Mean-field theory

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.^[1] 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,^[2] artificial intelligence, epidemic models,^[3] queueing theory,^[4] computer-network performance and game theory,^[5] as in the quantal response equilibrium^{[citation needed]}.

^ Chaikin, P. M.; Lubensky, T. C. (2007). Principles of condensed matter physics (4th print ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-79450-3.
^ Parr, Thomas; Sajid, Noor; Friston, Karl (2020). "Modules or Mean-Fields?" (PDF). Entropy. 22 (552): 552. doi:10.3390/e22050552. PMC 7517075. PMID 33286324. Retrieved 22 May 2020.
^ Boudec, J. Y. L.; McDonald, D.; Mundinger, J. (2007). "A Generic Mean Field Convergence Result for Systems of Interacting Objects". Fourth International Conference on the Quantitative Evaluation of Systems (QEST 2007) (PDF). p. 3. CiteSeerX 10.1.1.110.2612. doi:10.1109/QEST.2007.8. ISBN 978-0-7695-2883-0. S2CID 15007784.
^ Baccelli, F.; Karpelevich, F. I.; Kelbert, M. Y.; Puhalskii, A. A.; Rybko, A. N.; Suhov, Y. M. (1992). "A mean-field limit for a class of queueing networks". Journal of Statistical Physics. 66 (3–4): 803. Bibcode:1992JSP....66..803B. doi:10.1007/BF01055703. S2CID 120840517.
^ Lasry, J. M.; Lions, P. L. (2007). "Mean field games" (PDF). Japanese Journal of Mathematics. 2: 229–260. doi:10.1007/s11537-007-0657-8. S2CID 1963678.

[1] Chaikin, P. M.; Lubensky, T. C. (2007). Principles of condensed matter physics (4th print ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-79450-3.

[2] Parr, Thomas; Sajid, Noor; Friston, Karl (2020). "Modules or Mean-Fields?" (PDF). Entropy. 22 (552): 552. doi:10.3390/e22050552. PMC 7517075. PMID 33286324. Retrieved 22 May 2020.

[3] Boudec, J. Y. L.; McDonald, D.; Mundinger, J. (2007). "A Generic Mean Field Convergence Result for Systems of Interacting Objects". Fourth International Conference on the Quantitative Evaluation of Systems (QEST 2007) (PDF). p. 3. CiteSeerX 10.1.1.110.2612. doi:10.1109/QEST.2007.8. ISBN 978-0-7695-2883-0. S2CID 15007784.

[4] Baccelli, F.; Karpelevich, F. I.; Kelbert, M. Y.; Puhalskii, A. A.; Rybko, A. N.; Suhov, Y. M. (1992). "A mean-field limit for a class of queueing networks". Journal of Statistical Physics. 66 (3–4): 803. Bibcode:1992JSP....66..803B. doi:10.1007/BF01055703. S2CID 120840517.

[5] Lasry, J. M.; Lions, P. L. (2007). "Mean field games" (PDF). Japanese Journal of Mathematics. 2: 229–260. doi:10.1007/s11537-007-0657-8. S2CID 1963678.

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