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

${\mathrm{g}}_N^{(2)}(r_1,r_2)=V^2\frac{\int ...\int e^{-\beta \mathrm{\Phi }(r_1,...,r_N)}\mathrm{d}{r}_3...\mathrm{d}{r}_N}{\int e^{-\beta \mathrm{\Phi }(r_1,...,r_N)\mathrm{d}{r}_1...\mathrm{d}{r}_N}}$

Exact convolution equation for $g(r)$

See Eq. 5.10 of Ref. 1:

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

See also

• 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)