Internal energy

The internal energy is given by

$U=-T^{2}\left.{\frac {\partial (A/T)}{\partial T}}\right\vert _{N,V}=k_{B}T^{2}\left.{\frac {\partial \log Z(T)}{\partial T}}\right\vert _{N,V}$

where $A$ is the Helmholtz energy function and $Z$ is the partition function.