Pair distribution function

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 $\mathrm{\Phi }(r)$, the two particle distribution function is defined as

${\mathrm{g}}_N^{(2)}({\mathbf{r}}_1,{\mathbf{r}}_2)=V^2\frac{\int ...\int e^{-\beta \mathrm{\Phi }({\mathbf{r}}_1,...,{\mathbf{r}}_N)}\mathrm{d}{\mathbf{r}}_3...\mathrm{d}{\mathbf{r}}_N}{\int e^{-\beta \mathrm{\Phi }({\mathbf{r}}_1,...,{\mathbf{r}}_N)}\mathrm{d}{\mathbf{r}}_1...\mathrm{d}{\mathbf{r}}_N}$

where $\beta :=1/(k_{B}T)$, where $k_B$ is the Boltzmann constant.

Exact convolution equation for $\mathrm{g}(r)$

See Eq. 5.10 of Ref. 1:

$\ln \mathrm{g}(r_{12})+\frac{\mathrm{\Phi }(r_{12})}{k_{B}T}-E(r_{12})=n\int (\mathrm{g}(r_{13})-1-\ln \mathrm{g}(r_{13})-\frac{\mathrm{\Phi }(r_{13})}{k_{B}T}-E(r_{13}))(\mathrm{g}(r_{23})-1)\mathrm{d}{\mathbf{r}}_3$

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

See also

• Radial distribution function
• Compressibility equation
• Pressure equation
• Energy equation

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)