Pressure equation

For particles acting through two-body central forces alone one may use the thermodynamic relation

$p=-\left.{\frac {\partial A}{\partial V}}\right\vert _{T}$

Using this relation, along with the Helmholtz energy function and the canonical partition function, one arrives at the so-called pressure equation (also known as the virial equation):

$p^{*}={\frac {\beta p}{\rho }}={\frac {pV}{Nk_{B}T}}=1-\beta {\frac {2}{3}}\pi \rho \int _{0}^{\infty }\left({\frac {{\rm {d}}\Phi (r)}{{\rm {d}}r}}~r\right)~{\rm {g}}(r)r^{2}~{\rm {d}}r$

where $\beta =1/k_{B}T$, $\Phi (r)$ is a central potential and ${\rm {g}}(r)$ is the pair distribution function.