U = ⟨ E ⟩ = 1 Z ( β , N ) ∑ α E α exp ( − β E α ) = − ∂ β ln ( Z ( β , N ) ) {\displaystyle U=\left\langle E\right\rangle ={\frac {1}{Z\left(\beta ,N\right)}}\sum \limits _{\alpha }{{{E}_{\alpha }}\exp \left(-\beta {{E}_{\alpha }}\right)}=-{{\partial }_{\beta }}\ln \left(Z\left(\beta ,N\right)\right)}
