Mean-field theory

In physics and probability theory, mean field theory (MFT also known as self-consistent field theory) studies the behaviour of large and complex stochastic models by studying a simpler model. Such models consider a large large number of small interacting individuals who interact with each other. 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 ideas first appeared in physics in the work of Pierre Curie and Pierre Weiss to describe phase transitions. Approaches inspired by these ideas have seen applications in epidemic models, queueing theory, computer network performance and game theory.

A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases (random field theory, 1D Ising model). The n-body system is replaced by a 1-body 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 Bragg-Williams approximation, models on Bethe lattice, Landau theory, Pierre-Weiss 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. This reduces any multi-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 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 "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, in the formalism of fluctuations, MFT provides a convenient launch-point to studying first or second order fluctuations.

In general, dimensionality plays a strong role in determining whether a mean-field 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 long-range 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, graphical models theory and artificial intelligence.

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

${\mathcal {H}}={\mathcal {H}}_{0}+\Delta {\mathcal {H}}$

has the following upper bound:

$F\leq F_{0}\ {\stackrel {\mathrm {def} }{=}}\ \langle {\mathcal {H}}\rangle _{0}-TS_{0}$

where $S_{0}$ is the entropy and where the average is taken over the equilibrium ensemble of the reference system with Hamiltonian ${\mathcal {H}}_{0}$. In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as

${\mathcal {H}}_{0}=\sum _{i=1}^{N}h_{i}\left(\xi _{i}\right)$

where $\left(\xi _{i}\right)$ 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.,

${\mathcal {H}}=\sum _{(i,j)\in {\mathcal {P}}}V_{i,j}\left(\xi _{i},\xi _{j}\right)$

where ${\mathcal {P}}$ is the set of pairs that interact, the minimizing procedure can be carried out formally. Define ${\rm {Tr}}_{i}f(\xi _{i})$ 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

$F_{0}=\,\!$	${\rm {Tr}}_{1,2,..,N}{\mathcal {H}}(\xi _{1},\xi _{2},...,\xi _{N})P_{0}^{(N)}(\xi _{1},\xi _{2},...,\xi _{N})$
	$+kT\,{\rm {Tr}}_{1,2,..,N}P_{0}^{(N)}(\xi _{1},\xi _{2},...,\xi _{N})\log P_{0}^{(N)}(\xi _{1},\xi _{2},...,\xi _{N})$

where $P_{0}^{(N)}(\xi _{1},\xi _{2},...,\xi _{N})$ is the probability to find the reference system in the state specified by the variables $(\xi _{1},\xi _{2},...,\xi _{N})$. This probability is given by the normalized Boltzmann factor

$P_{0}^{(N)}(\xi _{1},\xi _{2},...,\xi _{N})={\frac {1}{Z_{0}^{(N)}}}e^{-\beta {\mathcal {H}}_{0}(\xi _{1},\xi _{2},...,\xi _{N})}=\prod _{i=1}^{N}{\frac {1}{Z_{0}}}e^{-\beta h_{i}\left(\xi _{i}\right)}\ {\stackrel {\mathrm {def} }{=}}\ \prod _{i=1}^{N}P_{0}^{(i)}(\xi _{i})$

where $Z_{0}$ is the partition function. Thus

$F_{0}=\sum _{(i,j)\in {\mathcal {P}}}{\rm {Tr}}_{i,j}V_{i,j}\left(\xi _{i},\xi _{j}\right)P_{0}^{(i)}(\xi _{i})P_{0}^{(j)}(\xi _{j})+kT\sum _{i=1}^{N}{\rm {Tr}}_{i}P_{0}^{(i)}(\xi _{i})\log P_{0}^{(i)}(\xi _{i}).$

In order to minimize we take the derivative with respect to the single degree-of-freedom probabilities $P_{0}^{(i)}$ using a Lagrange multiplier to ensure proper normalization. The end result is the set of self-consistency equations

$P_{0}^{(i)}(\xi _{i})={\frac {1}{Z_{0}}}e^{-\beta h_{i}^{MF}(\xi _{i})}\qquad i=1,2,..,N$

where the mean field is given by

$h_{i}^{MF}(\xi _{i})=\sum _{\{j|(i,j)\in {\mathcal {P}}\}}{\rm {Tr}}_{j}V_{i,j}\left(\xi _{i},\xi _{j}\right)P_{0}^{(j)}(\xi _{j}).$

Applications

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

Ising Model

Consider the Ising model on an N-dimensional cubic lattice. The Hamiltonian is given by

$H=-J\sum _{\langle i,j\rangle }s_{i}s_{j}-h\sum _{i}s_{i}$

where the $\sum _{\langle i,j\rangle }$ indicates summation over the pair of nearest neighbors $\langle i,j\rangle$, and $s_{i}=\pm 1$ and $s_{j}$ are neighboring Ising spins.

Let us transform our spin variable by introducing the fluctuation from its mean value $m_{i}\equiv \langle s_{i}\rangle$. We may rewrite the Hamiltonian:

$H=-J\sum _{\langle i,j\rangle }(m_{i}+\delta s_{i})(m_{j}+\delta s_{j})-h\sum _{i}s_{i}$

where we define $\delta s_{i}\equiv s_{i}-m_{i}$; 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 mean-field approximation consists in neglecting this second order fluctuation term. These fluctuations are enhanced at low dimensions, making MFT a better approximation for high dimensions.

$H\approx H^{MF}\equiv -J\sum _{\langle i,j\rangle }(m_{i}m_{j}+m_{i}\delta s_{j}+m_{j}\delta s_{i})-h\sum _{i}s_{i}$

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

$H^{MF}=-J\sum _{\langle i,j\rangle }\left(m^{2}+2m(s_{i}-m)\right)-h\sum _{i}s_{i}$

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

$H^{MF}={\frac {Jm^{2}Nz}{2}}-\underbrace {(h+mJz)} _{h^{eff}}\sum _{i}s_{i}$

where $z$ 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 $h^{eff}=h+Jzm$ which is the sum of the external field $h$ 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 $d$, $z=2d$).

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

$Z=e^{-\beta Jm^{2}Nz/2}\left[2\cosh \left({\frac {h+mJz}{k_{B}T}}\right)\right]^{N}$

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 non-zero solutions: $m=\pm m_{0}$. The system is ferromagnetic.

$T_{c}$ is given by the following relation: $T_{c}={\frac {Jz}{k_{B}}}$. 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 metal-superconductor transition. In this case, the analog of the magnetization is the superconducting gap $\Delta$. Another example is the molecular field of a liquid crystal that emerges when the Laplacian of the director field is non-zero.

Extension to Time-Dependent Mean Fields

Main article: Dynamical Mean Field Theory

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.

See also

• Dynamical Mean Field Theory
• Mean field game theory