The mean field, and the extremum of the free energy
The mean field apporoximation is this: In the calculation of the Boltzmann weight, in the Hamiltonian, terms of the order more than one in the deviation from the mean are neglected; using the resulted Boltzmann weight, which is a function of the mean, the mean is calculated and is equated with the mean which was used in the Boltzmann weight. This is a self-consistency equation for the mean.
Instead of the self-consistency equation, one can calculate the partition function and hence free energy, in terms of the mean, and take the condition governing the mean that the free energy is extremized.
The relation of these two approches with each other is studied.
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