What is Mean field theory: Definition and 12 Discussions

In physics and probability theory, mean-field theory (aka MFT or rarely 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. In MFT, 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 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.

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  1. josephsanders

    High Energy Literature for introduction to O(N) vector model

    TL;DR Summary: Looking for literature on O(N) vector model Hello, We have been going over the O(N) vector model in my QFT class but the notes are not very detailed and we are not using a textbook. Does anyone know of a good QFT book which goes over this material? I have a copy of Shrednicki...
  2. J

    A Mean field theory in the Heisenberg model

    I have a problem about Heisenberg model.When applying mean field approximation,why does the average of spin flip terms be zero(<S+>=<S-> = 0 )? Thanks
  3. B

    A Entanglement in Ultracold Chemistry and Physics

    I'm looking for work published discussing the relationship between von Neumann entropy as well as entanglement with regard to chemical reactivity in the ultracold temperature scales. An article published under the title "Ultracold chemistry and its reaction kinematics" discussed this...
  4. Leonardo Machado

    A Walecka mean field theory, Help to find its M*

    Hello friends. I'm trying to compute an EoS to walecka model of barion interaction, but I'm having trouble to solve this equation by bisection. M*=M-gs²*nb/ms² where nb= (M*)*( kf*Ef- (M*)²* ln (kf+Ef)/M*) , using Ef= sqrt( kf²+(M*)²) and Cs²= gs² M² / ms² = 267.1 I'm using J. D. Walecka...
  5. Xenosum

    Mean Field Theory for Fermions/Bosons

    I'm not really sure if this counts as a homework problem (I was reluctant to post in that section since they evidently force you to ensure you've used the template, even though it's not very applicable here) so much as a general misunderstanding of mean field theory. So, in Michale Plischke and...
  6. B

    Mean field solution for potts model

    Homework Statement The mean-field equation for the three-state Potts model H= -J∑δσiδσj can be derived as follows using this: a) show that H is equivalent to -J∑Si.Sj where Si=(1 0) , (-1/2 √3/2 ) , (-1/2 -√3/2) b) putting H0= (H0 H'0) show the mean field equation become...
  7. H

    Self consistency and mean field theory

    Homework Statement I am a little confused about the how self consistency conditions work and I was wondering if in the following case I have correctly understood the details? Homework Equations [/B] Say we have a harmonic oscillator with a perturbation...
  8. B

    Origin of number of bounds in ising model

    what is the origin of the number of bound (N z /2 ) in the calculating of average in ensemble in Mean Field for the Ising model?
  9. C

    Statistical Mechanics Mean Field Model

    I don't think I've fully grasped the underlying ideas of this class, so at the moment I'm just sort of flailing for equations to plug stuff into... Homework Statement Show that in the mean field model, M is proportional to H1/3 at T=Tc and that at H=0, M is proportional to (Tc - T)1/2...
  10. maverick280857

    Learning Dynamical Mean Field Theory

    Hi, I'm a masters student trying to apply DMFT to problems involving transport in strongly correlated systems. I have a cursory understanding of the physics behind the Hubbard model, which is to say, I have spent some time with it in a quantum many body theory course. However, I now want to...
  11. T

    Why Is Fourier Transform Used in Diagonalizing Operators in Mean Field Theory?

    Hi... I hope somebody can help me... Studying mean field theory in a passage it was necessary to calcolate the inverse of this operator defined on Z^2: $A(I,K)=-J\sum_e \delta(I,K-e)+1/(\beta)*\delta(I,K)$ where I,K pass all ZxZ and the sum on $e$ is a sum on the for basis vectors...
  12. S

    Mean Field Theory: Get Book Recommendations

    I have just started reading up on mean field theory. Any book you could refer??