Ornstein-Zernike relation: Difference between revisions
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Notation: | Notation: | ||
* | *<math>g(r)</math> is the [[pair distribution function]]. | ||
* | *<math>\Phi(r)</math> is the [[pair potential]] acting between pairs. | ||
* | *<math>h(1,2)</math> is the [[total correlation function]] <math>h(1,2) \equiv g(r) -1</math>. | ||
*$c(1,2)$ 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 $\gamma ({\bf r}) \equiv h({\bf r}) - c({\bf r})$. | ||
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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) | ||
<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> | ||
\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) | 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 $h(1,2)$, 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." | ||
is due in part to the | |||
but also to the | |||
Notice that this equation is basically a convolution, {\it i.e.} | Notice that this equation is basically a convolution, {\it i.e.} | ||
\begin{equation} | \begin{equation} | ||
Revision as of 15:03, 20 February 2007
Notation:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(r)} is the pair distribution function.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(r)} is the pair potential acting between pairs.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(1,2)} is the total correlation function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h=h[c]}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h[c]} denotes a functional of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} . This relation is exact. This is complemented by the closure relation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=c[h]}
Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} depends on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} depends on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} . Because of this Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma ({\bf r}) \equiv h({\bf r}) - c({\bf r}) = \rho \int h({\bf r'})~c(|{\bf r} - {\bf r'}|) {\rm d}{\bf r'}} 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 direct correlation between 1 and 2, represented by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(1,2)}
, but also to the indirect correlation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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).