Pair distribution function

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For a fluid of N particles, enclosed in a volume V at a given temperature T (canonical ensemble) interacting via the `central' intermolecular pair potential Φ(r), the two particle distribution function is defined as

{\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 β: = 1 / (kBT), where kB is the Boltzmann constant.

[edit] Exact convolution equation for g(r)

See Eq. 5.10 of Ref. 1:

\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

where, i.e. r_{12} = |{\mathbf r}_2 - {\mathbf r}_1|.

[edit] See also

[edit] References

  1. J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics 28 pp. 169-199 (1965)
  2. 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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