Editing Radial distribution function
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==Density Expansion of the radial distribution function== | ==Density Expansion of the radial distribution function== | ||
The | 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> | :<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 | 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. | 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 | 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 | 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} | :<math>{\rm g}_1 (r_{12})= \int f (r_{13}) f(r_{23}) ~{\rm d}r_3</math> | ||
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+ 2\psi (r_{12}) + \frac{1}{2} \chi (r_{12})</math> | + 2\psi (r_{12}) + \frac{1}{2} \chi (r_{12})</math> | ||
where <math>r_{ik}</math> is the distance <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]] | is the [[Mayer f-function]] | ||
:<math>\left. f(r) \right. = e^{-\beta | :<math>\left. f(r) \right. = e^{-\beta U(r)} -1</math> | ||
and | and | ||
:<math>\varphi (r_{12}) = \int f (r_{13}) f (r_{24}) f (r_{34}) ~ {\rm d} | :<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} | :<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} | :<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== | ==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]] | [[category: statistical mechanics]] |