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:

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

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

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 \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.

References

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