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' [[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})}{ | :<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>. | ||
Revision as of 18:28, 14 February 2008
For a fluid of particles, enclosed in a volume at a given temperature (canonical ensemble) interacting via the `central' intermolecular pair potential , the two particle distribution function is defined as
where , where is the Boltzmann constant.
Exact convolution equation for
See Eq. 5.10 of Ref. 1:
where, i.e. .
