Pair distribution function - Revision history

Carl McBride: /* References */ Added a reference

2008-02-26T10:39:16Z

References: Added a reference

← Older revision		Revision as of 12:39, 26 February 2008
Line 19:		Line 19:
	==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)]
			#[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (2003)]
	[[category: statistical mechanics]]		[[category: statistical mechanics]]

Carl McBride at 17:28, 14 February 2008

2008-02-14T17:28:23Z

← Older revision		Revision as of 19:28, 14 February 2008
Line 1:		Line 1:
	For a fluid of <math>N</math> particles, enclosed in a volume <math>V</math> at a given temperature <math>T</math>		For a fluid of <math>N</math> particles, enclosed in a volume <math>V</math> at a given [[temperature]] <math>T</math>
	([[canonical ensemble]]) interacting via the `central' [[intermolecular pair potential]] <math>\Phi(r)</math>, the two particle distribution function is defined as		([[canonical ensemble]]) interacting via the `central' [[intermolecular pair potential]] <math>\Phi(r)</math>, the two particle distribution function is defined as

	:<math>{\rm g}_N^{(2)}({\mathbf r}_1,{\mathbf r}_2)= V^2 \frac{\int ... \int e^{-\beta \Phi({\mathbf r}_1,...,{\mathbf r}_N)}{\rm d}{\mathbf r}_3...{\rm d}{\mathbf r}_N}{\int e^{-\beta \Phi({\mathbf r}_1,...,{\mathbf r}_N)}{\rm d}{\mathbf r}_1...{\rm d}{\mathbf r}_N}</math>		:<math>{\rm g}_N^{(2)}({\mathbf r}_1,{\mathbf r}_2)= V^2 \frac{\int ... \int e^{-\beta \Phi({\mathbf r}_1,...,{\mathbf r}_N)}{\rm d}{\mathbf r}_3...{\rm d}{\mathbf r}_N}{\int e^{-\beta \Phi({\mathbf r}_1,...,{\mathbf r}_N)}{\rm d}{\mathbf r}_1...{\rm d}{\mathbf r}_N}</math>

	where <math>\beta = 1/(k_BT)</math>, where <math>k_B</math> is the [[Boltzmann constant]].		where <math>\beta := 1/(k_BT)</math>, where <math>k_B</math> is the [[Boltzmann constant]].
	==Exact convolution equation for <math>g(r)</math>==		==Exact convolution equation for <math>{\mathrm g}(r)</math>==
	See Eq. 5.10 of Ref. 1:		See Eq. 5.10 of Ref. 1:

	:<math>\ln g(r_{12}) + \frac{\Phi(r_{12})}{kT} - E(r_{12}) = n \int \left(g(r_{13}) -1 - \ln g(r_{13}) - \frac{\Phi(r_{13})}{kT} - E(r_{13}) \right)(g(r_{23}) -1) ~{\rm d}{\mathbf r}_3</math>		:<math>\ln {\mathrm g}(r_{12}) + \frac{\Phi(r_{12})}{k_BT} - E(r_{12}) = n \int \left({\mathrm g}(r_{13}) -1 - \ln {\mathrm g}(r_{13}) - \frac{\Phi(r_{13})}{k_BT} - E(r_{13}) \right)({\mathrm g}(r_{23}) -1) ~{\rm d}{\mathbf r}_3</math>

	where, ''i.e.'' <math>r_{12} = \|{\mathbf r}_2 - {\mathbf r}_1\|</math>.		where, ''i.e.'' <math>r_{12} = \|{\mathbf r}_2 - {\mathbf r}_1\|</math>.

Nice and Tidy at 15:09, 10 July 2007

2007-07-10T15:09:02Z

← Older revision		Revision as of 17:09, 10 July 2007
Line 8:		Line 8:
	See Eq. 5.10 of Ref. 1:		See Eq. 5.10 of Ref. 1:

	:<math>\ln g(r_{12}) + \frac{\Phi(r_{12})}{kT} - E(r_{12}) = n \int \left(g(r_{13}) -1 - \ln g(r_{13}) - \frac{\Phi(r_{13})}{kT} - E(r_{13}) \right)(g(r_{23}) -1) ~{\rm d}~~r_3~~</math>		:<math>\ln g(r_{12}) + \frac{\Phi(r_{12})}{kT} - E(r_{12}) = n \int \left(g(r_{13}) -1 - \ln g(r_{13}) - \frac{\Phi(r_{13})}{kT} - E(r_{13}) \right)(g(r_{23}) -1) ~{\rm d}{\mathbf r}_3</math>

	where <math>r_{12} = \|{\mathbf r}_2 - {\mathbf r}_1\|</math>.		where, ''i.e.'' <math>r_{12} = \|{\mathbf r}_2 - {\mathbf r}_1\|</math>.
	==See also==		==See also==
	*[[Radial distribution function]]		*[[Radial distribution function]]

Nice and Tidy at 14:15, 10 July 2007

2007-07-10T14:15:04Z

← Older revision		Revision as of 16:15, 10 July 2007
Line 2:		Line 2:
	([[canonical ensemble]]) interacting via the `central' [[intermolecular pair potential]] <math>\Phi(r)</math>, the two particle distribution function is defined as		([[canonical ensemble]]) interacting via the `central' [[intermolecular pair potential]] <math>\Phi(r)</math>, the two particle distribution function is defined as

	:<math>{\rm g}_N^{(2)}(~~r_1~~,~~r_2~~)= V^2 \frac		:<math>{\rm g}_N^{(2)}({\mathbf r}_1,{\mathbf r}_2)= V^2 \frac{\int ... \int e^{-\beta \Phi({\mathbf r}_1,...,{\mathbf r}_N)}{\rm d}{\mathbf r}_3...{\rm d}{\mathbf r}_N}{\int e^{-\beta \Phi({\mathbf r}_1,...,{\mathbf r}_N)}{\rm d}{\mathbf r}_1...{\rm d}{\mathbf r}_N}</math>
	{\int ... \int e^{-\beta \Phi(~~r_1~~,...,~~r_N~~)}{\rm d}~~r_3~~...{\rm d}~~r_N~~}
	{\int e^{-\beta \Phi(~~r_1~~,...,~~r_N~~){\rm d}~~r_1~~...{\rm d}~~r_N~~}}</math>

	where <math>\beta = 1/(k_BT)</math>, where <math>k_B</math> is the [[Boltzmann constant]].		where <math>\beta = 1/(k_BT)</math>, where <math>k_B</math> is the [[Boltzmann constant]].
Line 12:		Line 10:
	:<math>\ln g(r_{12}) + \frac{\Phi(r_{12})}{kT} - E(r_{12}) = n \int \left(g(r_{13}) -1 - \ln g(r_{13}) - \frac{\Phi(r_{13})}{kT} - E(r_{13}) \right)(g(r_{23}) -1) ~{\rm d}r_3</math>		:<math>\ln g(r_{12}) + \frac{\Phi(r_{12})}{kT} - E(r_{12}) = n \int \left(g(r_{13}) -1 - \ln g(r_{13}) - \frac{\Phi(r_{13})}{kT} - E(r_{13}) \right)(g(r_{23}) -1) ~{\rm d}r_3</math>

			where <math>r_{12} = \|{\mathbf r}_2 - {\mathbf r}_1\|</math>.
	==See also==		==See also==
	*[[Radial distribution function]]		*[[Radial distribution function]]

Carl McBride at 15:46, 26 June 2007

2007-06-26T15:46:42Z

← Older revision		Revision as of 17:46, 26 June 2007
Line 1:		Line 1:
	For a fluid of <math>N</math> particles, enclosed in a volume <math>V</math> at a given temperature <math>T</math>		For a fluid of <math>N</math> particles, enclosed in a volume <math>V</math> at a given temperature <math>T</math>
	([[canonical ensemble]]) interacting via the `central' potential <math>\Phi(r)</math>, the two particle distribution function is defined as		([[canonical ensemble]]) interacting via the `central' [[intermolecular pair potential]] <math>\Phi(r)</math>, the two particle distribution function is defined as

	:<math>{\rm g}_N^{(2)}(r_1,r_2)= V^2 \frac		:<math>{\rm g}_N^{(2)}(r_1,r_2)= V^2 \frac
	{\int ... \int e^{-\beta \Phi(r_1,...,r_N)}{\rm d}r_3...{\rm d}r_N}		{\int ... \int e^{-\beta \Phi(r_1,...,r_N)}{\rm d}r_3...{\rm d}r_N}
	{\int e^{-\beta \Phi(r_1,...,r_N){\rm d}r_1...{\rm d}r_N}}</math>		{\int e^{-\beta \Phi(r_1,...,r_N){\rm d}r_1...{\rm d}r_N}}</math>

			where <math>\beta = 1/(k_BT)</math>, where <math>k_B</math> is the [[Boltzmann constant]].
	==Exact convolution equation for <math>g(r)</math>==		==Exact convolution equation for <math>g(r)</math>==
	See Eq. 5.10 of Ref. 1:		See Eq. 5.10 of Ref. 1:

Carl McBride: /* See also */

2007-06-26T15:43:52Z

Carl McBride at 16:11, 30 May 2007

2007-05-30T16:11:09Z

← Older revision		Revision as of 18:11, 30 May 2007
Line 5:		Line 5:
	{\int ... \int e^{-\beta \Phi(r_1,...,r_N)}{\rm d}r_3...{\rm d}r_N}		{\int ... \int e^{-\beta \Phi(r_1,...,r_N)}{\rm d}r_3...{\rm d}r_N}
	{\int e^{-\beta \Phi(r_1,...,r_N){\rm d}r_1...{\rm d}r_N}}</math>		{\int e^{-\beta \Phi(r_1,...,r_N){\rm d}r_1...{\rm d}r_N}}</math>
			==Exact convolution equation for <math>g(r)</math>==
			See Eq. 5.10 of Ref. 1:

			:<math>\ln g(r_{12}) + \frac{\Phi(r_{12})}{kT} - E(r_{12}) = n \int \left(g(r_{13}) -1 - \ln g(r_{13}) - \frac{\Phi(r_{13})}{kT} - E(r_{13}) \right)(g(r_{23}) -1) ~{\rm d}r_3</math>


	==See also==		==See also==
Line 10:		Line 15:
	*[[Pressure equation]]		*[[Pressure equation]]
	*[[Energy equation]]		*[[Energy equation]]
			==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)]

			[[category: statistical mechanics]]

Carl McBride: New page: For a fluid of $N$ particles, enclosed in a volume $V$ at a given temperature $T$ (canonical ensemble) interacting via the `central' potential $\P...$

2007-05-22T13:12:19Z

New page: For a fluid of <math>N</math> particles, enclosed in a volume <math>V</math> at a given temperature <math>T</math> (canonical ensemble) interacting via the `central' potential <math>\P...

New page

For a fluid of <math>N</math> particles, enclosed in a volume <math>V</math> at a given temperature <math>T</math>
([[canonical ensemble]]) interacting via the `central' potential <math>\Phi(r)</math>, the two particle distribution function is defined as

:<math>{\rm g}_N^{(2)}(r_1,r_2)= V^2 \frac
{\int ... \int e^{-\beta \Phi(r_1,...,r_N)}{\rm d}r_3...{\rm d}r_N}
{\int e^{-\beta \Phi(r_1,...,r_N){\rm d}r_1...{\rm d}r_N}}</math>

==See also==
*[[Compressibility equation]]
*[[Pressure equation]]
*[[Energy equation]]

← Older revision		Revision as of 17:43, 26 June 2007
Line 12:		Line 12:

	==See also==		==See also==
			*[[Radial distribution function]]
	*[[Compressibility equation]]		*[[Compressibility equation]]
	*[[Pressure equation]]		*[[Pressure equation]]
	*[[Energy equation]]		*[[Energy equation]]

	==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]]		[[category: statistical mechanics]]