Verlet modified: Difference between revisions
Carl McBride (talk | contribs) m (New page: The '''Verlet modified''' (1980) (Ref. 1) closure for hard sphere fluids, in terms of the cavity correlation function, is (Eq. 3) :<math>y(r) = \gamma (r) - A \frac{...) |
Carl McBride (talk | contribs) (Tidy up a bit.) |
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The '''Verlet modified''' (1980) | 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> | :<math> Y(r) = \gamma (r) - \left[ \frac{A \gamma^2(r)/2}{1+ B \gamma(r) /2} \right]</math> | ||
where | where the [[radial distribution function]] is expressed as (Eq. 1) | ||
:<math> | :<math>{\mathrm g}(r) = e^{-\beta \Phi(r)} + Y(r)</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== | ||
<references/> | |||
[[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)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y(r) = \gamma (r) - \left[ \frac{A \gamma^2(r)/2}{1+ B \gamma(r) /2} \right]}
where the radial distribution function is expressed as (Eq. 1)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathrm g}(r) = e^{-\beta \Phi(r)} + Y(r)}
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:
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Y(r)=\gamma (r)-{\frac {A}{2}}\gamma ^{2}(r)\left[{\frac {1+\lambda \gamma (r)}{1+\mu \gamma (r)}}\right]}
with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= 0.80} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda= 0.03496} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = 0.6586} where the radial distribution function for hard spheres is written as (Eq. 1)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathrm g}(r) = \exp[Y(r)] ~~~~ \mathrm{for} ~~~~ r \ge d}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is the hard sphere diameter.