Compressibility equation
The compressibility equation () can be derived from the density fluctuations of the grand canonical ensemble (Eq. 3.16 in Ref. 1). For a homogeneous system:
![{\displaystyle kT\left.{\frac {\partial \rho }{\partial P}}\right\vert _{T}=1+\rho \int h(r)~{\rm {d}}r=1+\rho \int [{\rm {g}}^{(2)}(r)-1]{\rm {d}}r={\frac {\langle N^{2}\rangle -\langle N\rangle ^{2}}{\langle N\rangle }}=\rho k_{B}T\chi _{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9500b9cded7df21fe7a8024dd8225f6b72d19cfc)
where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\rm {g}}^{(2)}(r)} is the pair distribution function. For a spherical potential
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{kT}}\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.