Editing Radial distribution function

==Density Expansion of the radial distribution function==
The  radial distribution function of a compressed gas may be expanded in powers of the density (Ref. 1)

:<math>\left. {\rm g}(r) \right. = e^{-\beta \Phi(r)} (1 + \rho {\rm g}_1 (r) + \rho^2 {\rm g}_2 (r) + ...)</math>

where <math>\rho</math> is the number of molecules per unit volume. The 
function <math>{\rm g}(r)</math> is normalized to the value 1 for large distances.
As is known, <math>{\rm g}_1 (r)</math>, <math>{\rm g}_2 (r)</math>, ... can be expressed by 
[[cluster integrals]] in which the position of of two particles is kept fixed.
In classical mechanics, and on the assumption of additivity of intermolecular forces, one has

:<math>{\rm g}_1 (r_{12})= \int f (r_{13}) f(r_{23}) ~{\rm d}r_3</math>


:<math>{\rm g}_2 (r_{12})= \frac{1}{2}({\rm g}_1 (r_{12}))^2 + \varphi (r_{12})
+ 2\psi (r_{12}) + \frac{1}{2} \chi (r_{12})</math>

where <math>r_{ik}</math> is the distance <math>|r_i -r_k|</math>, where <math>f(r)</math>
is the [[Mayer f-function]]

:<math>\left. f(r) \right. = e^{-\beta U(r)} -1</math>

and

:<math>\varphi (r_{12}) = \int  f (r_{13})  f (r_{24})  f (r_{34}) ~ {\rm d}r_3 {\rm d}r_4</math>

:<math>\psi (r_{12})  = \int  f (r_{13}) f (r_{23})  f (r_{24}) f (r_{34}) ~ {\rm d}r_3 {\rm d}r_4</math>

:<math>\chi (r_{12})  = \int  f (r_{13}) f (r_{23}) f (r_{14}) f (r_{24}) f (r_{34}) ~ {\rm d}r_3 {\rm d}r_4</math>
==References==
#[http://dx.doi.org/10.1103/PhysRev.85.777  B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]

[[category: statistical mechanics]]