Energy equation

The energy equation is given, in classical thermodynamics, by

\[\left. \frac{\partial U}{\partial V} \right\vert_T = T \left. \frac{\partial p}{\partial T} \right\vert_V -p \]

and in statistical mechanics it is obtained via the thermodynamic relation

\[U = \frac{\partial (A/T)}{\partial (1/T)}\]

and making use of the Helmholtz energy function and the canonical partition function one arrives at

\[\frac{U^{\rm ex}}{N}= \frac{\rho}{2} \int_0^{\infty} \Phi(r)~{\rm g}(r)~4 \pi r^2~{\rm d}r\] where \(\Phi(r)\) is a two-body central potential, \(U^{\rm ex}\) is the excess internal energy per particle, and \({\rm g}(r)\) is the radial distribution function.