Compressibility equation: Difference between revisions
Jump to navigation
Jump to search
mNo edit summary |
mNo edit summary |
||
| Line 2: | Line 2: | ||
:<math> | :<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> | |||
where <math>{\rm g}^{(2)}(r)</math> is the [[pair distribution function]]. | 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}{ | :<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> | ||
Revision as of 16:24, 10 July 2007
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:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b626fb4f5dfe92b1f9f04c6e54ea0bd0060b1c)
where is the pair distribution function and is the Boltzmann constant. For a spherical potential
Note that the compressibility equation, unlike the energy and pressure equations, is valid even when the inter-particle forces are not pairwise additive.