Pressure equation

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

$p=-{\frac{\partial A}{\partial V}|}_T$

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):

$p^{*}=\frac{\beta p}{\rho }=\frac{pV}{N{k}_{B}T}=1-\beta \frac{2}{3}\pi \rho \underset{0}{\overset{\mathrm{\infty }}{\int }}(\frac{\mathrm{d}\mathrm{\Phi }(r)}{\mathrm{d}r}r)\mathrm{g}(r)r^2\mathrm{d}r$

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