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 $⟨e^{\lambda \varphi }⟩\ge e^{⟨\lambda \varphi ⟩}$ for random variable $\varphi$ by using the convexity of the exponential: $e^{\lambda \varphi }\ge 1+\lambda \varphi$. Then consider a classical Hamiltonian $H$ whose exact scaled Helmholtz free energy is given by $\beta F=-\ln \mathrm{Tr}{e}^{-\beta H}$. Let $\rho$ be any normalized density function, and use the inequality above to show that an approximate scaled free energy $F_{\rho }$ defined by

$$\beta F_{\rho }=\mathrm{Tr}\rho \beta H+\mathrm{Tr}\rho \ln \rho$$

is bounded below by the exact free energy: $\beta F\le \beta F_{\rho }$. Show that minimizing $\beta F_{\rho }$ by finding the zero of the functional derivative

$$\frac{\delta \beta F_{\rho }}{\delta \rho }=0$$