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
\[\frac{1}{k_BT} \left.\frac{\partial p}{\partial \rho}\right\vert_{T} = 1 - \rho \int_0^{\infty} c(r) ~4 \pi r^2 ~{\rm d}r \equiv 1- \rho \hat{c}(0) \equiv \frac{1}{1+\rho \hat{h}(0)} \equiv \frac{1}{ 1 + \rho \int_0^{\infty} h(r) ~4 \pi r^2 ~{\rm d}r}\]
Note that the compressibility equation, unlike the energy and pressure equations, is valid even when the inter-particle forces are not pairwise additive.
