Editing Mean-field theory (section)

{{Short description|Approximation of physical behavior}}
{{Refimprove|date=September 2022}}
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 (statistics)|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 [[fundamental interaction|interactions]] to any one body with an average or effective interaction, sometimes called a ''molecular field''.<ref>{{cite book |title=Principles of condensed matter physics |last1=Chaikin |first1=P. M. |last2=Lubensky |first2=T. C. |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-79450-3 |edition=4th print |location=Cambridge}}</ref> 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]],<ref>{{cite journal |last1=Parr |first1=Thomas |last2=Sajid |first2=Noor |last3=Friston |first3=Karl |title=Modules or Mean-Fields? |journal=Entropy |date=2020 |volume=22 |issue=552 |page=552 |doi=10.3390/e22050552 |pmid=33286324 |pmc=7517075 |bibcode= |url=https://res.mdpi.com/d_attachment/entropy/entropy-22-00552/article_deploy/entropy-22-00552.pdf |access-date=22 May 2020|doi-access=free }}</ref> [[artificial intelligence]], [[epidemic model]]s,<ref>{{Cite book |url=http://www.cs.toronto.edu/~marbach/ENS/leboudec.pdf |title=Fourth International Conference on the Quantitative Evaluation of Systems (QEST 2007) |last1=Boudec |first1=J. Y. L. |last2=McDonald |first2=D. |last3=Mundinger |first3=J. |year=2007 |isbn=978-0-7695-2883-0 |pages=3 |chapter=A Generic Mean Field Convergence Result for Systems of Interacting Objects |doi=10.1109/QEST.2007.8 |citeseerx=10.1.1.110.2612|s2cid=15007784 }}</ref> [[queueing theory]],<ref>{{Cite journal |last1=Baccelli |first1=F. |last2=Karpelevich |first2=F. I. |last3=Kelbert |first3=M. Y. |last4=Puhalskii |first4=A. A. |last5=Rybko |first5=A. N. |last6=Suhov |first6=Y. M. |year=1992 |title=A mean-field limit for a class of queueing networks |journal=Journal of Statistical Physics |volume=66 |issue=3–4 |pages=803 |bibcode=1992JSP....66..803B |doi=10.1007/BF01055703 |s2cid=120840517 }}</ref> [[Network performance|computer-network performance]] and [[mean field game theory|game theory]],<ref>{{Cite journal |last1=Lasry |first1=J. M. |last2=Lions |first2=P. L. |author-link2=Pierre-Louis Lions |year=2007 |title=Mean field games |journal=Japanese Journal of Mathematics |volume=2 |pages=229–260 |doi=10.1007/s11537-007-0657-8 |s2cid=1963678 |url=https://basepub.dauphine.fr//bitstream/123456789/2263/1/Cahier_Chaire_2.pdf}}</ref> as in the [[quantal response equilibrium]]{{Citation Needed|date=June 2022}}.