Pair distribution function: Difference between revisions
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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} | :<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]] | ||
Revision as of 16:09, 10 July 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 (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)}({\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}}
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 \beta = 1/(k_BT)} , 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
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}{\mathbf r}_3}
where, i.e. 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 r_{12} = |{\mathbf r}_2 - {\mathbf r}_1|} .