Ornstein-Zernike relation: Difference between revisions

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Notation:
<div style="border:1px solid #f3f3ff; padding-left: 0.5em !important; background-color: #f3f3ff; border-width: 0 0 0 1.4em; clear:right; float:right;">
*<math>g(r)</math> is the [[pair distribution function]].
Notation used:
*<math>\Phi(r)</math> is the [[pair potential]] acting between pairs.
*<math>g(r)</math> is the [[Pair distribution function | pair distribution function]].
*<math>h(1,2)</math> is the [[total correlation function]] <math>h(1,2) \equiv  g(r) -1</math>.
*<math>\Phi(r)</math> is the [[Intermolecular pair potential  | pair potential]] acting between pairs.
*<math>c(1,2)</math> is the [[direct correlation function]].
*<math>h(1,2)</math> is the [[Total correlation function | total correlation function]].
*<math>\gamma (r)</math> is the [[indirect correlation function | indirect]] (or ''series'' or  ''chain'') correlation function <math>\gamma (r) \equiv  h(r) - c(r)</math>.
*<math>c(1,2)</math> is the [[Direct correlation function | direct correlation function]].
*<math>y(r_{12})</math> is the [[cavity correlation function]]<math>y(r)  \equiv g(r) /e^{-\beta \Phi(r)}</math>
*<math>\gamma (r)</math> is the [[Indirect correlation function | indirect]] (or ''series'' or  ''chain'') correlation function.
*<math>B(r)</math> is the [[Closures | bridge]] function.
*<math>y(r_{12})</math> is the [[Cavity correlation function | cavity correlation function]].
*<math>\omega(r)</math> is the [[thermal potential]], <math>\omega(r) \equiv \gamma(r) + B(r)</math>.
*<math>B(r)</math> is the [[ bridge function]].
*<math>f(r)</math> is the [[Mayer <math>f</math>-function]], defined as <math>f(r) \equiv  e^{-\beta \Phi(r)} -1</math>.
*<math>\omega(r)</math> is the [[Thermal potential | thermal potential]].
*<math>f(r)</math> is the [[Mayer f-function]].
</div>


 
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:
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.
Line 17: Line 18:
:<math>c=c\left[h\right]</math>
:<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>.
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]].
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 \& McDonald \S 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)
(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>
:<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 OZ relation becomes (\cite{KNAW_1914_17_0793} Eq. 6)
If the system is both homogeneous and isotropic, the Ornstein-Zernike  relation becomes (Eq. 6 of Ref. 1)


<math>\gamma (r) \equiv  h(r) - c(r) = \rho \int  h(r')~c(|r - r'|) dr'</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 \& McDonald \S 5.2 p. 107)
 
``...describes the fact that the ''total'' correlation between particles 1 and 2, represented by <math>h(1,2)</math>,  
In words, this equation (Hansen and McDonald, section 5.2 p. 107)
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,
:"...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."
:<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.''
Line 36: Line 36:
(here truncated at the fourth iteration):
(here truncated at the fourth iteration):


<math>h(r) = c(r)  + \rho \int c(|r - r'|)  c(r')  dr'
+ \rho^2  \int \int  c(|r - r'|)  c(|r' - r''|)  c(r'')  dr''dr' 
+ \rho^3 \int\int\int  c(|r - r'|) c(|r' - r''|) c(|r'' - r'''|) c(r''')  dr'''dr''dr'
+ \rho^4 \int \int\int\int  c(|r - r'|) c(|r' - r''|) c(|r'' - r'''|) c(|r''' - r''''|) h(r'''')  dr'''' dr'''dr''dr'</math>


''etc.''
:<math>h({\mathbf r}) = c({\mathbf r})  + \rho \int c(|{\mathbf r} - {\mathbf r'}|)  c({\mathbf r'})  {\rm d}{\mathbf r'}</math>
Diagrammatically this expression can be written as \cite{PRA_1992_45_000816}:
 
\begin{figure}[H]
:::::<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>
\begin{center}  
 
\includegraphics[clip,height=30pt,width=350pt]{oz_diag.eps}
:::::<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>
\end{center}  
 
\end{figure}
:::::<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>
\noindent
 
:::::''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.
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 Mayer bonds passing through increasing
point to another. An `uphill path' is a sequence of [[Mayer f-function |Mayer bonds]] passing through increasing
particle labels.
particle labels.
The OZ relation can be derived by performing a functional differentiation  
The Ornstein-Zernike relation can be derived by performing a functional differentiation  
of the grand canonical distribution function (HM check this).
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}(k) 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==
<references/>
'''Related reading'''
*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 &sect; 3.5
*[http://doi.org/10.1063/1.4972020 Yan He, Stuart A. Rice, and Xinliang Xu "Analytic solution of the Ornstein-Zernike relation for inhomogeneous liquids", Journal of Chemical Physics  '''145''' 234508 (2016)]
[[Category: Integral equations]]

Latest revision as of 14:17, 21 December 2016

Notation used:

The Ornstein-Zernike relation integral equation [1] is given by:

where denotes a functional of . This relation is exact. This is complemented by the closure relation

Note that depends on , and depends on . Because of this 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)

If the system is both homogeneous and isotropic, the Ornstein-Zernike relation becomes (Eq. 6 of Ref. 1)

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 , is due in part to the direct correlation between 1 and 2, represented by , but also to the indirect correlation, , propagated via increasingly large numbers of intermediate particles."

Notice that this equation is basically a convolution, i.e.

(Note: the convolution operation written here as is more frequently written as ) This can be seen by expanding the integral in terms of (here truncated at the fourth iteration):


etc.

Diagrammatically this expression can be written as [2]:

where the bold lines connecting root points denote functions, the blobs denote 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 bonds passing through increasing particle labels. The Ornstein-Zernike relation can be derived by performing a functional differentiation of the grand canonical distribution function.

Ornstein-Zernike relation in Fourier space[edit]

The Ornstein-Zernike equation may be written in Fourier space as ([3] Eq. 5):

The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly to:


Note:


References[edit]

  1. 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)
  2. James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A 45 pp. 816-824 (1992)
  3. 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)

Related reading