Carl McBride at 16:27, 30 May 2007

2007-05-30T16:27:52Z

← Older revision		Revision as of 18:27, 30 May 2007
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	leaving one with the task of finding the so called [[bridge function]] <math>B[h(r)]</math>		leaving one with the task of finding the so called [[bridge function]] <math>B[h(r)]</math>
	rather than the entire closure relation.		rather than the entire closure relation.
			==See also==
			*[[List of closures for integral equations]]
	==References==		==References==
	#[http://dx.doi.org/10.1143/PTP.23.1003 Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics '''23''' pp. 1003-1027 (1960)]		#[http://dx.doi.org/10.1143/PTP.23.1003 Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics '''23''' pp. 1003-1027 (1960)]
	[[category: integral equations]]		[[category: integral equations]]

Carl McBride: New page: When there are more unknowns than there are equations then one requires a '''closure relation'''. There are two basic types of closure schemes. ====Truncation Schemes==== In truncation sc...

2007-05-30T16:22:10Z

New page: When there are more unknowns than there are equations then one requires a '''closure relation'''. There are two basic types of closure schemes. ====Truncation Schemes==== In truncation sc...

New page

When there are more unknowns than there are equations then
one requires a '''closure relation'''.
There are two basic types of closure schemes.
====Truncation Schemes====
In truncation schemes, higher order moments are arbitrarily
assumed to vanish, or simply prescribed in terms of lower moments.
Truncation schemes can often provide quick insight into fluid systems,
but always involve uncontrolled approximation.
====Asymptotic schemes====
Asymptotic schemes depend on the
rigorous exploitation of some small parameter.
They have the advantage of being systematic, and
providing some estimate of the error involved in the closure.
On the other hand, the asymptotic approach to
closure is mathematically very demanding.
====General form====
The closure relation can be written in the general form

:<math>\left. c(r) \right. = f [ \gamma (r) ]</math>

Morita and Hiroike eased this task for a closure relation (see Ref. 1)
by providing the formally exact closure formula:

:<math>\left. g(r) \right. = \exp[-\beta \Phi(r) + h(r) -c(r) +B(r)]</math>

or written using the [[cavity correlation function]]

:<math>\ln y(r) \equiv h(r) -c(r) +B(r)</math>

leaving one with the task of finding the so called [[bridge function]] <math>B[h(r)]</math>
rather than the entire closure relation.
==References==
#[http://dx.doi.org/10.1143/PTP.23.1003 Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics '''23''' pp. 1003-1027 (1960)]
[[category: integral equations]]

Closure relations - Revision history

Carl McBride at 16:27, 30 May 2007

Carl McBride: New page: When there are more unknowns than there are equations then one requires a '''closure relation'''. There are two basic types of closure schemes. ====Truncation Schemes==== In truncation sc...