Editing Mean-field theory (section)

====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 [[Helmholtz free energy|free energy]].<ref>{{cite journal |last1=Sakthivadivel |first1=Dalton A R |title=Magnetisation and Mean Field Theory in the Ising Model |journal=SciPost Physics Lecture Notes |date=Jan 2022 |volume=35 |pages=1–16 |doi=10.21468/SciPostPhysLectNotes.35 |s2cid=237623181 |url=https://scipost.org/SciPostPhysLectNotes.35|doi-access=free |arxiv=2102.00960 }}</ref> The first step is choosing a more tractable approximation of the true Hamiltonian. Using a non-interacting or effective field Hamiltonian, 

:<math> -m \sum_i s_i </math>,

the variational free energy is

:<math> F_V = F_0 + \left \langle \left( -J \sum s_i s_j - h \sum s_i \right) - \left(-m\sum s_i\right) \right \rangle_0. </math>

By the Bogoliubov inequality, simplifying this quantity and calculating the magnetisation function that [[Mathematical optimization|minimises]] the variational free energy yields the best approximation to the actual magnetisation. The minimiser is
:<math> m = J\sum\langle s_j \rangle_0 + h, </math>
which is the [[Ensemble average (statistical mechanics)|ensemble average]] of spin. This simplifies to
:<math> m = \text{tanh}(zJ\beta m) + h. </math>

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.