Editing Ornstein-Zernike relation
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Notation: | |||
Notation | *<math>g(r)</math> is the [[pair distribution function]]. | ||
*<math>g(r)</math> is the [[ | *<math>\Phi(r)</math> is the [[pair potential]] acting between pairs. | ||
*<math>\Phi(r)</math> is the [[ | *<math>h(1,2)</math> is the [[total correlation function]] <math>h(1,2) \equiv g(r) -1</math>. | ||
*<math>h(1,2)</math> is the [[ | *<math>c(1,2)</math> is the [[direct correlation function]]. | ||
*<math>c(1,2)</math> is the [[ | *<math>\gamma (r)</math> is the [[indirect correlation function | indirect]] (or ''series'' or ''chain'') correlation function <math>\gamma (r) \equiv h(r) - c(r)</math>. | ||
*<math>\gamma (r)</math> is the [[ | *<math>y(r_{12})</math> is the [[cavity correlation function]]<math>y(r) \equiv g(r) /e^{-\beta \Phi(r)}</math> | ||
*<math>y(r_{12})</math> is the [[ | *<math>B(r)</math> is the [[Closures | bridge]] function. | ||
*<math>B(r)</math> is the [[ bridge | *<math>\omega(r)</math> is the [[thermal potential]], <math>\omega(r) \equiv \gamma(r) + B(r)</math>. | ||
*<math>\omega(r)</math> is the [[ | *<math>f(r)</math> is the [[Mayer]] <math>f</math>-function, defined as <math>f(r) \equiv e^{-\beta \Phi(r)} -1</math>. | ||
*<math>f(r)</math> is the [[Mayer f-function | |||
</ | |||
The '''Ornstein-Zernike relation''' integral equation | |||
The '''Ornstein-Zernike relation''' (OZ) integral equation is | |||
:<math>h=h\left[c\right]</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. | ||
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:<math>c=c\left[h\right]</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]]. | ||
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 and McDonald, section 5.2 p. 106) For a system in an external field, the | (Hansen and McDonald, section 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 (Ref. 1Eq. 6) | ||
<math>\gamma (r) \equiv h(r) - c(r) = \rho \int h(r')~c(|r - r'|) dr'</math> | |||
In words, this equation (Hansen and McDonald, section 5.2 p. 107) | In words, this equation (Hansen and McDonald, section 5.2 p. 107) | ||
``...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, | |||
<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.'' | ||
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(here truncated at the fourth iteration): | (here truncated at the fourth iteration): | ||
<math>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'</math> | |||
''etc.'' | |||
Diagrammatically this expression can be written as (Ref. 2): | |||
[[Image:oz_diag.png]] | |||
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 | ||
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). | ||
== | ==OZ equation in Fourier space== | ||
The | The OZ equation may be written in Fourier space as (Eq. 5 in Ref. 3): | ||
:<math>\hat{\gamma} = ( | :<math>\hat{\gamma} = (I - \rho \hat{c})^{-1} \hat{c} \rho \hat{c}</math> | ||
The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly | The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly | ||
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:<math>\gamma (r) = \frac{1}{2 \pi^2 r} \int_0^\infty k~\sin (kr) \hat{\gamma}( | :<math>\gamma (r) = \frac{1}{2 \pi^2 r} \int_0^\infty k~\sin (kr) \hat{\gamma}(r) dk</math> | ||
Note: | Note: | ||
:<math>\hat{h}(0) = \int h(r) | :<math>\hat{h}(0) = \int h(r) dr</math> | ||
:<math>\hat{c}(0) = \int c(r) | :<math>\hat{c}(0) = \int c(r) dr</math> | ||
==References== | ==References== | ||
#[L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)] | |||
#[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816 - 824 (1992)] | |||
''' | #[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)] | ||
#[Hansen and MacDonald "Theory of Simple Liquids"] | |||
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