one has the exact result ∂ ∂ F ( λ λ ) ∂ ∂ λ λ = ⟨ ∂ ∂ U ( λ λ ) ∂ ∂ λ λ ⟩ λ λ This can either be directly verified from definitions or seen from the limit of the above Gibbs-Bogoliubov inequalities when A = λ λ + , B = λ λ − − . we can therefore write