Internal energy

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The internal energy of a system is given by the sum of the kinetic energy and the potential energy. This excludes the external energy of the system, for example motion of the center of mass of the system, or the presence of an external field.

The internal energy in classical thermodynamics is given by

\left. U \right.=TS-pV

and in statistical mechanics 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.