Gibbs--Bogoliubov--Feynman inequality

For an exact dimensionless Hamiltonian $H$ and trial Hamiltonian $H_0$, the free energies obeywhere $F_0=-\log Z_0$ and the expectation uses the trial Gibbs distribution $Z_0^{-1}{e}^{-H_0}$.

The identity $e^{-F}=e^{-F_0}⟨e^{-(H-H_0)}{⟩}_0$ and the Jensen inequality for the convex exponential function give $\log ⟨e^{-(H-H_0)}{⟩}_0\ge -⟨H-H_0{⟩}_0$. Taking minus the logarithm proves the bound.