Compressibility equation

The compressibility equation ($\chi$) can be derived from the density fluctuations of the grand canonical ensemble (Eq. 3.16 in Ref. 1). For a homogeneous system:

$k_{B}T\left.{\frac {\partial \rho }{\partial p}}\right\vert _{T}=1+\rho \int h(r)~{\rm {d}}{\mathbf {r} }=1+\rho \int [{\rm {g}}^{(2)}({\mathbf {r} })-1]{\rm {d}}{\mathbf {r} }={\frac {\langle N^{2}\rangle -\langle N\rangle ^{2}}{\langle N\rangle }}=\rho k_{B}T\chi _{T}$

where $p$ is the pressure, $T$ is the temperature, $h$ is the total correlation function, ${\rm {g}}^{(2)}(r)$ is the pair distribution function and $k_{B}$ is the Boltzmann constant.

For a spherical potential

${\frac {1}{k_{B}T}}\left.{\frac {\partial p}{\partial \rho }}\right\vert _{T}=1-\rho \int _{0}^{\infty }c(r)~4\pi r^{2}~{\rm {d}}r\equiv 1-\rho {\hat {c}}(0)\equiv {\frac {1}{1+\rho {\hat {h}}(0)}}\equiv {\frac {1}{1+\rho \int _{0}^{\infty }h(r)~4\pi r^{2}~{\rm {d}}r}}$

Note that the compressibility equation, unlike the energy and pressure equations, is valid even when the inter-particle forces are not pairwise additive.

References

1. J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics 28 pp. 169-199 (1965)