Helmholtz Free Energy - Bogoliubov Inequality

Bogoliubov Inequality

Computing the free energy is an intractable problem for all but the simplest models in statistical physics. A powerful approximation method is mean field theory, which is a variational method based on the Bogoliubov inequality. This inequality can be formulated as follows.

Suppose we replace the real Hamiltonian of the model by a trial Hamiltonian, which has different interactions and may depend on extra parameters that are not present in the original model. If we choose this trial Hamiltonian such that

where both averages are taken with respect to the canonical distribution defined by the trial Hamiltonian, then

where is the free energy of the original Hamiltonian and is the free energy of the trial Hamiltonian. By including a large number of parameters in the trial Hamiltonian and minimizing the free energy we can expect to get a close approximation to the exact free energy.

The Bogoliubov inequality is often formulated in a sightly different but equivalent way. If we write the Hamiltonian as:

where is exactly solvable, then we can apply the above inequality by defining

Here we have defined to be the average of X over the canonical ensemble defined by . Since defined this way differs from by a constant, we have in general

Therefore

And thus the inequality

holds. The free energy is the free energy of the model defined by plus . This means that

and thus: