Ornstein-Zernike relation: Difference between revisions

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


\noindent
 
The {\it Ornstein-Zernike relation} (OZ) integral equation is
The '''Ornstein-Zernike relation''' (OZ) integral equation is
\begin{equation}
:<math>h=h[c]</math>
h=h[c]
where <math>h[c]</math> denotes a functional of <math>c</math>. This relation is exact.
\end{equation}
where $h[c]$ denotes a functional of $c$. This relation is exact.\\
\end{equation}
where $h[c]$ denotes a functional of $c$. This relation is exact.\\
This is complemented by the closure relation
This is complemented by the closure relation
\begin{equation}
:<math>c=c[h]</math>
c=c[h]
Note that <math>h</math> depends on <math>c</math>, and <math>c</math> depends on <math>h</math>.
\end{equation}
Because of this <math>h</math> must be determined [[self-consistently]].
Note that $h$ depends on $c$, and $c$ depends on $h$.
This need for self-consistency is characteristic of all many-body problems.
Because of this $h$ must be determined {\it self-consistently}.
(Hansen \& McDonald \S 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)
This need for self-consistency is characteristic of all many-body problems.\\
:<math>h(1,2) = c(1,2) + \int \rho^{(1)}(3) c(1,3)h(3,2) d3</math>
(Hansen \& McDonald \S 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)\\
\begin{equation}
h(1,2) = c(1,2) + \int \rho^{(1)}(3) c(1,3)h(3,2) {\rm d}3
\end{equation}
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 OZ relation becomes (\cite{KNAW_1914_17_0793} Eq. 6)
~\\
 
\fbox{\parbox{\columnwidth}{
<math></math>
\begin{equation}
\gamma ({\bf r}) \equiv  h({\bf r}) - c({\bf r}) = \rho \int  h({\bf r'})~c(|{\bf r} - {\bf r'}|) {\rm d}{\bf r'}
\gamma ({\bf r}) \equiv  h({\bf r}) - c({\bf r}) = \rho \int  h({\bf r'})~c(|{\bf r} - {\bf r'}|) {\rm d}{\bf r'}
\end{equation}
</math>
}}
~\\
In words, this equation (Hansen \& McDonald \S 5.2 p. 107)
In words, this equation (Hansen \& McDonald \S 5.2 p. 107)
``...describes the fact that the {\it total} correlation between particles 1 and 2, represented by $h(1,2)$,
``...describes the fact that the {\it total} correlation between particles 1 and 2, represented by $h(1,2)$,

Revision as of 14:56, 20 February 2007

Notation:

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


The Ornstein-Zernike relation (OZ) integral equation is

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 \& McDonald \S 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)

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

\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> In words, this equation (Hansen \& McDonald \S 5.2 p. 107) ``...describes the fact that the {\it total} correlation between particles 1 and 2, represented by $h(1,2)$, is due in part to the {\it direct} correlation between 1 and 2, represented by $c(1,2)$, but also to the {\it indirect} correlation, $\gamma (r)$, propagated via increasingly large numbers of intermediate particles."\\ Notice that this equation is basically a convolution, {\it i.e.} \begin{equation} h \equiv c + \rho h\otimes c \end{equation} (Note: the convolution operation written here as $ \otimes$ is more frequently written as $*$)\\ This can be seen by expanding the integral in terms of $h({\bf r})$ (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'} \\ &+& \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^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^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'} \end{eqnarray*} {\it 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).