many-body_problem.
- Mean_field_theory wikiPageWikiLink Mean_field_game_theory.
- Mean_field_theory wikiPageWikiLink Neuroscience.
- Mean_field_theory wikiPageWikiLink Partition_function_(statistical_mechanics).
- Mean_field_theory wikiPageWikiLink Phase_transition.
- Mean_field_theory wikiPageWikiLink Physics.
- Mean_field_theory wikiPageWikiLink Pierre_Curie.
- Mean_field_theory wikiPageWikiLink Pierre_Weiss.
- Mean_field_theory wikiPageWikiLink Pierre–Weiss_approximation.
- Mean_field_theory wikiPageWikiLink Probability_theory.
- Mean_field_theory wikiPageWikiLink Protein_structure_prediction.
- Mean_field_theory wikiPageWikiLink Queueing_theory.
- Mean_field_theory wikiPageWikiLink Random_field.
- Mean_field_theory wikiPageWikiLink Scheutjens–Fleer_theory.
- Mean_field_theory wikiPageWikiLink Self-consistent_mean_field_(biology).
- Mean_field_theory wikiPageWikiLink Side_chain.
- Mean_field_theory wikiPageWikiLink Statistical_mechanics.
- Mean_field_theory wikiPageWikiLink Stochastic.
- Mean_field_theory wikiPageWikiLink Superconductivity.
- Mean_field_theory wikiPageWikiLink Thermodynamic_free_energy.
- Mean_field_theory wikiPageWikiLink Two-body_problem.
- Mean_field_theory wikiPageWikiLinkText "Mean Field".
- Mean_field_theory wikiPageWikiLinkText "Mean field theory".
- Mean_field_theory wikiPageWikiLinkText "Mean field".
- Mean_field_theory wikiPageWikiLinkText "effective interaction".
- Mean_field_theory wikiPageWikiLinkText "external fields function".
- Mean_field_theory wikiPageWikiLinkText "mean field approach".
- Mean_field_theory wikiPageWikiLinkText "mean field approaches".
- Mean_field_theory wikiPageWikiLinkText "mean field approximation".
- Mean_field_theory wikiPageWikiLinkText "mean field approximations".
- Mean_field_theory wikiPageWikiLinkText "mean field theory".
- Mean_field_theory wikiPageWikiLinkText "mean field".
- Mean_field_theory wikiPageWikiLinkText "mean-field approximation".
- Mean_field_theory wikiPageWikiLinkText "mean-field theory".
- Mean_field_theory wikiPageWikiLinkText "mean-field type".
- Mean_field_theory wikiPageWikiLinkText "mean-field".
- Mean_field_theory wikiPageWikiLinkText "self-consistent field".
- Mean_field_theory wikiPageUsesTemplate Template:Main.
- Mean_field_theory wikiPageUsesTemplate Template:Refimprove.
- Mean_field_theory subject Category:Concepts_in_physics.
- Mean_field_theory subject Category:Statistical_mechanics.
- Mean_field_theory type Concept.
- Mean_field_theory type Mechanic.
- Mean_field_theory type Physic.
- Mean_field_theory type Redirect.
- Mean_field_theory comment "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.".
- Mean_field_theory label "Mean field theory".
- Mean_field_theory sameAs Q626011.
- Mean_field_theory sameAs Molekularfeldtheorie.
- Mean_field_theory sameAs תורת_שדה_ממוצע.
- Mean_field_theory sameAs 平均場近似.
- Mean_field_theory sameAs Teoria_pola_średniego.
- Mean_field_theory sameAs Teoria_de_campo_médio.
- Mean_field_theory sameAs m.02pkx4.
- Mean_field_theory sameAs Метод_самосогласованного_поля.
- Mean_field_theory sameAs Одноелектронне_наближення.
- Mean_field_theory sameAs Q626011.
- Mean_field_theory sameAs 平均场论.
- Mean_field_theory wasDerivedFrom Mean_field_theory?oldid=686655021.
- Mean_field_theory isPrimaryTopicOf Mean_field_theory.