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

From SklogWiki
Jump to: navigation, search
m (Sligh ttidy + Cite format of references)
Line 1: Line 1:
The exact solution for the [[Percus Yevick]] integral equation for the [[hard sphere model]]
+
The exact solution for the [[Percus Yevick]] [[Integral equations |integral equation]] for the [[hard sphere model]]
was derived by M. S. Wertheim in 1963 Ref. 1 (See also Ref. 2)
+
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>.
(and for mixtures by in Lebowitz 1964 Ref. 3).
+
 
The [[direct correlation function]] is given by (Ref. 1 Eq. 6)
+
The [[direct correlation function]] is given by (Eq. 6 of <ref name="wertheim1" > </ref> )
  
 
:<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>
Line 11: Line 11:
  
 
and <math>R</math> is the hard sphere diameter.
 
and <math>R</math> is the hard sphere diameter.
The equation of state is (Ref. 1 Eq. 7)
+
The [[Equations of state | equation of state]] is given by (Eq. 7 of <ref name="wertheim1" > </ref>)
  
 
:<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>  
  
Everett Thiele (1963  Ref. 4}) also studied this system,
+
where <math>\beta</math> is the [[inverse temperature]]. Everett Thiele also studied this system <ref>[http://dx.doi.org/10.1063/1.1734272  Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics, '''39'''  pp. 474-479 (1963)]</ref>,
 
resulting in (Eq. 23)
 
resulting in (Eq. 23)
  
Line 36: Line 36:
 
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=nkT\frac{(1+2x+3x^2)}{(1-x)^2}</math>
+
:<math>P=nk_BT\frac{(1+2x+3x^2)}{(1-x)^2}</math>
  
and the compressibility route is
+
and the [[Compressibility equation |compressibility]] route is
  
:<math>P=nkT\frac{(1+x+x^2)}{(1-x)^3}</math>
+
:<math>P=nk_BT\frac{(1+x+x^2)}{(1-x)^3}</math>
  
 
==References==
 
==References==
#[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)]
+
<references/>
#[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)]
+
 
#[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)]
 
#[http://dx.doi.org/10.1063/1.1734272  Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics, '''39'''  pp. 474-479 (1963)]
 
  
 
[[Category: Integral equations]]
 
[[Category: Integral equations]]

Revision as of 11:56, 5 April 2011

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] )

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}

where

\eta = \frac{1}{6} \pi R^3 \rho

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

\frac{\beta P}{\rho} = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}

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

\left.h_0(r)\right. = ar+ br^2 + cr^4

where (Eq. 24)

a = \frac{(2x+1)^2}{(x-1)^4}

and

b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}

and

c= \frac{x(2x+1)^2}{2(x-1)^4}

and where x=\rho/4.

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

P=nk_BT\frac{(1+2x+3x^2)}{(1-x)^2}

and the compressibility route is

P=nk_BT\frac{(1+x+x^2)}{(1-x)^3}

References