Canonical energy variance ($\mathrm{Var}(E)=k_{\mathrm{B}}{T}^2{C}_V$)

Differentiating the canonical partition function at fixed external parameters gives $⟨E⟩=-{\partial }_{\beta }\ln Z$ and ${\partial }_{\beta }^2\ln Z=⟨E^2⟩-⟨E{⟩}^2$. Consequently the heat capacity isThe identity assumes an energy spectrum independent of temperature and the appropriate fixed-volume canonical ensemble. It relates a thermal response to equilibrium variance.