The heat capacity

One way to measure the heat capacity at constant external field id from the definition:

$$C=\frac{\partial \langle E \rangle}{\partial T}.$$

Another way is to use the statistical fluctuations for the total energy in the canonical ensemble:

$$C=\frac{1}{k_BT^2}\left( \langle E^2 \rangle - \langle E \rangle ^2 \right).$$