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Ornstein-Zernike relation

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Notation used:

The Ornstein-Zernike relation integral equation [1] is given by:

h=h\left[c\right]

where h[c] denotes a functional of c. This relation is exact. This is complemented by the closure relation

c=c\left[h\right]

Note that h depends on c, and c depends on h. Because of this 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)

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)

\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 h(1,2), is due in part to the direct correlation between 1 and 2, represented by c(1,2), but also to the indirect correlation, \gamma (r), propagated via increasingly large numbers of intermediate particles."

Notice that this equation is basically a convolution, i.e.

h\equiv c+\rho h\otimes c

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


h({{\mathbf  r}})=c({{\mathbf  r}})+\rho \int c(|{{\mathbf  r}}-{{\mathbf  r}'}|)c({{\mathbf  r}'}){{\rm {d}}}{{\mathbf  r}'}
+\rho ^{2}\iint c(|{{\mathbf  r}}-{{\mathbf  r}'}|)c(|{{\mathbf  r}'}-{{\mathbf  r}''}|)c({{\mathbf  r}''}){{\rm {d}}}{{\mathbf  r}''}{{\rm {d}}}{{\mathbf  r}'}
+\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}'}
+\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]:

Oz diag.png

where the bold lines connecting root points denote c functions, the blobs denote 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[edit]

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

{\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:

{\hat  {\gamma }}(k)={\frac  {4\pi }{k}}\int _{0}^{\infty }r~\sin(kr)\gamma (r)dr


\gamma (r)={\frac  {1}{2\pi ^{2}r}}\int _{0}^{\infty }k~\sin(kr){\hat  {\gamma }}(k)dk

Note:

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


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

References[edit]

  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)

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