Assumptions behind Ornstein-Zernike correlation function

Question

Let $S(\mathbf q)$ be come correlation function in Fourier space ($\mathbf q$ = wavevector). In the study of condensed matter systems, I have often encountered the statements that a reasonable form for $S(\mathbf q)$ is a Lorentzian, i.e.

$$ S(\mathbf q) = \frac{S(0)}{1+(q\xi)^2} \tag{1} $$

where $q=|\mathbf q|$ and $\xi$ should be interpreted as a correlation length.

Authors usually refer to $(1)$ as the "Ornstein-Zernike" function, apparently after two papers (a), which unfortunately I wasn't able to find. Apparently, the two authors were discussing the problem of light scattering from a fluid in the vicinity of the liquid-gas transition as the critical point is approached (which I think is called "critical opalescence").

We find this kind of function in the study of magnetic systems, in which case $S$ is the magnetic susceptibility (b), or in the study of density fluctuation in polymer solutions (c).

I know that (1) is related to the Ornstein-Zernike recursive integral equation for the direct pair correlation function $c(\mathbf r,\mathbf r')$, which for a uniform and isotropic system takes in Fourier space the form (d):

$$ \tilde h( q) = \frac{\tilde c ( q)}{1-\rho \tilde c(q)} \tag{2} $$

where the "tilde" denotes the Fourier transform and $h(r)=g(r)-1$, with $g(r)$ the pair correlation function. I also know that the structure factor (sometimes called "scattering function"), which is nothing else than a response function for density fluctuations, is related to $h$ by

$$ S(\mathbf q) = 1 +\rho \tilde h(\mathbf q) \tag{3} $$

and that often it is assumed that it has the form $(1)$.

However, it is not clear to me under which assumption does $(1)$ follow from $(2)-(3)$ (even if I suspect that a small wavevector limit is involved).

In general, what I would like to know is: under which assumption can we say that a reasonable form for some correlation function in Fourier space is given by $(1)$?

A mathematically detailed treatment and pertinent references would be greatly appreciated.

PS: It may help to know that the real space functional form corresponding to $(1)$, i.e., its Fourier transform is, in 3D:

$$ \tilde S(\mathbf r) = \frac{\lambda}{r} e^{-r/\xi} $$

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(a): L. S. Ornstein and F. Zernike, Physik. Z., 19, 134 (1918); 27, 761 (1926)

(b): Chaikin P.M., Lubensky T.C. - Principles of Condensed Matter Physics

(c): Doi M., Edwards S.F. - The Theory of Polymer Dynamics

(d) Hansen J.P., McDonald I.R. - Theory of Simple Liquids

Answer 1

The mathematical assumption that Ornstein and Zernike made was that the direct correlation function is short ranged, in the sense that its second moment $$ \int d\mathbf{r} r^2 c(\mathbf{r}) $$ is finite, and as a consequence its Fourier transform has a Taylor series expansion in $q$ at low $q$, at least up to second order, which we may write $$ \tilde{c}(q) = \tilde{c}(0)[1- \alpha q^2] + \ldots $$ since any linear term in $q$ must vanish by symmetry. This leads almost immediately to your first equation, which is taken to describe the low-$q$ behaviour of $S(q)$: $$ S(q)=\frac{1}{1-\rho\tilde{c}(q)} \approx \frac{S(0)}{1+\xi^2q^2} $$ where $$ \xi^2=\alpha\frac{\rho\tilde{c}(0)}{1-\rho\tilde{c}(0)} =\alpha \rho\tilde{h}(0). $$ There's plenty of discussion of this in the literature on critical phenomena, for example section 3 of ME Fisher J Math Phys, 5, 944 (1964) and very briefly in section 9.2 of Rowlinson and Widom, Molecular theory of capillarity. It is very clearly explained in the last chapter on 'Phase Transitions' of the First Edition of Hansen and McDonald's book Theory of Simple Liquids. Unfortunately (IMHO) they dropped that chapter in later editions!

The point of this is that, although the correlation length that characterizes $h(r)$ will diverge as the critical point is approached, the $c(r)$ function may be assumed to remain of finite range, and your eqn (1) still applies, and may be used to discuss density fluctuations which give rise to critical opalescence. This assumption, that $c(r)$ remains short ranged even at the critical point, turns out not to be quite true, but it is a reasonable first approximation.