Editing Mean-field theory (section)

==Formal approach (Hamiltonian)==
The formal basis for mean-field theory is the [[Helmholtz free energy#Bogoliubov inequality|Bogoliubov inequality]]. This inequality states that the [[thermodynamic free energy|free energy]] of a system with Hamiltonian

: <math>\mathcal{H} = \mathcal{H}_0 + \Delta \mathcal{H}</math>

has the following upper bound:

: <math>F \leq F_0 \ \stackrel{\mathrm{def}}{=}\  \langle \mathcal{H} \rangle_0 - T S_0,</math>

where <math>S_0</math> is the [[entropy]], and <math>F</math> and <math>F_0</math> are [[Helmholtz free energy|Helmholtz free energies]]. The average is taken over the equilibrium [[Statistical ensemble (mathematical physics)|ensemble]] of the reference system with Hamiltonian <math>\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>\mathcal{H}_0 = \sum_{i=1}^N h_i(\xi_i),</math>

where <math>\xi_i</math> are the [[degrees of freedom (physics and chemistry)|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.,

: <math>\mathcal{H} = \sum_{(i,j) \in \mathcal{P}} V_{i,j}(\xi_i, \xi_j),</math>

where <math>\mathcal{P}</math> is the set of pairs that interact, the minimising procedure can be carried out formally. Define <math>\operatorname{Tr}_i f(\xi_i)</math> as the generalized sum of the observable <math>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>\begin{align}
 F_0 &= \operatorname{Tr}_{1,2,\ldots,N} \mathcal{H}(\xi_1, \xi_2, \ldots, \xi_N) P^{(N)}_0(\xi_1, \xi_2, \ldots, \xi_N) \\
     &+ kT \,\operatorname{Tr}_{1,2,\ldots,N} P^{(N)}_0(\xi_1, \xi_2, \ldots, \xi_N) \log P^{(N)}_0(\xi_1, \xi_2, \ldots,\xi_N),
\end{align}</math>

where <math>P^{(N)}_0(\xi_1, \xi_2, \dots, \xi_N)</math> is the probability to find the reference system in the state specified by the variables <math>(\xi_1, \xi_2, \dots, \xi_N)</math>. This probability is given by the normalized [[Boltzmann factor]]
: <math>\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(\xi_i)} \ \stackrel{\mathrm{def}}{=}\  \prod_{i=1}^N P^{(i)}_0(\xi_i),
\end{align}</math>

where <math>Z_0</math> is the [[Partition function (statistical mechanics)|partition function]]. Thus
:<math>\begin{align}
  F_0 &= \sum_{(i,j) \in \mathcal{P}} \operatorname{Tr}_{i,j} V_{i,j}(\xi_i, \xi_j) 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 minimise, we take the derivative with respect to the single-degree-of-freedom probabilities <math>P^{(i)}_0</math> using a [[Lagrange multiplier]] to ensure proper normalization. The end result is the set of self-consistency equations
: <math>P^{(i)}_0(\xi_i) = \frac{1}{Z_0} e^{-\beta h_i^{MF}(\xi_i)},\quad i = 1, 2, \ldots, N,</math>

where the mean field is given by
: <math>h_i^\text{MF}(\xi_i) = \sum_{\{j \mid (i,j) \in \mathcal{P}\}} \operatorname{Tr}_j V_{i,j}(\xi_i, \xi_j) P^{(j)}_0(\xi_j).</math>