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:

$k_{B}T{\frac{\partial \rho }{\partial p}|}_T=1+\rho \int h(r)\mathrm{d}\mathbf{r}=1+\rho \int [{\mathrm{g}}^{(2)}(\mathbf{r})-1]\mathrm{d}\mathbf{r}=\frac{⟨N^2⟩-⟨N{⟩}^2}{⟨N⟩}=\rho k_{B}T{\chi }_T$

where $p$ is the pressure, $T$ is the temperature, $h$ is the total correlation function, ${\mathrm{g}}^{(2)}(r)$ is the pair distribution function and $k_B$ is the Boltzmann constant.

For a spherical potential

$\frac{1}{k_{B}T}{\frac{\partial p}{\partial \rho }|}_T=1-\rho \underset{0}{\overset{\mathrm{\infty }}{\int }}c(r)4\pi r^2\mathrm{d}r\equiv 1-\rho \hat{c}(0)\equiv \frac{1}{1+\rho \hat{h}(0)}\equiv \frac{1}{1+\rho \underset{0}{\overset{\mathrm{\infty }}{\int }}h(r)4\pi r^2\mathrm{d}r}$

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)