Compressibility equation

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The compressibility equation ( $χ$ ) 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, $g (2) (r)$ 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.