Ornstein-Zernike relation: Difference between revisions

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The Ornstein-Zernike equation may be written in [[Fourier analysis |Fourier space]] as (<ref>[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)]</ref> Eq. 5):
The Ornstein-Zernike equation may be written in [[Fourier analysis |Fourier space]] as (<ref>[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)]</ref> Eq. 5):


:<math>\hat{\gamma} = ({\mathbf I} - \rho {\mathbf \hat{c}})^{-1} {\mathbf \hat{c}} \rho  {\mathbf \hat{c}}</math>
:<math>\hat{\gamma} = (\mathbf{I} - \rho \mathbf{\hat{c}})^{-1}   \mathbf{\hat{c}} \rho  \mathbf{\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

Revision as of 12:49, 14 May 2012

Notation used:

  • 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 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 y(r_{12})} is the cavity 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 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 f(r)} is the Mayer f-function.

The Ornstein-Zernike relation integral equation [1] is given 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=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 and McDonald, section 5.2 p. 106) For a system in an external field, the Ornstein-Zernike relation 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 Ornstein-Zernike relation becomes (Eq. 6 of Ref. 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 \gamma ({\mathbf r}) \equiv h({\mathbf r}) - c({\mathbf r}) = \rho \int h({\mathbf r'})~c(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}}

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 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({\mathbf r}) = c({\mathbf r}) + \rho \int c(|{\mathbf r} - {\mathbf r'}|) c({\mathbf r'}) {\rm d}{\mathbf 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 + \rho^2 \iint c(|{\mathbf r} - {\mathbf r'}|) c(|{\mathbf r'} - {\mathbf r''}|) c({\mathbf r''}) {\rm d}{\mathbf r''}{\rm d}{\mathbf 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 + \rho^3 \iiint c(|{\mathbf r} - {\mathbf r'}|) c(|{\mathbf r'} - {\mathbf r''}|) c(|{\mathbf r''} - {\mathbf r'''}|) c({\mathbf r'''}) {\rm d}{\mathbf r'''}{\rm d}{\mathbf r''}{\rm d}{\mathbf 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 + \rho^4 \iiiint c(|{\mathbf r} - {\mathbf r'}|) c(|{\mathbf r'} - {\mathbf r''}|) c(|{\mathbf r''} - {\mathbf r'''}|) c(|{\mathbf r'''} - {\mathbf r''''}|) h({\mathbf r''''}) {\rm d}{\mathbf r''''} {\rm d}{\mathbf r'''}{\rm d}{\mathbf r''}{\rm d}{\mathbf r'}}
etc.

Diagrammatically this expression can be written as [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 Ornstein-Zernike relation can be derived by performing a functional differentiation of the grand canonical distribution function.

Ornstein-Zernike relation in Fourier space

The Ornstein-Zernike equation may be written in Fourier space as ([3] Eq. 5):

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 \hat{\gamma} = (\mathbf{I} - \rho \mathbf{\hat{c}})^{-1} \mathbf{\hat{c}} \rho \mathbf{\hat{c}}}

The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly to:

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 \hat{\gamma} (k) = \frac{4 \pi}{k} \int_0^\infty r~\sin (kr) \gamma(r) dr}


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) = \frac{1}{2 \pi^2 r} \int_0^\infty k~\sin (kr) \hat{\gamma}(r) dk}

Note:

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 \hat{h}(0) = \int h(r) {\rm d}{\mathbf 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 \hat{c}(0) = \int c(r) {\rm d}{\mathbf r}}

References

  1. ↑ 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)
  2. ↑ James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A 45 pp. 816-824 (1992)
  3. ↑ 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)

Related reading

  • Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5
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