Compressibility equation

The compressibility equation (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \chi } ) can be derived from the density fluctuations of the grand canonical ensemble (Eq. 3.16 \cite{RPP_1965_28_0169}). For a homogeneous system:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm g}^{(2)}(r)} is the pair distribution function. For a spherical potential

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

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