Editing Compressibility equation
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The '''compressibility equation''' (<math>\chi</math>) can be derived from the density fluctuations of the [[grand canonical ensemble]] (Eq. 3.16 | The '''compressibility equation''' (<math>\chi</math>) can be derived from the density fluctuations of the [[grand canonical ensemble]] (Eq. 3.16 \cite{RPP_1965_28_0169}). | ||
For a homogeneous system: | |||
:<math> | :<math> 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</math> | ||
= \frac{ \langle N^2 \rangle - \langle N\rangle^2}{\langle N\rangle}=\rho k_B T \chi_T</math> | |||
where <math>{\rm g}^{(2)}(r)</math> is the [[par distribution function]]. | |||
For a spherical potential | For a spherical potential | ||
:<math>\frac{1}{ | :<math>\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}</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> | ||
Note that the compressibility equation, unlike the [[energy | Note that the compressibility equation, unlike the [[energy equatiomn | energy]] and [[pressure equation]]s, | ||
is valid even when the inter-particle forces are not pairwise additive. | is valid even when the inter-particle forces are not pairwise additive. | ||
==References== | ==References== | ||