Ornstein-Zernike relation: Difference between revisions

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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (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 $h(r)$ (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

[KNAW_1914_17_0793]
[PRA_1992_45_000816]

@@ 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)
 :<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>
@@ Line 42: / Line 42: @@
 ''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.
 An arrow pointing from left to right indicates an uphill path from one root
@@ Line 57: / Line 54: @@
 ==References==
+#[KNAW_1914_17_0793]
+#[PRA_1992_45_000816]

Ornstein-Zernike relation: Difference between revisions

Revision as of 15:25, 20 February 2007

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