Energy and specific heat in the canonical ensemble

The energy is proportional to the first derivative of the potential, Helmholtz' free energy. The corresponding variance is defined as $$\begin{equation*} \sigma_E^2=\langle E^2 \rangle-\langle E \rangle^2= \frac{1}{Z}\sum_{i=1}^M E_i^2e^{-\beta E_i}- \left(\frac{1}{Z}\sum_{i=1}^M E_ie^{-\beta E_i}\right)^2. \end{equation*}$$ If we divide the latter quantity with $kT^2$ we obtain the specific heat at constant volume $$\begin{equation*} C_V= \frac{1}{k_BT^2}\left(\langle E^2 \rangle-\langle E \rangle^2\right), \end{equation*}$$ which again can be related to the second derivative of Helmholtz' free energy.

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