In the special case of entropy, entropy is given by S ≡ ≡ − − k B ∑ ∑ s P s ln ⁡ ⁡ P s = k B ( ln ⁡ ⁡ Z + β β ⟨ ⟨ E ⟩ ⟩ ) = ∂ ∂ ∂ ∂ T ( k B T ln ⁡ ⁡ Z ) = − − ∂ ∂ A ∂ ∂ T where A is the Helmholtz free energy defined as A = U − TS, where U = ⟨ E ⟩ is the total energy and S is the entropy, so that A = ⟨ ⟨ E ⟩ ⟩ − − T S = − − k B T ln ⁡ ⁡ Z .