Verlet modified: Difference between revisions
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in terms of the [[cavity correlation function]], is (Eq. 3) | in terms of the [[cavity correlation function]], is (Eq. 3) | ||
:<math>y(r) = \gamma (r) - A \frac{1}{2} \gamma^2(r) \left[ \frac{1}{1+ B \gamma(r) /2} \right]</math> | :<math>\ln y(r) = \gamma (r) - A \frac{1}{2} \gamma^2(r) \left[ \frac{1}{1+ B \gamma(r) /2} \right]</math> | ||
where several sets of values are tried for ''A'' and ''B'' (Note, when ''A=0'' the [[HNC| hyper-netted chain]] is recovered). | where several sets of values are tried for ''A'' and ''B'' (Note, when ''A=0'' the [[HNC| hyper-netted chain]] is recovered). | ||
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by minimising the difference between the pressures obtained via the [[Pressure equation | virial]] and [[Compressibility equation | compressibility]] routes: | by minimising the difference between the pressures obtained via the [[Pressure equation | virial]] and [[Compressibility equation | compressibility]] routes: | ||
:<math>y(r) = \gamma (r) - A \frac{1}{2} \gamma^2(r) \left[ \frac{1+ \lambda \gamma(r)}{1+ \mu \gamma(r)} \right]</math> | :<math>\ln y(r) = \gamma (r) - A \frac{1}{2} \gamma^2(r) \left[ \frac{1+ \lambda \gamma(r)}{1+ \mu \gamma(r)} \right]</math> | ||
with <math>A= 0.80</math>, <math>\lambda= 0.03496</math> and <math>\mu = 0.6586</math>. | with <math>A= 0.80</math>, <math>\lambda= 0.03496</math> and <math>\mu = 0.6586</math>. | ||
Revision as of 18:00, 1 September 2015
The Verlet modified (1980) (Ref. 1) closure relation for hard sphere fluids, in terms of the cavity correlation function, is (Eq. 3)
![{\displaystyle \ln y(r)=\gamma (r)-A{\frac {1}{2}}\gamma ^{2}(r)\left[{\frac {1}{1+B\gamma (r)/2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6de2d826214ac4ce2cd372a3e1c3098be1bde7)
where several sets of values are tried for A and B (Note, when A=0 the hyper-netted chain is recovered). Later (Ref. 2) Verlet used a Padé (2/1) approximant (Eq. 6) fitted to obtain the best hard sphere results by minimising the difference between the pressures obtained via the virial and compressibility routes:
![{\displaystyle \ln y(r)=\gamma (r)-A{\frac {1}{2}}\gamma ^{2}(r)\left[{\frac {1+\lambda \gamma (r)}{1+\mu \gamma (r)}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a14b3c421a09553765c9edaf366b3833026cd078)
with , and .
