1. Origins

The ideas first appeared in physics (statistical mechanics) in the work of Pierre Curie^[1] and Pierre Weiss to describe phase transitions.^[2] MFT has been used in the Bragg–Williams approximation, models on Bethe lattice, Landau theory, Pierre–Weiss approximation, 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 solvable and open to calculation. Sometimes, MFT gives 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 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 to 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 perturbative results or Feynman diagrams that correct the mean field approximation.

2. Validity

In general, dimensionality plays a strong 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 not so much.

Heuristically in MFT, many interactions are replaced 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.

3. Formal Approach (Hamiltonian)

The formal basis for mean field theory is the Bogoliubov inequality. This inequality states that the free energy of a system with Hamiltonian

[math]\displaystyle{ \mathcal{H} = \mathcal{H}_0 + \Delta \mathcal{H} }[/math]

has the following upper bound:

[math]\displaystyle{ F \leq F_0 \ \stackrel{\mathrm{def}}{=}\ \langle \mathcal{H} \rangle_0 - T S_0 }[/math]

where [math]\displaystyle{ S_0 }[/math] is the entropy and [math]\displaystyle{ F }[/math] and [math]\displaystyle{ F_0 }[/math] are Helmholtz free energies. The average is taken over the equilibrium ensemble of the reference system with Hamiltonian [math]\displaystyle{ \mathcal{H}_{0} }[/math]. In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as

[math]\displaystyle{ \mathcal{H}_0 = \sum_{i=1}^N h_i \left( \xi_i \right) }[/math]

where [math]\displaystyle{ \left(\xi_i\right) }[/math] 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 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.,

[math]\displaystyle{ \mathcal{H} = \sum_{(i,j) \in \mathcal{P}} V_{i,j} \left( \xi_i, \xi_j \right) }[/math]

where [math]\displaystyle{ \mathcal{P} }[/math] is the set of pairs that interact, the minimizing procedure can be carried out formally. Define [math]\displaystyle{ \operatorname{Tr}_i f(\xi_i) }[/math] as the generalized sum of the observable [math]\displaystyle{ f }[/math] over the degrees of freedom of the single component (sum for discrete variables, integrals for continuous ones). The approximating free energy is given by

[math]\displaystyle{ \begin{align} F_{0} ={} &\operatorname{Tr}_{1,2,\ldots,N} \mathcal{H} \left(\xi_1,\xi_2,\ldots,\xi_N\right) P^{(N)}_0 \left(\xi_1,\xi_2,\ldots,\xi_N\right) \\ &{} + kT \,\operatorname{Tr}_{1,2,\ldots,N} P^{(N)}_0 \left(\xi_1,\xi_2,\ldots,\xi_N\right) \log P^{(N)}_0 \left(\xi_1,\xi_2,\ldots,\xi_N\right) \end{align} }[/math]

where [math]\displaystyle{ P^{(N)}_{0}(\xi_{1},\xi_{2},...,\xi_{N}) }[/math] is the probability to find the reference system in the state specified by the variables [math]\displaystyle{ (\xi_{1},\xi_{2},...,\xi_{N}) }[/math]. This probability is given by the normalized Boltzmann factor

[math]\displaystyle{ \begin{align} P^{(N)}_0(\xi_1,\xi_2,\ldots,\xi_N) & {}= \frac{1}{Z^{(N)}_0} e^{-\beta \mathcal{H}_0 (\xi_1, \xi_2, \ldots, \xi_N)} \\ & {}= \prod_{i=1}^N \frac{1}{Z_0} e^{-\beta h_i \left( \xi_i \right)} \ \stackrel{\mathrm{def}}{=}\ \prod_{i=1}^N P^{(i)}_0 (\xi_i) \end{align} }[/math]

where [math]\displaystyle{ Z_0 }[/math] is the partition function. Thus

[math]\displaystyle{ \begin{align} F_0 ={} &\sum_{(i,j)\in\mathcal{P}} \operatorname{Tr}_{i,j} V_{i,j} \left( \xi_i,\xi_j \right) P^{(i)}_0 (\xi_i) P^{(j)}_0 (\xi_j) \\ &{} + kT \sum_{i=1}^N \operatorname{Tr}_i P^{(i)}_0 (\xi_i) \log P^{(i)}_0 (\xi_i). \end{align} }[/math]

In order to minimize we take the derivative with respect to the single degree-of-freedom probabilities [math]\displaystyle{ P^{(i)}_0 }[/math] using a Lagrange multiplier to ensure proper normalization. The end result is the set of self-consistency equations

[math]\displaystyle{ P^{(i)}_0(\xi_i) = \frac{1}{Z_0} e^{-\beta h_i^{MF}(\xi_i)}\qquad i = 1, 2, \ldots, N }[/math]

where the mean field is given by

[math]\displaystyle{ h_i^{MF}\left(\xi_i\right) = \sum_{\{j|(i,j)\in\mathcal{P}\}} \operatorname{Tr}_j V_{i,j} \left( \xi_i,\xi_j \right) P^{(j)}_0 \left(\xi_j\right). }[/math]

4. Applications

Mean field theory can be applied to a number of physical systems so as to study phenomena such as phase transitions.^[3]

4.1. Ising Model

Consider the Ising model on a [math]\displaystyle{ d }[/math]-dimensional lattice. The Hamiltonian is given by

[math]\displaystyle{ H = -J \sum_{\langle i,j \rangle} s_i s_j - h \sum_i s_i }[/math]

where the [math]\displaystyle{ \sum_{\langle i,j \rangle} }[/math] indicates summation over the pair of nearest neighbors [math]\displaystyle{ \langle i,j \rangle }[/math], and [math]\displaystyle{ s_i = \pm 1 }[/math] and [math]\displaystyle{ s_j }[/math] are neighboring Ising spins.

Let us transform our spin variable by introducing the fluctuation from its mean value [math]\displaystyle{ m_i \equiv \langle s_i \rangle }[/math]. We may rewrite the Hamiltonian:

[math]\displaystyle{ H = -J \sum_{\langle i,j \rangle} \left(m_i + \delta s_i\right)\left(m_j + \delta s_j\right) - h \sum_i s_i }[/math]

where we define [math]\displaystyle{ \delta s_i \equiv s_i - m_i }[/math]; 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 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. These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.

[math]\displaystyle{ H \approx H^{MF} \equiv -J \sum_{\langle i,j \rangle} \left(m_i m_j + m_i \delta s_j + m_j \delta s_i \right) - h \sum_i s_i }[/math]

Again, the summand can be reexpanded. In addition, we expect that the mean value of each spin is site-independent, since the Ising chain is translationally invariant. This yields

[math]\displaystyle{ H^{MF} = -J \sum_{\langle i,j \rangle} \left( m^2 + 2m(s_i - m) \right) - h \sum_i s_i }[/math]

The summation over neighboring spins can be rewritten as [math]\displaystyle{ \sum_{\langle i,j \rangle} = \frac{1}{2} \sum_i \sum_{j\in nn(i)} }[/math] where [math]\displaystyle{ nn(i) }[/math] means 'nearest-neighbor of [math]\displaystyle{ i }[/math]' and the [math]\displaystyle{ 1/2 }[/math] prefactor avoids double-counting, since each bond participates in two spins. Simplifying leads to the final expression

[math]\displaystyle{ H^{MF} = \frac{J m^2 N z}{2} - \underbrace{(h + m J z)}_{h^\text{eff}} \sum_i s_i }[/math]

where [math]\displaystyle{ z }[/math] 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 [math]\displaystyle{ h^\text{eff} = h + J z m }[/math] which is the sum of the external field [math]\displaystyle{ h }[/math] 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 [math]\displaystyle{ d }[/math], [math]\displaystyle{ z = 2 d }[/math]).

Substituting this Hamiltonian into the partition function, and solving the effective 1D problem, we obtain

[math]\displaystyle{ Z = e^{-\frac{\beta J m^2 Nz}{2}} \left[2 \cosh\left(\frac{h + m J z}{k_BT} \right)\right]^N }[/math]

where [math]\displaystyle{ N }[/math] 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 [math]\displaystyle{ m }[/math] as a function of [math]\displaystyle{ h^{\mathrm{eff}} }[/math].

We thus have two equations between [math]\displaystyle{ m }[/math] and [math]\displaystyle{ h^\text{eff} }[/math], allowing us to determine [math]\displaystyle{ m }[/math] as a function of temperature. This leads to the following observation:

• for temperatures greater than a certain value [math]\displaystyle{ T_c }[/math], the only solution is [math]\displaystyle{ m = 0 }[/math]. The system is paramagnetic.
• for [math]\displaystyle{ T \lt T_c }[/math], there are two non-zero solutions: [math]\displaystyle{ m = \pm m_0 }[/math]. The system is ferromagnetic.

[math]\displaystyle{ T_c }[/math] is given by the following relation: [math]\displaystyle{ T_c = \frac{J z}{k_B} }[/math]. This shows that MFT can account for the ferromagnetic phase transition.

4.2. Application to Other Systems

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 [math]\displaystyle{ \Delta }[/math].
• 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).
• To determine the elastic properties of a composite material.

5. Extension to Time-dependent Mean Fields

In mean-field theory, the mean field appearing in the single-site problem is a scalar or vectorial time-independent quantity. However, this need not always be 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.