Exact solution of the Percus Yevick integral equation for hard spheres: Difference between revisions

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(Added section concerning the Carnahan-Starling equation of state)
m (I replaced x by \eta in all the eqs. refered to Thiele's work. This is correct because the "dimensionless number density" (eq.6 of Thiele) divided by 4 is the packing fraction. I also modify the cross ref to Wertheim1 ref. to solve a problem with it.)
 
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was derived by M. S. Wertheim in 1963 <ref name="wertheim1" >[http://dx.doi.org/10.1103/PhysRevLett.10.321  M. S. Wertheim "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres", Physical Review Letters '''10''' 321 - 323 (1963)]</ref> (see also <ref>[http://dx.doi.org/10.1063/1.1704158  M. S. Wertheim "Analytic Solution of the Percus-Yevick Equation", Journal of Mathematical Physics, '''5''' pp. 643-651 (1964)]</ref>), and for [[mixtures]] by Joel Lebowitz in 1964 <ref>[http://dx.doi.org/10.1103/PhysRev.133.A895  J. L. Lebowitz, "Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres", Physical Review '''133''' pp. A895 - A899 (1964)]</ref>.
was derived by M. S. Wertheim in 1963 <ref name="wertheim1" >[http://dx.doi.org/10.1103/PhysRevLett.10.321  M. S. Wertheim "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres", Physical Review Letters '''10''' 321 - 323 (1963)]</ref> (see also <ref>[http://dx.doi.org/10.1063/1.1704158  M. S. Wertheim "Analytic Solution of the Percus-Yevick Equation", Journal of Mathematical Physics, '''5''' pp. 643-651 (1964)]</ref>), and for [[mixtures]] by Joel Lebowitz in 1964 <ref>[http://dx.doi.org/10.1103/PhysRev.133.A895  J. L. Lebowitz, "Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres", Physical Review '''133''' pp. A895 - A899 (1964)]</ref>.


The [[direct correlation function]] is given by (Eq. 6 of <ref name="wertheim1" > </ref> )
The [[direct correlation function]] is given by (Eq. 6 of <ref name="wertheim1" /> )


:<math>C(r/R) = - \frac{(1+2\eta)^2 - 6\eta(1+ \frac{1}{2} \eta)^2(r/R) + \eta(1+2\eta)^2\frac{(r/R)^3}{2}}{(1-\eta)^4}</math>
:<math>C(r/R) = - \frac{(1+2\eta)^2 - 6\eta(1+ \frac{1}{2} \eta)^2(r/R) + \eta(1+2\eta)^2\frac{(r/R)^3}{2}}{(1-\eta)^4}</math>
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and <math>R</math> is the hard sphere diameter.
and <math>R</math> is the hard sphere diameter.
The [[Equations of state | equation of state]] is given by (Eq. 7 of <ref name="wertheim1" > </ref>)
The [[Equations of state | equation of state]] is given by (Eq. 7 of <ref name="wertheim1" />)


:<math>\frac{\beta P}{\rho} = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math>  
:<math>\frac{\beta P}{\rho} = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math>  
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where (Eq. 24)
where (Eq. 24)


:<math>a = \frac{(2x+1)^2}{(x-1)^4}</math>
:<math>a = \frac{(2\eta+1)^2}{(\eta-1)^4}</math>


and
and


:<math>b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}</math>
:<math>b= - \frac{12\eta + 12\eta^2 + 3\eta^3}{2(\eta-1)^4}</math>


and
and


:<math>c= \frac{x(2x+1)^2}{2(x-1)^4}</math>
:<math>c= \frac{\eta(2\eta+1)^2}{2(\eta-1)^4}</math>
 
and where <math>x=\rho/4</math>.


The [[pressure]] via the pressure route (Eq.s 32 and 33) is
The [[pressure]] via the pressure route (Eq.s 32 and 33) is


:<math>P=nk_BT\frac{(1+2x+3x^2)}{(1-x)^2}</math>
:<math>P=nk_BT\frac{(1+2\eta+3\eta^2)}{(1-\eta)^2}</math>


and the [[Compressibility equation |compressibility]] route is
and the [[Compressibility equation |compressibility]] route is


:<math>P=nk_BT\frac{(1+x+x^2)}{(1-x)^3}</math>
:<math>P=nk_BT\frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math>


==A derivation of the Carnahan-Starling equation of state==
==A derivation of the Carnahan-Starling equation of state==

Latest revision as of 16:25, 22 January 2018

The exact solution for the Percus Yevick integral equation for the hard sphere model was derived by M. S. Wertheim in 1963 [1] (see also [2]), and for mixtures by Joel Lebowitz in 1964 [3].

The direct correlation function is given by (Eq. 6 of [1] )

where

and is the hard sphere diameter. The equation of state is given by (Eq. 7 of [1])

where is the inverse temperature. Everett Thiele also studied this system [4], resulting in (Eq. 23)

where (Eq. 24)

and

and

The pressure via the pressure route (Eq.s 32 and 33) is

and the compressibility route is

A derivation of the Carnahan-Starling equation of state[edit]

It is interesting to note (Ref [5] Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the exact solution via the compressibility route, to one third via the pressure route, i.e.

The reason for this seems to be a slight mystery (see discussion in Ref. [6] ).

References[edit]