= Canonical energy variance
{title2=$\operatorname{Var}(E)=k_{\rm B}T^2C_V$}
Differentiating the <canonical partition function> at fixed external parameters gives $\langle E\rangle=-\partial_\beta\ln Z$ and $\partial_\beta^2\ln Z=\langle E^2\rangle-\langle E\rangle^2$. Consequently the <heat capacity> is
$$
C_V=\frac{\operatorname{Var}(E)}{k_{\rm B}T^2}\ge0.
$$
The identity assumes an <energy> spectrum independent of <temperature> and the appropriate fixed-volume <canonical ensemble>. It relates a thermal response to equilibrium <variance>.
Back to article page
