mean-field theory

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In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
and
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themsel ...
) 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 A statistic (singular) or sample statistic is any quantity computed from values in a Sample (statistics), sample which is considered for a statistical purpose. Statistical purposes include Estimation, estimating a Statistical population, populat ...
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''. This reduces any
many-body problem The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provi ...
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 Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
, graphical models,
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,
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,
epidemic model Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious, ...
s,
queueing theory Queueing theory is the mathematical study of waiting lines, or wikt:queue, queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research b ...
, computer-network performance and
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, as in the quantal response equilibrium.

# Origins

The idea first appeared in physics (
statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...
) in the work of
Pierre Curie Pierre Curie ( , ; 15 May 1859 – 19 April 1906) was a French physicist, a pioneer in crystallography, magnetism, piezoelectricity, and radioactivity. In 1903, he received the Nobel Prize in Physics with his wife, Marie Curie, and Henri Becquere ...
and Pierre Weiss to describe
phase transitions In chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, stru ...
. MFT has been used in the Bragg–Williams approximation, models on Bethe lattice,
Landau theory Landau theory in physics is a theory that Lev Landau introduced in an attempt to formulate a general theory of continuous (i.e., second-order) phase transitions. It can also be adapted to systems under externally-applied fields, and used as a qua ...
, Pierre–Weiss approximation, Flory–Huggins solution theory, and Scheutjens–Fleer theory.
Systems A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
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 The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discr ...
). 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, 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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other ar ...
of the interaction terms in the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used ...
can sometimes at best produce perturbation results or
Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduce ...
s that correct the mean-field approximation.

# Validity

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 A polymer (; Greek ''wikt:poly-, poly-'', "many" + ''wikt:-mer, -mer'', "part") is a Chemical substance, substance or material consisting of very large molecules called macromolecules, composed of many Repeat unit, repeating subunits. Due to t ...
). 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)

The formal basis for mean-field theory is the Bogoliubov inequality. This inequality states that the free energy of a system with Hamiltonian : $\mathcal = \mathcal_0 + \Delta \mathcal$ has the following upper bound: : $F \leq F_0 \ \stackrel\ \langle \mathcal \rangle_0 - T S_0,$ where $S_0$ is the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, and $F$ and $F_0$ are Helmholtz free energies. The average is taken over the equilibrium ensemble of the reference system with Hamiltonian $\mathcal_0$. In the special case that the reference Hamiltonian is that of a non-interacting system and can thus be written as : $\mathcal_0 = \sum_^N h_i\left(\xi_i\right),$ where $\xi_i$ 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., : $\mathcal = \sum_ V_\left(\xi_i, \xi_j\right),$ where $\mathcal$ is the set of pairs that interact, the minimising procedure can be carried out formally. Define $\operatorname_i f\left(\xi_i\right)$ 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 :$\begin F_0 &= \operatorname_ \mathcal\left(\xi_1, \xi_2, \ldots, \xi_N\right) P^_0\left(\xi_1, \xi_2, \ldots, \xi_N\right) \\ &+ kT \,\operatorname_ P^_0\left(\xi_1, \xi_2, \ldots, \xi_N\right) \log P^_0\left(\xi_1, \xi_2, \ldots,\xi_N\right), \end$ where $P^_0\left(\xi_1, \xi_2, \dots, \xi_N\right)$ is the probability to find the reference system in the state specified by the variables $\left(\xi_1, \xi_2, \dots, \xi_N\right)$. This probability is given by the normalized Boltzmann factor : $\begin P^_0\left(\xi_1, \xi_2, \ldots, \xi_N\right) &= \frac e^ \\ &= \prod_^N \frac e^ \ \stackrel\ \prod_^N P^_0\left(\xi_i\right), \end$ where $Z_0$ is the partition function. Thus :$\begin F_0 &= \sum_ \operatorname_ V_\left(\xi_i, \xi_j\right) P^_0\left(\xi_i\right) P^_0\left(\xi_j\right) \\ &+ kT \sum_^N \operatorname_i P^_0\left(\xi_i\right) \log P^_0\left(\xi_i\right). \end$ In order to minimise, we take the derivative with respect to the single-degree-of-freedom probabilities $P^_0$ using a
Lagrange multiplier In mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is ge ...
to ensure proper normalization. The end result is the set of self-consistency equations : $P^_0\left(\xi_i\right) = \frac e^,\quad i = 1, 2, \ldots, N,$ where the mean field is given by : $h_i^\text\left(\xi_i\right) = \sum_ \operatorname_j V_\left(\xi_i, \xi_j\right) P^_0\left(\xi_j\right).$

# Applications

Mean field theory can be applied to a number of physical systems so as to study phenomena such as
phase transitions In chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, stru ...
.

## Ising model

### Formal derivation

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. The first step is choosing a more tractable approximation of the true Hamiltonian. Using a non-interacting or effective field Hamiltonian, :$-m \sum_i s_i$, the variational free energy is :$F_V = F_0 + \left \langle \left\left( -J \sum s_i s_j - h \sum s_i \right\right) - \left\left(-m\sum s_i\right\right) \right \rangle_0.$ 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 :$m = J\sum\langle s_j \rangle_0 + h,$ which is the
ensemble average In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scienc ...
of spin. This simplifies to :$m = \text\left(zJ\beta m\right) + h.$ 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

Consider the
Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discr ...
on a $d$-dimensional lattice. The Hamiltonian is given by : $H = -J \sum_ s_i s_j - h \sum_i s_i,$ where the $\sum_$ indicates summation over the pair of nearest neighbors $\langle i, j \rangle$, and $s_i, s_j = \pm 1$ are neighboring Ising spins. Let us transform our spin variable by introducing the fluctuation from its mean value $m_i \equiv \langle s_i \rangle$. We may rewrite the Hamiltonian as : $H = -J \sum_ \left(m_i + \delta s_i\right) \left(m_j + \delta s_j\right) - h \sum_i s_i,$ where we define $\delta s_i \equiv s_i - m_i$; this is the ''fluctuation'' of the spin. If we expand the right 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: : $H \approx H^\text \equiv -J \sum_ \left(m_i m_j + m_i \delta s_j + m_j \delta s_i\right) - h \sum_i s_i.$ 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 : $H^\text = -J \sum_ \big\left(m^2 + 2m\left(s_i - m\right)\big\right) - h \sum_i s_i.$ The summation over neighboring spins can be rewritten as $\sum_ = \frac \sum_i \sum_$, where $nn\left(i\right)$ 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 : $H^\text = \frac - \underbrace_ \sum_i s_i,$ where $z$ is the
coordination number In chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structu ...
. At this point, the Ising Hamiltonian has been ''decoupled'' into a sum of one-body Hamiltonians with an ''effective mean field'' $h^\text = h + J z m$, 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 $d$, $z = 2 d$). 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
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s. In particular, we can obtain the magnetization $m$ as a function of $h^\text$. We thus have two equations between $m$ and $h^\text$, allowing us to determine $m$ as a function of temperature. This leads to the following observation: * For temperatures greater than a certain value $T_\text$, the only solution is $m = 0$. The system is paramagnetic. * For $T < T_\text$, there are two non-zero solutions: $m = \pm m_0$. The system is ferromagnetic. $T_\text$ is given by the following relation: $T_\text = \frac$. 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 as in the following cases: * To study the metal– superconductor transition. In this case, the analog of the magnetization is the superconducting gap $\Delta$. * The molecular field of a
liquid crystal Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal may flow like a liquid, but its molecules may be oriented in a crystal-like way. The ...
that emerges when the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where ...
of the director field is non-zero. * To determine the optimal
amino acid Amino acids are organic compound In chemistry, organic compounds are generally any chemical compounds that contain carbon-hydrogen or carbon-carbon chemical bond, bonds. Due to carbon's ability to Catenation, catenate (form chains with ot ...
side chain In organic chemistry Organic chemistry is a subdiscipline within chemistry involving the science, scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that ...
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. Variationally minimisation like mean field theory can be also be used in statistical inference.

# Extension to time-dependent mean fields

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 The Hubbard model is an Approximation, approximate model used to describe the transition between Conductor (material), conducting and Electrical insulation, insulating systems. It is particularly useful in solid-state physics. The model is named ...
to study the metal–Mott-insulator transition.