Editing Compressibility equation
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:<math>k_B T \left.\frac{\partial \rho }{\partial | :<math>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</math> | = \frac{ \langle N^2 \rangle - \langle N\rangle^2}{\langle N\rangle}=\rho k_B T \chi_T</math> | ||
where | where <math>{\rm g}^{(2)}(r)</math> is the [[pair distribution function]] and <math>k_B</math> is the [[Boltzmann constant]]. | ||
For a spherical potential | For a spherical potential | ||
:<math>\frac{1}{k_BT} \left.\frac{\partial | :<math>\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}</math> | \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}</math> | ||