Verlet modified - Revision history

Carl McBride: Tidy up a bit.

2015-09-10T15:17:31Z

Tidy up a bit.

← Older revision		Revision as of 17:17, 10 September 2015
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	The '''Verlet modified''' (1980) ~~(Ref. 1)~~ [[Closure relations \| closure relation]] 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>~~\ln 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
			:<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:		by minimising the difference between the pressures obtained via the [[Pressure equation \| virial]] and [[Compressibility equation \| compressibility]] routes:

	:<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>		:<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)

	~~with <math>A= 0.80</math>,~~ <math>\~~lambda~~= ~~0.03496</math> and <math>~~\~~mu = 0.6586~~</math>.		:<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]]

158.49.50.230 at 17:00, 1 September 2015

2015-09-01T17:00:42Z

← Older revision		Revision as of 19:00, 1 September 2015
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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>.

Carl McBride at 17:14, 26 June 2007

2007-06-26T17:14:32Z

← Older revision		Revision as of 19:14, 26 June 2007
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	The '''Verlet modified''' (1980) (Ref. 1) closure for [[hard sphere model \| hard sphere]] fluids,		The '''Verlet modified''' (1980) (Ref. 1) [[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)

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	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).
	Later (Ref. 2) Verlet used a Padé (2/1) approximant (Eq. 6) fitted to obtain the best [[hard sphere model \| hard sphere]] results		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:		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>y(r) = \gamma (r) - A \frac{1}{2} \gamma^2(r) \left[ \frac{1+ \lambda \gamma(r)}{1+ \mu \gamma(r)} \right]</math>

Carl McBride at 13:29, 21 March 2007

2007-03-21T13:29:06Z

← Older revision		Revision as of 15:29, 21 March 2007
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	The '''Verlet modified''' (1980) (Ref. 1) closure for [[~~Hard~~ sphere \| hard sphere]] fluids,		The '''Verlet modified''' (1980) (Ref. 1) closure 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)

Carl McBride at 13:28, 21 March 2007

2007-03-21T13:28:37Z

← Older revision		Revision as of 15:28, 21 March 2007
Line 5:		Line 5:

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

Carl McBride at 13:27, 21 March 2007

2007-03-21T13:27:58Z

← Older revision		Revision as of 15:27, 21 March 2007
Line 5:		Line 5:

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

Nice and Tidy at 16:30, 27 February 2007

2007-02-27T16:30:11Z

← Older revision		Revision as of 18:30, 27 February 2007
Line 5:		Line 5:

	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).
	Later (Ref. 2) ~~(1981)~~ Verlet used a Padé (2/1) approximant (Eq. 6) fitted to obtain the best [[Hard sphere \| hard sphere]] results		Later (Ref. 2) Verlet used a Padé (2/1) approximant (Eq. 6) fitted to obtain the best [[Hard sphere \| hard sphere]] results
	by minimising the difference between the pressures obtained via the virial and compressibility routes:		by minimising the difference between the pressures obtained via the virial and compressibility routes:

Nice and Tidy at 11:11, 27 February 2007

2007-02-27T11:11:50Z

← Older revision		Revision as of 13:11, 27 February 2007
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	#[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)]		#[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)]
	#[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)]		#[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]]

Carl McBride at 15:02, 26 February 2007

2007-02-26T15:02:19Z

← Older revision		Revision as of 17:02, 26 February 2007
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	:<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) - 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]] is recovered).		where several sets of values are tried for ''A'' and ''B'' (Note, when ''A=0'' the [[HNC\| hyper-netted chain]] is recovered).
	Later (Ref. 2) (1981) Verlet used a Padé (2/1) approximant (Eq. 6) fitted to obtain the best [[Hard sphere \| hard sphere]] results		Later (Ref. 2) (1981) Verlet used a Padé (2/1) approximant (Eq. 6) fitted to obtain the best [[Hard sphere \| hard sphere]] results
	by minimising the difference between the pressures obtained via the virial and compressibility routes:		by minimising the difference between the pressures obtained via the virial and compressibility routes:

Carl McBride: /* References */

2007-02-26T14:28:35Z

References

← Older revision		Revision as of 16:28, 26 February 2007
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	==References==		==References==
	#[~~MP_1980_41_0183~~]		#[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)]
	#[~~MP_1981_42_1291~~]		#[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)]