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. 2)

:<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 and <math>\Phi(r)</math>
is the [[intermolecular pair potential]]. 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 diagrams | 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 \Phi(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/10.1063/1.1723737     John G. Kirkwood and Elizabeth Monroe Boggs "The Radial Distribution Function in Liquids", Journal of Chemical Physics '''10''' pp. 394-402 (1942)]
#[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]]
@@ Line 1: / Line 1: @@
-The '''radial distribution function''' is a special case of the  [[pair distribution function]] for an isotropic system.
-A [[Fourier analysis | Fourier transform]] of the radial distribution function results in the [[structure factor]], which is experimentally measurable. The following plot is of a typical radial distribution function for the monatomic [[Lennard-Jones model |Lennard-Jones]] liquid.
-[[Image:LJ_rdf.png|center|450px|Typical radial distribution function for the monatomic Lennard-Jones liquid.]]
 ==Density Expansion of the radial distribution function==
-The  '''radial distribution function''' of a compressed gas may be expanded in powers of the density <ref>[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)]</ref>
+The  '''radial distribution function''' of a compressed gas may be expanded in powers of the density (Ref. 2)
 :<math>\left. {\rm g}(r) \right. = e^{-\beta \Phi(r)} (1 + \rho {\rm g}_1 (r) + \rho^2 {\rm g}_2 (r) + ...)</math>
@@ Line 14: / Line 11: @@
 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}{\mathbf r}_3</math>
+:<math>{\rm g}_1 (r_{12})= \int f (r_{13}) f(r_{23}) ~{\rm d}r_3</math>
@@ Line 20: / Line 17: @@
 + 2\psi (r_{12}) + \frac{1}{2} \chi (r_{12})</math>
-where <math>r_{ik}</math> is the distance <math>|{\mathbf r}_i -{\mathbf r}_k|</math>, where <math>f(r)</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]]
@@ Line 27: / Line 24: @@
 and
-:<math>\varphi (r_{12}) = \int  f (r_{13})  f (r_{24})  f (r_{34}) ~ {\rm d}{\mathbf r}_3 {\rm d}{\mathbf r}_4</math>
+:<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}{\mathbf r}_3 {\rm d}{\mathbf 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}{\mathbf r}_3 {\rm d}{\mathbf 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>
-==See also==
-*[[Pair distribution function]]
-*[[Total correlation function]]
 ==References==
-<references/>
+#[http://dx.doi.org/10.1063/1.1723737     John G. Kirkwood and Elizabeth Monroe Boggs "The Radial Distribution Function in Liquids", Journal of Chemical Physics '''10''' pp. 394-402 (1942)]
-;Related reading
+#[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)]
-*[http://dx.doi.org/10.1063/1.1723737     John G. Kirkwood and Elizabeth Monroe Boggs "The Radial Distribution Function in Liquids", Journal of Chemical Physics '''10''' pp. 394-402 (1942)]
-*[http://dx.doi.org/10.1063/1.1703948 J. L. Lebowitz and J. K. Percus "Asymptotic Behavior of the Radial Distribution Function", Journal of Mathematical Physics '''4''' pp. 248-254 (1963)]
-*[http://dx.doi.org/10.1063/1.1725652  B. Widom "On the Radial Distribution Function in Fluids", Journal of Chemical Physics '''41''' pp. 74-77 (1964)]
-*[http://dx.doi.org/10.1143/JPSJ.12.326 Kazuo Hiroike "Radial Distribution Function of Fluids I", Journal of the Physical Society of Japan '''12''' pp. 326-334 (1957)]
-*[http://dx.doi.org/10.1143/JPSJ.12.864 Kazuo Hiroike "Radial Distribution Function of Fluids II", Journal of the Physical Society of Japan '''12''' pp. pp. 864-873 (1957)]
-*[http://dx.doi.org/10.1080/00268970701678907 J. G. Malherbe and W. Krauth "Selective-pivot sampling of radial distribution functions in asymmetric liquid mixtures", Molecular Physics '''105''' pp. 2393-2398 (2007)]
-*[http://doi.org/10.1063/1.4973804 Sergey V. Sukhomlinov, Martin H. Müser "Determination of accurate, mean bond lengths from radial distribution functions", Journal of Chemical Physics '''146''' 024506 (2017)]
 [[category: statistical mechanics]]