Ornstein-Zernike relation

From SklogWiki

Notation:

$g (r)$ is the pair distribution function.
$Φ(r)$ is the pair potential acting between pairs.
$h (1,2)$ is the total correlation function.
$c (1,2)$ is the direct correlation function.
$γ(r)$ is the indirect (or series or chain) correlation function.
$y (r 12)$ is the cavity correlation function.
$B (r)$ is the bridge function.
$ω(r)$ is the thermal potential.
$f (r)$ is the Mayer f-function.

The Ornstein-Zernike relation (OZ) integral equation is

$h=h\left[c\right]$

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 OZ 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 OZ relation becomes (Ref. 1Eq. 6)

$\gamma ({\mathbf r}) \equiv h({\mathbf r}) - c({\mathbf r}) = \rho \int h({\mathbf r'})~c(|{\mathbf r} - {\mathbf r'}|) {\rm d}{\mathbf r'}$

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,  
 $γ(r)$ , 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 (Ref. 2):

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 OZ relation can be derived by performing a functional differentiation of the grand canonical distribution function (HM check this).