Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 344 / 1 / c / Solution

This is the Gibbs--Bogoliubov--Feynman inequality. In units with $k_{B}T=1$, $H$ is the exact dimensionless Hamiltonian, here the full interacting free-energy functional, and $H_0$ is a freely chosen trial Hamiltonian. Its partition function and free energy areThe notationmeans an expectation in the trial Gibbs ensemble. The bound is therefore