Ornstein-Zernike relation: Difference between revisions

Revision as of 16:21, 20 February 2007

Notation:

$g(r)$ is the pair distribution function.
$\Phi (r)$ is the pair potential acting between pairs.
$h(1,2)$ is the total correlation function $h(1,2)\equiv g(r)-1$ .
$c(1,2)$ is the direct correlation function.
$\gamma (r)$ is the indirect (or series or chain) correlation function $\gamma (r)\equiv h(r)-c(r)$ .
$y(r_{12})$ is the cavity correlation function $y(r)\equiv g(r)/e^{-\beta \Phi (r)}$
$B(r)$ is the bridge function.
$\omega (r)$ is the thermal potential, $\omega (r)\equiv \gamma (r)+B(r)$ .
$f(r)$ is the [[Mayer $f$ -function]], defined as $f(r)\equiv e^{-\beta \Phi (r)}-1$ .

The Ornstein-Zernike relation (OZ) integral equation is

h=h\left[c\right]

where $h[c]$ denotes a functional of $c$ . This relation is exact. This is complemented by the closure relation

c=c\left[h\right]

Note that $h$ depends on $c$ , and $c$ depends on $h$ . Because of this $h$ must be determined self-consistently. 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)

h(1,2)=c(1,2)+\int \rho ^{(1)}(3)c(1,3)h(3,2)d3

If the system is both homogeneous and isotropic, the OZ relation becomes (\cite{KNAW_1914_17_0793} Eq. 6)

$\gamma (r)\equiv h(r)-c(r)=\rho \int h(r')~c(|r-r'|)dr'$ 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  $h(1,2)$ , 
is due in part to the direct correlation between 1 and 2, represented by  $c(1,2)$ , but also to the indirect correlation,

\gamma (r)

, propagated via increasingly large numbers of intermediate particles."

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

h\equiv c+\rho h\otimes c

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

$h(r)=c(r)+\rho \int c(|r-r'|)c(r')dr'+\rho ^{2}\int \int c(|r-r'|)c(|r'-r''|)c(r'')dr''dr'+\rho ^{3}\int \int \int c(|r-r'|)c(|r'-r''|)c(|r''-r'''|)c(r''')dr'''dr''dr'+\rho ^{4}\int \int \int \int c(|r-r'|)c(|r'-r''|)c(|r''-r'''|)c(|r'''-r''''|)h(r'''')dr''''dr'''dr''dr'$

etc. Diagrammatically this expression can be written as \cite{PRA_1992_45_000816}: \begin{figure}[H] \begin{center} \includegraphics[clip,height=30pt,width=350pt]{oz_diag.eps} \end{center} \end{figure} \noindent where the bold lines connecting root points denote $c$ functions, the blobs denote $h$ 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 OZ relation can be derived by performing a functional differentiation of the grand canonical distribution function (HM check this).

@@ Line 12: / Line 12: @@
 The '''Ornstein-Zernike relation''' (OZ) integral equation is
-:<math>h=h[c]</math>
+:<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[h]</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>.
 Because of this <math>h</math> must be determined [[self-consistently]].
@@ Line 23: / Line 23: @@
 If the system is both homogeneous and isotropic, the OZ relation becomes (\cite{KNAW_1914_17_0793} Eq. 6)
-<math>\gamma ({\bf r}) \equiv  h({\bf r}) - c({\bf r}) = \rho \int  h({\bf r'})~c(|{\bf r} - {\bf r'}|) {\rm d}{\bf r'}</math>
+<math>\gamma (r) \equiv  h(r) - c(r) = \rho \int  h(r')~c(|r - r'|) dr'</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>,
   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.''
-<math>h  \equiv c  + \rho h\otimes c </math>
+:<math>h  \equiv c  + \rho h\otimes c </math>
-(Note: the convolution operation written here as $ \otimes$ is more frequently written as $*$)\\
+(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 $h({\bf r})$
+This can be seen by expanding the integral in terms of <math>h(r)</math>
 (here truncated at the fourth iteration):
-\begin{eqnarray*}
-h({\bf r}) = c({\bf r})  &+& \rho \int c(|{\bf r} - {\bf r'}|)  c({\bf r'})  {\rm d}{\bf r'} \\
+<math>h(r) = c(r)  + \rho \int c(|r - r'|)  c(r')  dr'
-&+& \rho^2  \int \int  c(|{\bf r} - {\bf r'}|)   c(|{\bf r'} - {\bf r''}|)  c({\bf r''})   {\rm d}{\bf r''}{\rm d}{\bf r'} \\
++ \rho^2  \int \int  c(|r - r'|)   c(|r' - r''|)  c(r'')   dr''dr'
-&+& \rho^3 \int\int\int  c(|{\bf r} - {\bf r'}|) c(|{\bf r'} - {\bf r''}|) c(|{\bf r''} - {\bf r'''}|) c({\bf r'''})   {\rm d}{\bf r'''}{\rm d}{\bf r''}{\rm d}{\bf r'}\\
++ \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(|{\bf r} - {\bf r'}|) c(|{\bf r'} - {\bf r''}|) c(|{\bf r''} - {\bf r'''}|) c(|{\bf r'''} - {\bf r''''}|) h({\bf r''''})  {\rm d}{\bf r''''} {\rm d}{\bf r'''}{\rm d}{\bf r''}{\rm d}{\bf r'}
++ \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>
-\end{eqnarray*}
-{\it etc.}\\
+''etc.''
 Diagrammatically this expression can be written as  \cite{PRA_1992_45_000816}:
 \begin{figure}[H]
@@ Line 49: / Line 49: @@
 \end{figure}
 \noindent
-where the bold lines connecting root points denote $c$ functions, the blobs denote $h$ 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
 point to another. An `uphill path' is a sequence of Mayer bonds passing through increasing
-particle labels.\\
+particle labels.
 The OZ relation can be derived by performing a functional differentiation
 of the grand canonical distribution function (HM check this).
+==References==

Ornstein-Zernike relation: Difference between revisions

Revision as of 16:21, 20 February 2007

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