Ornstein-Zernike relation: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
mNo edit summary
mNo edit summary
Line 21: Line 21:
(Hansen \& McDonald \S 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)
(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 OZ relation becomes (\cite{KNAW_1914_17_0793} Eq. 6)
If the system is both homogeneous and isotropic, the OZ relation becomes (Ref. 1Eq. 6)


<math>\gamma (r) \equiv  h(r) - c(r) = \rho \int  h(r')~c(|r - r'|) dr'</math>
<math>\gamma (r) \equiv  h(r) - c(r) = \rho \int  h(r')~c(|r - r'|) dr'</math>
Line 42: Line 42:


''etc.''
''etc.''
Diagrammatically this expression can be written as  \cite{PRA_1992_45_000816}:
Diagrammatically this expression can be written as  (Ref. 2):
\begin{figure}[H]
 
\begin{center}
[[Image:oz_diag.png]]
\includegraphics[clip,height=30pt,width=350pt]{oz_diag.eps}
 
\end{center}
\end{figure}
\noindent
where the bold lines connecting root points denote <math>c</math> functions, the blobs denote <math>h</math> 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
Line 57: Line 54:


==References==
==References==
#[KNAW_1914_17_0793]
#[PRA_1992_45_000816]

Revision as of 15:25, 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)

If the system is both homogeneous and isotropic, the OZ relation becomes (Ref. 1Eq. 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 (Ref. 2):

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

  1. [KNAW_1914_17_0793]
  2. [PRA_1992_45_000816]
Retrieved from "http://www.sklogwiki.org/SklogWiki/index.php?title=Ornstein-Zernike_relation&oldid=220"