Exact solution of the Percus Yevick integral equation for hard spheres: Difference between revisions
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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 | 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 ( | 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> | ||
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and <math>R</math> is the hard sphere diameter. | and <math>R</math> is the hard sphere diameter. | ||
The equation of state is ( | 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 | 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) | ||
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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= | :<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= | :<math>P=nk_BT\frac{(1+x+x^2)}{(1-x)^3}</math> | ||
==References== | ==References== | ||
<references/> | |||
[[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] )
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
and where .
The pressure via the pressure route (Eq.s 32 and 33) is
and the compressibility route is
References
- ↑ 1.0 1.1 1.2 M. S. Wertheim "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres", Physical Review Letters 10 321 - 323 (1963)
- ↑ M. S. Wertheim "Analytic Solution of the Percus-Yevick Equation", Journal of Mathematical Physics, 5 pp. 643-651 (1964)
- ↑ J. L. Lebowitz, "Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres", Physical Review 133 pp. A895 - A899 (1964)
- ↑ Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics, 39 pp. 474-479 (1963)