Editing Ornstein-Zernike relation
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Notation: | |||
Notation | |||
*<math>g(r)</math> is the [[Pair distribution function | pair distribution function]]. | *<math>g(r)</math> is the [[Pair distribution function | pair distribution function]]. | ||
*<math>\Phi(r)</math> is the [[Intermolecular pair potential | pair potential]] acting between pairs. | *<math>\Phi(r)</math> is the [[Intermolecular pair potential | pair potential]] acting between pairs. | ||
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*<math>\omega(r)</math> is the [[Thermal potential | thermal potential]]. | *<math>\omega(r)</math> is the [[Thermal potential | thermal potential]]. | ||
*<math>f(r)</math> is the [[Mayer f-function]]. | *<math>f(r)</math> is the [[Mayer f-function]]. | ||
The '''Ornstein-Zernike relation''' integral equation | |||
The '''Ornstein-Zernike relation''' (OZ) integral equation is | |||
:<math>h=h\left[c\right]</math> | :<math>h=h\left[c\right]</math> | ||
where <math>h[c]</math> denotes a functional of <math>c</math>. This relation is exact. | where <math>h[c]</math> denotes a functional of <math>c</math>. This relation is exact. | ||
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Because of this <math>h</math> must be determined self-consistently. | Because of this <math>h</math> must be determined self-consistently. | ||
This need for self-consistency is characteristic of all many-body problems. | This need for self-consistency is characteristic of all many-body problems. | ||
(Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the | (Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7) | ||
:<math>h(1,2) = c(1,2) + \int \rho^{(1)}(3) c(1,3)h(3,2) d3</math> | :<math>h(1,2) = c(1,2) + \int \rho^{(1)}(3) c(1,3)h(3,2) d3</math> | ||
If the system is both homogeneous and isotropic, the | If the system is both homogeneous and isotropic, the OZ relation becomes (Ref. 1Eq. 6) | ||
:<math>\gamma ({\mathbf r}) \equiv h({\mathbf r}) - c({\mathbf r}) = \rho \int h({\mathbf r'})~c(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}</math> | :<math>\gamma ({\mathbf r}) \equiv h({\mathbf r}) - c({\mathbf r}) = \rho \int h({\mathbf r'})~c(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}</math> | ||
In words, this equation (Hansen and McDonald, section 5.2 p. 107) | In words, this equation (Hansen and McDonald, section 5.2 p. 107) | ||
``...describes the fact that the ''total'' correlation between particles 1 and 2, represented by <math>h(1,2)</math>, | |||
is due in part to the ''direct'' correlation between 1 and 2, represented by <math>c(1,2)</math>, but also to the ''indirect'' correlation, | |||
<math>\gamma (r)</math>, propagated via increasingly large numbers of intermediate particles." | |||
Notice that this equation is basically a convolution, ''i.e.'' | Notice that this equation is basically a convolution, ''i.e.'' | ||
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:::::''etc.'' | :::::''etc.'' | ||
Diagrammatically this expression can be written as | Diagrammatically this expression can be written as (Ref. 2): | ||
:[[Image:oz_diag.png]] | :[[Image:oz_diag.png]] | ||
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where the bold lines connecting root points denote <math>c</math> functions, the blobs denote <math>h</math> functions. | where the bold lines connecting root points denote <math>c</math> functions, the blobs denote <math>h</math> functions. | ||
An arrow pointing from left to right indicates an uphill path from one root | An arrow pointing from left to right indicates an uphill path from one root | ||
point to another. An `uphill path' is a sequence of | point to another. An `uphill path' is a sequence of Mayer bonds passing through increasing | ||
particle labels. | particle labels. | ||
The | The OZ relation can be derived by performing a functional differentiation | ||
of the | of the grand canonical distribution function (HM check this). | ||
== | ==OZ equation in Fourier space== | ||
The Ornstein-Zernike equation may be written in [[Fourier analysis |Fourier space]] as ( | The Ornstein-Zernike equation may be written in [[Fourier analysis |Fourier space]] as (Eq. 5 in Ref. 3): | ||
:<math>\hat{\gamma} = (\mathbf | :<math>\hat{\gamma} = ({\mathbf I} - \rho {\mathbf \hat{c}})^{-1} {\mathbf \hat{c}} \rho {\mathbf \hat{c}}</math> | ||
The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly | The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly | ||
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:<math>\gamma (r) = \frac{1}{2 \pi^2 r} \int_0^\infty k~\sin (kr) \hat{\gamma}( | :<math>\gamma (r) = \frac{1}{2 \pi^2 r} \int_0^\infty k~\sin (kr) \hat{\gamma}(r) dk</math> | ||
Note: | Note: | ||
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==References== | ==References== | ||
#L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914) | |||
#[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816-824 (1992)] | |||
#[http://www.essaywriters.net/ freelance writer] | |||
'''Related reading''' | '''Related reading''' | ||
*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5 | *Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5 | ||
[[Category: Integral equations]] | [[Category: Integral equations]] |