Compressibility equation

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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 \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, {\rm 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.


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