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urces unsourced material may be challenged and removed find sources mean field theory news newspapers books scholar jstor september 2022 learn how and when to remove this message 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 origins edit the idea first appeared in physics statistical mechanics in the work of pierre curie 6 and pierre weiss to describe phase transitions 7 mft has been used in the bragg williams approximation models on bethe lattice landau theory curie weiss law for magnetic susceptibility flory huggins solution theory and scheutjens fleer theory systems with many sometimes infinite degrees of freedom are generally hard to solve exactly or compute in closed analytic form except for some simple cases e g certain gaussian random field theories the 1d ising model often combinatorial problems arise that make things like computing the partition function of a system difficult mft is an approximation method that often makes the original problem to be solvable and open to calculation and in some cases mft may give very accurate approximations 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 that an mft system has no fluctuations but this coincides with the idea that one is replacing all interactions with a mean field quite often mft provides a convenient launch point for studying higher order fluctuations for example when computing the partition function studying the combinatorics of the interaction terms in the hamiltonian can sometimes at best produce perturbation results or feynman diagrams that correct the mean field approximation validity edit in general dimensionality plays an active role in determining whether a mean field approach will work for any particular problem there is sometimes a critical dimension above which mft is valid and below which it is not heuristically many interactions are replaced in mft by one effective interaction so if the field or particle exhibits many random interactions in the original system they tend to cancel each other out so the mean effective interaction and mft will be more accurate this is true in cases of high dimensionality when the hamiltonian includes long range forces or when the particles are extended e g polymers the ginzburg criterion is the formal expression of how fluctuations render mft a poor approximation often depending upon the number of spatial dimensions in the system of interest formal approach hamiltonian edit the formal basis for mean field theory is the bogoliubov inequality this inequality states that the free energy of a system with hamiltonian h h 0 δ h displaystyle mathcal h mathcal h _ 0 delta mathcal h has the following upper bound f f 0 d e f h 0 t s 0 displaystyle f leq f_ 0 stackrel mathrm def langle mathcal h rangle _ 0 ts_ 0 where s 0 displaystyle s_ 0 is the entropy and f displaystyle f and f 0 displaystyle f_ 0 are helmholtz free energies the average is taken over the equilibrium ensemble of the reference system with hamiltonian h 0 displaystyle mathcal h _ 0 in the special case that the reference hamiltonian is that of a non interacting system and can thus be written as h 0 i 1 n h i ξ i displaystyle mathcal h _ 0 sum _ i 1 n h_ i xi _ i where ξ i displaystyle xi _ i are 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 minimising the right side of the inequality the minimising 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 h i j p v i j ξ i ξ j displaystyle mathcal h sum _ i j in mathcal p v_ i j xi _ i xi _ j where p displaystyle mathcal p is the set of pairs that interact the minimising procedure can be carried out formally define tr i f ξ i displaystyle operatorname tr _ i f xi _ i as the generalized sum of the observable f displaystyle f over the degrees of freedom of the single component sum for discrete variables integrals for continuous ones the approximating free energy is given by f 0 tr 1 2 n h ξ 1 ξ 2 ξ n p 0 n ξ 1 ξ 2 ξ n k t tr 1 2 n p 0 n ξ 1 ξ 2 ξ n log p 0 n ξ 1 ξ 2 ξ n displaystyle begin aligned f_ 0 operatorname tr _ 1 2 ldots n mathcal h xi _ 1 xi _ 2 ldots xi _ n p_ 0 n xi _ 1 xi _ 2 ldots xi _ n kt operatorname tr _ 1 2 ldots n p_ 0 n xi _ 1 xi _ 2 ldots xi _ n log p_ 0 n xi _ 1 xi _ 2 ldots xi _ n end aligned where p 0 n ξ 1 ξ 2 ξ n displaystyle p_ 0 n xi _ 1 xi _ 2 dots xi _ n is the probability to find the reference system in the state specified by the variables ξ 1 ξ 2 ξ n displaystyle xi _ 1 xi _ 2 dots xi _ n this probability is given by the normalized boltzmann factor p 0 n ξ 1 ξ 2 ξ n 1 z 0 n e β h 0 ξ 1 ξ 2 ξ n i 1 n 1 z 0 e β h i ξ i d e f i 1 n p 0 i ξ i displaystyle begin aligned p_ 0 n xi _ 1 xi _ 2 ldots xi _ n frac 1 z_ 0 n e beta mathcal h _ 0 xi _ 1 xi _ 2 ldots xi _ n prod _ i 1 n frac 1 z_ 0 e beta h_ i xi _ i stackrel mathrm def prod _ i 1 n p_ 0 i xi _ i end aligned where z 0 displaystyle z_ 0 is the partition function thus f 0 i j p tr i j v i j ξ i ξ j p 0 i ξ i p 0 j ξ j k t i 1 n tr i p 0 i ξ i log p 0 i ξ i displaystyle begin aligned f_ 0 sum _ i j in mathcal p operatorname tr _ i j v_ i j xi _ i xi _ j p_ 0 i xi _ i p_ 0 j xi _ j kt sum _ i 1 n operatorname tr _ i p_ 0 i xi _ i log p_ 0 i xi _ i end aligned in order to minimise we take the derivative with respect to the single degree of freedom probabilities p 0 i displaystyle p_ 0 i using a lagrange multiplier to ensure proper normalization the end result is the set of self consistency equations p 0 i ξ i 1 z 0 e β h i m f ξ i i 1 2 n displaystyle p_ 0 i xi _ i frac 1 z_ 0 e beta h_ i mf xi _ i quad i 1 2 ldots n where the mean field is given by h i mf ξ i j i j p tr j v i j ξ i ξ j p 0 j ξ j displaystyle h_ i text mf xi _ i sum _ j mid i j in mathcal p operatorname tr _ j v_ i j xi _ i xi _ j p_ 0 j xi _ j applications edit mean field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions 8 ising model edit formal derivation edit the bogoliubov inequality shown above can be used to find the dynamics of a mean field model of the two dimensional ising lattice a magnetisation function can be calculated from the resultant approximate free energy 9 the first step is choosing a more tractable approximation of the true hamiltonian using a non interacting or effective field hamiltonian m i s i displaystyle m sum _ i s_ i the variational free energy is f v f 0 j s i s j h s i m s i 0 displaystyle f_ v f_ 0 left langle left j sum s_ i s_ j h sum s_ i right left m sum s_ i right right rangle _ 0 by the bogoliubov inequality simplifying this quantity and calculating the magnetisation function that minimises the variational free energy yields the best approximation to the actual magnetisation the minimiser is m j s j 0 h displaystyle m j sum langle s_ j rangle _ 0 h which is the ensemble average of spin this simplifies to m tanh z j β m h displaystyle m text tanh zj beta m h equating the effective field felt by all spins to a mean spin value relates the variational approach to the suppression of fluctuations the physical interpretation of the magnetisation function is then a field of mean values for individual spins non interacting spins approximation edit consider the ising model on a d displaystyle d dimensional lattice the hamiltonian is given by h j i j s i s j h i s i displaystyle h j sum _ langle i j rangle s_ i s_ j h sum _ i s_ i where the i j displaystyle sum _ langle i j rangle indicates summation over the pair of nearest neighbors i j displaystyle langle i j rangle and s i s j 1 displaystyle s_ i s_ j pm 1 are neighboring ising spins let us transform our spin variable by introducing the fluctuation from its mean value m i s i displaystyle m_ i equiv langle s_ i rangle we may rewrite the hamiltonian as h j i j m i δ s i m j δ s j h i s i displaystyle h j sum _ langle i j rangle m_ i delta s_ i m_ j delta s_ j h sum _ i s_ i where we define δ s i s i m i displaystyle delta s_ i equiv s_ i m_ i this is the fluctuation of the spin if we expand the right side we obtain one term that is entirely dependent on the mean values of the spins and independent of the spin configurations this is the trivial term which does not affect the statistical properties of the system the next term is the one involving the product of the mean value of the spin and the fluctuation value finally the last term involves a product of two fluctuation values the mean field approximation consists of neglecting this second order fluctuation term h h mf j i j m i m j m i δ s j m j δ s i h i s i displaystyle h approx h text mf equiv j sum _ langle i j rangle m_ i m_ j m_ i delta s_ j m_ j delta s_ i h sum _ i s_ i these fluctuations are enhanced at low dimensions making mft a better approximation for high dimensions again the summand can be re expanded in addition we expect that the mean value of each spin is site independent since the ising chain is translationally invariant this yields h mf j i j m 2 2 m s i m h i s i displaystyle h text mf j sum _ langle i j rangle big m 2 2m s_ i m big h sum _ i s_ i the summation over neighboring spins can be rewritten as i j 1 2 i j n n i displaystyle sum _ langle i j rangle frac 1 2 sum _ i sum _ j in nn i where n n i displaystyle nn i means nearest neighbor of i displaystyle i and the 1 2 displaystyle 1 2 prefactor avoids double counting since each bond participates in two spins simplifying leads to the final expression h mf j m 2 n z 2 h m j z h eff i s i displaystyle h text mf frac jm 2 nz 2 underbrace h mjz _ h text eff sum _ i s_ i where z displaystyle z is the coordination number at this point the ising hamiltonian has been decoupled into a sum of one body hamiltonians with an effective mean field h eff h j z m displaystyle h text eff h jzm which is the sum of the external field h displaystyle h and of the mean field induced by the neighboring spins it is worth noting that this mean field directly depends on the number of nearest neighbors and thus on the dimension of the system for instance for a hypercubic lattice of dimension d displaystyle d z 2 d displaystyle z 2d substituting this hamiltonian into the partition function and solving the effective 1d problem we obtain z e β j m 2 n z 2 2 cosh h m j z k b t n displaystyle z e frac beta jm 2 nz 2 left 2 cosh left frac h mjz k_ text b t right right n where n displaystyle n is the number of lattice sites this is a closed and exact expression for the partition function of the system we may obtain the free energy of the system and calculate critical exponents in particular we can obtain the magnetization m displaystyle m as a function of h eff displaystyle h text eff we thus have two equations between m displaystyle m and h eff displaystyle h text eff allowing us to determine m displaystyle m as a function of temperature this leads to the following observation for temperatures greater than a certain value t c displaystyle t_ text c the only solution is m 0 displaystyle m 0 the system is paramagnetic for t t c displaystyle t t_ text c there are two non zero solutions m m 0 displaystyle m pm m_ 0 the system is ferromagnetic t c displaystyle t_ text c is given by the following relation t c j z k b displaystyle t_ text c frac jz k_ b this shows that mft can account for the ferromagnetic phase transition application to other systems edit similarly mft can be applied to other types of hamiltonian as in the following cases to study the metal superconductor transition in this case the analog of the magnetization is the superconducting gap δ displaystyle delta to study the molecular field of a liquid crystal that emerges when the laplacian of the director field is non zero to determine the optimal amino acid side chain packing given a fixed protein backbone in protein structure prediction see self consistent mean field biology to determine the elastic properties of a composite material variational minimisations like mean field approximation also appear in statistical inference through variational bayesian methods extension to time dependent mean fields edit main article dynamical mean field theory in mean field theory the mean field appearing in the single site problem is a time independent scalar or vector quantity however this isn t always the case in a variant of mean field theory called dynamical mean field theory dmft the mean field becomes a time dependent quantity for instance dmft can be applied to the hubbard model to study the metal mott insulator transition see also edit dynamical mean field theory mean field game theory references edit 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 bibcode 2020entrp 22 552p 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 cite book cite uses deprecated parameter citeseerx help 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 ...