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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' potential <math>\Phi(r)</math>, the two particle distribution function is defined as |
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| :<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)}(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> |
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| where <math>\beta := 1/(k_BT)</math>, where <math>k_B</math> is the [[Boltzmann constant]].
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| ==Exact convolution equation for <math>{\mathrm g}(r)</math>==
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| See Eq. 5.10 of Ref. 1:
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| :<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>
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| where, ''i.e.'' <math>r_{12} = |{\mathbf r}_2 - {\mathbf r}_1|</math>.
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| ==See also== | | ==See also== |
| *[[Radial distribution function]]
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| *[[Compressibility equation]] | | *[[Compressibility equation]] |
| *[[Pressure equation]] | | *[[Pressure equation]] |
| *[[Energy equation]] | | *[[Energy equation]] |
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| ==References==
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| #[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)]
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| #[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)]
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| [[category: statistical mechanics]]
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