Verlet modified: Difference between revisions

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The '''Verlet modified''' (1980) (Ref. 1) closure for [[hard sphere model | hard sphere]] fluids,
The '''Verlet modified''' <ref>[http://dx.doi.org/10.1080/00268978000102671 Loup Verlet "Integral equations for classical fluids I. The hard sphere case", Molecular Physics '''41''' pp. 183-190 (1980)]</ref> [[Closure relations | closure relation]] for [[hard sphere model | hard sphere]] fluids,
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> Y(r) = \gamma (r) -   \left[ \frac{A \gamma^2(r)/2}{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 the [[radial distribution function]] is expressed as (Eq. 1)
Later (Ref. 2)  Verlet used a Padé (2/1) approximant (Eq. 6) fitted to obtain the best [[hard sphere model | hard sphere]] results
by minimising the difference between the pressures obtained via the virial and 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>{\mathrm g}(r)  = e^{-\beta \Phi(r)} + Y(r)</math>


with <math>A= 0.80</math>, <math>\lambda=  0.03496</math> and <math>\mu = 0.6586</math>.
and where several sets of values are tried for ''A'' and ''B''  (Note, when ''A=0'' the [[HNC| hyper-netted chain]] is recovered).


Later  Verlet used a Padé (2/1) approximant (<ref>[http://dx.doi.org/10.1080/00268978100100971 Loup Verlet "Integral equations for classical fluids II. Hard spheres again", Molecular Physics '''42''' pp. 1291-1302 (1981)]</ref> Eq. 6) fitted to obtain the best [[hard sphere model | hard sphere]] results
by minimising the difference between the pressures obtained via the [[Pressure equation | virial]] and [[Compressibility equation | compressibility]] routes:
:<math> Y(r) = \gamma (r) - \frac{A}{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>
where the radial distribution function for hard spheres is written as (Eq. 1)
:<math>{\mathrm g}(r)  =  \exp[Y(r)] ~~~~ \mathrm{for} ~~~~ r \ge d</math>
where <math>d</math> is the hard sphere diameter.
==References==
==References==
#[http://dx.doi.org/10.1080/00268978000102671 Loup Verlet "Integral equations for classical fluids I. The hard sphere case", Molecular Physics '''41''' pp. 183-190 (1980)]
<references/>
#[http://dx.doi.org/10.1080/00268978100100971 Loup Verlet "Integral equations for classical fluids II. Hard spheres again", Molecular Physics '''42''' pp. 1291-1302 (1981)]
 
 
[[Category: Integral equations]]
[[Category: Integral equations]]

Latest revision as of 16:17, 10 September 2015

The Verlet modified [1] closure relation for hard sphere fluids, in terms of the cavity correlation function, is (Eq. 3)

where the radial distribution function is expressed as (Eq. 1)

and where several sets of values are tried for A and B (Note, when A=0 the hyper-netted chain is recovered).

Later Verlet used a Padé (2/1) approximant ([2] Eq. 6) fitted to obtain the best hard sphere results by minimising the difference between the pressures obtained via the virial and compressibility routes:

with , and where the radial distribution function for hard spheres is written as (Eq. 1)

where is the hard sphere diameter.

References[edit]