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:
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 is the pressure, 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 T} is the temperature, 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 h} is the total correlation function, 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 and 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 k_B} is the Boltzmann constant.
For a spherical potential
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{k_{B}T}}\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.