Pair distribution function: Difference between revisions
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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: | ||
Revision as of 16:46, 26 June 2007
For a fluid of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} particles, enclosed in a volume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} at a given temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} (canonical ensemble) interacting via the `central' intermolecular pair potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(r)} , the two particle distribution function is defined as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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}}}
where , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant.
Exact convolution equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(r)}
See Eq. 5.10 of Ref. 1:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}