Ornstein-Zernike relation

2012-05-14T11:49:06Z

155.185.147.143: /* Ornstein-Zernike relation in Fourier space */ latex fix

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Notation used:
*<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>h(1,2)</math> is the [[Total correlation function | total correlation function]].
*<math>c(1,2)</math> is the [[Direct correlation function | direct correlation function]].
*<math>\gamma (r)</math> is the [[Indirect correlation function | indirect]] (or ''series'' or ''chain'') correlation function.
*<math>y(r_{12})</math> is the [[Cavity correlation function | cavity correlation function]].
*<math>B(r)</math> is the [[ bridge function]].
*<math>\omega(r)</math> is the [[Thermal potential | thermal potential]].
*<math>f(r)</math> is the [[Mayer f-function]].
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The '''Ornstein-Zernike relation''' integral equation <ref>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)</ref> is given by:
:<math>h=h\left[c\right]</math>
where <math>h[c]</math> denotes a functional of <math>c</math>. This relation is exact.
This is complemented by the closure relation
:<math>c=c\left[h\right]</math>
Note that <math>h</math> depends on <math>c</math>, and <math>c</math> depends on <math>h</math>.
Because of this <math>h</math> must be determined self-consistently.
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 Ornstein-Zernike relation 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>
If the system is both homogeneous and isotropic, the Ornstein-Zernike relation becomes (Eq. 6 of Ref. 1)

:<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)
:"...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.''
:<math>h \equiv c + \rho h\otimes c </math>

(Note: the convolution operation written here as <math>\otimes</math> is more frequently written as <math>*</math>)
This can be seen by expanding the integral in terms of <math>h(r)</math>
(here truncated at the fourth iteration):

:<math>h({\mathbf r}) = c({\mathbf r}) + \rho \int c(|{\mathbf r} - {\mathbf r'}|) c({\mathbf r'}) {\rm d}{\mathbf r'}</math>

:::::<math>+ \rho^2 \iint c(|{\mathbf r} - {\mathbf r'}|) c(|{\mathbf r'} - {\mathbf r''}|) c({\mathbf r''}) {\rm d}{\mathbf r''}{\rm d}{\mathbf r'}</math>

:::::<math>+ \rho^3 \iiint c(|{\mathbf r} - {\mathbf r'}|) c(|{\mathbf r'} - {\mathbf r''}|) c(|{\mathbf r''} - {\mathbf r'''}|) c({\mathbf r'''}) {\rm d}{\mathbf r'''}{\rm d}{\mathbf r''}{\rm d}{\mathbf r'}</math>

:::::<math>+ \rho^4 \iiiint c(|{\mathbf r} - {\mathbf r'}|) c(|{\mathbf r'} - {\mathbf r''}|) c(|{\mathbf r''} - {\mathbf r'''}|) c(|{\mathbf r'''} - {\mathbf r''''}|) h({\mathbf r''''}) {\rm d}{\mathbf r''''} {\rm d}{\mathbf r'''}{\rm d}{\mathbf r''}{\rm d}{\mathbf r'}</math>

:::::''etc.''

Diagrammatically this expression can be written as <ref>[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)]</ref>:

:[[Image:oz_diag.png]]

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
point to another. An `uphill path' is a sequence of [[Mayer f-function |Mayer bonds]] passing through increasing
particle labels.
The Ornstein-Zernike relation can be derived by performing a functional differentiation
of the [[Grand canonical ensemble |grand canonical]] distribution function.
==Ornstein-Zernike relation in Fourier space==
The Ornstein-Zernike equation may be written in [[Fourier analysis |Fourier space]] as (<ref>[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]</ref> Eq. 5):

:<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
to:

:<math>\hat{\gamma} (k) = \frac{4 \pi}{k} \int_0^\infty r~\sin (kr) \gamma(r) dr</math>

:<math>\gamma (r) = \frac{1}{2 \pi^2 r} \int_0^\infty k~\sin (kr) \hat{\gamma}(r) dk</math>

Note:

:<math>\hat{h}(0) = \int h(r) {\rm d}{\mathbf r}</math>

:<math>\hat{c}(0) = \int c(r) {\rm d}{\mathbf r}</math>

==References==
<references/>

'''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

[[Category: Integral equations]]

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Ornstein-Zernike relation