Editing Ornstein-Zernike relation
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Notation: | |||
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''' integral equation | |||
:<math>h=h | The '''Ornstein-Zernike relation''' (OZ) integral equation is | ||
:<math>h=h[c]</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. | ||
This is complemented by the closure relation | This is complemented by the closure relation | ||
:<math>c=c | :<math>c=c[h]</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 | (Hansen \& McDonald \S 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 (\cite{KNAW_1914_17_0793} Eq. 6) | ||
Notice that this equation is basically a convolution, | <math></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> | |||
(Note: the convolution operation written here as | In words, this equation (Hansen \& McDonald \S 5.2 p. 107) | ||
This can be seen by expanding the integral in terms of | ``...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): | (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} | |||
Diagrammatically this expression can be written as | \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 | |||
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). | ||