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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