Ornstein-Zernike relation: Difference between revisions

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The '''Ornstein-Zernike relation''' (OZ) integral equation is
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.
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[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>.
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]].
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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>\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)
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>,  
  ``...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,   
  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.''
<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):
(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}:
Diagrammatically this expression can be written as  \cite{PRA_1992_45_000816}:
\begin{figure}[H]
\begin{figure}[H]
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\end{figure}
\end{figure}
\noindent
\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
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 bonds passing through increasing
particle labels.\\
particle labels.
The OZ relation can be derived by performing a functional differentiation  
The OZ relation can be derived by performing a functional differentiation  
of the grand canonical distribution function (HM check this).
of the grand canonical distribution function (HM check this).
==References==

Revision as of 15:21, 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} .
  • 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)} is the direct 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 \gamma (r)} is the indirect (or series or chain) 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 \gamma (r) \equiv h(r) - c(r)} .
  • 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 y(r_{12})} is the cavity correlation functionFailed 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 y(r) \equiv g(r) /e^{-\beta \Phi(r)}}
  • 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 B(r)} is the bridge 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 \omega(r)} is the thermal potential, 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 \omega(r) \equiv \gamma(r) + B(r)} .
  • 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 f(r)} is the [[Mayer 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 f} -function]], defined as 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 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\left[c\right]}

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\left[h\right]}

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 (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 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 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, i.e.

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 \equiv c + \rho h\otimes c }

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

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(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 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} functions, the blobs denote 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} 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).

References

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