Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner E s = E s ( 0 ) + λ λ A s for all s then the expected value of A is ⟨ ⟨ A ⟩ ⟩ = ∑ ∑ s A s P s = − − 1 β β ∂ ∂ ∂ ∂ λ λ ln ⁡ ⁡ Z ( β β , λ λ ) .