Mean field theories can be derived using a variational approach in which the exact free energy is a lower bound of an approximate free energy; see Chaikin and Lubensky (1995). First prove the inequality \(\left\langle e^{\lambda \phi}\rightangle \geq e^{\langle\lambda \phiangle}\) for random variable \(\phi\) by using the convexity of the exponential: \(e^{\lambda \phi} \geq 1+\lambda \phi\). Then consider a classical Hamiltonian \(H\) whose exact scaled Helmholtz free energy is given by \(\beta F=-\ln \operatorname{Tr} e^{-\beta H}\). Let \(ho\) be any normalized density function, and use the inequality above to show that an approximate scaled free energy \(F_{ho}\) defined by

\[\beta F_{ho}=\operatorname{Tr} ho \beta H+\operatorname{Tr} ho \ln ho\]

is bounded below by the exact free energy: \(\beta F \leq \beta F_{ho}\). Show that minimizing \(\beta F_{ho}\) by finding the zero of the functional derivative

\[\frac{\delta \beta F_{ho}}{\delta ho}=0\]

gives \(ho=\exp (-\beta H) / \operatorname{Tr} \exp (-\beta H)\) and the exact free energy \(\beta F=-\ln \operatorname{Tr} e^{-\beta H}\). Use a Lagrange multiplier term of the form \(\lambda \operatorname{Tr} \delta ho\) to keep the density normalized.