Mean field theory
Encyclopedia
Mean field theory is a method to analyse physical systems with multiple bodies. 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
in graphical models theory in artificial intelligence
.
of a system with Hamiltonian
has the following upper bound:
where
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 non-interacting 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 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.,
where
is the set of pairs that interact, the minimizing procedure can be carried out formally. Define 
as the generalized sum of the observable 
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 
. This probability is given by the normalized Boltzmann factor
where
is the partition function
. Thus
In order to minimize we take the derivative with respect to the single degree-of-freedom probabilities
using a Lagrange multiplier to ensure proper normalization. The end result is the set of self-consistency equations
where the mean field is given by
on an N-dimensional cubic lattice. The Hamiltonian is given by
where the
indicates summation over the pair of nearest neighbors 
, and 
and
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 mean-field 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 site-independent, since the Ising chain is translationally invariant. This yields
The summation over neighboring spins can be rewritten as
where 
means 'nearest-neighbor of 
' and the 
prefactor avoids double-counting, since each bond participates in two spins. Simplifying leads to the final expression
where
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 
which is the sum of the external field 
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 
, 
).
Substituting this Hamiltonian into the partition function, and solving the effective 1D problem, we obtain
where
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 exponent
s. In particular, we can obtain the magnetization
as a function of 
.
We thus have two equations between
and 
, allowing us to determine 
as a function of temperature. This leads to the following observation:
is given by the following relation: 
.
This shows that MFT can account for the ferromagnetic phase transition.
. Another example is the molecular field of a liquid crystal
that emerges when the Laplacian of the director field is non-zero.
(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.
Random field
A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....
theory, 1D Ising model
Ising model
The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
). 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
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
of the system) is the treatment of combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
generated by the interaction terms in the Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
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
Bethe lattice
A Bethe lattice or Cayley tree , introduced by Hans Bethe in 1935, is a connected cycle-free graph where each node is connected to z neighbours, where z is called the coordination number. It can be seen as a tree-like structure emanating from a central node, with all the nodes arranged in shells...
, Landau theory
Landau theory
Landau theory in physics was introduced by Lev Landau in an attempt to formulate a general theory of second-order phase transitions. He was motivated to suggest that the free energy of any system should obey two conditions: that the free energy is analytic, and that it obeys the symmetry of the...
, 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
Classical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
, 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
Ginzburg criterion
Mean field theory gives sensible results as long as we are able to neglect fluctuations in the system under consideration. The Ginzburg Criterion tells us quantitatively when mean field theory is valid...
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
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, it has more recently been applied elsewhere, for example in inference
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
in graphical models theory in artificial intelligence
Artificial intelligence
Artificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...
.
Formal approach
The formal basis for mean field theory is the Bogoliubov inequality. This inequality states that the free energyThermodynamic free energy
The thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...
of a system with Hamiltonian
has the following upper bound:
where
is the entropyEntropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
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 non-interacting system and can thus be written aswhere
is shorthand for the degrees of freedomDegrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
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.,
where
is the set of pairs that interact, the minimizing procedure can be carried out formally. Define as the generalized sum of the observable 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 . This probability is given by the normalized Boltzmann factorBoltzmann factor
In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...
where
is the partition functionPartition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
. Thus
In order to minimize we take the derivative with respect to the single degree-of-freedom probabilities
using a Lagrange multiplier to ensure proper normalization. The end result is the set of self-consistency equationswhere 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.Ising Model
Consider the Ising modelIsing model
The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
on an N-dimensional cubic lattice. The Hamiltonian is given by
where the
indicates summation over the pair of nearest neighbors , and and
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
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
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 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 site-independent, since the Ising chain is translationally invariant. This yields
The summation over neighboring spins can be rewritten as
where means 'nearest-neighbor of ' and the prefactor avoids double-counting, since each bond participates in two spins. Simplifying leads to the final expressionwhere
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 which is the sum of the external field 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 , ).Substituting this Hamiltonian into the partition function, and solving the effective 1D problem, we obtain
where
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 exponentCritical exponent
Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...
s. In particular, we can obtain the magnetization
as a function of .We thus have two equations between
and , allowing us to determine as a function of temperature. This leads to the following observation:
- for temperatures greater than a certain value
, the only solution is 
. The system is paramagnetic. - for
, there are two non-zero solutions: 
. The system is ferromagnetic.
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 metal-superconductor transition. In this case, the analog of the magnetization is the superconducting gap. Another example is the molecular field of a liquid crystalLiquid crystal
Liquid crystals are a state of matter that have properties between those of a conventional liquid and those of a solid crystal. For instance, an LC may flow like a liquid, but its molecules may be oriented in a crystal-like way. There are many different types of LC phases, which can be...
that emerges when the Laplacian of the director field is non-zero.
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 TheoryDynamical mean field theory
Dynamical Mean Field Theory is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in Density Functional Theory and usual band structure calculations, breaks down...
(DMFT), the mean-field becomes a time-dependent quantity. For instance, DMFT can be applied to the Hubbard model
Hubbard model
The Hubbard model is an approximate model used, especially in solid state physics, to describe the transition between conducting and insulating systems...
to study the metal-Mott insulator transition.