Compressibility equation - Revision history

Carl McBride at 17:24, 14 February 2008

2008-02-14T17:24:09Z

← Older revision		Revision as of 19:24, 14 February 2008
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	:<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}		:<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 <math>{\rm g}^{(2)}(r)</math> is the [[pair distribution function]] and <math>k_B</math> is the [[Boltzmann constant]].		where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math>h</math> is the [[total correlation function]], <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 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)		:<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>

Nice and Tidy at 14:24, 10 July 2007

2007-07-10T14:24:06Z

← Older revision		Revision as of 16:24, 10 July 2007
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	:<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>		:<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}{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)		:<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>

Nice and Tidy at 16:33, 25 May 2007

2007-05-25T16:33:19Z

← Older revision		Revision as of 18:33, 25 May 2007
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	==References==		==References==
	#[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]		#[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]
			[[category: statistical mechanics]]

Carl McBride at 13:27, 22 May 2007

2007-05-22T13:27:11Z

← Older revision		Revision as of 15:27, 22 May 2007
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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 ~~\cite{RPP_1965_28_0169}~~).		The '''compressibility equation''' (<math>\chi</math>) can be derived from the density fluctuations of the [[grand canonical ensemble]] (Eq. 3.16 in Ref. 1). For a homogeneous system:
	For a homogeneous system:


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	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==
			#[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]

Carl McBride at 13:21, 22 May 2007

2007-05-22T13:21:48Z

← Older revision		Revision as of 15:21, 22 May 2007
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	:<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>		:<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>

	where <math>{\rm g}^{(2)}(r)</math> is the [[~~par~~ distribution function]].		where <math>{\rm g}^{(2)}(r)</math> is the [[pair distribution function]].
	For a spherical potential		For a spherical potential

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	\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 ~~equatiomn~~ \| energy]] and [[pressure equation]]s,		Note that the compressibility equation, unlike the [[energy equation \| 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==

Carl McBride: New page: The '''compressibility equation''' ( $\chi$ ) can be derived from the density fluctuations of the grand canonical ensemble (Eq. 3.16 \cite{RPP_1965_28_0169}). For a homogene...

2007-05-22T11:26:13Z

New page: 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 homogene...

New page

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> 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>

where <math>{\rm g}^{(2)}(r)</math> is the [[par distribution function]].
For a spherical potential

:<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>

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.
==References==