 Mean field theory

Mean field theory (MFT, also known as selfconsistent field theory) is a method to analyse physical systems with multiple bodies. A manybody system with interactions is generally very difficult to solve exactly, except for extremely simple cases (random field theory, 1D Ising model). The nbody system is replaced by a 1body 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 BraggWilliams approximation, models on Bethe lattice, Landau theory, PierreWeiss 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.^{[1]} This reduces any multibody problem into an effective onebody 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 "zerothorder" 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 launchpoint to studying first or second order fluctuations.
In general, dimensionality plays a strong role in determining whether a meanfield 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 longrange 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 in graphical models theory in artificial intelligence.
Contents
Formal approach
The formal basis for mean field theory is the Bogoliubov inequality. This inequality states that the free energy of a system with Hamiltonian
has the following upper bound:
where S_{0} is the entropy and where the average is taken over the equilibrium ensemble of the reference system with Hamiltonian . In the special case that the reference Hamiltonian is that of a noninteracting system and can thus be written as
where is shorthand for 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 minimizing the right hand side of the inequality. The minimizing reference system is then the "best" approximation to the true system using noncorrelated 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.,
where is the set of pairs that interact, the minimizing procedure can be carried out formally. Define Tr_{i}f(ξ_{i}) as the generalized sum of the observable 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
where is the probability to find the reference system in the state specified by the variables (ξ_{1},ξ_{2},...,ξ_{N}). This probability is given by the normalized Boltzmann factor
where Z_{0} is the partition function. Thus
In order to minimize we take the derivative with respect to the single degreeoffreedom probabilities using a Lagrange multiplier to ensure proper normalization. The end result is the set of selfconsistency equations
where the mean field is given by
Applications
Mean field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions.^{[2]}
Ising Model
Consider the Ising model on an Ndimensional cubic lattice. The Hamiltonian is given by
where the indicates summation over the pair of nearest neighbors , and and s_{j} are neighboring Ising spins.
Let us transform our spin variable by introducing the fluctuation from its mean value . We may rewrite the Hamiltonian:
where we define ; this is the fluctuation of the spin. If we expand the right hand 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 partition function 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 meanfield approximation consists in neglecting this fluctuation term. These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.
Again, the summand can be reexpanded. In addition, we expect that the mean value of each spin is siteindependent, since the Ising chain is translationally invariant. This yields
The summation over neighboring spins can be rewritten as where nn(i) means 'nearestneighbor of i' and the 1 / 2 prefactor avoids doublecounting, since each bond participates in two spins. Simplifying leads to the final expression
where z is the coordination number. At this point, the Ising Hamiltonian has been decoupled into a sum of onebody Hamiltonians with an effective meanfield h^{eff} = h + Jzm which is the sum of the external field h and of the meanfield 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, z = 2d).
Substituting this Hamiltonian into the partition function, and solving the effective 1D problem, we obtain
where 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 as a function of h^{eff}.
We thus have two equations between m and h^{eff}, allowing us to determine m as a function of temperature. This leads to the following observation:
 for temperatures greater than a certain value T_{c}, the only solution is m = 0. The system is paramagnetic.
 for T < T_{c}, there are two nonzero solutions: . The system is ferromagnetic.
T_{c} is given by the following relation: . This shows that MFT can account for the ferromagnetic phase transition.
Application to other systems
Similarly, MFT can be applied to other types of Hamiltonian to study the metalsuperconductor transition. In this case, the analog of the magnetization is the superconducting gap Δ. Another example is the molecular field of a liquid crystal that emerges when the Laplacian of the director field is nonzero.
Extension to TimeDependent Mean Fields
Main article: Dynamical Mean Field TheoryIn meanfield theory, the mean field appearing in the singlesite problem is a scalar or vectorial timeindependent quantity. However, this need not always be the case: in a variant of meanfield theory called Dynamical Mean Field Theory (DMFT), the meanfield becomes a timedependent quantity. For instance, DMFT can be applied to the Hubbard model to study the metalMott insulator transition.
References
 ^ Chaikin, P. M.; Lubensky, T. C. (2007). Principles of condensed matter physics (4th print ed.). Cambridge: Cambridge University Press. ISBN 9780521794503.
 ^ HE Stanley (1971). "Mean field theory of magnetic phase transitions". Introduction to phase transitions and critical phenomena. Oxford University Press. ISBN 0195053168.
See also
Categories: Statistical mechanics
 Fundamental physics concepts

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